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QUANTUM STATISTICAL MECHANICS

Selected Works of N N Bogolubov

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N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I

World Scientific

QUANTUM STATISTICAL MECHANICS

Selected Works of N N Bogolubov

N N Bogolubov, JrMoscow State University, Russia

9205_9789814612517_tp.indd 2 27/6/14 3:36 pm

Published by

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Library of Congress Cataloging-in-Publication DataBogoliubov, N. N. (Nikolai Nikolaevich), 1909–1992, author. Quantum statistical mechanics : selected works of N.N. Bogolubov / N.N. Bogolubov, Jr., Moscow State University, Russia. pages cm Includes bibliographical references and index. ISBN 978-9814612517 (hardcover : alk. paper) 1. Quantum statistics. 2. Statistical mechanics. I. Bogoliubov, N. N. (Nikolai Nikolaevich), author. II. Title. QC174.4.B65 2014 530.13'3--dc23 2014015288

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

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FOREWORD

Part I of the Selected Works by N.N. Bogolubov contains some of his paperson statistical mechanics, a field in which he has obtained a number offundamental results.

The name of Bogolubov has been inseparably linked with the creation ofthe modern theory of non-ideal quantum macrosystems. His formulation forsuch important physical phenomena as superfluidity and superconductivityformed the basis of this theory. During the 1940s Bogolubov produced aseries of papers dealing with these problems. He developed the method ofapproximate second quantization which has been considered to be one ofthe basic tools of quantum statistics. The new method has made possible,in particular, the discovery of a very important physical phenomenon, thestabilization of a condensate in non-ideal gases at temperatures close to zero.

The phenomenon of superfluidity was discovered in 1938 by one ofthe most prominent Soviet physicists, Academician P.L. Kapitsa. It wasfound that at temperatures close to absolute zero the viscosity of 2Hewas equal to zero. A new type of energy spectrum was discovered whoseinvestigation became the main task in the study of properties of matterat low temperatures. However, the dynamical nature of the spectrum wasobscure for a long time. It was not clear whether this phenomenon couldbe interpreted within the usual quantum mechanical scheme for the pairinteraction of individual particles.

In 1947 in his classic work “On the Theory of Superfluidity”, Bogolubovgave a brilliant physical explanation for the phenomenon of superfluidity.He brought out the dominant role of the interaction of correlated pairsof particles with opposite momenta in the formation of the ground state,whereas an ideal gas does not possess this property. In this way,Bogolubov constructed a special transformation of Bose amplitudes, which

v

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vi FOREWORD

was thereafter called the Bogolubov transformation. As a result of theseinvestigations, a consistent microscopic theory of superfluidity was builtthat describes the energy spectrum for a superfluid system and explains therelation between the superfluid and normal states.

A valuable contribution has been made by Bogolubov to the theory ofsuperconductivity. He showed that the same type of excitations that occurin the superfluids also occur in superconductors, in which a decisive part isplayed by the interaction of the electrons with lattice oscillations. Duringconstruction of the microscopic theory of superconductivity it was found in1957 that the above-mentioned mathematical methods were also useful forstudying this phenomenon. It has been established that between superfluidityand superconductivity there is a deep physical and mathematical analogy.In brief, it can be said that superconductivity is superfluidity of electrons inmetal.

Bogolubov has investigated in detail the hydrodynamic stage in theevolution of classic many-particle systems. One of the works devoted tothis field,“Hydrodynamics Equations in Statistical Mechanics”, publishedin 1948, is reproduced in this volume. Later he generalized the methodof constructing kinetic equations for quantum systems and applied it tostudying superfluid liquids.

Studies of degeneracy in systems led Bogolubov in 1961 to the formulationof the method of quasi-averages in his work “Quasi-Averages in problems ofstatistical mechanics”. This method has proved to be a universal tool forsystems whose ground states become unstable under small perturbations.The subsequent development of statistical mechanics and quantum fieldtheory demonstrated the fruitfulness of the concept of quasi-averagesand of the idea contained in them of spontaneous symmetry breaking.Bogolubov himself successfully applied this method, for example, to problemsof superfluid hydrodynamics in the paper “On the Hydrodynamics of aSuperfluid Liquid”, published in 1963.

The influence of the studies on spontaneous symmetry breaking inmacroscopic systems on elementary particle physics and quantum field theorywas emphasized by S. Weinberg in his Nobel address, in which he reportedthat at the beginning of the 1960s he became acquainted with an ideathat first appeared in solid theory and was then introduced into particlephysics by people who worked in both of these branches of physics. Thiswas the idea of symmetry breaking, according to which the Hamiltonianand the commutation relations of quantum theory can possess an exact

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FOREWORD vii

symmetry but, nevertheless, the physical states need not correspond to therepresentations of the symmetry. In particular, it may happen that thesymmetry of the Hamiltonian is not the symmetry of the vacuum. Thesubsequent systematic development of these ideas in quantum filed theory hasled to the construction of the theory of electromagnetic and weak interactions,for which S.L. Glashow, A. Salam and S. Weinberg received the Nobel Pricein 1979.

In connection with the formulation of the concept of quasi-averages,Bogolubov also proved a fundamental theorem on 1/q2 singularities,according to which elementary excitations with energy, which vanishes inthe long-wave limit, arise in systems with spontaneous symmetry breaking.In other words, there are massless excitations — quanta of photon or phonontype — whose exchange leads to an interaction of infinite range. Soon afterthis, a similar result was obtained in quantum field theory.

Bogolubov’s concept of quasi-averages has also provided the foundationfor the modern theory of phase transitions. It should be mentioned thatfor problems of statistical mechanics Bogolubov’s studies on the theories ofsuperfluidity, superconductivity and quasi-averages produced the basis forthe development of the methods of variational inequalities and majorizingestimates for systems of many interacting particles with spontaneouslybroken symmetry.

Part II is devoted to methods for studying model Hamiltonians forproblems in quantum statistical mechanics. In this part methods areproposed for solving certain problems in statistical physics which containfour-fermion interaction. It has been possible, via “approximating (trial)Hamiltonians”, to distinguish a whole class of exactly soluble systems.

An essential difference between the two types of problem with positiveand negative four-fermion interaction is discovered and examined. Thedetermination of exact solutions for the free energies, single-time and many-time correlation functions, T -products and Green’s functions is treated foreach type of problem.

I express my sincere thanks to Academician N.N. Bogolubov for valuableremarks. The material of this part was the subject of theoretical seminarsand lectures at Moscow State University. I consider it my pleasant dutyto thank participants in the theoretical seminars at the V.A. SteklovMathematical Institute and the Theoretical Physics Laboratory of JointInstitute for Nuclear Research at Dubna for their interest and encouragementfor discussion. Some of the results of Part II were discussed in a theoretical

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viii FOREWORD

physics seminars at the E. T. H., Zurich. The author is thankful to Dr. AlanA. Dzhioev (BLTP, JINR, Dubna, Russia) for the help in presentation ofthe manuscript for publication and to Dr. Denis Blackmore (Departmentof Mathematic Science and Center for Applied mathematic and statistics,New Jersey Institute of Technology, USA) for checking the English of themanuscript.

N.N. Bogolubov, Jr.

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CONTENTS

Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v

PART I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 1On the Theory of Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . 3

CHAPTER 2Quasi-Averages in Problems of Statistical Mechanics . . . . . . . 21

Part A. QUASI-AVERAGES . . . . . . . . . . . . . . . . . . . . . . . 211. Green’s Functions, Defined with Regular Averages; Additive

Conservation Laws and Selection Rules . . . . . . . . . . . . . 212. Degeneracy of the Statistical Equilibrium States; Introduction

of Quasi-averages . . . . . . . . . . . . . . . . . . . . . . . . . 253. Principle of Correlation Weakening . . . . . . . . . . . . . . . 514. Particle Pair States . . . . . . . . . . . . . . . . . . . . . . . . 565. Certain Inequalities . . . . . . . . . . . . . . . . . . . . . . . 63

Part B. CHARACTERISTIC THEOREMS ABOUT THE 1/q2

TYPE INTERACTION IN THE THEORY OFSUPERCONDUCTIVITY OF BOSE ANDFERMI SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . 68

6. Symmetry Properties of Basic Green’s Functions for BoseSystems in the Presence of a Condensate . . . . . . . . . . . . 68

7. Model with a Condensate . . . . . . . . . . . . . . . . . . . . 738. The 1/q2 Theorem and its Application . . . . . . . . . . . . . 829. The 1/q2 Theorem for Fermi Systems . . . . . . . . . . . . . . 92

ix

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x CONTENTS

CHAPTER 3Hydrodynamics Equations in Statistical Mechanics . . . . . . . . 100

CHAPTER 4On the Hydrodynamics of a Superfluid Liquid . . . . . . . . . . . . 123

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1231. Preliminary Identities . . . . . . . . . . . . . . . . . . . . . . 1242. Hydrodynamic Equations for a Normal Liquid . . . . . . . . . 1303. Hydrodynamic Equations for a Superfluid . . . . . . . . . . . 1404. Variational Equations and Green’s Functions . . . . . . . . . 157

CHAPTER 5On the Model Hamiltonian of Superconductivity . . . . . . . . . . 168

1. Statement of the Problem . . . . . . . . . . . . . . . . . . . . 1682. General Properties of the Hamiltonian . . . . . . . . . . . . . 1713. Upper Bound for the Minimum Eigenvalue

of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 1764. Lower Bound for the Minimum Eigenvalue

of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 1805. Green’s Functions (Case ν > 0) . . . . . . . . . . . . . . . . . 1926. Green’s Functions (Case ν = 0) . . . . . . . . . . . . . . . . . 208

Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Appendix B. The Principle of Extinction of Correlations . . . . . . . . . . . . . . . 234

PART II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

CHAPTER 6Model Hamiltonians with Fermion Interaction . . . . . . . . . . . . 249

1. General Treatment of the Problem. SomePreliminary Results . . . . . . . . . . . . . . . . . . . . . . . 250

2. Calculation of the Free Energy for Model Systemwith Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . 257

3. Further Properties of the Expressions for the Free Energy . . 2694. Construction of Asymptotic Relations for the Free Energy . . 273

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CONTENTS xi

5. On the Uniform Convergence with Respect to θ of the FreeEnergy Function and on the Bounds for the Quantities δv . . 279

6. Properties of Partial Derivatives of the Free Energy Function.Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7. Rider to Theorem 3 and Construction of an AuxiliaryInequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

8. On the Difficulties of Introducing Quasi-Averages . . . . . . . 2889. A New Method of Introducing Quasi-Averages . . . . . . . . . 29210. The Question of the Choice of Sign for the Source-Terms . . . 29711. The Construction of Upper-Bound Inequalities in the

Case C = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306


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PART I

1


ii



CHAPTER 1

ON THE THEORY OF SUPERFLUIDITY

In the present paper an attempt is made to develop a consistent moleculartheory for the phenomenon of superfluidity without any assumptionsregarding the structure of the energy spectrum.

With this goal in mind it is more natural to proceed from the modelof a non-ideal Bose gas with weak interaction between particles. Similarattempts to account for superfluidity with the aid of the phenomenon ofdegeneracy of an ideal Bose gas have already been undertaken by L. Tisza andF. London, but these encountered active criticism. It was noted, for instance,that Helium II does not resemble an ideal gas at all, since its moleculesstrongly interact with each other. By the way, this objection can not beconsidered as crucial. Indeed, if the goal is to develop a molecular but nota phenomenological theory proceeding only from the common microscopicequations of quantum mechanics, then it is quite clear that any attempts toevaluate the properties of real liquids are hopeless. A molecular theory ofsuperfluidity can be expected only, at least at the first stage, to account forthe phenomenon itself proceeding from a simplified model.

In fact, the real objection to the above-mentioned criticism consists in thefact that, in a degenerate ideal Bose gas, particles in the ground state cannotbehave as superfluid ones, since nothing can prevent them from exchangingmomenta with excited particles in collisions and thereby undergoing frictionwhen moving through the fluid.

In the present paper we shall try to overcome this main difficulty andshow that under certain conditions in a weakly non-ideal Bose gas the socalled degenerate condensate can move without friction at a sufficiently lowvelocity relative to elementary excitations. It should be emphasized that inour theory these elementary excitations are a collective effect and cannot beidentified with separate molecules. The necessity of considering collective

3


4 N. N. BOGOLUBOV

elementary excitations instead of individual molecules was first stressed byLandau in his well known paper on the theory of superfluidity.

Let us consider a system of N identical monoatomic molecules enclosed in amacroscopic volume V and obeying the Bose statistics. As usual, we supposethat its Hamiltonian is of the form

H =∑

(i≤i≤N)

T (pi) +∑

(1≤i≤j≤N)

Φ(|qi − qj|)

where

T (pi) =|pi|22m

=∑

(1≤α≤3)

(pαi )2

2m

is the kinetic energy of the i-the molecule and Φ(|qi − qj|) is the potentialenergy of the pair (i, j).

Then we take advantage of the secondary quantization method to writethe basic equation in the form

i∂ψ

∂t= −

2

2m∆ψ +

∫Φ(|q − q′

j |)ψ†(q′)ψ(q′) dq′ ψ (1)

where

ψ =∑

f

afϕf (q), ψ† =∑

f

a†fϕ∗f (q).

Here af and a†f are conjugate operators with the usual commutation relations

afaf ′ − af ′af = 0,

afa†f ′ − a†f ′af = ∆f,f ′ =

0, f = f ′

1, f = f ′

and ϕf(q) is an orthonormal

∫ϕ∗

f(q)ϕf ′(q) dq = ∆f,f ′

complete set of functions.


SOME SELECTED WORKS 5

To simplify the problem, hereafter we shall make use of the set ofeigenfunctions of the momentum operator for a single particle

ϕf (q) =1

V 1/2exp

iqf

, qf =

∑(1≤α≤3)

fαqα.

For this set the operator Nf = a†af corresponds to the number of particleswith momentum f . For a finite value of the volume V , the vector f isapparently quantized. For instance, under the usual periodic boundaryconditions

fα =2πnα

l

where n1, n2, n3 are integers and l is the edge length of a cube of volume V .However, since here we shall deal with thermodynamics, i.e. bulk

properties, we should always imply the limit transition when the walls ofa vessel recede to infinity V → ∞, N → ∞ keeping the specific volumeν = V/N constant. Therefore, we shall finally proceed to a continuousspectrum replacing sums of the form∑

f

F (f )

by the integralsV

(2π)3

∫F (f) df .

Equations (1) are exact equations for the problem ofN bodies. To succeedin studying the motion of the system of molecules under consideration weshould make an approximation based on the assumption that the interactionenergy is small. In accordance with this assumption we shall considerthe potential Φ(r) to be proportional to a small parameter ε. Whichdimensionless ratio can be taken for ε will be clarified below. Now we onlynote that the assumption made corresponds, strictly speaking, to neglect ofthe finiteness of the molecular radius, since here we do not take into accountthe intensive increase of Φ(r) for small r that ensures the impermeabilityof the molecules. By the way, as we shall see below, the results which willbe obtained can be generalized to the case when one takes into account thatthe radius is finite.

Proceeding to formulating the approximation we note that, if there isno interaction, i.e. if the parameter ε exactly equals zero, then at zero


6 N. N. BOGOLUBOV

temperature we could put N0 = N , Nf = 0, (f = 0). However, in theconsidered case of a small ε and weakly excited states of the gas, theserelations are satisfied approximately in the sense that the major part ofthe molecules possess momenta close to zero. Of course, the choice of zeromomentum as a limit value for particles in the ground state corresponds to aspecific choice of framework, namely, one in which the condensate is at rest.

The above speculations are the basis for the following method forapproximate solution of Equation (1):

1) In view of the fact that N0 = a†0a0 is quite large compared with unity,the expression

a0a†0 − a†0a0 = 1

must be small compared with a0 and a†0 themselves. Therefore, we considera0 and a†0 as ordinary numbersa neglecting their non-commutativity.

2) Putting

ψ =a0√V

+ ϑ, ϑ =1√V

∑(f =0)

af expifq

we consider ϑ as a so called correction term of the first order and inEquation (1) we neglect terms beginning with a quadratic in ϑ, whichcorresponds to taking into account the weak excitations.

We then obtain basic approximate equations in the form

i∂ϑ

∂t= −

2

2m∆ϑ+

N0

VΦ0ϑ+

N0

V

∫Φ(|q − q′|)ϑ(q′) dq′

+a2

0

V

∫Φ(|q − q′|)ϑ(q′) dq′,

i∂a0

∂t=N0

VΦ0a0, (2)

where

Φ0 =

∫Φ(|q|) dq.

To proceed from the operator wave function ϑ to the creation and annihilationoperators af and a†f we use the Fourier expansion

Φ(|q − q′|) =∑

f

1

Vexp

i

(f(q − q′)

)ν(f ). (3)

aA similar remark was used by Dirac in his monograph “Foundation of QuantumMechanics” at the end of Section 3 named “Waves and Bose-Einstein particles.”



Due to the radial symmetry of the potential, the coefficients of the expansion

ν(f ) =

∫Φ(|q|) exp

−ifq

dq

depend on upon the length |f | of the vector f . We substitute Equation (1)into Equation (2) to obtain

i∂af

∂t=

T (f) + E0 +

N0

Vν(f )

af +

a20

Vν(f )a−f ,

E0 =N0

VΦ0.

If we put

af = exp(E0

it)bf , a0 = exp

(E0

i

)b, (4)

we can also write

i∂bf∂t

=T (f ) +

N0

Vν(f )

bf +

b2

Vν(f )b†−f ,

−i∂b†−f

∂t=

(b∗)2

Vν(f )bf +

T (f) +

N0

Vν(f )

b†−f . (5)

On solving this system of two equations with constant coefficients, we seethat the dependence of the operators bf and b†f on time is expressed by alinear combination of exponents of the form

exp(±E(f )

i

)where

E(f) =(2T (f)

N0

Vν(f ) + T 2(f)

)1/2

. (6)

Now we note that if

ν(0) =

∫Φ(|q|) dq > 0, (7)

then in the considered case of sufficiently small ε the expression inEquation (6) under the square root sign is positive. Thus. the operators bfand b†f turn out to be periodic functions of time. If, conversely, theinequality ν(0) < 0 holds, then for small momenta this expression is negative,and therefore the quantity E(f) becomes complex. Consequently, the


8 N. N. BOGOLUBOV

operators bf and b†f contain a real exponential increasing with time, so thatthe states with small Nf turn out to be unstable.

To ensure stability of weakly excited states, we shall consider below onlysuch types of interactions among the molecules for which the inequality (7)is valid. It is interesting to note that inequality (7) is simply the conditionfor thermodynamic stability of a gas at absolute zero. Indeed, at zerotemperature the free energy coincides with the internal energy, while theleading term in the expression of the internal energy is

E =N 2

2V

∫Φ(|q|) dq,

since the correction terms, for instance, the average kinetic energy, areproportional to higher powers of ε. Hence, the pressure p is given by

p = −∂E∂V

=N2

2V 2

∫Φ(|q|) dq =

ρ2

2m2

∫Φ(|q|) dq

where ρ = Nm/V is the mass density of the gas. Therefore, inequality (7)is equivalent to the condition of thermodynamic stability

∂p

∂ρ> 0.

Note, finally, that since here we take into account only the leading terms,we can write with the same degree of accuracy

E(f ) =(2T (f)

N

Vν(f ) + T 2(f )

)1/2

=( |f |2ν(f )

mν+

|f |44m2

)1/2

, (6′)

instead of Equation (6). Thus, for small momentab

E(f) =(ν(0)

mν

)1/2

|f |(1 + . . .) =(∂p∂ρ

)1/2

|f |(1 + . . .) (8)

where . . . denotes terms vanishing together with f .Let us now agree to take any square root we encounter with a positive

sign. Then for small momenta

E(f) = c|f |(1 + . . .) (9)

bIf we write the corresponding frequency E(f )/ and take the limit → 0, f/h = k, weobtain the classical A.A. Vlasov formula for the dependence of frequency on wave number.



where c is the velocity of sound at zero temperature. Conversely, forsufficiently large momenta one can expand E(f) in powers of ε and write

E(f ) =|f |22m

+1

νν(f ) + . . . .

Since the quantity ν(f ) vanishes with increasing |f |, for sufficiently largemomenta the energy E(f ) approaches the kinetic energy of a singlemolecule T (f ).

Let us now return to Equation (5) and introduce new mutually conjugateoperators ξf and ξ†f instead of bf and b†f by the relations

ξf =(bf − Lfb

†−f

)(1 − |Lf |2

)1/2

,

ξ†f =(b†f − L∗

fb−f

)(1 − |Lf |2

)1/2

, (10)

where Lf are numbers defined as follows

Lf =V b2

N20 ν(f )

E(f ) − T (f) − N0

Vν(f )

.

We have

|Lf |2 =(N0

Vν(f )

)2(E(f) + T (f ) +

N0

Vν(f )

)−2

,

1 − |Lf |2 = 2E(f )(E(f ) + T (f) +

N0

Vν(f )

)−1

. (11)

If we reverse Equations (10), we find

bf =(ξf + Lfξ

†−f

)(1 − |Lf |2

)1/2

,

b†f =(ξ†f + L∗

fξ−f

)(1 − |Lf |2

)1/2

. (12)

We substitute these relations into Equations (5) to obtain

i∂ξf∂t

= E(f)ξf , − i∂ξ†f∂t

= E(f )ξ†f . (13)

It is not difficult to check directly that the new operators satisfy the samecommutation relations as the operators af and a†f do

ξfξf ′ − ξf ′ξf = 0, ξfξ†f ′ − ξ†f ′ξf = ∆f,f ′ . (14)


10 N. N. BOGOLUBOV

This implies that the excited states of the considered system of moleculecan be thought of as an ideal gas of so called elementary excitations,or quasiparticles, with energies depending on momenta according theformula E = E(f). Just as the molecules have been described by theoperators af and a†f , these quasiparticles are described by the operators ξf

and ξ†f , and therefore they obey the Bose statistics. The operator

nf = ξ†fξf

represents the number of quasiparticles with momentum f .The above remark will be absolutely clear when we consider the total

energyH = Hkin +Hpot

where

Hkin =

2m

∫ψ†(q)∆ψ(q) dq,

Hpot =1

2

∫Φ(|q − q′|)ψ†(q)ψ†(q′)ψ(q)ψ(q′) dq dq′

=1

2V

∑f

ν(f )

∫exp

if(q − q′)

ψ†(q)ψ†(q′)ψ(q)ψ(q′) dq dq′.

For the kinetic energy we have

Hkin =∑

f

T (f)a†faf =∑

f

T (f)b†fbf .

We calculate the potential energy in accordance with the acceptedapproximation. Namely, in the expression

ψ†(q)ψ†(q′)ψ(q)ψ(q′) =( a†0√

V+ ϑ†(q)

)

×( a†0√

V+ ϑ†(q′)

)( a0√V

+ ϑ(q))( a0√

V+ ϑ(q′)

)

we omit the terms beginning with the cubic in ϑ and ϑ†. Then we have

Hpot =Φ0

1

2

N20

V+N0

V

∑f =0

b†fbf

+b2

2V

∑f =0

ν(f )b†fb†−f



+(b∗)2

2V

∑f =0

ν(f )bfb−f +N0

V

∑f =0

ν(f )b†fbf .

Since the following relations hold here∑f =0

b†fbf =∑f =0

Nf = N −N0,

in the accepted approximation we have

1

2

N20

V+N0

V

∑f =0

b†fbf =1

2

N2

V,

and therefore

H =N0

2VΦ0 +

b2

2V

∑f =0

ν(f )b†fb†−f +

(b∗)2

2V

∑f =0

ν(f )bfb−f

+N0

V

∑f =0

ν(f )b†fbf +∑

f

T (f)b†fbf .

We express the operators bf and b†f in terms of the operators ξf and ξ†f tofind finally

H = H0 +∑f =0

E(f )nf , nf = ξ†fξf , (15)

where

H0 =1

2

N2

V+ Φ0

∑f =0

1

2

(E(f ) − T (f ) − N0

Nν(f )

)

=1

2

N2

V+

V

2(2π)3

∫E(f) − T (f) − N0

Nν(f )

df . (16)

Thus, the total energy of non-ideal gas under consideration consists ofthe energy of the ground state H0 and the sum of individuals energies ofseparate quasiparticles. The quasiparticles do not, apparently, interact witheach other and form an ideal Bose gas.

It is easy to see that the absence of interaction between the quasiparticlesis due to the approximation used, where the terms beginning with a cubicin ξf and ξ†f have been omitted. Therefore, the result obtained is relevant


12 N. N. BOGOLUBOV

only for weakly excited states. Had we taken into account either the omittedcubic terms in the expression for energy or, respectively, the quadratic termsin Equations (13) as a small perturbation, we would find a weak interactionbetween quasiparticles which is responsible for approaching the state ofstatistical equilibrium in the system.

Proceeding to a study of the state of statistical equilibrium, we prove thatthe total momentum of quasiparticles

∑f fnf is conserved. With this aim

in view we consider the components of the total momentum of the system ofmolecules. We have

∑(1≤i≤N)

pαi =

∫ψ†(q)

−i∂ψ

†(q)

∂qα

dq =

∑f

fαa†faf =∑

f

fαb†fbf ,

and hence, due to transformation formulae (12), we see that

∑(1≤i≤N)

pαi =

∑f

fα(ξ†f + L∗

fξ−f)(ξf + Lfξ†−f)

1 − |Lf |2 .

But in view of the fact that Lf and L∗f are invariant with regard to the

replacement of f by −f , we can write

∑f

fαL∗

fξ−fξf

1 − |Lf |2 =∑

f

fαLfξ

†fξ

†−f

1 − |Lf |2 = 0,

∑f

fα|Lf |2ξ−fξ

†−f

1 − |Lf |2 =∑

f

fα|Lf |2(ξ†−fξ−f − 1)

1 − |Lf |2 = −∑

f

fα |Lf |21 − |Lf |2 ξ

†fξf ,

and therefore ∑(1≤i≤N)

pαi =

∑f

fαnf .

Thus, the total momentum of the system of molecules is equal to that ofthe system of quasiparticles. Since the total momentum of the system ofmolecule is conserved, the sum ∑

f

fnf

is really an integral of motion.



It is easy to see that the total number of quasiparticles∑

f nf is notconserved. They can be created and annihilated. Therefore, with the aidof the usual reasoning we see that in the state of statistical equilibrium theaverage occupation numbers nf (f = 0) are given by

nf =A exp

(E(f) − fu

θ

)− 1

−1

, A = 1, (17)

where θ is the temperature and u is an arbitrary vector. By the way, thelength of this vector must be limited from above. Indeed, since the averageoccupation numbers must be positive, for all f = 0 the following inequalitymust hold

E(f ) > fu

which yield the inequality

E(f ) > |f | |u|.But by virtue of the properties of the function E(f) established above, theratio

E(f)/|f |is a continuous positive function of |f | which takes the value c > 0 at |f | = 0and grows with |f |/2m as |f | → ∞. Therefore, the ratio considered has astrictly positive minimum value. Thus, the condition for positivity of thenumbers nf is equivalent to the inequality

|u| ≤ minE(f )/|f |. (18)

If for small momenta the function E(f) had decreased in proportion to thesquare root of the momentum but no to the momentum itself, as the kineticenergy of a molecule does, the right hand side of the inequality obtainedwould be equal to zero and zero would be the only possible value for u.However, in the case considered the u may be arbitrary, the only restrictionbeing that its length must be sufficiently short.

Note that formula (17) describes a distribution of momenta ofquasiparticles in the gas such that it moves as an entity with the velocity u.First we have chosen a framework such that in the ground state thecondensate, i.e. the system of molecules, is at rest. Had we proceededto a framework in which in the ground state the gas of quasiparticles isat rest, we would, conversely, observe the motion of the condensate with


14 N. N. BOGOLUBOV

velocity u. Since this relative motion is stationary motion in the state ofstatistical equilibrium in the absence of external fields, we see that it is notaccompanied by friction and is, therefore, the phenomenon of superfluidity.c

We have already seen that at small momenta the energy of thequasiparticle is approximately equal to c|f | where c is the velocity of sound.Therefore, at small momenta the quasiparticle is nothing but a phonon.With increasing momentum when the kinetic energy T (f) becomes largein comparison with the coupling energy of the molecule, the energy of thequasiparticle transforms continuously into the individual energy T (f ) of themolecule.

Thus, it is not possible to speak of the subdivision of the quasiparticlesinto two different sorts, phonons and rotons.

Now we consider the distribution of momenta in a system of moleculesfor the state of statistical equilibrium. We introduce a function w(f) byrequiring that Nw(f) df is the average number of particles with momentafrom the elementary volume df in the momentum space. This function is,obviously, normalized in the sense that

∫w(f) df = 1. (19)

Now let F (f) be an arbitrary continuous function of momentum. Thenthe average value of the dynamical variable∑

(i≤i≤N)

F (pi)

is

N

∫F (f)w(f) df . (20)

cIf we take a framework in which the condensate moves with velocity u, it is not difficultto see that the energy of the considered system of molecules is

H =∑

f

E(f ) − fu

nf + H0 + Mu2/2.

In view of Landau’s speculations from the above mentioned paper this expression makesthe property of superfluidity obvious. Indeed the appearance of elementary excitations isenergetically unfavorable, since it is accomplished by increase of energy.



On the other hand, this average is equal to

∑f

F (f)N f =V

(2π)3

∫F (f)Nf df . (21)

Thus, if we compare Equations (20) and (21), we find

w(f) =ν

(2π)3N f =

ν

(2π)3b†fbf .

Then expressing the operators bf and b†f in terms of ξf and ξ†f we obtain

w(f) =V

(2π)3

(1 − |Lf |2

)−1(ξ†f + L∗

fξ−f)(ξf + L∗fξ

†−f)

=ν

(2π)3

nf + |Lf |2(n−f + 1)

1 − |Lf |2 , (22)

where in view of Equation (17)

nf =

exp(E(f) − fu

θ

)− 1

−1

. (23)

The obtained expression (22) for the probability density is valid onlyfor f = 0. Therefore due to the normalization condition (19), the generalexpression for the probability density for the molecule momenta is

w(f) = Cδ(f) +ν

(2π)3

nf + |Lf |2(n−f + 1)

1 − |Lf |2 (24)

where Cδ(f ) is the Dirac delta-function and C is the number determined bythe equality

C = 1 − ν

(2π)3

∫nf + |Lf |2(n−f + 1)

1 − |Lf |2 df . (25)

The value of C is, obviously, equal to N0/N , since CN is the average numberof molecules with zero momentum.

In the formulae obtained, on the basis of Equation (11) we have

|Lf |21 − |Lf |2 =

(N0

Vν(f )

)2

2E(f )E(f ) + T (f) +

N0

Vν(f )

,


16 N. N. BOGOLUBOV

1

1 − |Lf |2 =E(f ) + T (f) +

N0

Vν(f )

2E(f), (26)

and consequently at zero temperature the momentum probability density is

w(f) = Cδ(f) +ν

(2π)3

(N0

Vν(f )

)2

2E(f)E(f ) + T (f) +

N0

Vν(f )

where

1 − C =ν

(2π)3

∫ (N0

Vν(f )

)2

2E(f)E(f ) + T (f) +

N0

Vν(f )

df . (27)

Thus, at zero temperature as well, only a part of the molecules possessexactly zero momentum. The rest are continuously distributed over the wholespectrum of momenta.

In view of above remarks, the approximate method used is applicableonly while the following inequality holds

(N −N0) = 1 − C 1.

Therefore, to ensure the smallness of the integral (27), the interactionbetween the molecules should be sufficiently small.

Now let us determine how the smallness of the interaction should beunderstood. We put

Φ(r) = Φm F( rr0

)where F (ρ) is a function such that it and its derivatives take values of theorder of unity for ρ ∼ 1 and vanish rapidly for ρ→ ∞. Then we have

ν(f ) = Φm r30ω

( |f |r0

)

where ω(x) is a function taking values of ∼ 1 for x ∼ 1 and vanishing rapidlyas x→ ∞.



If in equation (27) we proceed to dimensionless variables and reduce thethree-dimensional integral to a one-dimensional one, we find

N −N0

N=

ν

r30

η1

(2π)2

∞∫

0

ηω2(x)x dx

α(x)xα(x) + x2 + ηω(x) (28)

where

α(x) =(x2 + 2ηω(x)

)1/2,

η =r30N0

VΦm

(

2

2mr20

)−1

∼ r30

νΦm

(

2

2mr20

)−1

.

It is not difficult to see that for small η the integral in the right-hand side ofEquation (28) is of the order of

√η and the condition for applicability of the

method considered is given by the inequality

η 1,(ν/r3

0

)η3/2 1,

that isr30

νΦm

2

2mr20

,(r3

0

ν

)1/3

Φm 2

2mr20

. (29)

For temperatures different from zero, similar consideration of the generalformula (24) will result in an auxiliary condition for the weakness of theinteraction requiring that the temperature should be low compared with thetemperature of the λ-point.

We see that the condition for the smallness of interaction in the form ofexpression (29) automatically excludes the possibility for taking into accountshort-range repulsive forces, since for that it would be necessary to considerintensive growth of the function Φ(r) as r → ∞. However, it is not difficultto modify the results obtained here in order to extend them over the morerealistic case of a gas of low density with molecules of a finite radius.

Indeed, in our final formulae, the potential Φ(r) enters only the expression

ν(f ) =

∫Φ(|q|) exp

−ifq

dq (30)

proportional to the amplitude of the Born probability of pair collision. Sinceat low density the interaction between molecules is mainly realized via pair


18 N. N. BOGOLUBOV

collisions, expression (30) should be replacedd by an expression proportionalto the amplitude of the exact probability of pair collision. In other words,we should put

ν(f ) =

∫Φ(|f |)ϕ(q,f) dq, (31)

where ϕ(q,f) is the solution of the Schrodinger equation for the relativemotion of a pair of molecules

−2

m∆ϕ+

Φ(|q|) −E

ϕ = 0

which behaves like exp−ifq/ at infinity.The replacement of Equation (30) by Equation (31) in the expression for

E(f ) will result in formulae valid for gases of low density. Therefore, thecondition for existence of superfluidity ν(0) > 0 will be written, for instance,in the form ∫

Φ(|f |)ϕ(|q|) dq > 0 (32)

where ϕ(|q|) is the spherically symmetric solution of the equation

−2

m∆ϕ+ Φ(|q|)ϕ = 0

approaching unity at infinity.In order to connect the inequality (32) with the condition for the

thermodynamic stability, as this has been done above, we calculate theleading term in the expansion of the gas free energy at zero temperaturein powers of density. Since at zero temperature the free energy coincideswith the internal energy, we have the following expression for the energy permolecule

E = T +1

2ν

∫Φ(|q|)g(|q|) dq (33)

where T is the average kinetic energy of a molecule, g(r) is the moleculardistribution function, approaching unity as r → ∞.

On the other hand, but the virial theorem, the pressure ρ can bedetermined from the formula

pν =2

3T − 1

6

∫Φ′(|q|) |q| g(|q|) dq. (34)

dI was kindly informed of this important fact by L.D. Landau.



Now note that the leading term in the expression of the molecular distributionfunction at zero temperature in powers of the density is, obviously, equalto ϕ2(|q|). Therefore, omitting in Equations (33) and (34) the termsproportional to the square root of the density, we find

E = T =1

2ν

∫Φ(|q|)ϕ2(|q|) dq,

pν =2

3T − 1

6

∫Φ′(|q|)ϕ2(|q|) dq.

Then, taking into account that

pν = −ν ∂E∂ν

we obtain the equation for determining the leading term in the expressionfor T . After calculations we find

E =1

2ν

∫Φ(|q|)ϕ(|q|) dq =

ν(0)

2ν, p =

ν(0)

2ν2.

Thus, in the considered case of a gas of low density, the condition forthe existence of superfluidity (32) is equivalent to the usual condition forthermodynamic stability of a gas at zero temperature, i.e.,

∂p

∂ν< 0.

As an example, let us consider a model where the molecules are idealrigid spheres of diameter r0, so that

Φ(r) = +∞, r < r0,

Φ(r) = +o, r > r0.

After simple calculations we find

ν(0) = 4π

2r0m

.

If we suppose that there is weak attraction between spheres and put

Φ(r) = +∞, r < r0,


20 N. N. BOGOLUBOV

Φ(r) = εΦ0(r) < 0, r > r0,

where ε is a small parameter, we obtain, up to the terms of the order of ε2,

ν(0) = 4π

2r0m

+ 4π

∞∫

r0

r2Φ(r) dr.

Thus, in this model the appearance of superfluidity is due to the relationshipbetween the repulsion and attraction forces. The repulsion forces promotesuperfluidity, while the attraction forces hinder it.

Finally, it should be emphasized that it is apparently possible to proceedto consideration of realistic fluids in the frame of the theory developedhere, if such semi-phenomenological concepts as the free energy for weaklynonequilibrium states is used.


CHAPTER 2

QUASI-AVERAGES IN PROBLEMS OF STATISTICALMECHANICS

Part A. QUASI-AVERAGES

1. Green’s Functions, Defined with Regular Averages; AdditiveConservation Laws and Selection Rules

In modern statistical mechanics all newly developed methods involveobtaining an understanding and use of the methods of the quantum fieldtheory.

The introduction of Green’s functions is very fruitful, since, for example,with their help it is possible to generalize diagrammatic perturbationmethods in statistical mechanics and to perform partial summation ofexpressions. We shall, first of all, discuss the definition of Green’s functions.As is known these functions are expressed as linear forms in the averagevalues

〈. . .Ψ†(tj, xj) . . .Ψ(ts, ts) . . .〉 (1.1)

with coefficients made up of products of the step function θ(ti − tk). We willuse the following notation: x = (r, σ) represents all the space coordinates (r)and the series of discrete indices (σ), characterizing the spin of the particle,their type, etc; Ψ(t, x), Ψ†(t, x) – represent field operators in the Heisenbergpicture. These operators can be expressed in ”quasi-discrete” summation

Ψ(t, x) =1√V

∑(k)

akσ(t) ei(k·r);

Ψ†(t, x) =1√V

∑(k)

a†kσ(t) e−i(k·r) (1.2)

21


22 N. N. BOGOLUBOV

where a†kσ is the creation operator and akσ is the destruction operator, whichsatisfy the usual Bose or Fermi commutation relations. In these sums kα =2πnα/L; α = 1, 2, 3; nα – integer; V = L3 – the volume of the system.

The definition of the Green’s functions is independent of the natureof the system. They are linear forms of the averages of the type (1.1).The question of defining the Green’s function reduces to the definition ofexpression (1.1). Usually they are defined as averages with respect to theGibbs Grand Canonical ensemble, in accordance with which there alwaysappears the usual statistical mechanical limit V → ∞. That is,

〈. . .Ψ†(tj , xj) . . .Ψ(ts, ts) . . .〉

= limV →∞

Tr(. . .Ψ†(tj, xj) . . .Ψ(ts, ts) . . .e−

Hθ

)Tr e−

Hθ

(1.3)

where H is the total Hamiltonian of the system, and includes terms withchemical potential due to the conservation of the number of particles.

Let us agree to call the average values (1.1), which are defined bythe relationship (1.3), the regular averages, and the corresponding Green’sfunctions, the Green’s functions, constructed from regular averages.

Let us now draw our attention to the well known fact that the additivelaws of conservation lead to selection rules for regular averages and also forGreen’s functions.

For example we have the conservation law for the total number of particles

N =∑(k,σ)

a†kσakσ =∑(σ)

∫Ψ†Ψ dr,

so that [H,N ] = 0, where H is the total Hamiltonian of the system (includingthe term µN , where µ is the chemical potential. Whenever H = U†HU ,where U = eiϕN and ϕ is an arbitrary real number, then the Hamiltonian isinvariant under the gradient transformation of the 1st kind:

akσ → U †akσU = eiϕakσ.



Therefore we have

Tr(. . . a†kσ(t) . . . ak′σ′(t′) . . .e−H

θ

)= Tr

(. . . a†kσ(t) . . . ak′σ′(t′) . . .U †e−

Hθ U

)= Tr

(U. . . a†kσ(t) . . . ak′σ′(t′) . . .U †e−

Hθ

)= e−iϕn Tr

(. . . a†kσ(t) . . . ak′σ′(t′) . . .e−H

θ

)where n is the difference between the numbers of a and a† operatorsin the products . . . a†kσ(t) . . . ak′σ′(t′) . . .. From this, on the basis of thedefinition (1.3) we find

(1 − e−iϕn)〈. . . a†kσ(t) . . . ak′σ′(t′) . . .〉 = 0

and thus

〈. . . a†kσ(t) . . . ak′σ′(t′) . . .〉 = 0

if in the given product the number of creation operators is not equal to thenumber of destruction operators.

Since the Green’s functions are expressed as linear forms of regularaverages these same selection rules also hold for Green’s functions. Forexample,

〈T (. . . a†kσ(t) . . . ak′σ′(t′) . . .)〉 = 0, if n = 0.

Let us examine the selection rules derived from conservation of totalmomentum. The total momentum operator is

P =∑(k,σ)

a†kσakσ.

The law of conservation of total momentum gives

Tr[

P, U]e−

Hθ

= Tr

U

[e−

Hθ , P

] = 0.

Let us take

U = . . .Ψ†(tj, rj + ξ, σj) . . .Ψ(ts, rs + ξ, σs) . . .


24 N. N. BOGOLUBOV

and note that

∑α

1

i

∂

∂rα

〈U 〉 = 〈[ P, U ]〉 =Tr

[P, U

]e−

Hθ

Tr e−

Hθ

= 0.

Therefore, the average (1.1) do not change under the translation

rj → rj + ξ

where ξ is an arbitrary vector. Saying this differently the regularaverages (1.1) must be spatially homogeneous. Let us now make use of themomentum representation (1.2). We get

∑α

∂

∂rα

〈. . .Ψ†(tj, rj , σj) . . .Ψ(ts, rs, σs) . . .〉

=1

V n/2

∑(...kν , σν ...)

(∑α

kα

)〈. . . a†−kj , σj

(tj) . . . aks, σs(ts) . . .〉ei(k1r1+...+knrn) = 0

using translational invariance. This leads to the selection rule

〈. . . a†−kj , σj(tj) . . . aks, σs(ts) . . .〉 = 0, if k1 + . . .+ kn = 0.

Such relationships are also satisfied for the Green’s functions. We have forexample

〈T (. . . a†−kj , σj(tj) . . . aks, σs(ts) . . .)〉 = 0, if k1 + . . .+ kn = 0.

An analogous situation arises when the laws of conservation of total spin andother additive dynamical variables are taken into account.

The selection rules become more descriptive if one introduces adiagrammatic presentation of perturbation theory. For the purpose offormulating perturbation theory the total Hamiltonian is divided into twoparts, H0 and H1,

H = H0 +H1

where the expansion is carried out “in powers of H1”. As a rule,a Hamiltonian is picked for H0 which corresponds to the “ideal gas”without interaction, and all interactions are included in H1. Note that byseparating H0 it is guaranteed that the above mentioned conservation laws



for additive variables are also satisfied for the “dynamical system in zerothapproximation” characterized by the Hamiltonian H0. In such a separationof H0, exactly the same selection rules are obtained for the hierarchy of theexact Green’s functions as for the hierarchy of of the zeroth approximationGreen’s functions. Let us consider the diagrammatic presentation usingeither the general Feynman diagrams for the case of zero temperature orthe corresponding diagrams of Matsubara, C. Bloch for θ > 0. In both casesthe diagrams are characterized by lines and loops. In each one of the loopsthe conservation laws are satisfied, and all allowed lines likewise satisfy theselaws.

Assume for example, that we have a system of particles with spin σ =±1/2 with the total number of particles conserved. Also, assume thatthere exist total momentum conservation laws a conservation law for thetotal z component of the spin. Then all ”contractions” i.e. all the zerothapproximation Green’s functions of the type

a†pσa†p′σ′ ; apσap′σ′ ; apσa

†p′σ′ ; a†pσap′σ′

when p = p′ or σ = σ′, are exactly equal to zero.The only allowed lines of the particles under consideration will be the

lines which correspond to the contractions

apσa†pσ; a†pσapσ

conserving p and σ.The same situation exists for the “wide”, or “summed” lines correspond

to the exact Green’s functions. The exact Green’s functions of type

a†pσ, a†p′σ′ , apσ, ap′σ′ , apσ, a

†p′σ′ , a†pσ, ap′σ′

when p = p′ or σ = σ′, are all equal to zero. The only allowed lines willbe the “wide” lines characterized by the Green’s functions apσ, a

†pσ ,

a†pσ, apσ conserving p and σ.As demonstrated, the selection rules considerably simplify the topological

structure of the diagrams and the actual calculations.

2. Degeneracy of the Statistical Equilibrium States;Introduction of Quasi-averages

In applying the diagram technique one cannot forget that it is only a handypresentation of the ordinary perturbation theory, and that one encounters


26 N. N. BOGOLUBOV

similar difficulties and sometimes complex problems about the convergenceof the resultant expansion.

At the present time the convergence can be proven for only a numberof simplest models. For more realistic problems one can only assumethe presence of a certain correspondence between the real solutions andthe resultant formal expansions. Such formal expansions will be used, inparticular, for the formation of approximate solutions. A very effectiveprocedure here is the partial summation of the (in some sense) “main”terms, which is easily performed with the help of the diagram technique.If the perturbation is “weak enough” and is characterized by a smallparameter, the approximate solution take the form of asymptotic expressions.When the smallness parameter of the perturbation has a zero value, the“correctness” of the partial summations can be ascertained. Although themathematics of these procedures is not fully justified, nevertheless, in manyimportant problems one can obtain physically correct results, not only forthe asymptotic formulas, but also for results pertaining to the qualitativeproperties of exact solutions. However, in a number of cases, for examplein the theory of super conductivity and in the theory crystalline state, theordinary diagram technique does not lead to physically correct results. Inour opinion it would be not enough to limit the expansion by referring tosuch formal reasons as the lack of convergence, complexity of the analyticalstructure pertaining to the small parameter, etc. It follows that we haveto look for physically basic, constructive solutions of the newly appearingproblems.

Let us now turn our attention to the well known quantum mechanicalproblems of degeneracy. When investigating the problem of findingeigenfunctions in quantum mechanics one discovers that perturbation theoryin its regular form, developed for non-degenerate cases, cannot be directlyapplied to problems having degeneracy. It is essential to modify it first. Inproblems of statistical mechanics we always have a case of degeneracy dueto presence of the additive conservation laws.

However, at first glance it might seem that the degeneracy is notimportant and can in practice be neglected. Actually in these quantummechanical problems linear combination of different eigenfunctions cancorrespond to one eigenvalue of the energy. The eigenfunction in this casecontains undetermined constants.

In statistical mechanics the average value of any dynamical variable U



is always unambiguously determined:

〈U 〉 =Tr

(U e−

Hθ

)Tr e−

Hθ

.

It follows that the Green’s functions constructed from regular averages mustlikewise be unambiguously determined. From this it might seem that whenstudying statistical equilibrium, for example, with the help of the diagramtechnique, one need not take into account the presence of degeneracy.However, in reality, the situation is not so simple. In order to form anintuitive feeling for the nature of the problem here let us look at the case ofan ideal isotropic ferromagnetic.

For the definition let us assume a dynamical system, characterized by theHeisenberg model Hamiltonian

H = −1

2

∑(f1, f2)

I(f1 − f2)(Sf1 · Sf2) (2.1)

where (f) represents space points, corresponding to the sites of the

crystalline lattice (occupying volume V ), Sf is the spin vector with the usualcommutation rules, I(f1 − f2) is a non-negative number. For example wemay assume that I(f1 − f2) is greater that zero when the sites f1, f2 are“nearest neighbors”.

For this dynamical system each of the components of the total spin vectorS =

∑(f)

Sf is an integral of the motion. We also have

SxSy − SySx = iSz,

SySz − SzSy = iSx,

SzSx − SxSz = iSy.

From this it follows that

iTr(Sze

−Hθ

)= iTr

((SxSy − SySx)e

−Hθ

).

But, in so far as Sx commutes with H , we obtain

Tr(SySxe

−Hθ

)= Tr

(Sxe

−Hθ Sy

)= Tr

(SxSye

−Hθ

)


28 N. N. BOGOLUBOV

and thus

Tr(Sze

−Hθ

)= 0.

Similarly we find

Tr(Sxe

−Hθ

)= 0, Tr

(Sye

−Hθ

)= 0.

Introducing the magnetization vector,

M = µ1

V

∑(f)

Sf = µ1

VS

we have

Tr(

M e−Hθ

)= 0

and therefore:

〈 M 〉 = limV →∞

Tr(

M e−Hθ

)Tr e−

Hθ

= 0. (2.2)

The regular average of the vector M is equal to zero. This corresponds to theisotropy of this dynamical system with respect to the spin rotation group.

Let us note that expression (2.2) is correct for all temperatures θ, and inparticular, for temperatures below the Curie point.

Let us now investigate specifically this last case. As is known, when themagnitude of the magnetization vector is different form zero its directioncan be taken arbitrary. In this sense the statistical equilibrium state in thissystem is degenerate.

Now let us include an external magnetic field Be (B > 0, e2 = 1)changing the Hamiltonian (2.1) to the Hamiltonian

HBe → H +B(e · M )V. (2.3)

Then, taking into account the characteristic property of isotropicferromagnetic when the temperature is below the Curie point, we can see

〈 M 〉 = eMB.

Furthermore, MB will approach the limit, different from zero when theintensity B of the external magnetic field approaches zero. From the formal



point of view we have here an “instability” of regular averages. When theterm B(e · M )V is added to the Hamiltonian (2.1) with an infinitesimally

smalle B the average 〈 M 〉 obtains a limit which is different from zero, forexample,

em; where, m = limB→0

MB.

Let us now introduce the concept of “quasi-averages” for a dynamical systemwith the Hamiltonian (2.1).

Take any dynamical variable A, which is a linear combination of theproducts

Sα1

f1(t1) . . . S

αr

fr(tr)

and define the quasi-average ≺ A of this variable

≺ A = limB→0

〈A〉Be

where 〈A〉Be is the regular average of A with the Hamiltonian HBe. Inthis manner the presence of degeneracy in the problem is reflected in thedependence of the quasi-averages on the arbitrary direction e.

It is not difficult to see that

〈A〉 =

∫≺ A de. (2.4)

Now, it is understood that for the description of the case under consideration,(the degenerate statistical equilibrium state), the quasi-averages are moreconvenient, more “physical” than the regular averages. These latter expressthe same quasi-averages, only they are averaged in all directions of e.

Further note that the regular averages

〈Sα1

f1(t1) . . . S

αr

fr(tr)〉

must be invariant with respect to spin rotation group. The correspondingquasi-averages

≺ Sα1f1

(t1) . . . Sαrfr

(tr) (2.5)

will posses only the property of covariance; when there is a rotation ofthe spin a similar rotation must be made on the vector e so that theexpression (2.5) does not change.

eWhen we talk about an infinitesimally small B we always mean that first the statisticalmechanical limit V → ∞ is carried out and then B approaches zero.


30 N. N. BOGOLUBOV

In such a way the quasi-averages will not have the selection rules whichfor regular averages depended upon their invariance with respect to thespin rotation group. The arbitrary direction e, which is the direction ofthe magnetization vector, characterizes the degeneracy of the statisticalequilibrium state under consideration. In order to remove the degeneracythe direction e must be fixed. We will pick the λ-axis for this direction.Then all the quasi-averages will become definite numbers. Exactly the sametype of averages are encountered in the theory of ferromagnetism.

In other words, we can remove the degeneracy of the statisticalequilibrium state with respect to spin rotation group by including in theHamiltonian H the additional invariant member BMzV with an infinitelysmall B.

Let us now look at another example of degeneracy, this time turning tothe theory of the crystalline state. Consider a dynamical system with spin-less particles having a binary interaction characterized by a Hamiltonian ofthe ordinary type

H =∑(p)

( p2

2m− µ

)a†pap +

1

2V

∑(p1, p2, p ′

1, p ′2)

a†p1a†p2ap ′

1ap ′

2

× δ(p1 + p2 − p ′1 − p ′

2 )ν(|p1 − p ′1 |)p (2.6)

in which δ(p) is the discrete δ function, ν(p) is the Fourier transform ofthe interaction potential energy φ(r) of a pair of particles. Assume that thistype of interaction is such that our dynamical system must be in a crystallinestate when the temperature is low enough θ < θcr. Consider the observedparticle density ρ(r), which evidently, must be a periodic function of r withthe period of the crystal lattice. It would be natural to consider that ρ(r) isequal to the regular average of the operator density Ψ†(r)Ψ(r)

ρ(r) = 〈Ψ†(r)Ψ(r)〉 =1

V

∑(q)

∑(k)

〈a†kak+q〉

ei(q·r).

This, however, is not true.Actually, in the present case the total momentum

p =∑(k)

ka†kak

is conserved and thus, as mentioned in 1. we have the selection rule

〈a†kak+q〉 = 0, if q = 0,



from which follows

〈Ψ†(r)Ψ(r)〉 =1

V

∑(k)

〈a†kak+q〉 =N

V= const.

In such a way the value of the regular average of the operator densitycannot be equal to the periodic function ρ(r). It is clear that this situation isbrought about by the conservation of momentum in the statistical equilibriumstate considered. Actually the crystal lattice, as a whole, can be arbitraryplaced in space. In particular, our Hamiltonian possesses translationalinvariance, and thus the lattice can be arbitrary translated.

No special position of the crystal lattice is preferred in space, and whenwe take a regular average we thereby average over all possible positions of thislattice. In order to remove the degeneracy and introduce the quasi-averagewe must include in the Hamiltonian the term

ε

∫U(r)Ψ†(r)Ψ(r) dr; ε > 0, ε→ 0 (2.7)

corresponding to the infinitely small external filed εU(r). We will denote theresulting Hamiltonian by Hε. As U(r) we will take the periodic function of rwith the periodicity of the lattice in such a way that the external field εU(r)removes the degeneracy, thus fixing the position of our crystal in space.

In as much as we are inherently investigating only the physically stablecases, it is clear that the inclusion of the infinitely small external field canonly slightly change the physical properties of the dynamical system underconsideration. In as much as the position of the crystal is now fixed inspace, taking the regular average of the density operator Ψ†(r)Ψ(r) with theHamiltonian Hε (with infinitesimally small ε) actually produces the averagefor the system with the initial Hamiltonian H , but without the extra averageover the position of the whole crystal lattice in space.

In this way we will obtain the observed density distribution of particlesρ(r). Let us formally define quasi-averages by placing

≺ . . .Ψ†(tj , rj) . . .Ψ(ts, rs) . . . = limε→0

〈. . .Ψ†(tj, rj) . . .Ψ(ts, rs) . . .〉Hε.

Thus, as was just indicated

≺ Ψ†(r)Ψ(r) = ρ(r).


32 N. N. BOGOLUBOV

Noting, that

≺ Ψ†(r)Ψ(r) =1

V

∑(q)

∑(k)

≺ a†kak+q

ei(q·r)

we see that the quasi-averages

≺ a†kak′ ; k′ = k (2.8)

cannot all be equal zero.In this way the selection rules, determined by the law of conservation of

the total momentum, are not satisfied for these quasi-averages.Now note that the quasi-averages generally depend upon a series of

arbitrary parameters, for example, upon an arbitrary vector ξ. Actually,if we replace the function U(r) by an equally acceptable function U(r + ξ)then it is not difficult to show that the quasi-average (2.8) becomes

≺ a†kak′ ei(k−r ′)ξ.

The quasi-average become well defined, when we fix the function U(r). Upto this point we have investigated cases involving the degeneracy of thestatistical equilibrium state, connected with the law of conservation of thetotal spin or the total momentum. In both cases the degeneracy can beremoved and adequate physical quasi-averages can be introduced by includingthe appropriate infinitesimally small external field.

Let us now turn to those cases when the degeneracy is connected withthe law of conservation of the total number of particles. Let us start withelementary example of condensation of a Bose-Einstein ideal gas. In order toconveniently extract the condensate we shall take the ideal has Hamiltonianin the form

H = −µa†0a0 +∑|k|>ε

( k2

2m− µ

)a†kak, ε > 0. (2.9)

Here we shall let ε approach zero after taking the limit V → ∞. We shallfind that the average number for certain momentum states

N0 =

exp(−µθ

)− 1

−1

,

Nk =

exp(1

θ

( k2

2m− µ

))− 1

−1

, |k| > ε.



will become large. From this it is seen that µ < 0. Expressing the totalnumber of particles by N we obtain

N

V=N0

V+

1

V

∑|k|>ε

exp

(1

θ

( k2

2m− µ

))− 1

−1

,

µ = −θ ln(1 +

1

N0

).

(2.10)

Let us consider the Bose-Einstein condensation, where n0 = limV →∞

N0

Vis

the thermodynamic limit ofN0

Vand is different from zero. In this case, when

taking the limit in the expression (2.10), we shall find

n = limV →∞

N

V= n0 +

1

(2π)3

∫

|k|>ε

dk

expk2/2mθ

− 1.

Here, letting the “cut-off momentum” ε approach zero, we shall finally obtain

n = limV →∞

N

V= n0 +

1

(2π)3

∫dk

expk2/2mθ

− 1. (2.11)

Thus we obtain the condition of the condensation in its usual form,

1

(2π)3

∫dk

expk2/2mθ

− 1< n.

It is not hard to see that the operatora†0a0

Vis asymptotically equal to the C

number:a†0a0

V∼ n0. (2.12)

Let us consider the amplitudes

a†0√V,

a0√V

;

which commute with all amplitudes ak, a†k, k = 0. Since the commutator

[ a0√V,a†0√V

]=

1

V


34 N. N. BOGOLUBOV

is infinitesimally small (V → ∞) we can similarly consider the amplitudesas C numbers, while in view of (2.12)

a0√V

∼ √n0e

iα,a†0√V

∼ √n0e

−iα. (2.13)

The real phase angle α is arbitrary. This is due to the gradient invariance ofthe 1st type, specified by the law of conservation of the number of particles,and indicates the appearance of a degeneracy.

Let us consider the regular averages

⟨ a†0√V

⟩,

⟨ a0√V

⟩

and note that because of the selection rules they are exactly equal to zero.Note also that the regular averages include an additional averaging over theangle α. In order to introduce quasi-averages and to remove degeneracy weshall include the following term in the Hamiltonian H :

−ν(a†0eiϕ + a0e−iϕ)

√V , ν > 0.

Assume

Hν, ϕ = H − ν(a†0eiϕ + a0e

−iϕ)√V (2.14)

where ϕ is some fixed angle.To reduce (2.14) to a diagonal form we have to perform a canonical

transformation on the amplitudes a0, a†0, while keeping the other amplitudes

ak, a†k fixed,

a0 = −νµeiϕ

√V + a′0

a†0 = −νµe−iϕ

√V + a†′0 (2.15)

we find

H = −µa†′0 a′0 +∑( k2

2m− µ

)a†kak +

ν2

µV. (2.16)

Let us now assume that:

µ = − ν√n0

. (2.17)



Then we have

〈a′0〉ν, ϕ = 0, 〈a†0′〉ν, ϕ = 0,

〈a†0′a′0〉ν, ϕ =

exp( ν

θ√n0

)− 1

−1

,

〈a†k ′a′k〉ν, ϕ =

exp(1

θ

( ν√n0

+k2

2m

))− 1

−1

,

where 〈. . .〉ν, ϕ designates the average for the Hamiltonian Hν, ϕ. Because ofthis, on the basis of (2.15) and (2.17) we have

Nk =

exp(1

θ

( ν√n0

+k2

2m

))− 1

−1

,

N0 = n0V +

exp( ν

θ√n0

)− 1

−1

, (2.18)

andN

V=N0

V+

1

V

∑(k)

exp

(1

θ

( ν√n0

+k2

2m

))− 1

−1

. (2.19)

Due to the presence of the “compensation” termν√n0

in the exponent, we

no longer have to include the “cut-off” momentum ε. By direct transition tothe thermodynamic limit in expression (2.19) we find

n = n0 +1

(2π)3

∫exp

(1

θ

( ν√n0

+k2

2m

))− 1

−1

dk. (2.20)

Let us further note that⟨( a†0√V

−√n0 e−iϕ

)( a0√V

−√n0 eiϕ

)⟩ν, ϕ

=1

V〈a†0′a′0〉 = lim

V →∞1

V

exp

( ν

θ√n0

)− 1

−1

= 0.

Consequently we have, asymptotically

a0√V

∼ √n0e

iϕ,a†0√V

∼ √n0e

−iϕ,


36 N. N. BOGOLUBOV

i.e. for the system with a HamiltonianHν,ϕ the amplitudes for the condensateare asymptotically fixed C numbers. The results in (2.18), (2.19), (2.20) showthat, by performing the limit ν → 0 (after the limit V → ∞), we arrive atthe usual result of the theory of condensation for an ideal Bose gas.

Let us introduce the quasi-averages:

≺ . . . = limν→0

〈. . .〉ν,ϕ.

Then we have

≺ a0√V

=√n0 eiϕ, ≺ a†0√

V=

√n0 e−iϕ.

As we see, the selection rules specified by the particle conservation law are notsatisfied for quasi-averages. We also see that the quasi-averages depend uponthe phase angle ϕ which we can arbitrary fix. Let us choose ϕ = 0. Thenthe quasi-averages become specific values. In other words the degeneracy isremoved by adding to the Hamiltonian H the infinitesimally small term

−ν(a0 + a†0)√V .

In the present case the quasi-averages differ from the regular averages only inthe amplitudes of the condensate. This is due to the fact that we have an idealgas without interaction. With the presence of interactions this difference isextended to the other amplitudes. There will appear, for example, quasi-averages different from zero of the type

≺ aka−k .

Let us now examine a more complex example. Let us consider a modelsystem with the Hamiltonian

H =∑(f)

T (f)a†faf − 1

2V

∑(f, f ′)

λ(f)λ(f ′)a†fa†−fa−f ′a−f ′ , (2.21)

which is studied in conjunction with the theory of superconductivity [1, 2].Here we shall use the following notation:

f = (p, s), − f = (−p, −s), s = ±1,

T (f) =p2

2m− µ, µ > 0,



λ(f) =

Jε(s),∣∣∣ p2

2m− µ

∣∣∣ < ∆,

0,∣∣∣ p2

2m− µ

∣∣∣ > ∆,

where af , a†f are the usual Fermi amplitudes. This example is interesting

because it is not trivial. The equations of motion for the Hamiltonian (2.21)cannot be integrated exactly. However, asymptotically exact formulas (withV → ∞) can be obtained for the Green’s functions of all orders.

Let us briefly present results pertaining to the above problem which havebeen published [3, 4]. Let us choose an “approximate Hamiltonian”

H0 =∑(f)

T (f)a†faf − 1

2

∑(f)

λ(f)C∗afa−f + C a†fa

†−f

+

1

2|C|2V, (2.22)

in which C is a c-number (complex in general) defining the non-trivialsolution (C = 0) to the equation:

C =1

V

∑(f)

λ(f)〈a−faf〉(V )H0. (2.23)

Here:

〈. . .〉(V )H0

=Tr(. . .) e−H0/θ

Tr e−H0/θ.

In accordance with the previous notation, the regular average 〈. . .〉H isdefined as the limit

limV →∞

〈. . .〉(V )H0

Since H0 is a quadratic form with respect to the operators a, a†, to withina constant term we can reduce it to “diagonal form” by means of a linearcanonical transformation. We shall introduce, for this purpose new Fermi-amplitudes α and α†, defined by

αf = afuf + a†−fvf , α†f = a†fuf + a−fvf ,

where

uf =1√2

(1 +

T (f)

E(f)

)1/2

, vf =−ε(s)√

2

C

|C|(1 − T (f)

E(f)

)1/2

,


38 N. N. BOGOLUBOV

E(f) =

√λ2(f)|C|2 + T 2(f).

Then

H0 =∑(f)

E(f)α†fαf +K, (2.24)

where K is the constant

K =1

2V

|C|2 − 1

V

∑(f)

[E(f) − T (f)

].

From this result we obtain

〈α†fαf〉(V )

H0=

exp

(E(f)

θ

)+ 1

−1

,

and

〈α−fαf 〉(V )H0

= ufvf

1 − exp(E(f)

θ

)1 + exp

(E(f)

θ

)= −ufvv tanh

(E(f)

2θ

)=

λ(f)

2E(f)C · tanh

(E(f)

2θ

).

Using relationship (2.23) we obtain

1 − 1

2V

∑(f)

λ2(f)

2E(f)C · tanh

(E(f)

2θ

)C = 0.

In such a way the desired non-trivial solution for C is given by the equation

1 =1

V

∑(f)

λ2(f)√λ2(f)|C|2 + T 2(f)

tanh(√

λ2(f)|C|2 + T 2(f)

2θ

). (2.25)



Taking the limit V → ∞, we have

1 =1

(2π)3

∫λ2(p)dp√

λ2(p)|C|2 + T 2(p)tanh

(√λ2(p)|C|2 + T 2(p)

2θ

). (2.26)

As is known this equation has a solution for θ less that a certain θcr. We shalllook at only such a case (θ < θcr). Let us also note that the equation (2.25)(or (2.26)) determines only the coefficient |C|, and that the phase of Cremains arbitrary.

Let us examine the average

〈. . . a†fj(tj) . . . afs(ts) . . .〉(V )

H0(2.27)

formed form the product of any number of operators a and a† (in any order).Since the Fermi-amplitudes a and a† are linearly expressed through the Fermi-amplitudes α and α†

af = ufαf − vfα†−f , a†f = ufα

†f − vfα−f ,

in terms of which H0 has the diagonal form (2.24), we see that the theoremof Wick and Bloch is applicable for the calculation of the expression (2.27).With this theorem these expressions can be written in the form of the sumof the products of “simple contractions”

〈a†f(t)af (τ)〉(V )H0

= u2f

eiE(f)(t−τ)

1 + eE(f)/θ+ |v|2f

e−iE(f)(t−τ)

1 + e−E(f)/θ,

〈af(t)a†f (τ)〉(V )

H0= u2

f

e−iE(f)(t−τ)

1 + e−E(f)/θ+ |v|2f

eiE(f)(t−τ)

1 + eE(f)/θ,

〈a−f(t)af (τ)〉(V )H0

= ufvf

eiE(f)(t−τ)

1 + eE(f)/θ− e−iE(f)(t−τ)

1 + e−E(f)/θ

,

〈a†f(t)a†−f (τ)〉(V )H0

= ufv∗f

eiE(f)(t−τ)

1 + eE(f)/θ− e−iE(f)(t−τ)

1 + e−E(f)/θ

, (2.28)


40 N. N. BOGOLUBOV

The last two expressions depend not only on the magnitude of C but alsoon its phase. Therefore, in general, the expression (2.27) can depend uponthe phase of C. It will be shown that this dependence is very simple. TheHamiltonian H0 is invariant with respect to the substitution

af → eiϕaf , a†f → e−iϕa†f , C → e2iϕC,

in which ϕ is an arbitrary (real) angle. Because of this we have

〈. . . a†fj(tj) . . . afs(ts) . . .〉(V )

H0

∣∣∣C=eiα|C|

= 〈. . . a†fj(tj) . . . afs(ts) . . .〉(V )

H0

∣∣∣C=|C|

e−inα/2, (2.29)

where n is the difference between the number of creation operators andannihilation operators in the products considered. Clearly n here can beconsidered even, since with an odd n we have the identity

〈. . . a†fj(tj) . . . afs(ts) . . .〉(V )

H0= 0.

Also note that in the case when n = 0 it follows from (2.29) that theaverage (2.27) does not depend on the phase of C. As is seen, theinvestigation of the system with the “approximate Hamiltonian” H0 iscompletely elementary. The corresponding equations of motion can beintegrated exactly. We have examined a system with the Hamiltonian H0

with the view toward proving [3, 4] the following important results:

If for the product

. . . a†fj(tj) . . . afs(ts) . . . (2.30)



the number n = 0, thenf

fIn order to clarify the basis of such a result, we present the following simpleconsideration. Note that the Hamiltonian H can be written in the form

H =∑(f)

T (f)a†faf − 12

∑(f)

λ(f)β†a−faf + a†fa†−fβ +

β†β2V, (1)

where

β =1V

∑(f)

λ(f)a−faf .

The equation of motion will then be:

idaf

dt= T (f)af − λ(f)a†−fβ,

ida†fdt

= −T (f)a†f + λ(f)β†af .

(2)

Further we note that

|βa†g − a†gβ| =∣∣∣ 2Vλ(g)a−g

∣∣∣ < 2V|λ(g)|,

|βag − agβ| = 0,

|β†β − ββ†| ≤ 2V

∣∣∣∑(f)

1Vλ2(f)

∣∣∣.In such a way all the commutators involving β, β† with themselves and with the operators

af , a†f are infinitesimally small having magnitudes of the order of1V

. Because of this, one

expects that the quantum operator nature of β and β† disappears in the limit V → ∞.Substituting in (a) and (b) the respective average values for β and β† we arrive at theproblem with the Hamiltonian H0(β = C). It is not hard to see that the operators β, β†

are very similar in their character to the operatorsa0√V

,a†0√V

found in the condensation

theory of a Bose gas. Both sets of operators have an arbitrary phase.In accordance with this situation the relations (2.31) are proven only for those

products (2.30) which do not depend upon this phase, i.e., which have n = 0. Themathematical proof is considerably simplified if we eliminate the arbitrariness of the phase,for example by including in the Hamiltonian H the term:

−νV (β + β†), ν > 0.


42 N. N. BOGOLUBOV

〈. . . a†fj(tj) . . . afs(ts) . . .〉(V )

H − 〈. . . a†fj(tj) . . . afs(ts) . . .〉(V )

H0→ 0, V → ∞.

(2.31)

Alternatively, the existence of the limit

limv→∞

〈. . . a†fj(tj) . . . afs(ts) . . .〉H0

is completely determined for any product (2.30). Further, if n = 0, thendue to selection rules with a Hamiltonian H which conserves the number ofparticles,

〈. . . a†fj(tj) . . . afs(ts) . . .〉(V )

H = 0,

and therefore

〈. . . a†fj(tj) . . . afs(ts) . . .〉H

=

〈. . . a†fj(tj) . . . afs(ts) . . .〉H0, n = 0,

0, n = 0.(2.32)

Likewise we can calculate the regular averages (2.32) of any order, and,consequently, we can calculate the Green’s functions for the model with theHamiltonian H. In addition, with any value of the number n, it can be provedthat:

≺ . . . a†fj(tj) . . . afs(ts) . . . H= 〈. . . a†fj

(tj) . . . afs(ts) . . .〉H0. (2.33)

Here, as before, the symbol ≺ . . . represent quasi-averages. As wasnoted earlier the second part of the equality (2.33) contains the factorexp(−inα/2). Because of this we have,

〈. . . a†fj(tj) . . . afs(ts) . . .〉H =

1

2π

2π∫

0

dα ≺ . . . a†fj(tj) . . . afs(ts) . . . H ,

i.e., the regular average is obtained from quasi-averages after the additionalaveraging over the arbitrary angle α is performed.



Just as in previously considered cases the quasi-averages can beintroduced by adding to the Hamiltonian infinitesimally small terms, whichremove the degeneracy. Let us take the Hamiltonian

Hν = H − ν

2

∑(f)

λ(f)a−faf + a†fa

†−f

, ν > 0, (2.34)

containing terms which remove the degeneracy with respect to gradientinvariance of the 1st type; that is terms which remove the conservation lawof the total number of particles.

Let us take the approximate Hamiltonian in the form

H0ν = H0 − ν

2

∑(f)


†−f

.

The quantity C introduced here is defined by the equation

C =1

V

∑(f)

λ(f)〈a−faf 〉(V )H0

ν

i.e.,

C =C + ν

2V

∑(f)

λ2(f)th

√λ2(f)(C + ν)2 + T 2(f)

2θ

√λ2(f)(C + ν)2 + T 2(f)

.

After the limit V → ∞, we obtain

C =C + ν

(2π)3

∫λ2(p)

th√

λ2(p)(C + ν)2 + T 2(p)

2θ

√λ2(p)(C + ν)2 + T 2(p)

dp.

We shall take for C that root of this equation which approaches the positiveroot of equation (2.26) when ν → ∞. Then one can prove that

〈. . . a†fj(tj) . . . afs(ts) . . .〉Hν = 〈. . . a†fj

(tj) . . . afs(ts) . . .〉H0ν.

Alternatively it is easy to be convinced that

〈. . . a†fj(tj) . . . afs(ts) . . .〉H0

ν→ 〈. . . a†fj

(tj) . . . afs(ts) . . .〉H0 , ν → 0


44 N. N. BOGOLUBOV

with C = |C|. Therefore

≺ . . . a†fj(tj) . . . afs(ts) . . . = lim

ν→0ν>0

〈. . . a†fj(tj) . . . afs(ts) . . .〉Hν

= 〈. . . a†fj(tj) . . . afs(ts) . . .〉H0, C = |C|.

If we had taken the Hamiltonian Hν,ϕ instead of Hν where

Hν, ϕ = H − ν∑(f)

λ(f)eiϕ a†fa

†−f + e−iϕa−faf

, ν > 0,

then we would have obtainedg

≺ . . . a†fj(tj) . . . afs(ts) . . . = lim

ν→0ν>0

〈. . . a†fj(tj) . . . afs(ts) . . .〉Hν

= 〈. . . a†fj(tj) . . . afs(ts) . . .〉H0, C = eiϕ|C|.

Thus, as one could have expected in the present case quasi-averages dependon the arbitrary phase angle ϕ. It is also essential that for the quasi-averageshere the selection rules, which are specified by the law of conservation of thenumber of particles, are not satisfied. In order to have well determined valuesfor quasi-averages we have to somehow fix this angle. Assume that ϕ = 0,i.e. let us agree to remove the degeneracy by including in the HamiltonianH infinitely small terms of the type:

−ν2

∑(f)


†−f

. (2.35)

Such a choice of the phase angle is convenient in that it make the values ofall the “simultaneous” quasi-averages of the type

≺ . . . a†fj(tj) . . . afs(ts) . . .

gAs is seen the regular average

〈. . . a†fj(tj) . . . afs(ts) . . .〉H =

12π

∫≺ . . . a†fj

(tj) . . . afs (ts) . . . H dϕ

suffers a discontinuity when we add to the Hamiltonian H infinitely small termsrepresenting pair sources:

−ν2

∑(f)

λ(f)eiϕa−faf + e−iϕa†fa

†−f



real. Also note that the result will not change if these additional terms (2.35)are written in more general form

−ν∑(f)

w(f)a−faf + a†fa

†−f

, ν > 0, (2.36)

where w(f) is a real, non-trivial, and fairly regular function.

In the above we dealt with products of field functions in the momentumrepresentation. A similar situation arises from products of field functions

Ψ(t, r, s) =∑(p)

ap, s(t)ei(pr),

Ψ†(t, r, s) =∑(p)

a†p, s(t)e−i(pr),

in the coordinate representation.

We have for example

〈Ψ†(t1, r1, s1)Ψ†(t2, r2, s2)Ψ

†(t′2, r′

2 , s′2)Ψ(t′1, r

′1 , s

′1)〉

= 〈Ψ†(t1, r1, s1)Ψ†(t2, r2, s2)Ψ

†(t′2, r′

2 , s′2)Ψ(t′1, r

′1 , s

′1)〉H0

= F (t1 − t′1, r1 − r ′1 )F (t2 − t′2, r2 − r ′

2 )δ(s1 − s′1)δ(s2 − s′2)

− F (t2 − t′1, r2 − r ′1 )F (t1 − t′2, r1 − r ′

2 )δ(s2 − s′1)δ(s1 − s′2)

+ Φ(t1 − t2, r1 − r2)Φ(t′1 − t′2, r′

1 − r ′2 )

× ∈(s1) ∈(s′1)δ(s1 + s2)δ(s′1 + s′2) (2.37)

where

F (t, r) =1

(2π)3

∫e−i(pr)

u2

p eiE(p)t

1 + eE(p)/θ+

v2p e−iE(p)t

1 + e−E(p)/θ

dp,

Φ(t, r) =1

2(2π)3

∫e−i(pr)

√1 − T 2(p)

E2(p)

e−iE(p)t

1 + e−E(p)/θ− eiE(p)t

1 + eE(p)/θ

dp.

(2.38)


46 N. N. BOGOLUBOV

We also have

F (t1 − t′1, r1 − r ′1 )δ(s1 − s′1) = 〈Ψ†(t1, r1, s1)Ψ(t′1, r

′1 , s

′1)〉

=≺ Ψ†(t1, r1, s1)Ψ(t′1, r′

1 , s′1)

Φ(t1 − t2, r1 − r2) ∈(s1)δ(s1 + s2)

=≺ Ψ†(t1, r1, s1)Ψ†(t2, r2, s2) =≺ Ψ†(t2, r2, s2)Ψ(t1, r1, s1)

(2.39)

In the above we have investigated a number of examples of degeneracy ofstates of statistical equilibrium. In all of these cases such special statesof statistical equilibrium were realized when the temperatures were belowa certain critical temperature (θ < θc). For temperatures above θc thereappears a phase change which leads to the “normal” non-degenerate state.

In the above examples the degeneracy was dependent upon the presenceof additive conservation law, or (which is the same) upon the presence ofinvariance with respect to corresponding transformation groups. Let usemphasize that not all the conservation laws in a given system producedegeneracy. That is, in the third and fourth examples, the degeneracy of thestatistical equilibrium states depended only on the conservation law of thenumber of particles. In the corresponding quasi-averages only those selectionrules were violated which were specified by this very law. The selection rulesspecified by other additive conservation laws, for example, by the law ofconservation of momentum and spin (in the fourth example) were left intact.

In the second example the degeneracy depended only upon the law ofconservation of momentum. The selection rules, specified, for example, bythe law of conservation of the number of particles, were not violated here.

We could increase the number of such examples by investigating cases ofdegeneracy in connection with other groups or simultaneously with severaltransformation groups. However, we shall not stay to consider there pointsbut shall turn to the general investigation, introducing the correspondinggeneral calculations.

Let us consider a specific microscopic system with a Hamiltonian H . Wenow add infinitesimally small terms to H , which correspond to external fieldsor sources which violate the additive conservation laws. In this manner weobtain a specific Hamiltonian Hν, ν → 0. Then, if all the average values

〈A〉, A = . . .Ψ†(tj, xj) . . .Ψ(ts, xs) . . . (2.40)

are changed only by an infinitesimal amount, we will say that the stateof statistical equilibrium being considered is not degenerate. Alternatively,



if some of the averages (2.40) are changed by a finite amount when thetransition is from H to an infinitesimally changed Hamiltonian Hν , we shallsay that the state of statistical equilibrium is degenerate. It is obvious that weshall limit ourselves to observing only stable systems in as much as only theyhave physical meaning. Because of this the infinitesimally small variationδH = Hν −H of the Hamiltonian can produce only an infinitesimally smallchange in those values which actually characterize the real physical propertyof the system.

For cases of degeneracy it is convenient to introduce instead of the regularaverages the following quasi-averages

≺ A = limν→0

〈A〉Hν .

As we have already seen from the series of examples for quasi-average, it is notnecessary to fulfill all the selection rules specified by the additive conservationlaws. Let us note that when determining quasi-averages we must first takethe limit V → ∞, and then let ν approach zero.

As was previously noted, the infinitesimally small terms producing thedifference Hν − H are chosen in such a way as to violate the additiveconservation laws.

Generally speaking, however, it is not necessary to violate all such lawsin order to obtain the Hamiltonian H , which removes the degeneracy.

For example, let infinitesimal small terms which bring about a violationof some of the laws, produce only an infinitesimally change in 〈A〉Hν . Then itis clear that there is no need to violate these conservation laws and that Hν ,which possesses only terms which violate the rest of the conservation law,will suffice to remove the degeneracy.

In such a case for quasi-averages, just those selection rules which arespecified by these last mentioned conservation law will be violated.

Let us take, in particular, the usual dynamical model for the theory ofsuperconductivity, in which we deal with the continuum, and do not considerthe direct presence of the crystal lattice. In this model when the externalfields are absent one naturally expects total space homogeneity and that allthe averages

〈. . .Ψ†(tα, xα) . . .Ψ(tβ, xβ) . . .〉are translationally invariant.

In such a case the momentum conservation law will also hold for quasi-averages, and there is no reason to violate it in order to remove the


48 N. N. BOGOLUBOV

degeneracy.Let us also assume the presence of total spin homogeneity when the

conservation law of total spin holds for quasi-averages. Then we are leftonly with the conservation law for the number of particles to be violated. Insuch case we can assume:

Hν = H + ν∑

w(f)(a†fa†−f + a−faf), w(f) =∈(σ)v(p) (2.41)

where v(p) is a real function of momentum. To investigate the case whenthere is spin homogeneity, then we use the more general form:

Hν = H+ν∑

w(p, σ, σ′)a†pσa†−pσ′+w

∗(p, σ, σ′)a†−pσ′a†pσ+λ(p, σ, σ′)a†pσapσ′.

Let us now turn to the problem of applying different forms of perturbationtheory (in particular, the diagram techniques) to investigate degeneratestates of statistical equilibrium.

In order to remove difficulties which arise in the usual formalism discussedearlier in this section, we shall use the following general rule: In orderto use perturbation theory to investigate the degenerate states of statisticalequilibrium, we must first of all remove the degeneracy, that is, we mustwork not with Green’s functions which are constructed from regular averagessatisfying all the selection rules, but instead with Green’s functions which arebuilt up from quasi-averages which do not satisfy the some of these rules.

In such a way, the corresponding diagrams can include “anomalous” lineswhich are forbidden by the usual selection rules. For example, the diagrams

in the theory of the crystalline state which have the “normal” lines a†pap

that conserve momentum, will now also include the “anomalous” lines a†pap′

(p = p′) which do not conserve momentum.

Anomalous lines afa−f , a†fa

†−f , etc. also appear in diagrams in the theory

of superconductivity. We must keep in mind that these anomalous linescorrespond to ”dangerous” diagrams in that their sum gives a contribution inthe limit although their very presence is formally specified by infinitesimallysmall complementary terms in the Hamiltonian, Hν.

Because of this such lines must always be introduced into a calculationin a summed (even if only partially) form. One can introduce, for example,only totally summed anomalous lines, and, for the determination of theircorresponding anomalous Green’s functions, one can obtain an equationof the Dyson type. Actually when the calculation is carried out one can



generally drop the infinitesimally small complementary terms, whose onlyrole is to introduce quasi-averages instead of regular averages. In those caseswhen the Dyson equation referred to above has only a trivial solution (theanomalous Green’s function are identically equal to zero), then, obviously,there is no degeneracy. Degeneracy arises if the realh solution is non-trivial.

As was mentioned at the end of the first section perturbation theory isusually constructed by dividing the total Hamiltonian of the system intotwo parts: H = H0 + H1. The Hamiltonian H0 is selected to correspondto an “ideal gas” without interactions which possesses all those additiveconservation laws which the total Hamiltonian possesses.

Such an approach to the construction of perturbation theory can begeneralized for the investigation of degenerate states. In order that theanomalous (partially summed) Green’s functions appear immediately in thezeroth approximation, we add to H0 terms ∆, of the same type as theinfinitesimally small additional terms in Hν . Thus, for H0 + ∆ we removea series of additive conservation laws which hold true for the total H andwhich are “responsible for degeneracy”.

Let us writeH ′

0 = H0 + ∆; H ′1 = H1 − ∆.

Then, proceeding form the modified decomposition H = H ′0 + H ′

1 one canconstruct in the usual way degenerate perturbation theory using an expansionin powers of H ′

1. By the very choice of H ′0 in the zeroth approximation

we obtain the corresponding anomalous Green’s functions. Let us take,for example, the dynamical system which is investigated in the theory ofsuperconductivity where the degeneracy is removed by infinitely small termsof the type (2.41).

In the normal forms of the perturbation theory which do not take intoaccount the possibility of degeneracy the following term is included in H0∑

(f)

Te(k)a†faf (2.42)

which correspond to the “renormalization” of the kinetic energy term∑(f)

( k2

2m− µ

)a†faf . (2.43)

hWe speak of the real situation, keeping in mind that the equations can always have atrivial solution which does not satisfy the necessary physical restrictions (for example itmay have the wrong spectral structure).


50 N. N. BOGOLUBOV

For the calculations involving degeneracy we shall introduce for H ′0,

instead of (2.42), a more general quadratic form in the Fermi-amplitudes:

Ω =∑(f)

Te(f)a†faf − 1

2

∑(f)

w(f)(a†fa†−f + a−faf), w∗(f) = w(f). (2.43′)

We then must include in H ′1, in addition to the interaction terms, another

compensating expression:

∑(f)

( k2

2m− µ

)a†faf − Ω.

The arbitrary function w(f) should be chosen in such a way as to improve thedegree of approximation. For example, for obtaining the fist approximationone can choose w(f) on the basis that the corrections to this approximationi.e. 〈a−faf〉, would be zero; so that this anomalous average in the zerothapproximation would already be “summed” from the point of view of theusual first approximation. Let us note in conjunction with this, thatin the specific case of the model system, considered previously, with theHamiltonian (2.21) we can thus obtain an asymptotically exact solution.For this, it is only necessary to take for Te(k) its non-renormalized valuefrom (2.43) and assume:

w(f) = λ(f)∑(f ′)

λ(f ′)〈a−f ′af ′〉.

Then, in fact, the “zeroth approximation”, determined by the HamiltonianH ′

0, will give an asymptotically exact solution, and corrections of any orderwill be asymptotically equal to zero.

Let us note that Ω reduces to the diagonal form:∑(f)

E(f)α†faf + const, E(f) =

√T 2

e (k) + w2(f)

by means of the canonical u− v transformation:

αf = afuf + α†−fvf ,

α†f = a†fuf + α−fv

∗f .

(2.44)



Thus, for the construction of degenerate perturbation theory it is absolutelyequivalent to modify the expression H0, by the substitution, H0 → H ′

0, or touse as the Hamiltonian of zeroth approximation the Hamiltonian of the idealgas: ∑

(ν)

E(ν)α†νaν

in which the “new Fermi-amplitudes”, α, are coupled with the “old” by u−vtransformation.

In our first papers [2,5] on the theory of superconductivity we made musethe u − v transformation to obtain the correct modification of perturbationtheory. The last observation is general in character and does not apply onlyto the case of the quadratic form Ω (2.43) considered above. Actually, if wetake an arbitrary quadratic form

Ω =∑

A(f, f ′)a†faf ′+∑

C(f, f ′)afaf ′ +∑

C∗(f, f ′)a†f ′a†f ,

A∗(f, f ′) = A(f ′, f)(2.45)

requiring only that it be positive definite then by means of the general u− vtransformation:

af =∑(ν)

ufναν +∑(ν)

vfνα†ν

(2.45) can be reduced to a diagonal form∑(f)

E(f)α†faf + const.

In conclusion, we note that if one works with completely summed Green’sfunctions (with “thick lines” in diagrams of the Feynman type), then thefinal equations are invariant with respect to the special form H ′

0, and it onlynecessary to introduce into the diagrams the corresponding anomalous lines.The method of Green’s functions is especially convenient if we must takedamping into account, if we have to deal with higher approximations.

3. Principle of Correlation Weakening

In this paragraph we will try to formulate the intuitive concept, generallyaccepted in statistical mechanics, that the correlations between space distantparts of a macroscopic system in vanishingly small.


52 N. N. BOGOLUBOV

Consider the average:

F (t1, x1, . . . , tn, xn) = 〈. . .Ψ†(tj , xj) . . .Ψ†(ts, xs) . . .〉, x = (r, σ) (3.1)

and arbitrarily divide the set of arguments t1, x1; . . . , tn, xn into a series ofgroups:

. . . , tα, xα, . . ., . . . , tβ , xβ, . . ., . . . .

The asymptotic form of F will be considered with the time points, t1, . . . , tnfixed and the distances between the points r, from different groups, tendingtoward infinity. First of all we postulate that under the average, the fieldfunctions,

ϕ(t1, r1, σ1), ϕ(t2, r2, σ2), (ϕ = Ψ† or Ψ),

with t1 and t2 fixed and |r1 − r2| → ∞, will exactly commute or anticommuteamong themselves in the limit.

Then, in order to find the asymptotic form F , we can reorder the fieldfunctions ϕ(ti, xi) in expression (3.1) and thus, obtain the field functions foreach given group of arguments together in one set. We will thus have

F (t1, x1, . . . , tn, xn)−η〈U1(. . . , tα, xα, . . .)U2(. . . , tβ, xβ , . . .) . . .〉 → 0, η = ±1, (3.2)

where U1(. . . , tα, xα, . . .) represents the product of field functions witharguments from only the 1st group, and U2(. . . , tβ , xβ, . . .) represents thecorresponding product with arguments from only the 2nd group, etc. Thestatement made about the asymptotic commutation expresses, in our opinion,a universal law for real dynamical systems of statistical mechanics.

As is known in quantum field theory, all the field functions ϕ(t1, x1),ϕ(t2, x2) must exactly commute or anti-commute, if the four dimension vectort1 − t2, r1 − r2 is space-like. In problems of statistical mechanics, wherewe deal with interactions which are formally non-local this characteristicof commutation rules must be satisfied, at least approximately, and moreexactly as |r1 − r2| increases with fixed t1, t2.

Let us now turn to the investigation of the asymptotic structure of theexpression

〈U1(. . . , tα, xα, . . .)U2(. . . , tβ , xβ, . . .) . . .〉 (3.3)

in the limit of infinite spatial separation between the points r fromdifferent groups (the temporal arguments t1, . . . , tn being fixed. Since the



correlation between dynamical quantities U1, U2, . . . must become weakerand practically disappear for large enough distances, the correspondingasymptotic form for (3.3) breaks up into products of the form:

〈U1(. . . , tα, xα, . . .)〉〈U2(. . . , tβ, xβ, . . .)〉 . . . . (3.4)

Here it is necessary to specify the type of “averages” we are dealingwith in our formulation of the principle of correlation weakening. In thenondegenerate case the expressions 〈. . .〉, are obviously regular averages.However, one should note that in the case where the state of statisticalequilibrium is degenerate, the expressions 〈. . .〉, entering into our formulation,must be understood as quasi-averages. The formulation of the principle ofcorrelation weakening presented above is incorrect if one considers 〈. . .〉 asregular averages.

Let us now investigate the crystalline state once again. In this casewhen we refer to the correlation weakening between dynamical quantitiesU1, U2, . . . we intuitively mean that the crystal lattice, as a whole, is fixedin space. Even though the crystal position is arbitrary fixed the calculationof the averages of U1 and U2, etc. involve just this one fixed position. Inother words we now assume that all the averages considered here dependupon the same fixed position of the crystal lattice. Thus we are dealing withquasi-averages, and not with regular averages which are obtained from quasi-averages by an additional average over all possible positions and orientationsof the crystal lattice.

In other cases of degeneracy of the statistical equilibrium state similarsituations arise with parameters which are fixed in the same way for all partsof the system. As further examples we have either the magnetic moment(ferromagnetism) or the phase angle (superfluidity or superconductivity),etc.

Thus, in our formulation of the principle of correlation weakening itfollows that we should consider the expressions〈. . .〉 as quasi-averages.i Notethat we cannot prove exactly the principle of correlation weakening formacroscopic dynamical systems considered in statistical mechanics. We candevelop an exact proof only in a number of simple models such as in the

iSince in the degenerate cases we will always deal with quasi-averages and in non-degenerate cases the quasi-averages and regular averages coincide, we will no longer usethe special symbol ≺ . . . for denoting quasi-averages, but will use the symbol 〈. . .〉everywhere since this will no longer lead to misunderstanding.


54 N. N. BOGOLUBOV

models mentioned in the previous paragraph. For the general can we can useeither intuitive ideas or arguments borrowed from perturbation theory. Inthis respect the principle of correlation weakening is no different from othergenerally accepted important assumptions in statistical mechanics.

Thus, for example, the problem of the proof of a considerably simplerassumption; namely, the existence of the limit

− limV →∞

θ ln Tr e−H/θ

V

which represent the free energy per unit volume is in almost the samesituation. Thus, we will not investigate here the difficult mathematicalproblem of the formulation of the correlation weakening principle but wewill restrict ourselves to its physical implementation. Let us first examinethe application of this principle in the construction of a somewhat different,generally more “physical” definition, of the meaning of quasi-averages.

Consider, as an example, the case investigated in the theory ofsuperconductivity with a statistical equilibrium state where the degeneracydepends only upon the law of conservation of the number of particles. Letus examine the expression

〈Ψ†(t1, x1)Ψ†(t2, x2)Ψ(t′2, x

′2)Ψ(t′1, x

′1)〉. (3.5)

Since the operator

Ψ†(t1, x1)Ψ†(t2, x2)Ψ(t′2, x

′2)Ψ(t′1, x

′1)

conserves the number of particles, expression (3.5) will be a regular average.Let us increase without limit the distance between two groups of spatial

points (r1, r2) and (r′1, r′2) with time variable fixed. The on the basis of

the correlation weakening principle the expression (3.5) will approach theproduct

〈Ψ†(t1, x1)Ψ†(t2, x2)〉〈Ψ(t′2, x

′2)Ψ(t′1, x

′1)〉.

Proceeding from such an asymptotic decomposition of the regularaverage (3.5), we can now define the quasi-averages

〈Ψ†(t1, x1)Ψ†(t2, x2)〉, 〈Ψ(t′2, x

′2)Ψ(t′1, x

′1)〉.

By using the analogous procedure one can introduce quasi-averagesof higher order products of field functions. Previously we introduced



quasi-averages by adding infinitesimal terms to the Hamiltonian, withoutnecessarily having a clear physical meaning. Now, with the principle ofcorrelation weakening we are able to introduce quasi-averages by examiningthe asymptotic forms of the regular averages with the given and unalteredHamiltonian which corresponds to the dynamical system under consideration.

However, we must point out that the method involving infinitesimallysmall additions to the Hamiltonian is more convenient for a formal derivationof the generalized diagram technique, (using anomalous lines), in as much asit automatically reduces this problem to the previously solved one.

Let us examine a system of spinless Bose particles, in a spatially homo-geneous statistical equilibrium state, and consider the expression:

F (r1 − r2) = 〈Ψ†(t, r1)Ψ(t, r2)〉 = 〈Ψ†(r1)Ψ(r2)〉,Ψ(r) = Ψ(0, r).

(3.6)

Here, transformation to the momentum representation gives:

F (r1 − r2) =1

V

∑k

〈a†kak〉e−k(r1−r2). (3.7)

Therefore, in the Fourier integral

F (r) =

∫w(k)e−

k·rdk, (3.8)

the product w(k)dk expresses the number density of particles with momenta

in the infinitesimal momentum volume k. From this it follows that ρ =N

Vrepresents the particle number density

w(k) ≥ 0,

∫w(k)dk = ρ.

Let us further consider the case when a quiescent condensate is presentin the system. Then

w(k) = ρ0δ(k) + w1(k)

where w1(k) is a regular function characterizing the continuous momentumdistribution of the particles not located in the condensate and ρ0 is thenumber density of particle in the condensate. However, in as much as w1(k)is well-behaved we have∫

w1(k)e−k·r dk → 0, |r| → ∞,


56 N. N. BOGOLUBOV

and thus

〈Ψ†(r1)Ψ(r2)〉 = F (r1 − r2) = ρ0 +

∫w1(k)e

−k(r1−r2) dk → ρ0 = 0,

as |r1 − r2| → ∞. Therefore 〈Ψ(r1)〉 = 0.

On the other hand if the statistical equilibrium state was not degeneratewith respect to the law of conservation of the number of particles, then onthe strength of the selection rules corresponding to this law we would havehad the identity, 〈Ψ(r1)〉 = 0. Thus, for the systems with a condensate, theselection rules specified by the law of conservation of particle number are notsatisfied and this statistical equilibrium state will be degenerate.

One can show that an analogous situation also arises for fermi systemswhen a condensate of coupled pairs appears. It is now necessary to define themeaning of ”coupled pair”. We proceed to do this in the following paragraph.

4. Particle Pair States

We will investigate here the spatially homogeneous statistical equilibriumstates for macroscopic systems composed of identical Fermi particles. Forthese states let us try to clarify such ideas as “wave function of a pair ofparticles”, [7] “state of a pair of particles”, and in particular “coupled stateof a pair”, etc. These ideas have a clear meaning in the case where thedynamical system consists of two particles. We wish to generalize theseideas to systems of macroscopically large number of particles which interactone with another. With this goal in mind let us look at a pair correlationfunction (corresponding to one instant of time):

F (x1, x2; x′1, x

′2) = 〈Ψ†(x1)Ψ

†(x2)Ψ(x′2)Ψ†(x′1)〉. (4.1)

Using the Hermitian property

F (x′1, x′2; x1, x2) = F (x1, x2; x

′1, x

′2) (4.2)

we can expand F in the orthonormal system of eigenfunctions Ψν :

F (x1, x2; x′1, x

′2) =

∑(ν)

NνΨ∗ν(x1, x2)Ψν(x

′1, x

′2) (4.3)



with the normalization ∫∫

V

∣∣Ψν(x1, x2)∣∣2 dx1 dx2 = 1 (4.4)

where, generally. ∫

V

. . . dx =∑

σ

∫. . . dr.

In the case of a low density gas, to the first approximation Ψν(x1, x2) willbe the usual wave function of the two body problem (which is very naturalsince to the first approximation the action of the other particles upon thegiven pair of particles can be neglected).

Because of this analogy, we will call the eigenfunctions Ψν(x1, x2) thewave functions of pairs of particles. We will interpret the coefficients Nν asthe average number of pairs of particles in the state with wave function Ψν .From (4.1), (4.3), and (4.4) it follows thatj

〈N2 −N〉 =∑

ν

Nν

i.e. the sum of all the Nν represents the total number of pairs.Let also note that due to (4.1):

F (x2, x1; x′1, x

′2) = −F (x1, x2; x

′1, x

′2)

F (x1, x2; x′2, x

′1) = −F (x1, x2; x

′1, x

′2)

and thusΨν(x2, x1) + Ψν(x1, x2) = 0.

As is seen, the function Ψν must be antisymmetric just like the usual wavefunctions of two Fermi particles. Now, let us write the expansion (4.3)

jActually, we really have from (4.1), (4.3), (4.4):

∑ν

Nν =∫∫

V

〈Ψ†(x1)Ψ†(x2)Ψ(x2)Ψ†(x1)〉dx1 dx2

=∫∫

V

〈Ψ†(x1)Ψ†(x1)Ψ†(x2)Ψ(x2)〉dx1 dx2 −∫

V

〈Ψ†(x)Ψ(x)〉dx

while∫V

Ψ†(x)Ψ(x)dx = N .


58 N. N. BOGOLUBOV

in a more detailed form. We make use of the law of conservation ofmomentum and separate out from the wave function Ψν(x1, x2) a factorwhich corresponds to the motion of the center of mass with momentum q

Ψν(x1, x2) = Ψω,q(r1 − r2, σ1, σ2) expiq

(r1 + r22

).

Assume the index ν = (ω, q) includes the momentum q and, possibly, someother indices ω. Then the relation (4.3) takes the form:

F (x1, x2; x′1, x

′2) = 2

∑(ω, q)

Nω, qΨ∗ω, q(r1 − r2; σ1, σ2)Ψω, q(r

′1 − r ′

2 ; σ′1, σ

′2)

× expiq

(r ′1 + r ′

2 − r1 − r22

). (4.5)

Here, Nω,q represents the average number of particle pais in the state Ψω, q,where each pair is counted once (and not twice as before). From (4.4) thefollowing normalization occurs in (4.5)

∑σ1, σ2

∫ ∣∣Ψω, q(r; σ1, σ2)∣∣2 dr =

1

V. (4.6)

Let us now write the expansion (4.5) in integral form. We will switchto a more convenient normalization. Consider the wave function of apair, Ψω,q(r; σ1, σ2) for a given fixed momentum q. Since the correlationbetween particles in the pair must disappear for large enough distances r,the asymptotic form (r → ∞) of the considered functions is either equalto zero or becomes a plane wave corresponding to relative free motion withrelative momentum p. Let us look at the first possibility and assume in thiscase:

Ψω,q(r; σ1, σ2) =1√Vϕω,q(r; σ1, σ2)

so that ∑σ1, σ2

∫ ∣∣ϕω, q(r; σ1, σ2)∣∣2 dr = 1. (4.7)

Let us then say that ϕω, q represents a bound state of a particle pair, withtotal momentum q. The discrete index ω indicates, so to speak, the numberof the bound state.



Now let us examine the other possibility where the asymptotic formof Ψω,q is a plane wave representing the relative motion of the particles inthe pair with momentum p. Assume

ω = (p, j), Ψω,q(r; σ1, σ2) =1

Vϕp,q,j(r; σ1, σ2).

In this case we will say that ϕp,q,j represents the wave function of an unboundor “dissociated” state of a particle pair. For ϕp,q,j we have the normalizationin this situation:

∑σ1, σ2

1

V

∫ ∣∣ϕp,q,j(r; σ1, σ2)∣∣2 dr = 1. (4.8)

We can write the expansion (4.5) in the following form

F (x1, x2; x′1, x

′2) =2

∑(ω, q)

Nω, q

Vϕ∗

ω, q(r1 − r2; σ1, σ2)ϕω, q(r′

1 − r ′2 ; σ′

1, σ′2)

× expiq

(r ′1 + r ′

2 − r1 − r22

)+ 2

∑(p,q,j)

Np,q,j

V 2ϕ∗

p,q,j(r1 − r2; σ1, σ2)ϕp,q,j(r′

1 − r ′2 ; σ′

1, σ′2)

× expiq

(r ′1 + r ′

2 − r1 − r22

).

In the limit V → ∞, we go from the momentum sums to the correspondingintegrals and obtain

F (x1, x2; x′1, x

′2) = 2

∑(ω)

∫dq w(ω, q)ϕ∗

ω, q(r1 − r2; σ1, σ2)ϕω, q(r′

1 − r ′2 ; σ′

1, σ′2)

× expiq

(r ′1 + r ′

2 − r1 − r22

)+ 2

∑(j)

∫dp dq wj(p, q)ϕ

∗p,q,j(r1 − r2; σ1, σ2)ϕp,q,j(r

′1 − r ′

2 ; σ′1, σ

′2)

× expiq

(r ′1 + r ′

2 − r1 − r22

). (4.9)


60 N. N. BOGOLUBOV

As seen

w(ω, q)dq

in this formula represents the number density of bound state pairs withmomentum q in an infinitesimally small momentum volume dq;

wj(p, q)dq dq

represents the number density of unbound pairs with relative momentum pand center of mass momentum q in the infinitesimally small volumes dp anddq. Let us take any wave function of the bound state:

ϕω,q(r, σ1, σ2).

If the linear dimension, l, of that space region in which ϕω,q is essentiallylocalized is considerably smaller than the average distance, r, betweenparticles (from different pairs) in the macroscopic system, then it is naturalto say that the given ϕω,q corresponds to a molecule composed of two particleswhich is in the state ω and moves with the momentum q. In the case where lis of the same order of magnitude or larger than r, then we can add prefixesof “quasi” or “pseudo” to the word “molecule”.

Let us compare the integral representation (4.9) with the representationof the simple average,

F (x, x′) = 〈Ψ†(x)Ψ(x′) = ∆(σ − σ′)∫dq w(q) eir(r ′−r). (4.10)

We see that although (4.10) describes the distribution of the particles by“single particle states”, i.e. the plane waves, the formula (4.9) characterizesthe distribution of the particles by “pair states”. With the above correlationfunction we can introduce in similar manner the concept of wave functionsfor a group of three or more particles. [7]

We recall at this point that Schafroth in his early investigations proposedthe hypothesis, which was later completely verified, that the phenomenon ofsuperconductivity depends upon the formation of a condensate consisting ofquasi-molecules, formed from pairs of electrons in the system of conductionelectrons. In this connection consider the case of a fermion system (withthe usual spin 1/2), with a condensate of quasi-molecule pairs, which arefor example, in S states. In other words, we will consider the case where in



formula (4.9) we putk

w(ω, q) = ρ0∆(ω − ω0)δ(q) + w1(ω, q)

ϕω0,0(r, σ1, σ2) = ∈(σ1)∆(σ1 + σ2)1√2ϕ(r) (4.11)

where,

1. w1(ω, q) and wj(p, q) correspond to the usual continuous particle pairstate momentum distribution function.

2. ϕ(r) is a real, radially symmetric function, and due to (4.7) itsnormalization is: ∫

ϕ2(r) dr = 1.

We now write formula (4.9) in the following form

F (x1, x2; x′1, x

′2)

= ρ0∈(σ1)∈(σ′1)∆(σ1 + σ2)∆(σ′

1 + σ′2)ϕ(r1 − r2)ϕ(r ′

1 − r ′2 )

+ 2∑(ω)

∫dq w(ω, q)ϕ∗

ω, q(r1 − r2; σ1, σ2)ϕω, q(r′

1 − r ′2 ; σ′

1, σ′2)

× expiq

(r ′1 + r ′

2 − r1 − r22

)+ 2

∑(j)

∫dp dq wj(p, q)ϕ

∗p,q,j(r1 − r2; σ1, σ2)ϕp,q,j(r

′1 − r ′

2 ; σ′1, σ

′2)

× expiq

(r ′1 + r ′

2 − r1 − r22

). (4.12)

Here ρ0 represents the bound pair number density in the condensate. Notethat we neglect the problem of the existence of bound states which are notin the condensate. If such bound states do not exist we would then putw1(ω, q) = 0 in (4.12)

kHere ∆(S) is the discrete S-function:

∆(S) =

1, S = 0;0, S = 0.


62 N. N. BOGOLUBOV

Let us consider, for example, the model dynamical system, studied in thesection 2 and make the formula (2.37). For this system we obtain in thepresent notation:

F (x1, x2; x′1, x

′2)

=φ(r1 − r2)φ(r ′1 − r ′

2 )∈(σ1)∈(σ′1)∆(σ1 + σ2)∆(σ′

1 + σ′2)

+ F (r1 − r ′1 )F (r2 − r ′

2 )∆(σ1 − σ′1)∆(σ′

2 − σ′2)

− F (r2 − r ′1 )F (r1 − r ′

2 )∆(σ2 − σ′1)∆(σ′

1 − σ′2) (4.13)

where

φ(r) = φ(0, r); F (r) = F (0, r).

Substituting the integral representation,

F (r) =

∫w(k) e−i(k·r) dk

we bring (4.13) into the form (4.12).Note that in this case w1(ω, q) = 0, and the pair states ϕp,q,j are

regular plane waves. Consequently we have only one bound state with totalmomentum zero, and the rest of the pair states which have total momentumq = 0 will be the same as those for non-interacting particles. Such a result iscompletely natural since in our model system interactions are possible onlybetween particle pairs having total momentum equal to zero.

Let us now go back to the “general case of Schafroth” and apply theprinciple of correlation weakening. We break up the arguments of thefunction (4.1) into two groups

(x1, x2); (x′2, x′2)

and increase without limit the distance, r, between points from differentgroups. Then, due the principle of correlation weakening the correspondingasymptotic form for F will be:

〈Ψ†(x1)Ψ†(x2)〉〈Ψ(x′1)Ψ(x′2)〉.

Alternatively, from (4.12) we obtain for this asymptotic form the product,

ρ0∈(σ1)∈(σ′1)∆(σ1 + σ2)∆(σ′

1 + σ′2)ϕ(r1 − r2)ϕ(r ′

1 − r ′2 ).



Thus we can write

〈Ψ†(x1)Ψ†(x2)〉 =

√ρ0∈(σ1)∆(σ1 + σ2)ϕ(r1 − r2)

〈Ψ(x′1)Ψ(x′2)〉 =√ρ0∈(σ′

1)∆(σ′1 + σ′

2)ϕ(r′1 − r′2). (4.14)

Hence, we see that these quasi-averages are not zero and they do notsatisfy the selection rules which are specified by the law of conservation ofthe number of particles. Thus, if a condensate of quasi-molecules of pairsexists for this state of statistical equilibrium, then this statistical equilibriumstate will be degenerate. The degeneracy here is dependent upon the law ofconservation of the number of particles.

In conclusion note that the formulas (4.14) give a simple interpretation ofthe “anomalous quasi-averages” 〈Ψ†Ψ†〉, 〈ΨΨ〉. That is, these quasi-averagesare proportional to the wave function of quasi-molecule in the condensate.The normalization ∑

(σ2)

∫ ∣∣〈Ψ(x1)Ψ(x2)〉∣∣2dr2 = ρ0 (4.15)

gives the number density of such quasi-molecules.

5. Certain Inequalities

We now investigate averages of the product of two operators; 〈AB〉, asbilinear forms A and B (linear with respect to each of these operators).If symbol 〈. . .〉 represents a regular average then one can easily see that

〈AB〉∗ = 〈B†A†〉〈AA†〉 ≥ 0. (5.1)

In as much as the quasi-averages can be investigated can be investigatedas regular averages taken for the system with an infinitely small variationHamiltonian, then the same relations (5.1) hold for quasi-averages. Furtherif A(t) and B(t) are operators in the Heisenberg picture, then in the case ofregular averages the following spectral formulas can be proven

〈B(τ)A(t)〉 =

+∞∫

−∞

JA,B(ω) e−iω(t−τ) dω,


64 N. N. BOGOLUBOV

〈A(t)B(τ)〉 =

+∞∫

−∞

JA,B(ω) eω/θ e−iω(t−τ) dω, (5.2)

where the spectral density JA,B(ω) is a bilinear form with respect to theoperators A and B. Due to the argument just presented, the same formulasremain correct for quasi-averages.

Using the properties (5.1) and (5.2), we shall now establish certaininequalities which will be needed in the next chapter. Here the symbol 〈. . .〉can represent a quasi-averages as well as a regular average.

First of all let us prove that,

JA,A†(ω) ≥ 0. (5.3)

For this assume an arbitrary function, f(ω), which is sufficiently regularenough and which goes to zero at infinity. If we are able to prove that forevery such function

+∞∫

−∞

JA,A†(ω)|f(ω)|2dω ≥ 0 (5.4)

holds, then (5.3) will thereby established in as much as we can always localize|f(ω)|2 in as narrow a vicinity as needed, of any point ω0.

In order to prove the inequality (5.4) construct the function

h(t) =1

2π

+∞∫

−∞

f(ω) eiωt dω

and note that

f(ω) =

+∞∫

−∞

h(t) e−iωt dt.

Thus we have

+∞∫

−∞

JA,A†(ω)|f(ω)|2dω =

+∞∫

−∞

dt

+∞∫

−∞

dτ

+∞∫

−∞

dω JA,A†(ω)h(t)h∗(τ) e−iω(t−τ)



=

+∞∫

−∞

dt

+∞∫

−∞

dτ〈A†(τ)A(t)〉h(t)h∗(τ) = 〈U U †〉,

where

U =

+∞∫

−∞

A†(τ)h∗(τ) dτ, U † =

+∞∫

−∞

A(t)h(t) dt.

The inequality (5.4) follows from this using (5.1). Now we prove that

J∗A,B(ω) = JB†,A†(ω). (5.5)

We have

〈A†(τ)B†(t)〉 =

+∞∫

−∞

JB†,A†(ω)e−iω(t−τ) dω

and thus

〈A†(t)B†(τ)〉 =

+∞∫

−∞

JB†,A†(ω)eiω(t−τ) dω. (5.6)

Alternatively,

〈A†(t)B†(τ)〉 = 〈B(τ)A(t)〉∗

=+∞∫

−∞

JA,B(ω)e−iω(t−τ) dω∗

=

+∞∫

−∞

J∗A,B(ω)eiω(t−τ) dω.

(5.7)

By comparing (5.6) and (5.7) we obtain (5.5). Now let Z(A,B) be anarbitrary bilinear form of A, B possessing the properties,

Z(A,A†) ≥ 0,

Z(A,B)∗ = Z(B†, A†). (5.8)

We shall demonstrate that the following inequality always exists,

|Z(A,B)|2 ≤ Z(A,A†)Z(B,B†). (5.9)

For the prove, not that on the basis (5.8)

Z(xA + y∗B†, x∗A† + yB) ≥ 0,


66 N. N. BOGOLUBOV

where x, y are arbitrary c-numbers. From this, by expansion we obtain,

xx∗Z(A,A∗) + xyZ(A,B) + y∗x∗Z(B†, A†) + y∗yZ(B†, B) ≥ 0. (5.10)

If we take

x∗ = −Z(A,B), x = −Z(A,B)∗ = −Z(B†, A†), y = y∗ = Z(A,A†)

then we obtain,

−|Z(A,B)|2Z(A,A∗) + Z(A,A†)2Z(B†, B) ≥ 0.

Form this, if Z(A,A†) = 0, we obtain the inequality (5.9). It remains for usto show that if

Z(A,A†) = 0 (5.11)

then we also haveZ(A,B) = 0 (5.12)

For this purpose let us substitute (5.11) into (5.10). We then set

x∗ = −Z(A,B)R, x = −Z(B†, A†)R, y = y∗ = 1,

where R is an arbitrary positive number. We obtain

−2R|Z(A,B)|2 + Z(B†, B) ≥ 0. (5.13)

Let R approach infinity. Then, if (5.12) does not hold, the left side of (5.13)must approach −∞, which is not possible. This completes the proof of theinequality (5.9).

Now, note that the choice,

Z(A,B) = JA,B(ω)

satisfies the condition (5.8), since the relationships (5.3) and (5.5) holdfor JA,B. Thus, in this case, we can make use of the inequality (5.9) andwrite:

|JA,B|2 ≤ JA,A†(ω)JB†,B(ω). (5.14)

We can also take

Z(A,B) =1

2π

+∞∫

−∞

JA,B(ω)eω/θ − 1

ωdω (5.15)



since the conditions (5.8) are again satisfies because of (5.3), (5.5), and thepositive nature of the function

eω/θ − 1

ω.

Let us relate the function (5.15) to Green’s functions. We shall investigatethe following [8] retarded and advanced Green’s functions

A(t), B(τ) r= −iθ(t− τ)〈A(t)B(τ) −B(τ)A(t)〉, A(t), B(τ) a= iθ(τ − t)〈A(t)B(τ) − B(τ)A(t)〉. (5.16)

On the basis of the spectral representation, (5.2), it is clear that their Fouriertransforms, due to the nature of the step function, θ(t), will be respectively

A,B E+iε, A,B E−iε .

We see that, A,B E

is given by the following formula as a function of the complex variable E

A,B E=1

2π

+∞∫

−∞


E − ωdω. (5.17)

We can see from this that the expression (5.15) may be written as

− A,B E=0 . (5.18)

Thus the inequality, (5.9), takes the form

| A,B E=0 |2 ≤ A,A† E=0 B†, B E=0 (5.19)

in the present case. We will apply this result later.In conclusion, let us consider one important application of the Green’s

function of the type (5.18). Give the Hamiltonian H an infinitesimalincrement, δH (independent of time). The corresponding variation of theaverage of an operator A(t) will be given by, [8]

δ〈A〉 = 〈A〉H+δH − 〈A〉H = 2π A, δH E=0 . (5.20)


68 N. N. BOGOLUBOV

Part B. CHARACTERISTIC THEOREMS ABOUT THE1/q2 TYPE INTERACTION IN THE THEORYOF SUPERCONDUCTIVITY OF BOSE ANDFERMI SYSTEMS

6. Symmetry Properties of Basic Green’s Functions for BoseSystems in the Presence of a Condensate

Consider a dynamical system of identical spinless Bose particles with aHamiltonian of the form,

H = − 1

2m

∫

V

Ψ†∆Ψ dr − µ

∫

V

Ψ†Ψ dr + U(Ψ†,Ψ)

=∑( k2

2m− µ

)a†kak + U(Ψ†,Ψ), (6.1)

U(Ψ†,Ψ) =1

2

∫∫

V

φ(r1 − r2)Ψ†(r1)Ψ†(r2)Ψ(r2)Ψ(r2) dr1 dr2. (6.2)

Here ψ(r) is a real function of distance and represents the interaction energyof a pair of particle. In addition, we limit ourselves to a system at a giventemperature θ, with a Bose condensate.

As previously noted in section 3, the corresponding statistical equilibriumstate must be degenerate, in such a case. The degeneracy here depends uponthe law of conservation of the number of particles. In order to remove thedegeneracy consider the Hamiltonian

Hν = H − ν√V (a0 + a†0), (6.3)

which contains additional infinitesimal terms of the form,

−ν√V (a0 + a†0), ν > 0. (6.4)

We are assuming that other types of degeneracy do not existl and, thus, theintroduction of the term (6.4) is sufficient for the removal of the degeneracy.

In this way for quasi-averages 〈A(t)B(τ)〉, where A, B = a±k, a†±k (k =

0), the selection rules resulting from the law of conservation of momentum

lActually we are here assuming that our system is in a spatially homogeneous phasewith no molecules of two or more particles.



must be satisfied; but, the selection rules specified by the law of conservationof the number of particles can be violated.

We introduce the following Green’s Functions

A(t), B(τ) ret = −iθ(t− τ)〈A(t)B(τ) −B(τ)A(t)〉, A(t), B(τ) adv = iθ(τ − t)〈A(t)B(τ) − B(τ)A(t)〉, A(t), B(τ) c = −〈T (A(t)B(τ))〉

= −iθ(t− τ)〈A(t)B(τ)〉 + θ(τ − t)〈B(τ)A(t)〉. (6.5)

We determine their “energy representation” with the Fourier integrals:

A(t), B(τ) =

+∞∫

−∞

A,B e−iE(t−τ)dE. (6.6)

Using the spectral formulas,

〈B(τ)A(t)〉 =

+∞∫

−∞

JA,B e−iω(t−τ)dω,

〈A(t)B(τ)〉 =

+∞∫

−∞

JA,B eω/θe−iω(t−τ)dω,

we obtain

A,B retE =

1

2π

+∞∫

−∞

(eω/θ − 1)JA,B(ω)dω

E − ω + iε

A,B advE =

1

2π

+∞∫

−∞

(eω/θ − 1)JA,B(ω)dω

E − ω − iε

A,B cE =

1

2π

+∞∫

−∞

JA,B(ω) eω/θ

E − ω + iε− 1

E − ω + iε

dω.

For the special case of zero temperatures the spectral formulas for theseaverages can be written in the form,

〈A(t)B(τ)〉 =

+∞∫

0

IA,B(ω)e−iω(t−τ)dω


70 N. N. BOGOLUBOV

〈B(τ)A(t), 〉 =

0∫

−∞

IA,B(ω)e−iω(t−τ)dω.

We then have for the energy representation of the Green’s functions

A,B retE =

1

2π

+∞∫

−∞

∈(ω)IA,B(ω)

E − ω + iεdω

A,B advE =

1

2π

+∞∫

−∞

∈(ω)IA,B(ω)

E − ω − iεdω

A,B cE =

1

2π

+∞∫

−∞

∈(ω)IA,B(ω)

E − ω + iε∈(ω)dω,

where

∈(ω) =

+1, ω > 0,

−1, ω < 0.

Obviously, the retarded and advanced Green’s functions,

A,B retE , A,B adv

E ,

are boundary values of the function of the complex variable E,

A,B =1

2π

+∞∫

−∞


E − ωdω. (6.7)

In general the causal Green’s functions, A,B cE, possesses this property,

only in the limit of zero temperature. Now, note that from the definition (6.5)the following holds,

A(t), B(τ) c= B(τ), A(t) c

A(t), B(τ) ret= B(τ), A(t) adv .

From (6.6) we obtain,

A,B cE= B,Ac

−E (6.8)



and also, A,B ret

E = B,Aadv−E . (6.9)

By continuing relation (6.9) into the complex E plane, we can convinceourselves that for the complex function, (6.7), the following equality holds,

A,B E= B,A−E . (6.10)

Let us investigate the matrix Green’s function:

G(E, k) =

∣∣∣∣G11(E,K); G21(E,K)G12(E,K); G22(E,K)

∣∣∣∣ (6.11)

where

G11(E, k) = ak, a†k E , G21(E, k) = a†−k, a

†k E ,

G12(E, k) = ak, a−k E, G22(E, k) = a†−k, a−k E .(6.12)

For A,B E we mean either the function of the complex variable, (6.7),or causal Green’s function for real E. In both cases, due to (6.8) and (6.10),we have,

G22(E, k) = G11(−E,−k);Gαβ(E, k) = Gαβ(−E,−k); if α = β.

(6.13)

Now note that the Hamiltonian Hν is invariant with respect to the canonicaltransformation,

ak → a−k, a†k → a†−k.

Because of this, the averages

〈a†k(t)ak(τ)〉, 〈ak(t)a†k(τ)〉

〈ak(t)a−k(τ)〉, 〈a†−k(t)a†k(τ)〉

can not change under the transformation k → −k. Consequently, for theGreen’s functions we will also have

Gαβ(E, k) = Gαβ(E,−k). (6.14)

Note further, that since all the coefficients are real in the expression for theHamiltonian Hν , the corresponding equations of motion must be invariant


72 N. N. BOGOLUBOV

with respect to time inversion (t → −t, accompanied by the substitution ifor −i).

Thus, the average

〈a−k(τ)ak(t)〉 =

+∞∫

−∞

Jk(ω) e−iω(t−τ) dω (6.15)

does not change under the transformation t→ −t, τ → −τ , i→ −i. Becauseof this:

+∞∫

−∞

Jk(ω) e−iω(t−τ) dω =

+∞∫

−∞

J∗k (ω) e−iω(t−τ) dω.

From this it follows that the spectral intensity, Jk(ω), is a real function, i.e.

J∗k (ω) = Jk(ω). (6.16)

Thus from (6.15) we have

〈a†k(t)a†−k(τ)〉 = 〈a−k(τ)ak(t)〉∗ =

+∞∫

−∞

Jk(ω) eiω(t−τ) dω

and〈a†k(τ)a†−k(t)〉 = 〈a−k(τ)ak(t)〉.

The corresponding relation for Green’s functions is

a†k, a†−k E= a−k, ak E .

Or, in our notation we have,

G21(E, k) = G12(E, k). (6.17)

Let us now introduce the matrix, Σ(E, k),

Σ(E, k) =1

2πG−1(E, k) (6.18)

or,2πΣ(E, k)G(E, k) = 1 (6.19)



where 1 is the unit matrix. We can interpret the matrix Σ(E, k) as thetotal “mass operator”. In the particular case of zero temperature, whenthe Feynman diagram technique is applicable, Σ(E, k) represents the usual“self-energy” part.

It is also clear from the definition (6.18) that the elements Σαβ(E, k)always satisfy the same symmetry relations (6.13), (6.14), (6.17) asGαβ(E, k). Explicitly writing out the matrix equality, (6.19), we obtain:

Σ11(E, k)G11(E, k) + Σ12(E, k)G21(E, k) =1

2π,

Σ21(E, k)G11(E, k) + Σ22(E, k)G21(E, k) = 0.

However, in view of the above, we have

Σ21(E, k) = Σ12(E, k), Σ22(E, k) = Σ11(−E, k).Thus we can write

Σ11(E, k) ak, a†k E +Σ12(E, k) a†−k, a

†k E=

1

2π,

Σ21(E, k) ak, a†k E +Σ11(−E, k) a†−k, a

†k E= 0.

(6.20)

These functions Σαβ possess the following symmetry properties,

Σ11(E, k) = Σ11(E,−k), Σ12(E, k) = Σ12(E,−k);Σ12(E, k) = Σ12(−E, k). (6.21)

In view of this, we can obtain from (6.20) the following formulas which expressthe Green’s functions in terms of Σ11 and Σ12

ak, a†k E =

1

2π

Σ11(−E, k)Σ11(E, k)Σ11(−E, k) − Σ2

12(E, k)

a†−k, a†k E

k =0

= − 1

2π

Σ12(E, k)

Σ11(E, k)Σ11(−E, k) − Σ212(E, k)

.(6.22)

7. Model with a Condensate

We observe that since the coefficients in the hamiltonian Hν are real, theexpression 〈a0〉 is also real and thus

⟨ a0√V

⟩=

⟨ a†0√V

⟩. (7.1)


74 N. N. BOGOLUBOV

Let us consider the average,

⟨a†0a0

V

⟩= ρ0

and note that

ρ0 =1

V 2

∫∫V

〈Ψ†(r1)Ψ(r2)〉 dr1 dr2. (7.2)

Since V → ∞ here, it is clear that all contribution to the integral in (7.2)comes the region where the points r1 and r2 are infinitely separated. Thus,by applying the principle of correlation weakening, we obtain asymptotically,

ρ0 =1

V

∫V

〈Ψ†(r1), dr11

V

∫V

〈Ψ(r2)〉 dr2 =⟨ a†0√

V

⟩⟨ a0√V

⟩,

where from (7.1), we have

⟨ a0√V

⟩=

⟨ a†0√V

⟩=

√ρ0. (7.3)

Let us now look at expressions of the type,

〈. . . A(tα) . . . ϕ(tβ , rβ) . . .〉in which

A =a0√V,a†0√V

; ϕ = Ψ, Ψ†.

Let us apply the principle of correlation weakening to them as in (7.2). Wewill obtain the equality,m

〈. . . A(tα) . . . ϕ(tβ , rβ) . . .〉 =(√

ρ0

)l〈. . . ϕ(tβ, rβ) . . .〉where l is the number of A’s involved in the average. We introduce therelations,

ϕ = ηa0√V

+ Ψ1, ηa†0√V

+ Ψ†1

mOf course, equalities of this type are asymptotic in character and become exact onlyin the limit V → ∞. However, since we are always dealing here with limit relations, wewill not mention this explicitly.



η = 0, 1; Ψ1 =1√V

∑k =0

ak ei(k r).

Then on the basis of above result we find that in the calculation of averagesof the form:

〈. . . ϕ(tα, rα) . . .〉we can substitute the c-number

√N0, (where N0 = ρ0V ) for the operators a0

and a†0, involved in ϕ. Taking this property into account, we will show thatthe problem with the Hamiltonian Hν can be reduced to the problem withthe Hν(N0), which is obtained from the expression (6.3) for Hν with thesubstitution,

Ψ(r) → √ρ0 + Ψ1(r); Ψ†(r) → √

ρ0 + Ψ†1(r)

i.e., by the substitution of the c-number√N 0 for the operators a0 and a†0

in Hν. Let us now examine the system of “double time” Green’s functionsof the type

ϕ1(t, r1) . . . ϕ(t, rs);ϕ(τ, x1) . . . ϕ(τ, xm) retadv

=−θ(t− τ)θ(t− τ)

〈[ϕ1(t, r1) . . . ϕ(t, rs);ϕ(τ, x1) . . . ϕ(τ, xm)]〉 (7.4)

where ϕ1 = Ψ1, Ψ†1.

In order to obtain a chain of equations connecting these functions, let usexpress the derivative

i∂

∂t ϕ1(t, r1) . . . ϕ(t, rs);ϕ(τ, x1) . . . ϕ(τ, xm) (7.5)

using the equations of motion. We have for the Hamiltonian H

i∂Ψ(t, r)

∂t= D(t;r; Ψ,Ψ†) ≡

≡(− 1

2m∆ − µ

)Ψ(t, r) − ν +

∫φ(r − r ′)Ψ†(t, r ′)Ψ(t, r ′) dr ′ Ψ(t, r)

and, thus,

i∂Ψ1(t, r)

∂t= D(t;r; Ψ,Ψ†) − 1

V

∫

V

D(t;r; Ψ,Ψ†) dr,


76 N. N. BOGOLUBOV

− i∂Ψ†

1(t, r)

∂t= D†(t;r; Ψ,Ψ†) − 1

V

∫

V

D†(t;r; Ψ,Ψ†) dr.

Since D and D† will enter into expression for the derivatives, (7.5), only asaverages of the form

〈. . . ϕ1 . . .D . . . ϕ1〉, 〈. . . ϕ1 . . .D† . . . ϕ1〉

thus, we can perform in the expressions

D(t, r; Ψ,Ψ†), D†(t, r; Ψ,Ψ†),

the following substitution,

Ψ → √ρ0 + Ψ1, Ψ† → √

ρ0 + Ψ†1 .

Similarly, to deal with terms of the type

ϕ1(t, r1) . . .∂ϕ1(t, r1)

∂t. . . , . . . ϕ1(τ, xm) (7.6)

we will make use of equations of the type

i∂Ψ1(t, r)

∂t=D(t, r;

√ρ0 + Ψ1,

√ρ0 + Ψ†

1)

− 1

V

∫

V

D(t, r;√ρ0 + Ψ1,

√ρ0 + Ψ†

1) dr,

−i∂Ψ†1(t, r)

∂t=D†(t, r;

√ρ0 + Ψ1,

√ρ0 + Ψ†

1)

− 1

V

∫

V

D†(t, r;√ρ0 + Ψ1,

√ρ0 + Ψ†

1) dr.

(7.7)

With the help of these equations the terms (7.6) will be introduced throughdifferentiating Green’s functions of the type under consideration.

In expression (7.5), in addition to the sum of terms of type (7.6), therewill be present an additional “inhomogeneous member”,

δ(t− τ)〈[ϕ1(t, r1) . . . , . . . ϕ1(t, xm)]〉.



We will obtain these averages from averages of products containing no morethan s+m− z simultaneous field functions ϕ. These averages can again beexpressed through Green’s functions (of lower order) with the help of spectralrepresentations.

For their calculation it will be convenient to use a momentum energyrepresentation. Let us designate the Green’s functions in this representationby

GHν (E; p1, . . . ps; q1, . . . qm)

Then we can write the hierarchy of equations in the form

EGHν (E; p1, . . . ps; q1, . . . qm) = L (E; p1, . . . ps; q1, . . . qm; GHν )

where the L are expressions which depend upon the functions GHν ofdifferent orders.

Since the spectral representations are “universal”, then it is clear from theabove that only those terms in L which result from terms of the type (7.6)will depend upon the specific form of the Hamiltonian. In their evaluationwe have made use of equations (7.7). It is not hard to see, however, that theseequations are the exact equations of motion using the Hamiltonian Hν(N0).In such a way GHν satisfy the same hierarchy of equations as the correspondingGreen’s functions GHν(N0) with the Hamiltonian Hν(N0).

Alternatively, the “boundary conditions” for the functions of the complexvariable E

GHν , GHν(N0)

on the real E axis (a type of dispersion relation), which are defined by thespectral representations, are also identical.


78 N. N. BOGOLUBOV

From this we conclude thatn

GHν = GHν(N0) (7.8)

or

ϕ1(t, r1) . . . ϕ(t, rs);ϕ(τ, x1) . . . ϕ(τ, xm) Hν

= ϕ1(t, r1) . . . ϕ(t, rs);ϕ(τ, x1) . . . ϕ(τ, xm) Hν(N0) .(7.9)

Further, since averages of the type ϕ1(t, r1) . . . ϕ(t, rn) can be expressedin terms of our Green’s functions, then we will also have

〈. . .Ψ†(rα) . . .Ψ(rβ) . . .〉Hν = 〈. . . (√ρ0 + Ψ†1(rα)) . . . (

√ρ0 + Ψ1(rβ)) . . .〉Hν

= 〈. . . (√ρ0 + Ψ†1(rα)) . . . (

√ρ0 + Ψ1(rβ)) . . .〉Hν(N0).

(7.10)

From this follows the equality of the corresponding average energies, andthus, also the free energies for both dynamical systems.

Thus, we arrive at the conclusion that the investigation of a dynamicalsystem with the Hamiltonian Hν can be reduced to the investigation of a“model system with a condensate”, characterized by the Hamiltonian Hν(N0).Note here that in the model with a condensate the value N0 can be formallyconsidered as an “arbitrary” external parameter.

In order to obtain an equation for N0 we will again turn to the originalsystem with the Hamiltonian Hν and evaluate the expression,

0 = id

dt

⟨ a0√V

⟩=

⟨iV −1/2da0

dt

⟩(7.11)

nThe same conclusion would have been arrived at with the same method, if we hadinvestigated, the Schwinger “multi-time” Green’s functions of the type,

〈T (ϕ1(t1, r1) . . . ϕ1(tn, rn))〉instead of the “double time” Green’s functions, (7.4). The corresponding Schwingerhierarchy of equations would then have been obtained. The Green’s functions

〈T (. . . ϕ1(tα, rα) . . .)〉Hν

would again satisfy the same hierarchy of equations as

〈T (. . . ϕ1(tα, rα) . . .)〉Hν (N0).

The corresponding spectral properties defined by the structure of averages from the T -product would also be identical.



using the equation of motion (which “is missing” in the model system)

ida0

dt=

1

2V

∑(k′

1,k′2,k2)

φ(k′1) + φ(k′2)∆(k2 − k ′1 − k ′

2 )a†k2ak′

2ak′

1

− ν√V − µa0

where

φ(k) =

∫φ(r)ei(k·r) dr.

Now, separating amplitudes with zero momentum, we find

ida0

dt= − µa0 +

a†0a0a0

Vφ(0) − ν

√V

+1

V

∑(p =0)

φ(p) + φ(0)a†papa0 +1

V

∑(p =0)

φ(p)a†0apa−p

+1

2V

∑(p =0,p1 =0,p2 =0)

φ(p1) + φ(p2)∆(p− p1 − p2)a†pap1ap2 .

Substituting this equation into the right hand side of equation (7.11), weobtain

S = 0 (7.12)

where

S =⟨a†0a0a0

V 3/2

⟩Hν

− ν − µ⟨ a0√

V

⟩Hν

+1

V 3/2

∑(p =0)

φ(p) + φ(0)〈a†papa0〉Hν +1

V 3/2

∑(p =0)

φ(p)〈a†0apa−p〉Hν

+1

2V 3/2

∑(p =0,p1 =0,p2 =0)

φ(p1) + φ(p2)∆(p− p1 − p2)〈a†pap1ap2〉Hν .

In accordance with the above discussion we can substitute√N0 for a0 and

a†0, and the remaining averages,

〈a†p, ap〉Hν , 〈ap, a−p〉Hν , 〈a†p, ap1ap2〉Hν

can be replaced by the averages

〈a†p, ap〉Hν(N0), 〈ap, a−p〉Hν(N0), 〈a†p, ap1ap2〉Hν(N0).


80 N. N. BOGOLUBOV

We then obtain

S = ρ3/20 φ(0) − ν − µρ

1/20

+ρ

1/20

V

∑(p =0)

φ(p) + φ(0)〈a†pap〉Hν(N0) +ρ

1/20

V

∑(p =0)

φ(p)〈apa−p〉Hν(N0)

+1

2V 3/2

∑(p =0,p1 =0,p2 =0)

φ(p1) + φ(p2)∆(p− p1 − p2)〈a†pap1ap2〉Hν(N0).

(7.13)

On the other hand we have,

Hν(N0) = H(N0) − 2ν√N0V

1/2 (7.14)

where H(N0) is obtained fromH by the substitution of√N0 for the operators

a0 and a†0, i.e.

H(N0) = −µN0 +N2

0

2Vφ(0) +

∑(k =0)

( k2

2m− µ

)a†kak

+N0

V

∑(k =0)

φ(k) + φ(0)a†kak +N0

2V

∑(k =0)

φ(k)(aka−k + a†−ka†k)

+

√N0

2V

∑(k =0,k1 =0,k2 =0)

φ(k1) + φ(k2)∆(k − k1 − k2)a†k1a†k2ak

+

√N0

2V

∑(k =0,k1 =0,k2 =0)

φ(k1) + φ(k2)∆(k − k1 − k2)a†kak1ak2

+1

2V

∑(k1 =0,k2 =0,k′

1 =0,k′2 =0)

φ(k1 − k′1)∆(k1 + k2 − k′1 − k′2)

× a†k1a†k2ak′

2ak′

1. (7.15)

From this, using (7.12), we can write⟨∂Hν(N0)

∂N0

⟩Nν(N0)

=S + S∗

2√ρ0

=S√ρ0

= 0. (7.16)

Let us construct an expression for the free energy with theHamiltonian Nν(N0),

Fν(N0, µ, θ) = −θ ln Tr e−Hν(N0)

θ .



In this expression N0 is considered to be an arbitrary macroscopic parameter.We have, ⟨∂Hν(N0)

∂N0

⟩Nν(N0)

=∂Fν(N0, µ, θ)

∂N0

and thus equation (7.16) for the determination of N0 becomes,

∂Fν(N0, µ, θ)

∂N0= 0. (7.17)

This equation obviously agrees with thermodynamic consideration. In reality,since in the model system N0 is an outside parameter, its value for a given µand θ must be such as to minimize the free energy. Equation (7.17), in thisinterpretation, expresses the required condition for a minimum.

Let us now consider how the auxiliary parameter ν (which must approachzero only after the limit V → 0) is introduced into the calculation of ourmodel system. Note, first that since Hν differs from H(N0) only by thelast term, −2V

√N0V , then all the Green’s functions and averages of field

operator products will not, obviously, depend on ν for the given N0. Theywill be the same as for the system with the Hamiltonian H(N0). We have

Fν = F − 2ν√N0 V

1/2 (7.18)

where F = F (N0, µ, θ) represents the free energy of the system with theHamiltonian H(N0). Because of this equation (7.17) can be written in theform

∂F (N0, µ, θ)

∂N0

=ν√ρ

(7.19)

and we see that the parameter ν will enter the calculation only through N0.Since we must take the limit ν → 0, we finally obtain,

∂F (N0, µ, θ)

∂N0

= 0. (7.20)

The model with a condensate was first introduced in 1947 [9]. Theapproximate Hamiltonian investigated there was diagonalized by meansof u−v transformation. Green’s functions of the type (6.11) and the diagramtechnique were applied to it in 1958 [10]. This model has since been carefullyconsidered [11], more recently.


82 N. N. BOGOLUBOV

8. The 1/q2 Theorem and its Application

Let us return to the investigation of those Green’s functions which wereconsidered in section 6. We shall show that

| aq,a†q E=0 | ≥ const

q2,

|Σ11(0, q) − Σ12(0, q)| ≤ const q2.(8.1)

Gradient transformation will play important part in this proof. Our quasi-averages were not introduced by the gradient-invariant method. Thus, forcompleteness, we shall use the Hamiltonian Hν and first obtain a seriesof inequalities for the corresponding expressions based on regular averageswith H . We will then obtain formulas of the type (8.1) by taking thelimit ν → 0.

Let us now investigate the gradient transformation,

Ψ(r) → Ψ′(r) = e−iχ(r)Ψ(r),

Ψ†(r) → Ψ′†(r) = eiχ(r)Ψ†(r)(8.2)

and construct the “transformed” Hamiltonian,

H ′ν(Ψ

†,Ψ) = H ′ν(Ψ

′†,Ψ′). (8.3)

Note that,

〈. . .Ψ†(rα) . . .Ψ(rβ) . . .〉Hν =Tr

(. . .Ψ†(rα) . . .Ψ(rβ) . . .) e−

Hν(Ψ†,Ψ)θ

Tr e−

Hν(Ψ†,Ψ)θ

holds. From which, using (8.3) we have

〈. . .Ψ†(rα) . . .Ψ(rβ) . . .〉Hν = 〈. . .Ψ′†(rα) . . .Ψ′(rβ) . . .〉H′ν.

However, due to (8.3), we have directly,

〈. . .Ψ†(rα) . . .Ψ(rβ) . . .〉H′ν

= 〈. . .Ψ†(rα) . . .Ψ(rβ) . . .〉H′νexp i

(∑α

χ(rα) −∑

β

χ(rβ)).



Hence we find

〈. . .Ψ†(rα) . . .Ψ(rβ) . . .〉H′ν

= 〈. . .Ψ†(rα) . . .Ψ(rβ) . . .〉Hν exp i(∑

α

χ(rβ) −∑

β

χ(rα)). (8.4)

In this manner we obtain the following rule: in order to calculate the averageof products of field functions for the Hamiltonian H ′

ν we must take theseaverages for the Hamiltonian Hν and perform the gradient transformationsinverse to (8.2)

Ψ(r) → eiχ(r)Ψ(r),

Ψ†(r) → e−iχ(r)Ψ†(r).(8.5)

For our purpose it is sufficient to consider only the infinitesimal gradienttransformations, in which

χ(r) = δχ(r) =(eiq·r + e−iq·r

)δξ

where δξ is a real, infinitesimal quantity. In this case thetransformations (8.2) will take the form

Ψ′(r) = Ψ(r) − iΨ(r)δχ(r)

Ψ′†(r) = Ψ†(r) + iΨ†(r)δχ(r).

In the momentum representation this becomes,

a′k = ak − i(ak+q + ak−q)δξ

a′†k = a†k + i(a†k+q + a†k−q)δξ.(8.6)

Let us now examine

δHν = Hν(Ψ′†,Ψ′) −Hν(Ψ

†,Ψ).

Due to (6.2) we have,U(Ψ′†,Ψ′) = U(Ψ†,Ψ). (8.7)

Thus, we obtain

δHν = − 1

2m

∫

V

Ψ†(r)(p∂δχ

∂r+∂δχ

∂rp)dr − νV 1/2(δa0 + δa†0).


84 N. N. BOGOLUBOV

From this it follows that

δHν = Uaδξ (8.8)

where

Uq =q2

2µSq + iν(aq + a−q − a†q − a†−q)

Sq = −i∑(k)

(2k + q) · qq2

(a†k+qak − a†kak+q).(8.9)

Let us now obtain the increment,

〈Uq〉Hν+δHν − 〈Uq〉Hν .

For this purpose we need the quantities,

〈ak〉Hν+δHν , 〈a†k〉Hν+δHν , 〈a†k1ak2〉Hν+δHν .

Using the rule mentioned above we can calculate these quantities bysubstituting 〈. . .〉Hν for 〈. . .〉Hν+δHν and simultaneously subjecting theamplitudes a and a† to the gradient transformation inverse to (8.6), namely,

ak → ak + i(ak+q + ak−q)δξ

a′†k → a†k − i(a†k+q + a†k−q)δξ.

In this way we obtain,

〈ak〉Hν+δHν − 〈ak〉Hν = i〈ak+q + ak−q〉Hνδξ.However we have

〈ak〉Hν =

√N0, k = 0

0, k = 0

and thus we obtain

〈ak〉Hν+δHν − 〈ak〉Hν = i√N0∆(k + q) + ∆(k − q)δξ. (8.10)

Similarly we obtain,

〈a†k〉Hν+δHν − 〈a†k〉Hν = −i√N0∆(k − q) + ∆(k + q)δξ. (8.11)



We have, further

〈a†k1ak2〉Hν+δHν − 〈a†k1

ak2〉Hν = − i〈(a†k1+q + a†k1−q)ak2〉Hνδξ

+ i〈a†k1(ak2+q + ak2−q) Hνδξ.

Now, note that 〈a†p1ap2〉 = ∆(p1 − p2)Np1 where Np1 = 〈a†p1

ap2〉Hν . Wetherefore obtain,

〈a†k1ak2〉Hν+δHν − 〈a†k1

ak2〉Hν =i∆(k1 − k2 + q) + ∆(k1 − k2 − q)× (Nk1 −Nk2)δξ.

Making use of this equality, we find using (8.9),

〈Sq〉Hν+δHν − 〈Sq〉Hν = −4Nδξ

whereN =

∑(k)

Nk =∑(k)

〈a†kak〉Hν

represents the total number of particles in our system. Taking into account(8.10) and (8.11) we obtain,

〈Uq〉Hν+δHν − 〈Uq〉Hν = −4(N

q2

2m+ ν

√N0 · v1/2

)δξ,

〈aq〉Hν+δHν − 〈aq〉Hν = i√N0 δξ, (8.12)

〈aq − a†−q〉Hν+δHν−〈aq − a†−q〉Hν = 2i√N0 δξ.

Alternately, we can compute the increment 〈A〉Hν+δHν − 〈A〉Hν where A =Uq, aq, aq − a†−q using formula (5.20).

Taking into account (8.8), we obtain

〈Uq〉Hν+δHν − 〈Uq〉Hν = 2π Uq,Uq E=0 δξ,

〈aq〉Hν+δHν − 〈aq〉Hν = 2π aq,Uq E=0 δξ,

〈aq − a†−q〉Hν+δHν−〈aq − a†−q〉Hν = 2π (aq − a†−q),Uq E=0 δξ.

(8.13)

Comparing these formulas with (8.12) we see that,

Uq,Uq E=0= −2

π

(N

q2

2m+ ν

√N0 · v1/2

)(8.14)


86 N. N. BOGOLUBOV

and

| aq,Uq E=0 |2 =N0

4π2,

| (aq − a†−q),Uq E=0 |2 =N0

π2.

(8.15)

Let us now make use of inequality (5.19), in which we assume,

A = aq, aq − a†−q, ; B = Uq.

Note on the basis of (8.15), that Uq = U †q . We the obtain from (8.15),

N0

4π2≤ | aq, a

†q E=0 Uq,Uq E=0 |,

N0

4π2≤ | (aq − a†−q), (a

†q − a−q) E=0 Uq,Uq E=0 |.

From which, using (8.14), we will have

| aq, a†q E=0 | ≥ ρ0m

4π(q2ρ+ 2νm√ρ0)

,

| (aq − a†−q), (a†q − a−q) E=0 | ≥ ρ0m

(q2ρ+ 2νm√ρ0)

. (8.16)

Let us now take the limit ν → 0. As is seen, these expressions, in theneighborhood of q ∼ 0, have the characteristic behavior, constant q−2, as isseen from,

| aq, a†q E=0 | ≥

(ρ0m

4πρ

) 1

q2, (8.17)

| (aq − a†−q), (a†q − a−q) E=0 | ≥

(ρ0m

πρ

) 1

q2. (8.18)

We note that

(aq − a†−q), (a†q − a−q) E=0 = aq, a

†q E=0 − a†−q, a

†q E=0

− aq, a−q E=0 + a†−q, a−q E=0 .

Using the symmetry properties (6.13) and (6.17) gives

aq, a−q E=0= a†−q, a†q E=0, a†−q, a−q E=0= aq, a

†q E=0



and thus we have,

(aq − a†−q), (a†q − a−q) E=0 = 2 aq, a

†q E=0 −2 a†−q, a

†q E=0 .

(8.19)

Similarly, due to (6.22) we can write

aq, a†q E=0− a†−q, a

†q E=0

=1

2π

Σ11(0, q) + Σ12(0, q)

Σ211(0, q) − Σ2

12(0, q)=

1

2π

1

Σ11(0, q) − Σ12(0, q).

Thus, taking into account (8.18) and (8.19), we obtain,

∣∣∣ 1

Σ11(0, q) − Σ12(0, q)

∣∣∣ ≥ (ρ0m

ρ

) 1

q2

or,

|Σ11(0, q) − Σ12(0, q)| ≤ ρ

ρ0mq2. (8.20)

The inequalities (8.17) and (8.20) are just the inequalities (8.1) which wereto be proved.

Let us now consider some the applications. We write the spectral for-mulas,

〈aq(τ)a†q(t)〉 =

+∞∫

−∞

Jq(ω) e−iω(t−τ) dω,

〈a†q(τ)aq(t)〉 =

+∞∫

−∞

Jq(ω) eω/θ e−iω(t−τ) dω,

〈aqa†q〉 = − 1

2π

+∞∫

−∞

Jq(ω)eω/θ − 1

ωdω;

Jq(ω) ≥ 0. (8.21)

and note that,

eω/θ − 1

ω=

(1 + eω/θ)

ωthω

2θ≤ 1

2ω(1 + eω/θ).


88 N. N. BOGOLUBOV

Thus,

| aq, a†q E=0 | ≤ 1

4πθ

+∞∫

−∞

Jq(ω)(1 + eω/θ) dω

=1 + 2〈a†qaq〉

4πθ

holds. From this we have, on the basis of (8.17)

1 + 2〈a†qaq〉 ≥ mθ

q2

ρ0

ρ. (8.22)

Referring, for example, to (3.7) and (3.8) we see that

〈a†qaq〉 = (2π)3w(q) = (2π)3W1(q); q = 0.

We can also write

W1(q) ≥ 1

2(2π)3

mθq2

ρ0

ρ− 1

. (8.23)

From this it follows that the density of the continuous particle momentumdistribution approaches infinity as q−2 as q → 0.o This statement holdsonly for the case θ > 0. In order to obtain information about the situationwhen θ = 0 let us make use of the spectral formulas for this particular case(refer to section 6),

− aq, a†q E=0=

1

2π

+∞∫

−∞

Iq(ω)∈(ω)

ωdω =

1

2π

+∞∫

−∞

Iq(ω)dω

|ω|

2〈a†qaq〉 + 1 = 〈a†qaq + aqa†q〉 =

+∞∫

−∞

Iq(ω) dω,

Iq(ω) ≥ 0.

For small enough |q|, when one can speak of “elementary excitations”possessing a known energy, it is natural to assume that the spectral intensity

oIf we had investigated the auxiliary system with fixed ν > 0 and did note take intothe limit ν → 0, then this characteristic behavior would not have appeared.



Iq(ω) is almost equal to zero for |ω| < E(q). Here E(q) is the minimumenergy of the elementary excitation for the momentum q. Then we have

1

2π

+∞∫

−∞

Iq(ω)dω

|ω| ≤1

2πE(q)

+∞∫

−∞

Iq(ω) dω =2〈a†qaq〉 + 1

2πE(q)

and thus, on the basis of (8.17), we obtain

2〈a†qaq〉 + 1

2πE(q)≥ m

4πq2

ρ0

ρ

or

2(2π)3W1(q) ≥ E(q)

2q2

mρ0

ρ− 1.

If the elementary excitation spectra possesses a phonon character, E(q) =c|q|, then W1(q) approaches infinity as q → 0 not slower than const · |q|−1.

We will now show that by using our inequalities one can determine thecharacter of the excitation spectra. For this, let us turn to the relation (8.20),from which it follows that,

Σ11(0, 0) − Σ12(0, 0) = 0. (8.24)

Note that the equality for the case of zero temperature was first derived byHegenholtz and Pines using perturbation theory. In their work [11] a modelsystem with a condensate was considered and the diagram technique wasused for its investigation. They were able to show that the equalities (8.24)hold in any order of perturbation theory. The importance of this relationshiplies in its connection with the structure of the energy spectra of the perturbedsystem.

Let us consider the “secular” equation,

Σ11(E, k)Σ11(−E, k) − Σ212(E, k) = 0 (8.25)

and assume that the mass operator, Σ(E, k) is regular in the neighborhoodof the point E = 0, k = 0. We write (8.25) in the form

[Σ11(E, k) + Σ11(−E, k)2

]2

− Σ212(E, k) =

[Σ11(E, k) − Σ11(−E, k)2

]2

.

(8.26)


90 N. N. BOGOLUBOV

Note that due to the radial symmetry in our problem the function Σαβ(E, k2)will depend on k only the scalar k2.

Further, the left hand side of the equation is an even function of E. Dueto (8.24) it becomes zero when E = 0, k = 0. Thus for sufficiently small Eand k we can write[Σ11(E, k) + Σ11(−E, k)

2

]2

− Σ212(E, k) = βk2 + γE2

β, γ = const.

We also note, that the expression

Σ11(E, k) + Σ11(−E, k)2

will be an odd function of E. If we retain only the first order term, we willobtain, for sufficiently small E and k,

Σ11(E, k) + Σ11(−E, k)2

= αE, α = const.

In this way equations (8.25) and (8.26) yield,

α2E2 = βk2 + γE2.

let us exclude from this investigation the special cases when α2 − γ and βbecome zero. Then, we have

E2 = sk2, s = 0, s =β

α2 − γ.

We see that the magnitude, s, must be positive, since the pole of the Green’sfunction must lie on the real axis of the complex E-plane. Thus, for theexcitation energy, we obtain an “acoustical” dependency without gap,

E =√s |k|. (8.27)

From previous consideration, it is seen that equation (8.24) is relatedto the gradient invariance of the “potential energy” U . Thus, it is notsurprising that if we violate this invariance property then we also will violateequation (8.24) and will obtain formulas for E(k) which contain an energygap. This occurs, for example, when the investigating model system in which,



in the expression for U , only the interaction of pairs with opposite momentais kept. The same situation arises if we include a term which is not gradientinvariant, −ν√V (a0 + a†0), with a fixed ν > 0, in the Hamiltonian.

Let us illustrate this fact by a simple example where we keep only thelowest terms in the interaction, φ, for the case θ = 0. For the actualconstruction of such an approximation we will deal with a model system witha Hamiltonian Hν(N0). From the forms (7.14) and (7.15) of this Hamiltonianit is not difficult to note, that by taking into account only the first orderterms, we have,

Σ11(E, k) = E − k2

2m+ µ− ρφ(k) + φ(0),

Σ12(E, k) = −ρ0φ(k),

Fν = −µN0+N2

0

2Vφ(0) − 2ν

√N0V . (8.28)

Because of this equation (7.17) yields,

0 =∂Fν

∂N0= −µ+ ρ0φ(0) − ν√

ρ0.

Thus we haveΣ11(0, 0) − Σ12(0, 0) = − ν√

ρ0

< 0.

Let us further consider the secular equation (8.25). In the presentapproximation we obtain from (8.28),( k2

2m+ ρ0φ(k) +

ν√ρ0

)2

− (ρ0φ(k)

)2= E2.

From this we the formula for the energy of the elementary excitation

E(k) =

√ν2

ρ0

+ 2ν√ρ0

( k2

2m+ ρ0φ(k)

)+ ρ0φ(k)

k2

m+

( k2

2m

)2

.

which contains an energy gap.The gap disappears after taking the limit ν → 0, whereupon we arrive at

the usual expression,

E(k) =

√ρ0φ(k)

k2

m+

( k2

2m

)2

.


92 N. N. BOGOLUBOV

This spectrum has a quasi-acoustic character when k is small. Let us finallynote that the formulas “with a gap” in E(k) can be also obtained with ν = 0,if one “mismatches” the approximations. For example, we would obtain agap in E(k) if we used for Σαβ(E, k) the formulas of the first approximation,and substituted in the equation

∂Fν

∂N0

= 0

for Eν , the formulas of the second approximation.

9. The 1/q2 Theorem for Fermi Systems

Let us now turn to the derivation of the “1/q2 theorem” for the Fermi systemswhich were investigated in section 4. We consider the case when the systemhas a condensate of pair quasi-molecules, in S-state, (4.11), and when theHamiltonian has the usual form,

H =∑(σ)

∫Ψ†(r, σ)

( p2

2m− µ

)Ψ(r, σ) dr + U(Ψ†,Ψ). (9.1)

Here the expression U is gradient invariant. Note that the model of Frohlich,in which the electrons interact with the phonon field, is of the type whichwas considered. Actually, we can include the energy of the phonons andthe energy of their interaction with the electrons in U . Note that U mustbe invariant with respect to gradient transformation which act only on theFermi functions Ψ, Ψ†.

Let us return to the arguments of the previous paragraph and considerthem in detail. For brevity we will not introduce the infinitesimal termswhich remove the degeneracy into the expression for the Hamiltonian, andwe will agree to deal directly with the corresponding quasi-averages.

Thus, let us consider the infinitesimal gradient transformations,

Ψ(x) → Ψ′(x) = Ψ(x) − iΨ(x)δχ

Ψ†(x) → Ψ′†(x) = Ψ†(x) + iΨ†(x)δχ

δχ =(eiqr + e−iqr

)δ′ξ.

We construct the variation,

δH = H(Ψ′†,Ψ′) −H(Ψ†,Ψ).



We then have,

δH = − 1

2m

∫ ∑σ

Ψ†(r, σ)(p∂δξ

∂r+∂δξ

∂rp)Ψ(r, σ) dr.

From this follows

δH =q2

2mSqδχ (9.2)

where q is a given non-zero momentum and

Sq = −i∑(k,σ)

(2k + q) · qq2

(a†k+q,σak,σ − a†k,σak+q,σ). (9.3)

Now note that the increment 〈U 〉H+δH − 〈U 〉H can be computed by twomethods.

The direct calculations gives

〈U 〉H+δH − 〈U 〉H = 〈U ′ − U 〉H (9.4)

where U ′ is obtained from U with the inverse gradient transformation,

Ψ → Ψ + iΨδχ, Ψ† → Ψ† − iΨ†δχ.

In the momentum representation this becomes,

ak,σ → ak,σ + iak+q,σ + ak−q,σδξ,a†k,σ → a†k,σ − ia†k+q,σ + a†k−q,σδξ. (9.5)

Alternatively, we can make use of the formula (5.20) and write

〈U 〉H+δH − 〈U 〉H =πq2

m U , Sq E=0 dξ.

We therefore have,

〈U ′ − U 〉H =πq2

m U , Sq E=0 dξ. (9.6)

If we take U = V −1Sq, we then find

V −1/2Sq, V−1/2Sq E=0= −4ρm

πq2; ρ =

N

V. (9.7)


94 N. N. BOGOLUBOV

Let us introduce the operators,

βk =1√V

∑(σ1,σ2)

∈(σ1)∆(σ1 + σ2)

∫∫

V

Ψ(x2)Ψ(x1)ϑ(r1 − r2) e−ikr1+r2

2 dr1 dr2

(9.8)where ϑ(r) is a radially symmetric real function of r, which decreases rapidlyenough as r → ∞ and which satisfies the condition,

γ ≡∫ϕ(r)ϑ(r) dr = 0. (9.9)

We then have

〈V −1/2(β′q − β − q)〉

=i∑

(σ1,σ2)

∈(σ1)∆(σ1 + σ2)1

V

∫∫

V

eiqr2 + eiqr1 + e−iqr2 + e−iqr1

× 〈Ψ(x2)Ψ(x1)〉 e−ikr1+r2

2 dr1 dr2δξ

=i∑

(σ1,σ2)

∈(σ1)∆(σ1 + σ2)1

V

∫∫

V

2 cos(q(r1 − r2)

2

)

× 1 + e−iq(r1+r2)

〈Ψ(x2)Ψ(x1)〉 dr1 dr2δξand therefore, using (4.14) we obtain,

〈V −1/2(β ′q − βq)〉 = 4i

√ρ0

∫ϕ(r)ϑ(r) cos

(qr2

)dr δξ. (9.10)

From this, because the functions ϕ and ϑ are real we have,

〈V −1/2(β ′†−q − β†

−q)〉 = −4i√ρ0

∫ϕ(r)ϑ(r) cos

(qr2

)dr δξ.

Consequently,

〈V −1/2(β′q − β ′†

−q) − (βq − β†−q)〉 = 8i

√ρ0

∫ϕ(r)ϑ(r) cos

(qr2

)dr δξ. (9.11)

In the relations (9.6) we substitute,

U = V −1/2βq, V −1/2(βq + β†−q).



Then, due to (9.10) and (9.11) we will have,

| βq, V−1/2Sq E=0 |2 =

(4mγ(q)

πq2

)2

ρ0

| (βq − β†−q), V

−1/2Sq E=0|2 =(8mγ(q)

πq2

)2

ρ0 (9.12)

where

γ(q) =

∫ϕ(r)ϑ(r) cos

(qr2

)dr. (9.13)

Let us maximize the left hand side of equation (9.12) using theinequalities (5.19), in which we assume,

A = βq, (βq − β†−q); B = V −1/2Sq.

Since on the basis of (9.3), Sq = S†q , we arrive at inequalities of the form,

(4mγ(q)

πq2

)2

ρ0 ≤| βq, β†q E=0 V −1/2Sq, V

−1/2Sq E=0 |(8mγ(q)

πq2

)2

ρ0 ≤| (βq − β†−q), (β

†q − β−q) E=0

× V −1/2Sq, V−1/2Sq E=0 |.

From this, taking into account (9.7), we obtain

| βq, β†q E=0 | ≥ 4mγ2(q)

πq2

ρ0

ρ

(βq − β†−q), (β

†q − β−q) E=0 ≥ 16mγ2(q)

πq2

ρ0

ρ. (9.14)

Due to (9.9) and (9.13) we have

γ2(0) = γ2 > 0.

And thus the “1/q2 theorem” is proven for the present case. Apparentlyit is associated with the property of the energy spectrum of “collectiveexcitations”. Here we will not consider this problem, but will limit ourselvesto the application of the proven theorem for estimating the number of pairswith momentum q = 0 in the case when θ > 0.


96 N. N. BOGOLUBOV

Let us consider the spectral function

− βq, β†q E=0 =

1

2π

+∞∫

−∞

eω/θ − 1

ωJ(ω) dω, J(ω) > 0

〈βqβ†q + β†

qβq〉 =

+∞∫

−∞

(1 + eω/θ

)J(ω) dω. (9.15)

Taking into account that

eω/θ − 1

ω<

eω/θ + 1

2θ

we can write〈βqβ

†q + β†

qβq〉 ≥ 4πθ βq, β†q E=0 .

Because of this, using (9.14)

〈βqβ†q + β†

qβq〉 ≥ 16mθγ2(q)

q2

ρ0

ρ. (9.16)

Let us now estimate 〈βqβ†q + β†

qβq〉. For this purpose we introduce thequantity (9.8) in a somewhat more abstract form

βq =

∫K (x1, x2)Ψ(x2)Ψ(x1) dx1 dx2 (9.17)

where K (x2, x1) = −K (x1, x2). Let us make use of the commutationrelationships,

Ψ(x)Ψ†(x′) + Ψ†(x′)Ψ(x) = δ(x− x′).

We obtain

βqβ†q − β†

qβq =2

∫∣∣K (x1, x2)∣∣2dx1 dx2

− 4

∫∫

K (x1, x2)K∗(x′1, x2) dx2Ψ†(x′1)Ψ(x1) dx1 dx

′1.

By specifying the form of the function K so that (9.7) agrees with (9.8), weobtain

〈βqβ†q − β†

qβq〉 =4

∫ϑ2(r) dr



− 4

V

∑(p,σ)

〈a†pσapσ〉∫∫ϑ(r + r ′)ϑ(r ′) dr ′ e

i

(p− q

2

)rdr.

Thus, we have|〈βqβ

†q − β†

qβq〉| ≤ C

where

C = 4

∫ϑ2(r) dr + 4ρ

(∫ |ϑ(r)| dr )2.

Consequently, we obtain from (9.16)

〈β†qβq〉 ≥ 8mθγ2(q)

q2

ρ0

ρ− C

2. (9.18)

Let us now note that

〈β†qβq〉 =

∑(σ1 ,σ2,σ′

1,σ′2)

∈(σ1)∈(σ′1)∆(σ1 + σ2)∆(σ′

1 + σ′2)

× 1

V

∫〈Ψ†(x1)Ψ

†(x2)Ψ(x′2)Ψ(x′1)〉

× exp iq(r1 + r2 − r ′

1 − r ′2

2

)dr1, dr2 dr

′1 dr

′2 .

Utilizing expression (4.5), we find

〈β†qβq〉 = 2

∑(ω)

NωqV∣∣ ∑(σ1,σ2)

∈(σ1)∆(σ1+σ2)

∫Ψω,q(r, σ1, σ2)ϑ(r) dr

∣∣2. (9.19)

However, the functions Ψω,q in formula (4.5), possess the followingorthonormalization properties for a given q,

∑(σ1,σ2)

V

∫Ψω1,q(r, σ1, σ2)Ψω2,q(r, σ1, σ2) dr = δ(ω1 − ω2).

Thus, by applying the corresponding Bessel inequality we obtain∑(ω)

V∣∣ ∑(σ1,σ2)

∈(σ1)∆(σ1 + σ2)

∫Ψω,q(r, σ1, σ2)ϑ(r) dr

∣∣2

≤∑

(σ1,σ2)

∆(σ1 + σ2)

∫θ2(r) dr = 2

∫θ2(r) dr.


98 N. N. BOGOLUBOV

From this, using (9.19),

〈β†qβq〉 ≤ 4

(max(ω)

Nω,q)

∫θ2(r) dr

and thus, due to (9.18),

max(ω)

Nω,q ≥(

2mθγ2(q)∫θ2(r) dr

ρ0

ρ

)1

q2− C

8∫θ2(r) dr

. (9.20)

Let θ > 0 and let ζ2 be a positive quantity satisfying the inequality,

ζ2 ≤ 2mθγ2(q)∫θ2(r) dr

ρ0

ρ.

Then, for small enough q we obtain

max(ω)

Nω,q ≥ ζ2

q2.

Thus, for small enough momentum, q, there is a pair state (ω, q) suchthat the average number of particle existing in this state,

nq = Nω,q (9.21)

will satisfy the inequality

nq ≥ ζ2

q2. (9.22)

As can be seen, this is analogous tom the inequality (8.22), which wasdetermined for Bose system in the presence of a condensate.

In conclusion, let us emphasize that the inequality (9.22) is provenonly for the case when U(ψ†, ψ), which enters into expression (9.1) of theHamiltonian H , is gradient invariant. For the model system considered insection 2, U is not gradient invariant and, thus, it is not surprising that inthis case the inequality (9.22) is not satisfied.



References

1. J. Bardeen, L. Cooper and J. Schrieffer, Phys. Rev., 106, 162 (1957);Phys. Rev., 108, 1175 (1957).

2. N. N. Bogoliubov, D. N. Zubarev, and Yu. A. Tserkovnikov, Docl. Akad.Nauk SSSR, 117, 788 (1957); Sov. Phys. “Doklady”, English Transl.,2, 535, (1957).

3. N. N. Bogoliubov, D. N. Zubarev, and In. Tserkovnikov, Zh. Exper. iTeor. Fiz., 39, 120 (1960); Sov. Phys. JETP, English Transl., 12, 88(1961).

4. N. N. Bogoliubov, “On the model Hamiltonian in superconductivitytheory” preprint, (1960), see p. 167 this volume.

5. N. N. Bogoliubov, Zh. Exper. i Teor. Fiz., 34, 58 (1958); Sov. Phys.JETF, English Transl., 7, 41 (1958).

6. N. N. Bogoliubov, Docl. Acad. Nauk SSSR, 119, 52 (1958); Sov. Phys.“Doklady”, English Transl., 3, 279 (1958); Usp. Fiz. Nauk 67, 549(1959); Sov. Phys. – Usp., English Transl., 2, 236 (1959).

7. N. N. Bogoliubov, Lectures on quantum statistics, monograph, ed.“Radiansika Shkola”, Kiev, 1949; Lectures on Quantum Statistics,Vol. 1, Gordon and Breach, New York, (1967).

8. D. N. Zubarev, Usp. Fiz. Nauk 71, 71 (1960); Sov. Phys. –Usp., English Transl., 3, 320 (1960); V. L. Bonch-Bruevich andS. V. Tyablikov, The Green Function Method in Statistical Mechanics(English transl.) North Holland Pub. Co., Amsterdam (1962).

9. N. N. Bogoliubov, Izv. Akad. Nauk SSSR Ser. Fiz. 11, 77 (1947);Journal 7, 43 (1947).

10. S. T. Beliaev, Zh. Exper. i Theor. Fiz., 34, 417 (1958); Sov. Phys.JETP, English Transl., 7, 289 (1958).

11. H. Hugenholtz and D. Pines, Phys. Rev. 116, 489 (1959).


CHAPTER 3

HYDRODYNAMICS EQUATIONS IN STATISTICALMECHANICS

1. In the present paper the equations of hydrodynamics are derived onthe basis of classical mechanics of a system of molecules. As usual, werestrict ourselves to the simplest scheme and consider a system of a verylarge number N of identical monoatomic molecules enclosed in a macroscopicvolume V . We suppose also that the interaction between the molecules is dueto central forces described by the potential energy of a molecule pair Φ(r),which depends only on the distance between them.

We shall use the results and the notations of our preceding works [1, 2].We introduce the distribution function of dynamical variables of the moleculeset

Fs = Fs(t, q1, . . . , qs, p1, . . . ,ps), s = 1, 2, 3, . . . ,

in such a way that the expressions

1

V sFs dq1 . . . dqsdp1 . . . dps

give the probabilities of finding the coordinates and the momenta of the 1-st,. . ., s-th molecules from some arbitrary set of molecules in the infinitesimalvolumes dq1 . . . dqs, dp1 . . . dps of the coordinate and momentum spaces attime t.

Since our aim is to study only volume properties of the system, we restrictourselves to considering the leading asymptotic terms in the equation ofmotion for the function Fs that has been derived in the preceding works

∂Fs

∂t= [Hs, Fs] +

1

ν

∫[ ∑(i≤i≤s)

Φi,s+1, Fs+1

]dqs+1dps+1, (1)

100



where

Φi,s+1 = Φ(|qi − qs+1|), νV

N

and Hs denotes the Hamiltonian of a system of s isolated particles

Hs =∑

(i≤i≤s

)|pi|22m

+∑

(i≤i<j≤s)

Φi,j. (2)

Equation (1) must be augmented by the normalized conditions

Fs = limV →∞

1

V

∫

V

dqs+1

∫dps+1Fs+1,

limV →∞

1

V

∫

V

dq1

∫dp1F1 = 1 (3)

and by conditions of correlation weakening, which can be represented, forinstance, in the following form

S(s)−τ

Fs −

∏(i≤i≤s)

F1(t, qi,pi)→ 0, τ → +∞, (4)

where S(s)−τ denotes the operator replacing the coordinates

q1, . . . , qs

by

q1 −p1

mτ, . . . , qs −

ps

mτ,

respectively, with the values of the momenta remaining unaltered. Finally, itis clear that the functions Fs must be symmetric with regard to permutationof the molecule’s dynamic variables

PijFs = Fs (5)

where Pij is the operator replacing the variables (qi,pi) by (qj,pj) and viceversa.

2. Such dynamical variables as the particle number density σ(t, q),the particle number flux density J(t, q), and the internal energy flux


102 N. N. BOGOLUBOV

density E(t, q) are of special importance for the problems of hydrodynamics.These quantities can be represented as follows

σ(t, q) =∑

(i≤i≤s)

δ(qi − q),

J(t, q) =∑

(i≤i≤s)

Pi δ(qi − q),

E(t, q) =∑

(i≤i≤s)

p2i

2mδ(qi − q)

+1

2

∑(i≤i<j≤s)

Φ(|qi − qj |)δ(qi − q) + δ(qj − q)

. (6)

However, it should be emphasized that such a definition of quantities σ, Jand E is necessary only under microscopic consideration of the system’smotion. As far as the usual macroscopic hydrodynamics is concerned, it issufficient to know their average values, which can be determined with the aidof the function F1 and F2

σ =ρ =1

ν

∫F1(t, q,p) dp,

J =mρu =1

ν

∫q F1(t, q,p) dp,

E =1

ν

∫ |p|22m

F1(t, q,p) dp

+1

2ν2

∫Φ(|q − q′|)F2(t, q, q

′,p,p′) dq′dpdp′. (7)

Since, due to above definition, the quantity ρ(t, q)dq provides the averagenumber of molecules whose coordinates at time t are in the infinitesimalvolume dq, we can consider the quantity ρ as a distribution function for themolecules’ coordinates. The vector u is, obviously, the average velocity, theratio E/ρ is the average energy per molecule. If we subtract the kineticenergy of ordered motion from E/ρ, we obtain the average internal energy

ε = E/ρ− 1

2mu2.

From Equation (2) it is not difficult to obtain

ρε =1

ν

∫ |p −mu|22m

F1(t, q,p) dp



+1

2ν2

∫Φ(|q − q′|)F2(t, q, q

′,p,p′) dq′dpdp′. (8)

Our principle task is to derive the equations which describes evolution of themain hydrodynamics functions ρ, uα, and ε.p

3. In the theory of kinetic equations, in studying the distribution functionsapproaching their stationary form, which corresponds to the state ofstatistical equilibrium, particular attention is paid to the so called spatiallyhomogeneous case, when the functions F1(t, q,p) do not depend on q and thehigher correlation functions are invariant under the spatial transformations

q1 → q1 + q0, . . . , qs → qs + q0

(where q0 is arbitrary). It should be stressed that the equations consideredalways have a spatially homogeneous solution. This can be seen by takinginto account Equation (1) and noting the fact that, if there is spatialinhomogeneity at the initial moment, it still exists at later time t. It can alsobe established that in the spatially homogeneous case the hydrodynamicfunctions ρ, u, and ε are independent of t and do not vary with time.Thus, from the physical point of view, this case corresponds to completehomogeneity of the macroscopic spatial distribution of molecules.

In hydrodynamics, when introducing these functions we actually averagethem over a region the linear dimensions of which are large compared with theradius of intermolecular forces, and over an interval of time large comparedwith the so called molecular unit of time r0m/|p|av, where |p|av is the averagevalue of the modules of the molecule’s momentum. Thus, to obtain theusual macroscopic hydrodynamics we should consider distribution functionssufficiently close to the spatially homogeneous ones in order to ensure thatthe quantities ρ, u, and ε determined by Equations (7) and (8) beingconsidered as functions of q and t vary sufficiently smoothly with respectto the molecular scale r0, r0m/|p|av.

In this connection it is necessary to introduce some small parameter µ andto determine its physical significance. First of all, we consider the expression

Fs(t, q, q2 − q1+q, . . . , qs − q1 + q,p1, . . . ,ps)

= Fs(t, q, q1, . . . , qs,p1, . . . ,ps) (9)

pThe superscripts α, β, γ appearing after the vectors denote a Descartes components.


104 N. N. BOGOLUBOV

and note that

Fs(t, q, q1, . . . , qs,p1, . . . ,ps) = Fs(t, q, q1, . . . , qs,p1, . . . ,ps),

since expression (9) is spatially homogeneous with respect to q1, . . . , qs

Fs(t, q, q1+q0, . . . , qs + q0,p1, . . . ,ps)

= Fs(t, q1 + q0, q1, . . . , qs,p1, . . . ,ps)

Thus, the slower the variation of Fs(t, q, q1, . . . , qs,p1, . . . ,ps) withvarying q, the smaller is the variation of the functions Fs under thetranslations

q1 → q1 + q0, . . . , qs → qs + q0.

Therefore, restricting ourselves to distributions close to the spatiallyhomogeneous, we seek a solution for Fs in the form

Fs = fs(t, µq1, q1, . . . , qs,p1, . . . ,ps, µ),

F1 = f1(t, µq1,p1, . . . , µ), (10)

where µ is a small parameter and the functions

fs(t, ξ, q1, . . . , qs,p1, . . . ,ps, µ), s = 1, 2, 3 . . . (11)

are asymptotically regular in the vicinity of µ = 0. After taking the limit µ →0 expressions (10) yield distributions which are more closer to the spatiallyhomogeneous.

We next substitute expressions (10) into Equations (7) and (8) and takeinto account the fact that the functions ρ, u, and ε depend on q only throughthe product µq = ξ, and, therefore, are the smooth functions of q. Further,we note that the derivatives ∂ρ/∂t, ∂u/∂t, ∂ε/∂t are proportional to µ, sothat the functions ρ, u, and ε are smooth functions of t as well. In thisconnection the parameter µ introduced above can be physically interpretedas a number of the order of the ratio r0/l of the radius of intermolecularforces r0 to the length l characterizing the average size of macroscopicinhomogeneity. The equations of hydrodynamics to be derived below are ofobvious asymptotic character of an expansion in powers of the parameter µ.It should be stressed immediately that we need the expressions for the



hydrodynamic parameters ρ, u, and ε to be represented as functions of tand ξ. We substitute Equation (10) into Equations (7) and (8) to obtain

ρ(t, ξ) =1

ν

∫f1(t, ξ,p1) dp1, (12)

mρuα(t, ξ) =1

ν

∫pα

1 f1(t, ξ,p1) dp1, (13)

ρε(t, ξ) =1

ν

∫ |p1 −mu|22m

f1(t, ξ,p1) dp1

+1

2ν2

∫Φ(|q1 − q2|)f2(t, ξ, q1, q2,p1,p2) dq2dp1dp2. (14)

Since the function f2 is spatially homogeneous with respect to q1 and q2, itdepends on their difference q1 − q2

f2(t, ξ, q1, q2,p1,p2) = ϕ(t, ξ, q1 − q2,p1,p2), (15)

and hence the expression on the right-hand side of Equation (14) isindependent of q1 although the integration over this variable has not beenperformed.

Now on the basis of Equation (1) we obtain the equations determiningthe time evolution of the functions fs. Note that according to Equations (10)we can write

∂Fs

∂qj

=∂fs

∂qj

, j = 2, 3, . . . , s,

∂Fs

∂pj

=∂fs

∂pj

, j = 1, 2, 3, . . . , s,

and∂Fs

∂q1

=∂fs

∂q1

+ µ∂fs

∂ξ.

Consequently, Equations (1) give the main equations determining theevolution of the functions fs in the form

∂fs

∂t= − µ

∑(1≤α≤3)

pα1

m

∂fs

∂ξα+

[Hs, fs

]


106 N. N. BOGOLUBOV

+1

ν

∫[ ∑(1≤α≤3)

Φi,s+1, fs+1

]dqs+1dps+1, (16)

where the Poisson bracket acts only upon the canonical variables q1, . . . , qs,p1, . . . , ps and not upon the auxiliary variable ξ. If we take into accountthe identity ∫

∂Φ1,2

∂qα2

∂f2

∂pα2

dp2 = 0,

we can write the equation

∂f1

∂t= −µ

∑(1≤α≤3)

pα1

m

∂fs

∂ξα+

1

ν

∫ ∑(1≤α≤3)

∂Φ1,2

∂qα1

∂f2

∂pα1

dq2dp2 (17)

as a particular case of Equation (16).Now we turn to the symmetry conditions (5). In view of expressions (10)

it is obvious that

Pijfs = fs, i = 2, 3, . . . , s, j = 2, 3, . . . , s. (18)

Then we have

fs(t, µq1, q1, . . . , qs,p1, . . . ,ps)

= fs(t, µqj, qj , . . .q1, . . . , qs,pj , . . . ,p1, . . . ,ps)

= fs(t, µq1 + µ(qj − q1), qj, . . .q1, . . . , qs,pj, . . . ,p1, . . . ,ps)

Let us agree to consider P1j as an operator that acts only on the canonicalvariables but not on the variable ξ. Then we can write

fs(t, ξ, q1, . . . , qs,p1, . . . ,ps)

= P1jfs(t, ξ − µ(qj − q1), q1, . . . , qs,p1, . . . ,ps)

or

P1jfs(t, ξ, q1, . . . , qs,p1, . . . ,ps)

= fs(t, ξ − µ(qj − q1), q1, . . . , qs,p1, . . . ,ps). (19)

Hence we find

P1jfs =fs − µ∑

α

(qj − q1)α ∂fs

∂ξα



+µ2

2

∑αβ

(qj − q1)α(qj − q1)

β ∂2fs

∂ξα∂ξβ+ µ3 . . . (20)

Note now that the normalization conditions (3) and the correlation weakeningconditions (4) for functions fs can be represented in the form

fs = limV →∞

1

V

∫

V

dqs+1

∫dps+1 fs+1, (21)

S(s)−τ

fs(t, ξ, q1, . . . , qs,p1, . . . ,ps)

−∏

1≤i≤s

f1(t, ξ + µ(qi − q1),p1)→ 0, τ → +∞, (22)

where the operator S(s)−τ acts only on the canonical variables. The differential

equations (16) together with the symmetry (18), (20), normalization (21),and the correlation weakening condition (22) determine the distributionfunctions fs.

4. Now we proceed to calculate the derivatives

∂ρ

∂t,

∂(ρuα)

∂t,∂(ρε)

∂t.

We differentiate Equation (12) and take into account Equation (17) to obtain

∂ρ

∂t=

1

ν

∫∂f1(t, ξ,p1)

∂tdp1

= − µ∑

α

1

mν

∫pα

1

∂f1

∂ξαdp1 +

1

ν2

∑α

∫∂Φ1,2

∂qα1

∂f2

∂pα1

dp1dq2dp2.

Since ∫∂Φ1,2

∂qα1

∂f2

∂pα1

, dp1 = 0

and due to Equation (13)

1

mν

∫pα

1

∂f1

∂ξαdp1 =

∂

∂ξα

1

mν

∫pα

1f1 dp1 =∂(ρuα)

∂ξα,


108 N. N. BOGOLUBOV

we arrive at the usual continuity equation

∂ρ

∂t= −µ

∑α

∂(ρuα)

∂ξα. (23)

If we differentiate expression (13) and take into account Equation (17), weobtain

∂(ρuα)

∂t=

1

mν

∫pα

1

∂f1

∂tdp1 = −µ

∑β

1

m2ν

∂

∂ξβ

∫pα

1 pβ1 f1 dp1

+1

mν2

∑β

∫∂Φ1,2

∂qβ1

∂f2

∂pβ1

pα1 dp1dq2dp2,

whence after integrating the second term by parts, we find

∂(ρuα)

∂t= − µ

∑β

1

m2ν

∂

∂ξβ

∫pα

1 pβ1 f1 dp1

+1

mν2

∫∂Φ1,2

∂qα1

f2 dp1dq2dp2. (24)

Now we note that according to Equation (15) we have

∫∂Φ1,2

∂qα1

f2 dp1dq2dp2

=

∫∂Φ(|q1 − q2|)

∂qα1

ϕ(t, ξ, q1 − q2,p1,p2) dq2dp1dp2

=

∫∂Φ(|q|)∂qα

1

ϕ(t, ξ, q,p1,p2) dqdp1dp2,

and, consequently, changing the indices of the integration variables, we arriveat ∫

∂Φ1,2

∂qα1

f2 dp1dp2dq2

=

∫∂Φ(|q|)∂qα

1

ϕ(t, ξ, q,p2,p1) dqdp1dp2

= −∫∂Φ(|q|)∂qα

1

ϕ(t, ξ,−q,p2,p1) dqdp1dp2.



It is obvious that

ϕ(t, ξ,−(q1 − q2),p2,p1) = P1,2ϕ(t, ξ, q1 − q2,p2,p1),

and therefore we can write∫∂Φ1,2

∂qα1

f2 dp1dp2dq2 = −∫∂Φ1,2

∂qα1

(P1,2f2) dp1dp2dq2,

so that ∫∂Φ1,2

∂qα1

f2 dp1dp2dq2 =1

2

∫∂Φ1,2

∂qα1

f2 − P1,2f2

dp1dp2dq2.

Further, we take into account the symmetry property (20) to obtain

∫∂Φ1,2

∂qα1

f2 dp1dp2dq2 =µ

2

∑β

∫∂Φ1,2

∂qα1

(q2 − q1)β ∂f2

∂ξβdp1dp2dq2

+µ2

4

∑β,γ

∫∂Φ1,2

∂qα1

(q2 − q1)β(q2 − q1)

γ ∂2f2

∂ξβ∂ξγdp1dp2dq2 + µ3 + . . . (25)

Note also that according to Equations (12) and (13) we have

∫pα

1pβ1 f1 dp1 =

∫(p1 −mu)α(p1 −mu)β f1 dp1 +m2νuαuβρ. (26)

We substitute these relations into the right-hand side of Equation (24) tofind

∂(ρuα)

∂t= − µ

∑β

∂(ρuαuβ)

∂ξβ− µ

∑β

∂Tα,β

∂ξβ

+ µ2∑β,γ

∂2Tα,β,γ

∂ξβ∂ξγ+ µ3 + . . . (27)

where for brevity we have put

Tα,β ≡Tα,β(f1, f2) =1

mν2

∫(p1 −mu)α(p1 −mu)β f1 dp1

+1

2mν2

∫∂Φ(|q2 − q1|

∂qα1

(q2 − q1)β f2 dq2dp1dp2, (28)


110 N. N. BOGOLUBOV

Tα,β,γ ≡Tα,β,γ(f1, f2) =

+1

4mν2

∫∂Φ(|q2 − q1|

∂qα1

(q2 − q1)β(q2 − q1)

γ f2 dq2dp1dp2. (29)

Finally, in view of the continuity equation (23), Equation (27) can be alsorepresented in the form

ρ∂(uα)

∂t= − µ

∑β

ρuβ ∂(uα)

∂ξβ− µ

∑β

∂Tα,β

∂ξβ

+ µ2∑β,γ

∂2Tα,β,γ

∂ξβ∂ξγ+ µ3 + . . . (30)

At last, let us proceed to differentiate expression (14). If we notice thatthe following identity holds∫

f1

∂

∂t|p1 −mu|2 dp1 = −2

∑α

∂uα

∂t

∫f1(p

α1 −muα) dp1 = 0,

we obtain

∂(ρε)

∂t=

1

2mν

∫|p1 −mu|2 ∂f1

∂tdp1 +

1

2ν2

∫Φ1,2

∂f2

∂tdq2dp1dp2.

Consequently, making use of Equation (16) and (17) we find

∂(ρε)

∂t= − µ

2m2ν

∑α

∫|p1 −mu|2 ∂f1

∂ξαdp1

+1

2mν2

∑α

∫|p1 −mu|2∂Φ1,2

∂qα1

∂f2

∂pα1

dq2dp1dp2

+1

2ν2

∫Φ1,2

−µ

∑α

pα

m

∂f2

∂ξα+

[p21 + p2

2

2m+ Φ1,2, f2

]dq2dp1dp2

+1

2ν3

∫Φ1,2

[Φ1,3 + Φ2,3, f3

]dq2dq3dp1dp2dp3.

Due to the identities∫Φ1,2

[Φ1,3 + Φ2,3, f3

]dp1dp2dp3 = 0,



∫Φ1,2

[Φ1,2, f2

]dp1dp2 = 0

and in accordance with Equation (15)

[p21 + p2

2

2m, f2

]=

1

m

∑α

∂f2

∂qα2

(p1 − p2)α,

we can write

∂(ρε)

∂t= − µ

2m2ν

∑α

∫|p1 −mu|2 ∂f1

∂ξαdp1

− 1

2mν2

∑α

∫Φ1,2 p

α1

∂f2

∂pα1

dq2dp1dp2

+1

2mν2

∑α

∫Φ1,2

∂f2

∂qα2

(p1 − p2)α

+ |p1 −mu|2∂Φ1,2

∂qα1

∂f2

∂pα1

dq2dp1dp2 (31)

Note also that∫Φ1,2

∂f2

∂qα2

(p1 − p2)α + |p1 −mu|2∂Φ1,2

∂qα1

∂f2

∂pα1

dq2dp1dp2

=

∫−∂Φ1,2

∂qα2

f2(p1 − p2)α − ∂|p1 −mu|2

∂pα1

∂Φ1,2

∂qα1

f2

dq2dp1dp2 (32)

With the aid of permutations and the symmetry conditions obtained abovewe can see that the following relations hold

∫∂Φ1,2

∂qα1

2muα − pα

1 − pα2

f2 dq2dp1dp2

=

∫∂Φ1,2

∂qα1

muα − pα

1 + pα2

2

(f2 − P1,2f2

)dq2dp1dp2

= µ∑

β

∂Φ1,2

∂qα1

muα − pα

1 + pα2

2

∂f2

∂ξβ(q2 − q1)

β dq2dp1dp2

− µ2

2

∑β,γ

∂Φ1,2

∂qα1

muα − pα

1 + pα2

2

∂2f2

∂ξβ∂ξγ


112 N. N. BOGOLUBOV

× (q2 − q1)β(q2 − q1)

γ dq2dp1dp2 + µ3 + . . .

Thus Equations (31) and (32) yield

∂(ρε)

∂t= − µ

2m2ν

∑α

|p1 −mu|2 pα1

∂f1

∂ξαdp1

− µ

2m2ν

∑α

∫Φ1,2 p

α1

∂f2

∂ξαdq2dp1dp2

+µ

2m2ν

∑α,β

∫∂Φ1,2

∂qα1

muα − pα

1 + pα2

2

(q2 − q1)

β ∂f2

∂ξβdq2dp1dp2

− µ2

4mν2

∑α,β,γ

∂Φ1,2

∂qα1

muα − pα

1 + pα2

2

× (q2 − q1)β(q2 − q1)

γ ∂2f2

∂ξβ∂ξγdq2dp1dp2 + µ3 + . . .

We note also that∫|p1 −mu|2pα

1

∂f1

∂ξαdp1 =

∂

∂ξα

∫|p1 −mu|2pα

1 f1 dp1

+ 2∑

β

m∂uβ

∂ξα

∫(p1 −mu)β pα

1f1 dp1,

∫∂Φ1,2

∂qα1

muα − pα

1 + pα2

2

(q2 − q1)

β ∂f2

∂ξβdq2dp1dp2

=∂

∂ξβ

∫∂Φ1,2

∂qα1

muα − pα

1 + pα2

2

(q2 − q1)

β f2 dq2dp1dp2

−m∂uα

∂ξβ

∫∂Φ1,2

∂qα1

(q2 − q1)β f2 dq2dp1dp2,

and consequently

∂(ρε)

∂t= − µ

2m2ν

∑α

∂Xα

∂ξα− µ

2m2ν

∑α,β

∂uβ

∂ξαYα,β

− µ2

4mν2

∑α,β,γ

∂Φ1,2

∂qα1

(q2 − q1)β(q2 − q1)

γ



×muα − pα

1 + pα2

2

∂2f2

∂ξβ∂ξγdq2dp1dp2 + µ3 + . . . , (33)

where

Xα =1

m

∫|p1 −mu|2 pα

1 f1 dp1 +1

ν

∫Φ1,2p

α1 f1 dq2dp1dp2

− 1

ν

∑β

∫∂Φ1,2

∂qβ1

muβ − pβ

1 + pβ2

2

(q2 − q1)

αf2 dq2dp1dp2,

Yα,β = 2

∫(p1 −mu)βpα

1 f1 dp1 +m

ν

∫∂Φ1,2

∂qβ1

(q2 − q1)αf2 dq2dp1dp2.

We make use of Equations (12)-(14) as well as the identity

∫(p1 −mu)βuα f1 dp1 = uα

∫(p1 −mu)β f1 dp1 = 0,

to obtain

Xα = 2mνuαρε+1

m

∫|p1 −mu|2(p1 −mu)α f1 dp1

+1

ν

∫Φ1,2(p1 −mu)αf2 dq2dp1dp2

− 1

ν

∑β

∫∂Φ1,2

∂qβ1

muβ − pβ

1 + pβ2

2

(q2 − q1)

αf2 dq2dp1dp2.

If we put

Sα,β = Sα,β(f1, f2) =1

mν

∫(pβ

1 −muβ)(pα1 −muα)f1 dp1

+1

2ν2

∫∂Φ1,2

∂qβ1

(qα2 − qα

1 )f2 dq2dp1dp2 =Tα,β

m, (34)

Sα = Sα(f1, f2) =1

2m2ν

∫|p1 −mu|2(p1 −mu)α f1 dp1

+1

mν2

∫Φ1,2(p

α1 −muα)2f2 dq2dp1dp2


114 N. N. BOGOLUBOV

+1

2mν2

∑β

∫∂Φ1,2

∂qβ1

pβ1 + pβ

2

2−muβ

(q2 − q1)

αf2 dq2dp1dp2 (35)

R = R(f2) =1

4mν2

∑α,β,γ

∫∂Φ1,2

∂qα1

(q2 − q1)β(q2 − q1)

γ

×muα − pα

1 + pα2

2

∂2f2

∂ξβξαdq2dp1dp2, (36)

we can write

∂(ρε)

∂t= − µ

∑α

∂(uαρε)

∂ξα− µ

∑α

∂Sα

∂ξα

− µ∑α,β

∫∂uβ

∂ξαSα,β − µ2R+ µ3 . . . (37)

instead of Equation (33).

5. Having determined the derivatives of the main hydrodynamics variables,we now return to Equation (16), which is to be solved taking into accountthe symmetry conditions (18) and (20), the normalization condition (21),and the correlation weakening condition (22). It should be emphasized thatthe usual expansion in powers of µ

fs(t, ξ, q1, . . . , qs,p1, . . . ,ps) = f 0s (t, ξ, q1, . . . , qs,p1, . . . ,ps)

+ µf 1s (t, ξ, q1, . . . , qs,p1, . . . ,ps) + µ2 + . . . (38)

cannot result in obtaining the equations of hydrodynamics, with the aid ofwhich one could describe the evolution of hydrodynamics variables.

In fact, if we substitute Equation (38) into formulas (12)–(14), we find

ρ = ρ0 + µρ1 + . . . ,

uα = uα0 + µuα

1 + . . . ,

ε = ε0 + µε1 + . . . , (39)

where ρ0, uα0 , and ε0 are determined by Equations (12)-(14) where the

quantities f 0 are substituted for f , and ρ1, uα1 , and ε1 are determined by

the same equations where f 1 are substituted for f , etc. On the other hand,



if we substitute Equation (38) into Equation (16) and equate the coefficientsof identical power of µ, we obtain, for instance, for the leading terms

∂f 0s

∂t=

[Hs; f

0s

]+

1

ν

∫[ ∑1≤i≤s

Φi,s+1; f0s+1

]dqs+1dps+1

s = 1, 2, 3, . . . (40)

We note also that the symmetry conditions become

Pijf0s = f 0

s , P1jf0 = f0

s . (41)

Since the function f 0s is spatially homogeneous with regard to q1, . . . , qs and

there are no operators acting on the argument ξ, we find that the functions f 0s

vary in time according to the same law as the function Fs does in the caseof spatially homogeneous distribution. Therefore, the derivatives

∂ρ0

∂t,∂uα

0

∂t,∂ε0

∂t

must be identically equal to zero, since the formal expressions for them followEquations (23), (27), and (37) in proceeding to the spatially homogeneouscase, i.e. for µ = 0. Thus, ρ0, u

α0 , and ε0 do not vary in time t. We find that

in accordance with Equation (39) the variation of hydrodynamics functionsis entirely due to the evolution of the correction terms proportional to thesmall parameter.

However, since one can make use of the asymptotic expansion (38)preserving one or two terms only until the correction terms are smallcompared with the leading ones. Therefore, these expansions are valid onlyfor time intervals during which the quantities ρ, uα, and ε do not varysignificantly from their initial values. In other words, we are not able toderive the equations of hydrodynamics, i.e. equations which would describethe evolution of quantities ρ0, u

α0 , and ε0 for a sufficiently long period of time

during which these quantities change considerably. Apparently, the difficultyof using the usual expansion is trivial enough and is connected with theappearance of secular term, as is usual in such cases.

In order to formulate a proper method of expansion similar to that used innonlinear mechanics, we note that in the case under consideration one needsto know a particular solution fs(t, ξ, q1, . . . , qs,p1, . . . ,ps) of the equationsconsidered for which the derivatives ∂fs/∂t are of the order of magnitude


116 N. N. BOGOLUBOV

of µ from the beginning. Indeed, by introducing the small parameter µand considering the expressions for Fs in the form of Equation (10), weactually perform the spatial averaging over a region the dimensions ofwhich are sufficiently large by comparison with the molecular distances andinvestigate the solutions for which the derivative ∂fs/∂t is proportional to µand considered as averaged over time in a certain way. Then, in the firstapproximation, the functions

fs = f 0s (42)

satisfy the equations

[Hs; f

0s

]+

1

ν

∫[ ∑1≤i≤s

Φi,s+1; f0s+1

]dqs+1dps+1 = 0, (43)

the symmetry conditions (41), and the corresponding normalization andcorrelation weakening conditions. Note also that the functions f0

s arespatially homogeneous with regard to q1, . . . , qs, since the above relationscontain no operators acting on the variable ξ and this variable is containedin f 0

s only as a parameter. Equations (43) are those very equations whichdetermine the stationary spatially homogeneous distribution Fs. Suppose,as is valid for gases at usual densities, that there exists only one system ofstationary solutions, namely that which corresponds to statistical equilibriumand which id fully determined by the values of the temperature, temperature,and average velocity vector.q Then we look for the expression of f0

s in theform

f 0s = ϕs(ξ, q1, . . . , qs) exp

−

∑1≤i≤s

|pi − k|22mθ

, (44)

where k is a vector, and θ is the temperature. Since the function ϕs isspatially homogeneous, we have

f 01 = ϕ(ξ) exp

−|p1 − k|2

2mθ

. (45)

qFor amorphous solids such as glasses, there exist distributions which do not correspondto statistical equilibrium, since they have an infinitesimal relaxation velocity and,therefore, are stationary. On the other hand, for crystals which are not even in thestate of statistical equilibrium, the probability density depends on the the location of thecrystal lattice in space, and therefore it is necessary to fix a large number of parametersto determine the density.



Substituting expressions (44) and (45) into Equation (40) we obtain

∂ϕs

∂qα1

+1

θ

∂Us

∂qα1

ϕs +1

θ

(2πmθ)3/2

ν

∫∂Φ1,s+1

∂qα1

ϕs+1 dqs+1 = 0 (46)

whereUs =

∑(1≤i≤j≤s)

Φi,j

and the functions ϕs are assumed to be symmetric with regard to thevariables q1, . . . , qs. Moreover, taking into account the correlation weakeningcondition for f 0

s and formula (45) we see that the functions ϕs(ξ, q1, . . . , qs)tend to ϕ(ξ)s as the distance q1, . . . , qs tends to infinity.

In order to obtain the corresponding expressions for f 0s as well as for the

coefficients of the expansion

fs = f0s + µf 1

s + . . . , (47)

we impose the following auxiliary condition: performing the expansion weshould not expand the functions ρ, uα, and ε in powers of µ. In other words,we determine the coefficients of expansion (47) so that the following relationshold

ρ =1

ν

∫f 0

1 dp1, mρuα =1

ν

∫pα

1 f01 dp1,

ρε =1

ν

∫ |p1 −mu|22m

f 01 dp1 +

1

2ν2

∫Φ1,2f

02 dq2dp1dp2, (48)

and

1

ν

∫f1

1 dp1 = 0,1

ν

∫pα

1f11 dp1 = 0,

1

ν

∫ |p1 −mu|22m

f11 dp1 +

1

2ν2

∫Φ1,2f

12 dq2dp1dp2 = 0. (49)

Then Equations (44) and (45) imply that

kα = muα, ϕ(ξ) =νρ

(2πmθ)3/2. (50)

Puttingϕs(ξ, q1, . . . , qs)

(ϕ(ξ))s= χs,


118 N. N. BOGOLUBOV

we find that the functions χs satisfy the equations

∂χs

∂qα1

+1

θ

∂Us

∂qα1

χs +1

θρ

∫∂Φ1,s+1

∂qα1

χs+1 dqs+1 = 0. (51)

We note also that the functions χs are symmetric and spatially homogeneouswith regard to q1, . . . , qs. Moreover, χ1 = 1 while χs (s = 2 , 2, . . .)approaches unity when the distance between the molecules increasecontinuously. Thus, the functions χs are nothing but the distributionfunction for the coordinates for the molecule complex considered in the firstchapter of the monograph [2] (in this monograph these functions are denotedby Fs(q1, . . . , qs)). In the case under consideration, the particle numberdensity 1/ν is apparently replaced by ρ.

One can obtain explicit expressions for χs, for instance, with the aidof expansions in powers of ρ. In this case one does not need the explicitexpressions for χs. It is only necessary that the quantities χ2, χ3, . . .be determined by the number ρ and that they be invariant with regard totranslation and reflection transformations in the space of q. Therefore, weput

χs = χs(q1, . . . , qs, ρ),

χ1 = 1, χ2 = g(|q1 − q2|, ρ). (52)

We see that g(r, ρ) is the usual molecular distribution function, whichdepends on the distance r between two molecules and on the averagedensity ρ.

With the aid of the formulae obtained we write Equation (44) in the form

f 0s =

(νρ)s

(2πmθ)3s/2χs(q1, . . . , qs, ρ) exp

−

∑1≤i≤s

|pi −mu|22mθ

. (53)

We substitute this expression into formula (48) to obtain

ε =3

2θ + ρ

∫Φ(|q|) g(|q|, ρ) dq = ε(ρ, θ). (54)

It should be stressed that this expression is nothing but the thermodynamicaverage energy per molecule expressed in terms of the density and thetemperature θ.



If one expresses the parameter θ in terms of ε and ρ with the aid ofEquation (54), the expressions for f s

0 are completely determined by the valuesof ρ, u, and ε as arbitrary functions of χ. Keeping in mind Equations (23),(30), and (37) we see that in the first approximation the derivatives ∂ρ/∂t,∂ε/∂t, and ∂u/∂t are equal to zero, and therefore ρ, u, and ε can beconsidered as arbitrary constants with respect to the variable t.

6. Let us now proceed to the final formulation of the correspondingmethod for asymptotic expansion of the function fs in powers of the smallparameter µ.

Making use of the methods of nonlinear mechanics, which are thecorresponding generalization of Langrange’s method of the variation ofarbitrary constants, we look for an expression for fs in the form

fs =f 0s (ξ, q1, . . . , qs,p1, . . . ,ps|ρ,u, ε)+ µf 1

s (ξ, q1, . . . , qs,p1, . . . ,ps|ρ,u, ε) + µ2 + . . . , (55)

where ρ, u, and ε are solutions to the equations

∂ρ

∂t= µA1(ξ|ρ,u, ε) + µ2A2(ξ|ρ,u, ε) + µ3 + . . . ,

∂uα

∂t= µBα

1 (ξ|ρ,u, ε) + µ2Bα2 (ξ|ρ,u, ε) + µ3 + . . . , α = 1, 2, 3,

∂(ρε)

∂t= µC1(ξ|ρ,u, ε) + µ2C2(ξ|ρ,u, ε) + µ3 + . . . . (56)

In this relations, f 0s , f

1s , . . .; A1, A2, . . .; Bα

1 , Bα2 , . . .; C1, C2, . . .

are unknown functions of ρ, u and ε which depend on ξ and whichare to be chosen in such a way that, after substituting the solutions ofEquations (56) into expression (55), it satisfies Equations (16) and all theauxiliary conditions imposed on the functions fs, including conditions (48)and (49). It is clear that expansions (55) and (56) are of a solelyformal character and only the first one or two terms have practicalsignificance. Therefore, the above formulae should be understood as thosewhose asymptotic approximation, obtained neglecting some power of theparameter µ, satisfy the corresponding equations up to this power.

If we take into account Equation (23) we find directly

A1 = −∑

α

∂(ρuα)

∂ξα, A2 = A3 = . . . = 0, (57)


120 N. N. BOGOLUBOV

i.e. the first equation of (56) is really the continuity equation. ThenEquations (30) and (37) yield

Bα1 = −

∑β

ρuβ ∂uβ

∂ξβ−

∑β

∂Tα,β(f 01 , f

02 )

∂ξβ,

Bα2 = −

∑β

∂Tα,β(f 11 , f

12 )

∂ξβ+

∑β,γ

∂2Tα,β,γ(f02 )

∂ξβ∂ξγ

C1 = −∑

α

∂(uαρε)

∂ξα−

∑α

∂Sα(f01 , f

02 )

∂ξα−

∑α,β

∂uβ

∂ξαSα,β(f0

1 , f02 ),

C2 = −∑

α

∂Sα(f 01 , f

02 )

∂ξα−

∑α,β

∂uβ

∂ξαSα,β(f 0

1 , f02 ) − R(f0

2 ). (58)

On the other hand, making use of the considerations of the precedingsection it is not difficult to see that the first term in expansion (55) isdetermined by formula (53). Therefore the quantities Bα

1 and C1 can beeasily found. Therefore, we are now able to show that the equations of thefirst approximation

ρ∂uα

∂t= µBα

1 ,∂(ρε)

∂t= µC1, (59)

which follow Equation (56) after neglecting the terms of the second order ofsmallness are the ordinary equations of the hydrodynamics of an ideal liquid.

Indeed, we substitute expressions (53) into Equation (58) and make useof the notations (28), (34), and (35) for Tα,β, Sα, and Sα,β to obtain

Bα1 = −

∑β

ρuβ ∂uα

∂ξα− 1

m

∂P

∂ξα

C1 = −∑

α

∂(uαρε)

∂ξα−−

∑α

∂uα

∂ξαP, (60)

where

P = ρθ − ρ2

C

∫∂Φ(|q|)∂|q| g(|q|, ρ) dq. (61)

Thus, the equations in the first approximation (59) being reconsidered interms of the spatial variable q = µ−1ξ take the form

mρ∂uα

∂t+

∑β

∂uα

∂qβuβ

= − ∂P

∂qα, (62)



∂(ρε)

∂t+

∑α

∂(uαρε)

∂qα+ P

∑α

∂uα

∂ξα= 0, (63)

where mρ is the mass density. As we see (one may consult Equation (1.7) inthe monograph [2]) the quantity P given by Equation (61) is the pressure,while relation (61) itself is the equation of state in the case when anexplicit expression for the molecular distribution function g is known. Thus,Equation (62) is the ordinary equation of state of an ideal liquid.

To transform Equation (63) to the usual form we introduce the free energyof the system per molecule

Ψ = Ψ(ρ, θ).

Since the quantities P and ε are determined with aid of the Gibbs canonicaldistribution, we have

P = ρ2 ∂Ψ

∂ρ, ε = Ψ − θ

∂Ψ

∂θ. (64)

If we use the continuity equation, from Equation (63) we obtain

ρ(∂ε∂t

+∑

α

uα ∂ε

∂qα

)+ P

∑α

∂uα

∂qα= 0,

or

ρ∂ε

∂ρ

(∂ρ∂t

+∑

α

uα ∂ρ

∂qα

)+ ρ

∂ε

∂θ

(∂θ∂t

+∑

α

uα ∂θ

∂qα

)+ P

∑α

∂uα

∂qα= 0.

We then use the continuity equation again to find

ρ∂ε

∂θ

(∂θ∂t

+∑

α

uα ∂θ

∂qα

)+

(P − ρ2 ∂ε

∂ρ

)∑α

∂uα

∂qα= 0. (65)

After determining the entropy

S = −∂Ψ∂θ

, (66)

Equation (64) yields

P − ρ2 ∂ε

∂ρ= −ρ2θ

∂S

∂ρ, ρ

∂ε

∂θ= ρθ

∂S

∂θ.


122 N. N. BOGOLUBOV

Hence, Equation (65) becomes

∂S

∂θ

(∂θ∂t

+∑

α

uα ∂θ

∂qα

)− ρ

∂S

∂ρ

∑α

∂uα

∂qα= 0.

or∂S

∂θ

(∂θ∂t

+∑

α

uα ∂θ

∂qα

)+∂S

∂ρ

(∂ρ∂t

+∑

α

∂uα

∂qα

)= 0.

Equation (63) can be written finally as follows:

∂S

∂t+

∑α

uα ∂S

∂qα= 0. (67)

We see that this is the usual adiabatic equation (see, for instance, paper [3]).Thus, we have shown that the equations of the first approximation

are the ordinary equations of hydrodynamics for an ideal liquid. It is alsopossible to show that the equations of the second order, which are obtainedfrom Equation (56) by neglecting terms of the order of magnitude µ3, arethe equations of a viscous liquid taking into account heat transfer processes.

References

1. Bogolubov, N.N. (1946) Journal of Experimental and TheoreticalPhysics, 16, 681.Bogolubov, N.N. (1946) Journal of Experimental and TheoreticalPhysics, 16, 691,

2. Bogolubov, N.N. (1946) Problems of a Dynamical Theory in StatisticalPhysics , Moscow, Gostekhizdat.

3. Landau, L.D., Lifshitz, E.M. (1944) Mechanics of Continuous MediaMoscow, Gostekhizdat.


CHAPTER 4

ON THE HYDRODYNAMICS OF A SUPERFLUIDLIQUID

Introduction

The object of these lectures is to derive the hydrodynamic equations for asuperfluid liquid from the equations of motion of a system of identical Boseparticles, and to obtain in this way a “hydrodynamic approximation” for theGreen Functions. To simplify the presentation we shall consider our systemto be an ideal liquid.

We shall consider a system of identical Bose particles with pairinteractions, with a Hamiltonian which in the representation of secondquantization has the the form:r

H = +1

2m

∫∇ψ†(t, r)∇ψ(t, r) dr− λ

∫ψ†(t, r)ψ(t, r) dr

+1

2

∫Φ(r − r′)ψ†(t, r)ψ†(t, r′)ψ(t, r′)ψ(t, r) dr dr ′

+

∫η(t, r)ψ†(t, r) + η∗(t, r)ψ(t, r) + U(t, r)ψ†(t, r)ψ(t, r) dr.

Here λ is a constant, and ψ†(t, r), ψ(t, r) are Bose operators in theHeisenberg representation, with the usual commutation relations. The readerwill have noticed that in addition to the usual terms, our Hamiltonianincludes additional terms corresponding to “particle sources” η(t, r), η∗(t, r)and an external field U(t, r); η, η∗ and U are given c-number functions of rand t. The introduction of these “external sources” is necessary because we

rWe use a system of units in which = 1. When r is the argument of a function wewrite it simply as r.

123


124 N. N. BOGOLUBOV

intend to obtain an expression for the Green’s function by varying the usualhydrodynamic averages with respect to them.

1. Preliminary Identities

To derive the hydrodynamic equations we shall need a set of identities forthe time derivatives of the following “local” quantities:

ρ(t, r) = 〈ψ†(t, r)ψ(t, r)〉, φ(t, r) = 〈ψ(t, r)〉,

jα(t, r) =i

2

⟨∂ψ†(t, r)∂rα

ψ(t, r) − ψ†(t, r)∂ψ(t, r)

∂rα

⟩, (α = 1, 2, 3)

ρ(t, r)ε(t, r) = − 1

4m

⟨(∆ψ†(t, r))ψ(t, r) + ψ†(t, r)(∆ψ(t, r))

⟩+

1

2

∫Φ(r − r′)〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)ψ(t, r)〉 dr ′, (1.1)

which represents respectively the mean particle number density (ρ), the meancurrent (j) and the mean energy per particle (ε). In equation (1.1) thepointed brackets denote an average taken with respect to some statisticaloperator; in general the latter need not correspond to statistical equilibrium.

To calculate the time derivatives of the above quantities we use theequations of motion, which for our hamiltonian have the form:

i∂ψ(t, r)

∂t= − λψ(t, r) − ∆

2mψ(t, r)

+

∫Φ(r − r′)ψ†(t, r′)ψ(t, r′) dr ′ ψ(t, r)

+ U(t, r)ψ(t, r) + η(t, r),

i∂ψ†(t, r)

∂t=λψ†(t, r) +

∆

2mψ†(t, r)

− ψ†(t, r)∫

Φ(r − r′)ψ†(t, r′)ψ(t, r′) dr ′

− U(t, r)ψ†(t, r) − η∗(t, r). (1.2)

Differentiating the first equations (1.1), we get

∂ρ(t, r)

∂t=⟨∂ψ†(t, r)

∂tψ(t, r) + ψ†(t, r)

∂ψ(t, r)

t

⟩



=i

2m

⟨−(∆ψ†(t, r))ψ(t, r) + ψ†(t, r)(∆ψ(t, r))⟩

+ i

∫Φ(r − r′)

⟨ψ†(t, r)ψ†(t, r′)ψ(t, r′)ψ(t, r)

− ψ†(t, r)ψ†(t, r′)ψ(t, r′)ψ(t, r)⟩dr ′

+ iη∗(t, r)φ(t, r) − iη(t, r)φ∗(t, r).

But since ∑α

∂

∂rα

⟨∂ψ†(t, r)∂rα

ψ(t, r) − ψ†(t, r)∂ψ(t, r)

∂rα

⟩=⟨(∆ψ†(t, r))ψ(t, r) − ψ†(t, r)(∆ψ(t, r))

⟩and the term containing Φ is identical zero, we get as our final result:

m∂ρ(t, r)

∂t+∑

α

∂jα(t, r)

∂rα

= im[η∗(t, r)φ(t, r) − η(t, r)φ∗(t, r)]. (1.3)

In exactly the same way we get for the current density:s

∂jα∂t

=1

2

⟨( ∂

∂rα

i∂ψ†

∂t

)ψ +

∂ψ†

∂rα

i∂ψ

∂t− i

∂ψ†

∂t

∂ψ

∂rα

− ψ†( ∂

∂rα

i∂ψ

∂t

)⟩

=1

4m

⟨( ∂

∂rα

∆ψ†)ψ − ∂ψ†

∂rα

(∆ψ) − (∆ψ†)∂ψ

∂rα

+ ψ†( ∂

∂rα

∆ψ)⟩

− 1

2

⟨( ∂

∂rα

∫Φ(r − r′)ψ†(t, r)ψ†(t, r′)ψ(t, r′) dr ′

)ψ

− ∂ψ†

∂rα

∫Φ(r − r′)ψ†(t, r′)ψ(t, r′) dr ′ψ(t, r)

+ ψ†( ∂

∂rα

∫Φ(r − r′)ψ†(t, r′)ψ(t, r′) dr ′ψ(t, r)

)−

∫Φ(r − r′)ψ†(t, r)ψ†(t, r′)ψ(t, r′) dr ′∂ψ(t, r)

∂rα

⟩

+1

2

⟨∂ψ†

∂rα

η − ∂η∗

∂rα

ψ + η∗∂ψ

∂rα

− ψ† ∂η∂rα

⟩− 〈ψ†ψ〉 ∂U

∂rα

+1

2

⟨( ∂

∂rα

λψ†)ψ − ∂ψ†

∂rα

λψ − λψ† ∂ψ∂rα

+ ψ†( ∂

∂rα

λψ)⟩.

sFrom now on we omit the arguments t, r of ψ, ψ†, erc., where there is no risk ofambiguity.


126 N. N. BOGOLUBOV

Using the definitions (1.1) and simplifying the expressions somewhat, we find

∂jα∂t

=1

4m

∂

∂rα

∆ρ− 1

2m

∑β

⟨∂ψ†

∂rα

∂ψ

∂rβ

+∂ψ†

∂rβ

∂ψ

∂rα

⟩

−∫∂Φ(r − r′)

∂rα

〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)ψ(t, r)〉 dr ′

+ ρ∂

∂rα(λ− U) +

1

2

(∂φ∗

∂rαη +

∂φ

∂rαη∗ − φ∗ ∂η

∂rα− φ

∂η∗

∂rα

).

It is convenient to introduce a new quantity Dt(r, r − r′)

〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)ψ(t, r)〉 ≡ Dt(r, r′ − r) = Dt(r

′, r − r′)

=1

2Dt(r

′, r − r′) +Dt(r, r′ − r).

Using this quantity and changing the variable of integration to R = r − r ′,we can write the equation for the time derivative of j in the form:

∂jα∂t

=1

4m

∂

∂rα∆ρ− 1

2m

∑β

⟨∂ψ†

∂rα

∂ψ

∂rβ+∂ψ†

∂rβ

∂ψ

∂rα

⟩

−∫∂Φ(R)

∂Rα

Dt(r,−R) +Dt(r −R,R dR

+ ρ∂

∂rα

(λ− U) +1

2

(∂φ∗

∂rα

η +∂φ

∂rα

η∗ − φ∗ ∂η∂rα

− φ∂η∗

∂rα

). (1.4)

Lastly, we have the identity

∂(ρε)

∂t= − 1

m

⟨(∆∂ψ†

∂t

)ψ + (∆ψ†)

∂ψ

∂t+∂ψ†

∂t(∆ψ) + ψ†

(∆∂ψ

∂t

)⟩+

1

2

∫Φ(r − r′)

∂

∂t〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)ψ(t, r)〉 dr ′.

We proceed to calculate the last term:

i∂

∂t〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)ψ(t, r)〉

=1

2m〈(∆ψ†(t, r))ψ†(t, r′)ψ(t, r′)ψ(t, r)〉 +

1

2m〈ψ†(t, r)(∆ψ†(t, r′))

× ψ(t, r′)ψ(t, r)〉 − 1

2m〈ψ†(t, r)ψ†(t, r′)(∆ψ(t, r′))ψ(t, r)〉



+1

2m〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)(∆ψ(t, r))〉 −

∫Φ(r − r′)〈ψ†(t, r)

× ψ†(t, r1)ψ(t, r1)ψ†(t, r′)ψ(t, r′)ψ(t, r)〉 dr ′ +

∫Φ(r − r1)

× 〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)ψ†(t, r1)ψ(t, r1)ψ(t, r)〉 dr ′ +∫

Φ(r′ − r1)

× 〈ψ†(t, r)ψ†(t, r′)ψ†(t, r1)ψ(t, r1)ψ(t, r′)ψ(t, r)〉 dr ′ +∫

Φ(r′ − r1)

× 〈ψ†(t, r)ψ†(t, r′)ψ†(t, r1)ψ(t, r1)ψ(t, r′)ψ(t, r)〉 dr ′

+ η(r, t)〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)〉 − η∗(t, r)〈ψ†(t, r′)ψ(t, r)ψ(t, r)〉+ η(t, r′)〈ψ†(t, r)ψ†(t, r′)ψ(t, r)〉 − η∗(t, r′)〈ψ†(t, r)ψ(t, r′)ψ(t, r)〉.

Using the fact that ψ†(t, r)ψ(t, r) commutes with ψ†(t, r′)ψ(t, r′) we can writethis equation as:

∂

∂t〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)ψ(t, r)〉

=i

2m

⟨ψ†(t, r)ψ†(t, r′)(∆ψ(t, r′))ψ(t, r) + ψ†(t, r)ψ†(t, r′)ψ(t, r′)

× (∆ψ(t, r)) − (∆ψ†(t, r)ψ†(t, r′)ψ(t, r′)ψ(t, r) − ψ†(t, r)

× (∆ψ†(t, r′))ψ(t, r′)ψ(t, r)⟩

+ iη∗(t, r)〈ψ†(t, r′)ψ(t, r′)ψ(t, r)〉+ iη∗(t, r′)〈ψ†(t, r)ψ(t, r′)ψ(t, r)〉 − iη(t, r)〈ψ†(t, r)

× ψ†(t, r′)ψ(t, r′)〉 − iη(t, r′)〈ψ†(t, r)ψ†(t, r′)ψ(t, r)〉.

Substituting this into equation for ∂(ρε)/∂t and using the equations ofmotion, we find:

∂(ρε)

∂t=

i

8m2

⟨(∆∆ψ†)ψ − (∆ψ†)(∆ψ) + (∆ψ†)(∆ψ) − ψ†(∆∆ψ)

+i

4m

⟨−(∆

∫Φ(r − r′)ψ†(t, r)ψ†(t, r′)ψ(t, r′) dr ′

)ψ(t, r)

+

∫Φ(r − r′)[(∆ψ†(t, r))ψ†(t, r′)ψ(t, r′)ψ(t, r) − ψ†(t, r)

× ψ†(t, r′)ψ(t, r′)(∆ψ(t, r))]dr ′ + ψ†(t, r)(∆

∫Φ(r − r′)ψ†(t, r′)

× ψ(t, r′)ψ(t, r) dr ′)⟩

+i

2m

∫Φ(r − r′)

⟨ψ†(t, r)ψ†(t, r′)


128 N. N. BOGOLUBOV

× (∆ψ(t, r′))ψ(t, r) + ψ†(t, r)ψ†(t, r′)ψ(t, r′)(∆ψ(t, r))

− (∆ψ†(t, r))ψ†(t, r′)ψ(t, r′)ψ(t, r) − ψ†(t, r)(∆ψ†(t, r′))

× ψ(t, r′)ψ(t, r′)⟩dr ′ +

i

2

∫Φ(r − r′)η∗(t, r′)〈ψ†(t, r)ψ(t, r′)

× ψ(t, r)〉 + η∗(t, r)〈ψ†(t, r′)ψ(t, r′)ψ(t, r)〉 − η(t, r)

× 〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)〉 − η(t, r′)〈ψ†(t, r)ψ†(t, r′)

× ψ(t, r)〉dr ′ − i

4m

⟨(∆η∗)ψ − η∗∆ψ − (∆ψ†)η − ψ†(∆η)

⟩− i

2m

∑α

[ ∂

∂rα

(U − λ)⟨∂ψ†

∂rα

ψ − ψ† ∂ψ∂rα

⟩]. (1.5)

Now we notice that∑β

∂

∂rβ

( ∂

∂rβ

δψ†)ψ − ∆ψ† ∂ψ

∂rβ

− ψ† ∂

∂rβ

∆ψ +∆ψ†

∂rβ

∆ψ

= (∆∆ψ†)ψ + ψ†(∆∆ψ)

=∑

β

∂2

∂r2β

(∆ψ†)ψ − ψ†(∆ψ) − 2∑

β

∂

∂rβ

(∆ψ†)

∂ψ

∂rβ

− ∂ψ†

∂rβ

(∆ψ).

We can also transform the term containing Φ; we have

−(∆

∫Φ(r − r′)ψ†(t, r)ψ†(t, r′)ψ(t, r′) dr ′

)ψ(t, r)

+ (ψ†(t, r))∫

Φ(t− r′)ψ†(t, r′)ψ(t, r′) dr ′ ψ(t, r)

+ ψ†(t, r)(∆

∫Φ(r − r′)ψ†(t, r′)ψ(t, r′) dr′

)ψ(t, r)

−∫

Φ(r − r′)ψ†(t, r)ψ†(t, r′)ψ(t, r′) dr ′ (∆ψ(t, r))

= −2∑

β

∫∂Φ(r − r′)

∂rβ

∂ψ†(t, r)∂rβ

ψ†(t, r′)ψ(t, r′) dr ′ ψ(t, r)

+ 2∑

β

∫∂Φ(r − r′)

∂rβ

ψ†(t, r)ψ†(t, r′)ψ(t, r′)∂ψ(t, r)

∂rβ

and also ∫Φ(r − r′)ψ†(t, r)ψ†(t, r′)ψ(t, r′) dr ′ ∆ψ(t, r)



−∫

Φ(r − r′)(∆ψ†(t, r))ψ†(t, r′)ψ(t, r′) dr ′ ψ(t, r)

=∑

β

∫Φ(r − r′)

∂

∂rβ

ψ†(t, r)ψ†(t, r′)ψ(t, r′)

∂ψ(t, r)

∂rβ

− ∂ψ†(t, r)∂rβ

ψ†(t, r′)ψ(t, r′)ψ(t, r)dr ′

=∑

β

∂

∂rβ

∫Φ(r − r′)

ψ†(t, r)ψ†(t, r′)ψ(t, r′)

∂ψ(t, r)

∂rβ

− ∂ψ†(t, r)∂rβ

ψ†(t, r′)ψ(t, r′)ψ(t, r)dr ′ −

∑β

∫∂Φ(r − r′)

∂rβψ†(t, r)ψ†(t, r′)ψ(t, r′)

∂ψ(t, r)

∂rβ− ∂ψ†(t, r)

∂rβψ†(t, r′)

× ψ(t, r′)ψ(t, r)dr ′.

We also get in a similar way

∫Φ(r − r′)ψ†(t, r)ψ†(t, r′)(∆ψ(t, r′)) − (∆ψ†(t, r′))ψ(t, r′)ψ(t, r)dr ′

=∑

β

∫∂Φ(r − r′)

∂rβ

ψ†(t, r)ψ†(t, r′)

∂ψ†(t, r′)∂r′β

− ∂ψ†(t, r′)∂r′β

ψ(t, r)dr ′.

Substituting of these equations into (1.5) finally gives us

∂(ρε)

∂t=

i

8m2∆〈(∆ψ†)ψ − ψ†(∆ψ)〉 +

i

4m

∑β

∂

∂rβ

− 1

m

×⟨(∆ψ†)

∂ψ

∂rβ

− ∂ψ†

∂rβ

(∆ψ)⟩

+

∫Φ(r − r′)

⟨ψ†(t, r)ψ†(t, r′)

× ψ(t, r′)∂ψ(t, r)

∂rβ

− ∂ψ†(t, r)∂rβ

ψ†(t, r′)ψ(t, r′)ψ(t, r)⟩dr ′

+i

4m

∑β

∫∂Φ(r − r′)

∂rβ

⟨ψ†(t, r)ψ†(t, r′)ψ(t, r′)

∂ψ(t, r)

∂rβ

− ∂ψ†(t, r)∂rβ

ψ†(t, r′)ψ(t, r′)ψ(t, r) + ψ†(t, r)[ψ†(t, r′)

∂ψ†(t, r′)∂r′β


130 N. N. BOGOLUBOV

− ∂ψ†(t, r′)∂r′β

ψ(t, r)]ψ(t, r)

⟩dr ′ − 1

m

∑β

jβ∂

∂rβ

(U − λ)

+i

4m(η∆φ∗ − η∗∆φ+ φ∗∆η − φ∆η∗) +

i

2

∫Φ(r − r′)

× η∗(t, r)〈ψ†(t, r′)ψ(t, r′)ψ(t, r)〉 + η∗(t, r′)〈ψ†(t, r)ψ(t, r′)

× ψ(t, r)〉 − η(t, r′)〈ψ†(t, r)ψ†(t, r′)ψ(t, r)〉 − η(t, r)

× 〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)〉 dr ′.

If we introduce the quantity

G(α)t (r, r − r′) =

i

4m

⟨ψ†(t, r′)

(ψ†(t, r)

∂ψ(t, r)

∂rα

− ∂ψ†(t, r)∂rα

ψ(t, r))ψ(t, r′)

⟩(1.6)

and go over to a new variable R = r ′ − r in some of the integrals, theidentity (1.6) takes the form

∂(ρε)

∂t=

i

8m2∆〈(∆ψ†)ψ − ψ†(∆ψ)〉 +

∑β

∂

∂rβ

i

4m2

×⟨∂ψ†

∂rβ

(∆ψ) − (∆ψ†)∂ψ

∂rβ

⟩+

∫Φ(R)G

(β)t (r, R) dR

+∑

β

∫∂Φ(R)

∂Rβ

× G(β)t (r,−R) +G

(β)t (r −R,R) dR− 1

m

∑β

jβ∂

∂rβ

(U − λ)

+i

4m(η∆φ∗ − η∗∆φ+ φ∗∆η − φ∆η∗) +

i

2

∫Φ(r − r′)

× η∗(t, r) × 〈ψ†(t, r′)ψ(t, r′)ψ(t, r)〉 + η∗(t, r′)〈ψ†(t, r)ψ(t, r′)ψ(t, r)〉− η(t, r′)〈ψ†(t, r)ψ†(t, r′)ψ(t, r)〉 − η(t, r)〈ψ†(t, r)ψ†(t, r′)ψ(t, r′)〉 dr ′.

(1.7)

2. Hydrodynamic Equations for a Normal Liquid

We shall now proceed to derive the hydrodynamic equations for a normal(non-superfluid) liquid. Actually, this problem has already been treated in



the work of K. P. Gurov [1], and the only reason for dwelling on it here isthat is will form the basis of a subsequent generalization to the superfluidcase. For the purpose of this section the sources are unimportant and so weshall take

η = η† = 0.

Consider, first of all, the statistical equilibrium state of the normal liquid,which is characterized by the usual parameters: the particle number densityρ, temperature θ and velocity v characterizing the motion of the liquidas a whole. The dependence on the velocity v is trivial; by using thetransformation of the field operators

ψ → ψ eimv r

we can express the mean values in the state with velocity v

〈. . .〉ρ,θ,v

in terms of the mean values〈. . .〉ρ,θ,0

in the statistical equilibrium state of the liquid at rest. For example

j = mρv,

ε = ε0 +mv2

2= ε(ρ, θ) +

mv2

2(2.1)

where ε(ρ, θ) is the mean energy per particle in the statistical equilibriumstate of the liquid at rest.

In what follows we shall have to deal with averages of the general type:

U = 〈(D1ψ†(t, r))(D2ψ(t, r))〉ρ,θ,v

B = 〈(D1ψ†(t, r))(D2ψ

†(t, r′))(D3ψ(t, r′))(D4ψ(t, r))〉ρ,θ,v

where the Dν are linear combinations of constants and differential operatorswith respect to the spatial variables (cf., for example, the expressions for Dt

and G(α)t ; in particular, Dν may be simply equal to unity.

In view of the total spatial homogeneity of the state of statisticalequilibrium, it is easy to se that

U = U (ρ, θ, v),


132 N. N. BOGOLUBOV

B = B(ρ, θ, v/r − r′)

U(t, r) = U = const,

where neither U nor B can depend on the value of U ; this follows from thefact that all dependence on U can be eliminated by the gauge transformation

ψ → eiUtψ

while U and B are invariant under this transformation.We now proceed to consider the type of non-equilibrium processes

discussed in hydrodynamics, namely those for which non-equilibriumquantities of the type

U (t, r) = 〈(D1ψ†(t, r))(D2ψ(t, r))〉ρ,θ,v

B(t, r − r′) = 〈(D1ψ†(t, r))(D2ψ

†(t, r′))(D3ψ(t, r′))(D4ψ(t, r))〉ρ,θ,v (2.2)

and, of course, also the external field U(t, r) us sufficiently slowly varyingunder translations in space and time. We shall assume that the deviationsform statistical equilibrium are asymptotically damped out, so that we candefine, at least as an order of magnitude, some relaxation time T and meanfree path l. The non-equilibrium processes to be considered, that, are thosefor which the quantities (2.2), and U(t, r), change sufficiently slowly underthe translations

t→ t+ t0; r → r + t0, r′ → r′ + r0; |t0| ≈ T ; |r0| ≈ l,

and there are only small deviations from “local statistical equilibrium” inthe neighborhood of any given point. To put it more precisely, we assumethat quantities of the type (2.2) differ by a sufficiently small amount fromthe corresponding equilibrium values

U (ρ(t, r), θ(t, r), v(t, r)), B(ρ(t, r), θ(t, r), v(t, r)/r− r′) (2.3)

and that the difference tends to zero with vanishing gradients of ρ, θ, v and u.In equation (2.3) θ(t, r) and v(t, r) are defined (implicitly) by equations (2.1)in terms of the current j and mean energy ε at the point in question.

The assumption that for processes with sufficiently small gradients ofρ, θ, v and u the liquid in each small region is in a state of approximatelocal statistical equilibrium is an essential condition for the transition to



the hydrodynamics equations to be possible. For instance, if we were toconsider a dynamical system with no interactions at all (i.e., a system if non-interacting bosons) then of course we could consider processes characterizedby arbitrary small gradients of ρ, j, ε, etc., and could formally introduce localvalues of v and θ. However, in this case quantities of the type (2.2) couldnot, in general, be expressed even approximately in terms of their “quasi-equilibrium” values (2.3). This is what makes the hydrodynamics equationsinapplicable to a system of completely non-interacting particles. We mightalso remark that it is essential to the usual (Hilbert-Chapman-Enskog)derivation of the hydrodynamic equations for gases from the Boltzmannequation that this equations has solutions representing states close to localequilibrium, so that an expansion in powers of the gradients is legitimate.

After these preliminary remarks we return to our assumption about thenature of the nonequilibrium process considered. To formulate it in a formsuitable for calculation it is convenient to introduce a small parameter µwhich measures the “slowness” of the process and the deviation from isotropy.We shall represent averages of the type (2.2), and also the external field, inthe form

U(t, r) = ˜U(µt, µr) = U(τ, ξ),

U (t, r) =˜U (µt, µr;µ) = ˜U(τ, ξ;µ),

B(t, r) =˜B(µt, µr, r − r′;µ) =

˜B(τ, ξ, R, µ),

t = µt, ξ = µr, R = r − r ′. (2.4)

In particular, we write

ρ(t, r) = ˜ρ(τ, ξ), j(t, r) = j(τ, ξ), ε(t, r) = ˜ε(τ, ξ).

The representation (2.4) is simply a formalization of our assumption aboutthe slowness of the time rate of change and the weakness of the deviationfrom local isotropy.

To formulate the assumption of weak deviation from local statisticalequilibrium, we write

˜U (τ, ξ;µ) =˜U (0)(τ, ξ) + µ

˜U (1)(τ, ξ) + µ2 ˜U (2)(τ, ξ) + . . . ,

˜B(τ, ξ, R;µ) =˜B(0)(τ, ξ, R) + µ

˜B(1)(τ, ξ, R) + µ2 ˜B(2)(τ, ξ, R) + . . . ,

˜U (0)(τ, ξ) =˜U (˜ρ(τ, ξ), ˜θ(τ, ξ), ˜v(τ, ξ)),


134 N. N. BOGOLUBOV

˜B(0)(τ, ξ, R) =˜B(˜ρ(τ, ξ),

˜θ(τ, ξ), ˜v(τ, ξ), R), (2.5)

where˜θ and ˜v are defined by the relations

v =1

m ˜ρ

j,

˜ε = ε(˜ρ,˜θ) +

m˜v2

2(2.6)

and postulate that˜U (1),

˜B(1) are linear in the gradients of ˜ρ, ˜θ, ˜v, ˜u etc.Therefore, (2.5) is equivalent to the assumption that quantities of the types˜U ,

˜B may be expanded in powers of the gradients of ρ, v, θ and u. (Actuallythe possibility of such an expansion in intimately connected with the validityof a Boltzmann-like kinetic equation).

Now we may proceed directly to the derivation of the hydrodynamicequations. We shall have to deal with functionst Dt(r, R), G

(α)t (r, R)

belonging to the general type B. Let us write:

Dt(r, R) =˜Dτ(ξ, R;µ),

G(α)t (r, R) = ˜G(α)

τ (ξ, R;µ),

or in abbreviation notation:

Dt(r, R) =˜Dτ(ξ, R),

G(α)t (r, R) = ˜G(α)

τ (ξ, R).

Then, going over from the variables (t, r) to our new variables (τ, ξ) in theidentities of the preceding section (equation (1.3), (1.4), and (1.7)) we get:

m∂ ˜ρ

∂τ+∑

β

∂˜j(τβ, ξ)

∂ξβ= 0, (2.7)

∂˜jα(τ, ξ)

∂τ=

1

2m

∑β

∂

∂ξβ

δαβµ

2∆ξ

2˜ρ(τ, ξ) −

⟨∂ψ†

∂rα

∂ψ

∂rβ

+∂ψ†

∂rβ

∂ψ

∂rα

⟩

tTranslator’s Note: Dt is the same Dt introduced in section 1.



− 1

2µ

∫∂Φ(R)

∂Rα

˜Dτ (ξ,−R) +˜Dτ (ξ − µR,R) dR

− ˜ρ(τ, ξ)∂

∂ξα

˜U(τ, ξ), (2.8)

∂(˜ρ˜ε)

∂τ=iµ

8m2∆ξ〈(∆ψ†)ψ − ψ†(∆ψ)〉

+∑

β

∂

∂ξβ

i

4m2

⟨∂ψ†

∂rβ∆ψ − (∆ψ†)

∂ψ

∂rβ

⟩+

∫Φ(R)G(β)

τ (ξ, R) dR

+1

µ

∑β

∫∂Φ(R)

∂Rβ

˜G(β)τ (ξ,−R) + ˜G(β)

τ (ξ − µR,R) dR

+1

m

∑β

˜jβ∂ ˜U

∂ξβ. (2.9)

In equation (2.8) the quantity˜Dτ(ξ − µR,R) is multiplied by the factor

∂Φ(R)/∂Rα, which decreases rapidly for large R. Expanding˜D in powers of

µR, we get for the value of the relevant term in (2.8)

∫∂Φ(R)

∂Rα

˜Dτ (ξ,−R) +

˜Dτ (ξ, R) − µ∑

β

Rβ∂

˜Dτ (ξ, R)

∂ξβ

+µ2

2

∑β,γ

RαRγ∂2 ˜Dτ (ξ, R)

∂ξβ∂ξγ+ 0

((µr)3

)dR

= −µ∑

β

∫∂Φ(R)

∂Rα

Rβ∂

˜Dτ (ξ, R)

∂ξβdR+ 0(µ3).

Here we have used the fact that ∂Φ(R)/∂Rα is an odd function of R, and also

that˜Dτ (ξ, R) is an even function of R to zeroth order in µ. Thus, keeping

only terms of order µ, we get the equation

∂˜jα∂τ

=1

2m

∑β

∂

∂ξβ

−⟨∂ψ†

∂rα

∂ψ

∂rβ+∂ψ†

∂rβ

∂ψ

∂rα

⟩


136 N. N. BOGOLUBOV

+1

2

∑β

∂

∂ξβ

∫∂Φ(R)

∂Rα

Rβ˜Dτ (ξ, R) dR− ˜ρ

∂ ˜U

∂ξα

or∂˜jα∂τ

=∑

β

∂Tαβ

∂ξβ− ˜ρ

∂ ˜U

∂ξα(2.10)

where

Tαβ(τ, ξ) = − 1

2m

⟨∂ψ†

∂rα

∂ψ

∂rβ

+∂ψ†

∂rβ

∂ψ

∂rα

⟩

+1

2

∫∂Φ(R)

∂Rα

Rβ˜Dτ (ξ, R) dR. (2.11)

Performing an analogous expansion in powers of µ for the quantities ˜G(β)τ (ξ−

µR,R) in equation (2.9), we get to the first order in µ:

∂(˜ρ˜ε)

∂τ=

i

8m2µ∆ξ〈(∆ψ†)ψ − ψ†(∆ψ)〉 +

∑β

∂Iβ∂ξβ

− 1

m

∑β

˜jβ∂ ˜U

∂ξβ. (2.12)

Here Iα is given by the expression

Iα = − i

4m2

⟨(∆ψ†)

∂ψ

∂rα

− ∂ψ†

∂rα

∆ψ⟩−

∫Φ(R)G(α)

τ (ξ, R) dR

−∑

β

∫∂Φ(R)

∂Rβ

Rα˜G(β)

τ (ξ, R) dR

+µ

2

∑β,γ

∂

∂ξγ

∫∂Φ(R)

∂RβRα R

˜G(β)τ (ξ, R) dR. (2.13)

Now, the quantities Tαβ, Iα are composed of terms of the types U andB [cf. (2.2)]. and therefore we may expand them according to (2.5); in factwe get

Tαβ = T(0)αβ + µT

(1)αβ + . . .

Iα = I(0)α + µI(1)

α + . . . (2.14)

T(0)αβ = Tαβ(˜ρ, ˜θ, ˜v); I(0)

α = Iα(˜ρ, ˜θ, ˜v) (2.15)



Tαβ(˜ρ, ˜θ, ˜v) and Iα(˜ρ, ˜θ, ˜v) are to be obtained from expressions (2.11)and (2.13) respectively by replacing the averages

〈. . .〉by the corresponding equilibrium averages

〈. . .〉ρ,θ,v.

Taking terms of order µ into account in (2.14) would give us the viscous-liquid approximation. In as much as this work is restricted to the ideal-liquidapproximation, it will be sufficient to consider only the terms T

(0)αβ and I

(0)α .

Let us put these terms in a more convenient form. First of all, it isadvantageous to express the average

〈. . .〉ρ,θ,v

in terms of〈. . .〉ρ,θ,0.

This we can do by performing the transformation

ψ → ψeimv r.

We find⟨∂ψ†

∂rα

∂ψ

∂rβ+∂ψ†

∂rβ

∂ψ

∂rα

⟩ρ,θ,v

=⟨(∂ψ†

∂rα− imvαψ

†)( ∂ψ

∂rβ+ imvβψ

)+(∂ψ†

∂rβ− imvβψ

†)

×( ∂ψ∂rα

+ imvαψ)⟩

ρ,θ,v+⟨∂ψ†

∂rα

∂ψ

∂rβ

+∂ψ†

∂rβ

∂ψ

∂rα

⟩ρ,θ,0

+ 2m2vαvβρ,

since terms containing only one derivative vanish in the equilibrium statewith v = 0 because of the reflection invariance of this state. Therefore,noticing that for v = 0 Tαβ can only be an isotropic tensor, we get

Tαβ(ρ, θ, v) = −mρvαvβ − δαβP(ρ, θ), (2.16)

where for arbitrary α (= 1, 2, 3)

−P(ρ, θ) = − 1

m

⟨∂ψ†

∂rα

∂ψ

∂rα

⟩ρ,θ,0

+1

2

∫∂Φ(R)

∂Rα

Rα D(R|ρ, θ) dR, (2.17)


138 N. N. BOGOLUBOV

whereD(r − r′|ρ, θ) = 〈ψ†(r)ψ†(r′)ψ(r′)ψ(r)〉ρ,θ,0.

We shall now prove that if F (ρ, θ) is the free energy per particle and Nis the total number of particles,

P(ρ, θ) = −∂NF(

NV, θ)

∂V= ρ2∂F (ρ, θ)

∂ρ, (2.18)

i.e., that P is just the usual thermodynamic pressure. To establish this,let us suppose we subject the volume to a linear transformation along, say,the α-th coordinate axis: rα → Lrα, rβ → rβ for α = β. Let HL be theHamiltonian of the “stretched” system:

HL =1

2m

∫ψ†(r)p2

Lψ(r) dr +1

2

∫ΦL(r − r′)ψ†(r)ψ†(r′)ψ(r′)ψ(r) dr dr ′,

where

p2L = − 1

L2

∂2

∂r2α

−∑α=β

∂2

∂r2β

,

and ΦL(R) is obtained from Φ(R) by the substitution Rα → LRα, Rβ → Rβ

for α = β. Then∂NF

∂V=

1

V

⟨(∂HL

∂L

)L=1

⟩ρ,θ,0

,

or explicitly

∂NF

∂V=

1

m

⟨ψ†(r)

∂2

∂r2α

ψ(r)⟩

ρ,θ,0+

1

2

∫∂Φ(R)

∂Rα

Rα D(R|ρ, θ) dR

= − 1

m

⟨∂ψ†

∂rα

∂ψ

∂rα

⟩ρ,θ,0

+1

2

∫∂Φ(R)

∂Rα

Rα D(R|ρ, θ) dR,

which proves the relation (2.18).Substituting of (2.14), (2.15), and (2.16) in (2.10) finally gives:

∂˜jα∂τ

=∑

β

∂

∂ξβ

(m ˜ρ˜vα

˜vβ + δαβP(˜ρ,˜θ))− ˜ρ

∂ ˜U

∂ξα. (2.19)

A similar transformation may be carried out for the energy densityequation (2.12). Notice fist of all that

G(α)(r−r′|ρ, θ, v) =



=i

4m〈ψ†(r′)[ψ†(r)imvαψ(r) + ψ†(r)imvαψ(r)]ψ(r′)〉ρ,θ,0

= −vα

2D(r − r′|ρ, θ).

Now we transform the first term in expression (2.13) for I(0)α in the same way

as we did the corresponding terms for T(0)αβ , remembering that the average is

taken over a state of statistical equilibrium:

− i

4m

⟨∑β

(∂2ψ†

∂r2β

∂ψ

∂rα

+∂ψ†

∂rα

∂2ψ

∂rβ

)⟩ρ,θ,v

= −m2vα

∑β

v2β〈ψ†ψ〉ρ,θ,0 − 1

2m

∑β

vβ

⟨∂ψ†

∂rα

∂ψ

∂rβ

+∂ψ†

∂rβ

∂ψ

∂rα

⟩ρ,θ,0

+1

4mvα

∑β

∂

∂rβ

⟨∂ψ†

∂rβψ + ψ† ∂ψ

∂rβ

⟩ρ,θ,0

− vα

2m

∑β

⟨∂ψ†

∂rβ

∂ψ

∂rβ

⟩ρ,θ,0

= −ρvαmv2

2− vα

m

⟨∂ψ†

∂rα

∂ψ

∂rα

⟩ρ,θ,0

− vα

2m

∑β

⟨∂ψ†

∂rβ

∂ψ

∂rβ

⟩ρ,θ,0

We thereby obtain:

Iα(ρ, θ, v) = − ρvαmv2

2− vα

m

⟨∂ψ†

∂rα

∂ψ

∂rα

⟩ρ,θ,0

− vα

2m

∑β

⟨∂ψ†

∂rβ

∂ψ

∂rβ

⟩ρ,θ,0

− vα

2

∫Φ(R)D(R|ρ, θ) dR+

vα

2

∫∂Φ(R)

∂Rα

RαD(R|ρ, θ) dR.

On the other hand,

1

2m

∑β

⟨∂ψ†

∂rβ

∂ψ

∂rβ

⟩ρ,θ,0

+1

2

∫Φ(R)D(R|ρ, θ) dR

= ρε(ρ, θ) = ρF (ρ, θ) − θ

∂F (ρ, θ)

∂θ

,

and therefore, using (2.17), we get

Iα(ρ, θ, v) = −vα

ρ(ε(ρ, θ) +

mv2

2

)+ P(ρ, θ)

.


140 N. N. BOGOLUBOV

Equation (2.12) therefore leads to the following expression in ourapproximation

∂(˜ρ˜ε)

∂τ= −

∑α

∂

∂ξα˜vα

˜ρ(ε(˜ρ,

˜θ) +

m˜v2

2

)+ P(˜ρ,

˜θ)− 1

m

∑α

˜jα∂ ˜U

∂ξα, (2.20)

where the definition of the local velocity and temperature is given by (2.6).Finally, let us introduce the entropy

S(ρ, θ) = −∂F (ρ, θ)

∂θ.

Then by combining the equations obtained above we can easily get in placeof (2.20) an equation for the entropy density:

∂ ˜s

∂τ+∑

α

˜vα

∂ ˜s

∂ξα= 0. (2.21)

In equations (2.7), (2.19) and (2.21) we can now go back to the originalvariables t, r; obviously this is achieved simply by substituting t, r for τ, ξeverywhere and removing the ≈ signs. Thus we finally arrive at the usualhydrodynamic equations for a normal liquid.

3. Hydrodynamic Equations for a Superfluid

A superfluid liquid is characterized by the presence of a non-vanishingexpectation value 〈ψ(r, t)〉 = φ(t, r) = 0, even for vanishingly smallsource terms. Thus the equations of motion for the local quantitiesalready considered must be supplemented by an equation of motion for thequantity φ. Such an equation may be obtained by taking expectation valuesin the equation of motion for ψ; it reads

i∂φ(t, r)

∂t= −λφ(t, r) − ∆

2mφ(t, r)

+

∫Φ(r − r′)〈ψ†(t, r′)ψ(t, r′)ψ(t, r)〉 dr ′ + U(t, r)φ(t, r) + η(t, r).



In this equation φ is complex. For our purpose it is more convenient dodeal with two real quantities, the modulus and phase of φ: φ = a exp(iχ).We obtain equations for a and χ by direct substitution into equation for φ

∂χ

∂t=λ+

∆a

2ma− 1

2m

(∂χ∂r

)2

− U(t, r) +ζ∗ + ζ

2a

− 1

2a2

∫Φ(R)Xt(r, R) +X∗(r, R) dR, (3.1)

where

Xt(r, r′ − r) = 〈ψ†(t, r′)ψ(t, r′)ψ(t, r)〉ψ∗(t, r),

χ(t, r) = η(t, r) e−iχ(t,r),

and for a(t, r):

i∂a2

∂t= − i

m

a2∆χ+ 2

∑β

∂χ

∂rβ

∂a

∂rβ

a

+

∫Φ(R)Xt(r, R) −X∗(r, R) dR+ a(ζ − ζ∗). (3.2)

In what follows we shall consider frequently use the superfluid velocity v(α)s =

1

m

∂χ

∂rα

, which, according to (3.1), satisfies the equation

m∂v

(α)s

∂t=∂

∂r

∆a

2maα− mv2

s

2− U − ζ + ζ∗

2a

− 1

2a2

∫Φ(R)Xt(r, R) −X∗(r, R) dR

. (3.3)

Consider first of all a statistical equilibrium state of the superfluid liquidwith u = 0, η = 0. Whereas the statistical equilibrium state of a normalliquid is characterized by ρ, θ and a single velocity v, we now have statesin general must be described by two two velocities, e.g. the velocity of thecondensate vs and that of the “normal component” vn. We shall denoteaverages taken over such a statistical equilibrium state by the symbol

〈. . .〉vs,vn


142 N. N. BOGOLUBOV

suppressing the indices ρ, θ for the sake of conciseness. We shall find itconvenient to express this type of average in terms of the averages

〈. . .〉v′s,0; where v ′s = vs − vn;

corresponding to a state with zero “normal” velocity. To do so we canobviously perform a Galilean transformation on the field operators:

ψ → ψ exp(imvn r).

Let us start, then, by considering the state characterized by ρ, θ and vs

with vn = 0. In this case:

a = conts = a(ρ, θ, u); F = F (ρ, θ, u); u2 =v2

s

2,

and the chemical potential has the form

∂(FN)

∂N= F + ρ

∂F

∂ρ= Λ(ρ, θ, u).

Let us introduce the current j by the equation

jα = ρ∂F

∂v(α)s

= ρ∂F

∂uv(α)

s ,

and put

ρ∂F

∂u= ρsm,

so that

jα = mρs v(α)s .

Clearly ρs ≤ ρ.It will now be shown that the definition of j just introduces is equivalent

to the usual definition. This may conveniently be done by transformingto a system of coordinates moving with velocity vs and then taking allaverages only over states with vs = 0, vn = −vs; this maneuver will notof course change the numerical volume of the averaged quantityu. In the new

uTranslator’s Note: We are simply calculating the old quantities by going to a newrepresentation.



coordinate system the Hamiltonian, as a function of the new operators ψ,has the form

H =1

2m

∫ ∑α

(∂ψ†

∂rα− imv(α)

s ψ†)( ∂ψ

∂rα+ imv(α)

s ψ)dr

+

∫Φ(r − r′)ψ†(r)ψ†(r′)ψ(r′)ψ(r) dr dr ′.

Calculating ρ∂F

∂v(α)s

in this representation, we find

ρ∂F

∂v(α)s

=1

V

⟨ ∂H

∂v(α)s

⟩0,−vs

=i

2V

∫⟨(∂ψ†

∂rα

− imv(α)s ψ†

)ψ − ψ†

( ∂ψ∂rα

+ imv(α)s ψ

)⟩0,−vs

dr

=i

2V

∫⟨∂ψ†

∂rαψ − ψ† ∂ψ

∂rα

⟩vs,0

dr =i

2

⟨∂ψ†


∂rα

⟩vs,0

= jα.

The last expression is just the usual definition of j.Going back now to (3.1), we observe that in a state of thermodynamic

equilibrium X = X(r − r′|ρ, θ, vs). Equation (3.1) gives us:v

1

2a2

∫Ψ(R)X(R|ρ, θ, vs) +X∗(R|ρ, θ, vs) dR = Λ(ρ, θ, u) − mv2

s

2.

Hence, using (3.2) (and remembering that a = const., ζ = 0), we get

1

a2

∫Ψ(R)X(R|ρ, θ, vs) dR =

1

a2

∫Ψ(R)X∗(R|ρ, θ, vs) dR

= Λ(ρ, θ, u) − mv2s

2. (3.4)

Now we proceed to calculate the tensor [cf. (1.4)]

Tαβ(ρ, θ, vs) = − 1

2m

⟨∂ψ†

∂rα

∂ψ

∂rβ

+∂ψ†

∂rβ

∂ψ

∂rα

⟩vs,0

+1

2

∫∂Φ(R)

∂RαRβ D(R|ρ, θ, vs) dR

vTranslator’s Note: The constant λ in equations (1.3), (3.1) is here identified with thechemical potential Λ(ρ, θ, u).


144 N. N. BOGOLUBOV

in the equilibrium state (vs, 0). From general symmetry consideration itfollows that, since the only direction characterized the system is that of vs,Tαβ is of the form

Tαβ = A(ρ, θ, u)v(α)s v(β)

s + δαβB(ρ, θ, u),

where A and B are scalars. As proved above (section 2) the diagonal elementTαα is equal to the derivative

1

V

∂(NF )

∂L

∣∣∣L=1

,

where F (L) is the free energy of the system when “stretched” by a factor Lalong the α axis. Therefore,

ρ∂F

∂L

∣∣∣L=1

= Tαα = A(v(α)s )2 +B.

On the other hand

ρ∂F

∂L

∣∣∣L=1

= ρ2∂F

∂ρ− ρ

∂F

∂L(v(α)

s )2 = −P(ρ, θ, u) −mρs(v(α)s )2.

Therefore,A = −mρs, B = −P(ρ, θ, u),

and so, finally:Tαβ = −mρsv

(α)s v(β)

s − δαβP(ρ, θ, u).

Now consider states characterized by two velocities vs, vn. Notice first ofall that the expressions

X(r − r′|ρ, θ, vs, vn) = 〈ψ†(r′)ψ(r′)ψ(r)〉vs,vn〈ψ†(r)〉vs,vn ,

D(r − r′|ρ, θ, vs, vn) = 〈ψ†(r)ψ†(r′)ψ(r′)ψ(r)〉vs,vn

are invariant with respect to Galilean transformations

ψ → ψ exp(imv r),

and therefore can be functions only of the relative velocity of superfluid andnormal components:

X(r − r′|ρ, θ, vs, sn) = X(r − r′|ρ, θ, vs − vn),



D(r − r′|ρ, θ, vs, sn) = D(r − r′|ρ, θ, vs − vn).

Consider the expression for jα:

jα =i

2

⟨∂ψ†


∂rα

⟩vs,vn

=i

2

⟨(∂ψ†

∂rα

− imv(α)n ψ†

)ψ − ψ†

( ∂ψ∂rα

+ imv(α)n ψ

)⟩vs−vn,0

=mv(α)n ρ+mρs(v

(α)s − v(α)

n ) = mρsv(α)s +mρnv

(α)n .

In the last expression we have introduced the “density of the normalcomponent” ρn = ρ− ρs. Now let us calculate the stress tensor:

Tαβ(ρ, θ, vs, vn) =1

2

∫∂Φ(R)

∂Rα

Rβ D(R|ρ, θ, vs, vn) dR

− 1

2m

⟨(∂ψ†

∂rα− imv(α)

n ψ†)( ∂ψ

∂rβ+ imv(β)

n ψ)

+(∂ψ†

∂rβ

− imv(β)n ψ†

)( ∂ψ∂rα

+ imv(α)n ψ

)⟩vs−vn,0

= − 1

2m

⟨∂ψ†

∂rα

∂ψ

∂rβ

+∂ψ†

∂rβ

∂ψ

∂rα

⟩vs−vn,0

+1

2

∫∂Φ(R)

∂Rα

× Rβ D(R|ρ, θ, vs − vn) dR+iv

(α)n

2

⟨ψ† ∂ψ∂rβ

− ∂ψ†

∂rβψ⟩

vs−vn,0

− iv(β)n

2

⟨∂ψ†

∂rβ

ψ − ψ† ∂ψ∂rβ

⟩vs−vn,0

−mρv(α)n v(β)

n

= − δαβP(ρ, θ, u) −mρsv(α)s v(β)

s −mρnv(α)n v(β)

n . (3.5)

Here and subsequently u =(vs − vn)2

2.

Next we transform the expression for the average energy:

ρε(ρ, θ, vs, vn) =1

2

∫Φ(R) D(R|ρ, θ, vs − vn, 0) dR

− 1

4m

⟨[( ∂∂r

− imvn

)2

ψ]ψ + ψ†

[( ∂∂r

+ imvn

)2

ψ]⟩

vs−vn,0

=ρE(ρ, θ, u) +mρv2

n

2+mρs(vs − vn)vn. (3.6)


146 N. N. BOGOLUBOV

Here, as usual, E(ρ, θ, u) = F − θ∂F

∂θ.

We also need to find an expression for the vector Iα in the statisticalequilibrium state specified by vn and vs. Since

G(α)(r − r′|ρ, θ, vs, vn)

=i

4m

⟨ψ†(t, r′)

ψ†(t, r)

∂ψ(t, r)

∂rα

− ∂ψ†(t, r)∂rα

ψ(t, r)ψ(t, r′)

⟩vs,vn

=i

4m

⟨ψ†(t, r′)

ψ†(t, r)

(∂ψ(t, r)

∂rα

+ imv(α)n ψ(t, r)

)

−(∂ψ†(t, r)

∂rα

− imv(α)n ψ†(t, r)

)ψ(t, r)

ψ(t, r′)

⟩vs−vn,0

=G(α)(r − r′|ρ, θ, vs − vn) − v(α)

2D(r − r′|ρ, θ, vs − vn)

we get for Iα itself

Iα(ρ, θ, vs, vn) = − i

4m2

⟨(∆ψ†)

∂ψ

∂rα− ∂ψ†

∂rα∆ψ

⟩vs,vn

+

∫Φ(R)G(α)(R|ρ, θ, vs, vn) dR−

∑β

∫∂Φ(R)

∂RβRαG

(β)(R|ρ, θ, vs, vn) dR

= − i

4m2

⟨[( ∂∂r

− imvn

)2

ψ†]( ∂ψ∂rα

+ imv(α)n ψ

)−(∂ψ†

∂rα

+ imv(α)n ψ†

)×[( ∂∂r

− imvn

)2

ψ]⟩

vs,vn,0+

∫Φ(R)G(α)(R|ρ, θ, vs − vn) dR

− v(α)

2

∫Φ(R) D(R|ρ, θ, vs − vn) dR

−∑

β

∫∂Φ(R)

∂RβRαG

(β)(R|ρ, θ, vs − vn) dR

+1

2

∑β

∫∂Φ(R)

∂Rβ

Rα v(β)n D(R|ρ, θ, vs − vn) dR

=i

4m2

⟨(∆ψ†)

∂ψ

∂rα

− ∂ψ†

∂rα

∆ψ⟩

vs−vn,0+

∫Φ(R)G(α)(R|ρ, θ, vs − vn) dR

−∑

β

∫∂Φ(R)

∂RβRαG

(β)(R|ρ, θ, vs − vn) dR



+1

4mv(α)

n 〈(∆ψ†)ψ + ψ†(∆ψ)〉vs−vn,0 − 1

2m

∑β

v(β)n

×⟨∂ψ†

∂rβ

∂ψ

∂rα

+∂ψ†

∂rα

∂ψ

∂rβ

⟩vs−vn,0

+1

2

∑β

v(β)n

∫∂Φ(R)

∂Rβ

× RαD(R|ρ, θ, vs − vn) dR− v(α)n

2

∫Φ(R)D(R|ρ, θ, vs − vn) dR

− mv2n

2ρv(α)

n − i

2v(α)

n

∑β

v(β)n

⟨∂ψ†

∂rβ

ψ − ψ† ∂ψ∂rβ

⟩vs−vn,0

+i

4v2

n

⟨ψ† ∂ψ∂rβ

− ∂ψ†

∂rβψ⟩

vs−vn,0.

Using the definition of Iα(ρ, θ, vs − vn, 0) formulae (3.6) and (3.5) and thedefinition of jα we get:

Iα(ρ, θ, vs, vn) =Iα(ρ, θ, vs − vn, 0) − v(α)n ρE(ρ, θ, u)

+∑

β

v(β)n Tαβ(ρ, θ, vs − vn, 0) −

∑β

v(α)n v(β)

n mρs(v(β)s − v(β)

n )

− mv2n

2ρs(v

(α)s − v(α)

n ) − mv2n

2v(α)

n ρ

=Iα(ρ, θ, vs − vn, 0) − v(α)n

[ρE +

mv2n

2ρ+ P

+mρsvn(vs − vn) −mρs(v(α)s − v(α)

n )(vs vn − v2

n

2

)]. (3.7)

Let us write

Iα(ρ, θ, vs − vn, 0) = [−Λ(ρ, θ, u)ρs + A(ρ, θ, u)](v(α)s − v(α)

n ). (3.8)

Below it will be shown that A ≡ 0.So far we have assumed that U = 0. We should, of course, get the same

expressions for jα, X, Tαβ and Iα in the case U = const., since all theseexpressions are invariant under the gauge transformation

ψ → ψ e−iUt.

After these preliminary transformations of the equilibrium quantities wecan go on to discuss the derivation of the hydrodynamic equations for a


148 N. N. BOGOLUBOV

superfluid. We shall use a natural generalization of the procedure expoundedin section 2.

We introduce, as on page 133, a small parameter and put

ξ = µr, τ = µt.

We then write

ρ(t, r) = ˜ρ(τ, ξ), j(t, r) =j(τ, ξ), ε(t, r) = ˜ε(τ, ξ),

and also

U(t, r) = ˜U(τ, ξ),

vs(t, r) = i

∂

∂r〈ψ(t, r)〉

m〈ψ(t, r)〉 = vs(τ, ξ),

a(t, r) =√

〈ψ†(t, r)〉〈ψ(t, r)〉 = |〈ψ(t, r)〉| = ˜a(τ, ξ) = 0.

It should be emphasized that the assumption that vs is a slowly varyingfunction of t and r (which obviously implies that the denominator in theexpression for vs does not vanish, i.e., a = 0) restricts us to consideringirrotational flow:

rotvs = 0.

The case when 〈ψ(t, r)〉 can vanish has been investigated by S. V. Iordanskii[2]. To obtain an expression for the Green’s function we actually only needthe hydrodynamic equations in the “acoustic” case, when all the quantitiesintroduced above differ from their equilibrium values by a vanishingly smallamount. In this case we have

vs = δvs =−i

m〈ψ〉vs,vn

⟨∂φ∂r

⟩, a = a0 + δa,

and hence everywhererotvs = 0, a = 0.

Thus, in calculating the Green’s functions we have the condition ofirrotational flow fulfilled automatically.

Notice also that since the gradients with respect to t and r of ρ, vs etc.,are quantities of first order in µ, equation (1.3) implies that we must take η(or equivalently, ζ) proportional to µ. Let us therefore write

ζ(t, r) = µ ˜ζ(τ, ξ).



To simplify the notation we shall from here on omit the index ≈ on functionsof τ and ξ, provided there is no danger of confusion.

Now, starting from the expressions for ρ(τ, ξ), j(τ, ξ), ε(τ, ξ) and vs(τ, ξ)(all of which are defined as microscopic averages) we can introduce quantitiesθ(τ, ξ) and vn(τ, ξ), which together with ρ(τ, ξ) and vs(τ, ξ) describe the “localquasi-equilibrium state”. Thus, we take the functions

ε(ρ, θ, vs, vn); ρs(ρ, θ, u) =1

mρ∂F (ρ, θ, u)

∂u; u =

(vn − vs)2

2;

ρn(ρ, θ, u) = ρ− ρs(ρ, θ, u),

which express ε, ρs and ρ in statistical equilibrium, and define functions

θ(τ, ξ), vn(τ, ξ) (3.9)

by the relation

ε(τ, ξ) = ε(ρ, θ, vs, vn),

j(τ, ξ) = ρs(ρ, θ, u)vs + ρn(ρ, θ, u)vn. (3.10)

Once we have introduced the functions (3.9) in this way, we can also definequantities ρs(τ, ξ), ρn(τ, ξ) by the relations

ρs(τ, ξ) = ρs(ρ, θ, u), ρn(τ, ξ) = ρn(ρ, θ, u).

Now we formulate our assumption that for the non-equilibrium processconsidered the local state of the liquid is only slightly different from the localquasi-equilibrium state. Following the procedure of section 2, we write

a(τ, ξ) = a(ρ, θ, u) + µa(1)(τ, ξ) + . . . ,

Xτ (ξ, R) = X(R|ρ, θ, vs, vn) + µX(1)(τ, ξ) + . . .

= X(R|ρ, θ, vs − vn) + µX(1)(τ, ξ) + . . . ,

Tαβ(τ, ξ) = Tαβ(ρ, θ, vs, vn) + µT(1)αβ (τ, ξ) + . . . ,

Iα(τ, ξ) = Iα(ρ, θ, vs, vn) + µI(1)α (τ, ξ) + . . . (3.11)

and substitute these expansions, along with (3.10), in equations (1.3), (1.4),(1.7)w, and (3.3), after expressing the latter in terms of variables τ, ξ. In

wWe transform equations (1.4) and (1.7) into forms analogous to (2.10) and (2.12)


150 N. N. BOGOLUBOV

accordance with our ideal-liquid approximation we shall neglect correctionterms of order µ.

Consider, for example, equation (3.3). We have

m∂v

(α)s

∂τ=

∂

∂ξα

µ2 ∆ξa

2ma− mv2

s

2− U − µ

ζ∗ + ζ

2a

− 1

2a2

∫Φ(R)[Xτ (ξ, R) +X∗

τ (ξ, R)] dR,

and therefore, in our approximation,

m∂v

(α)s

∂τ= − ∂

∂ξα

mv2s

2+ U +

1

2a2(ρ, θ, u)

∫Φ(R)[X(R|ρ, θ, vs − vn)

+X∗(R|ρ, θ, vs − vn)] dR.

Hence, using (3.4), we finally get:

m∂v

(α)s

∂τ= − ∂

∂ξα

mv2s

2− m

2(vs − vn)2 + Λ(ρ, θ, u) + U

u =(vs − vn)2

2. (3.12)

In the same way we get from (1.3), (1.4), (1.7), and (3.8)

∂ρ

∂τ+∑

α

∂(ρsv(α)s + ρnv

(α)n )

∂ξα= ia(ζ∗ − ζ), (3.13)


(α)n )

∂τ=∑

β

∂Tαβ(ρ, θ, vs, vn)

∂ξβ+ imv(α)

s (ζ∗− ζ)a− ρ∂U

∂ξα, (3.14)

∂ρε(ρ, θ, vs, vn)

∂τ=∑

β

∂Iβ(ρ, θ, vs, vn)

∂ξβ−∑

β

(ρsv(α)s + ρnv

(α)n )

∂U

∂ξβ

+ ia(ζ∗ − ζ)mv2

s

2+ Λ(ρ, θ, µ) − (vs − vn)2

2m. (3.15)



Substituting in equations (3.14) and (3.15) expressions (3.5)-(3.8) forTαβ(ρ, θ, vs, vn), ε(ρ, θ, vs, vn) and Iβ(ρ, θ, vs, vn) we get:

m∂(ρsv

(α)s + ρnv

(α)n )

∂τ+∑

β

∂(ρsv(α)s v

(β)s + ρnv

(α)n v

(β)n )

∂ξα

= −∂P∂ξα

− ρ∂U

∂ξα+ imav(α)

s (ζ∗ − ζ), (3.16)

∂

∂τ

[ρE(ρ, θ, u) +

mρv2n

2+mρs(vs − vn)vn

]+∑

β

∂

∂ξβ

×v(β)

n

[ρmv2n

2+ ρsmvn(vs − vn) + ρE + P

]+ (v(β)

s − v(β)n )ρs

×[Λ +m

(vsvn − v2

n

2

)]+∑

β

∂U

∂ξβ(ρsvs + ρnvn) − ia(ζ∗ − ζ)

×[Λ +m

(vsvn − v2

n

2

)]= −

∑β

∂

∂ξβ(v(β)

s − v(β)n )A. (3.17)

Now we go back in equations (3.12), (3.13), (3.16), and (3.17) from

the auxiliary variables τ, ξ to our original variables t, r, This gives us thefollowing system of hydrodynamic equations:

m∂v

(α)s

∂t+

∂

∂rα

m(vsvn − v2

n

2

)+ Λ + U

= 0, (3.18a)

∂ρ

∂t+∑

β


(α)n )

∂rβ

− ia(ζ∗ − ζ) = 0, (3.18b)

m∂(ρsv

(α)s + ρnv

(α)n )

∂t+∑

β

∂(ρsv(α)s v

(β)s + ρnv

(α)n v

(β)n )

∂rβ

+∂P

∂rα+ ρ

∂U

∂rα− imav(α)

s (ζ∗ − ζ) = 0, (3.18c)

∂

∂t

[ρE +

mρv2n

2+mρs(vs − vn)vn

]


152 N. N. BOGOLUBOV

+∑

β

∂

∂rβ

v(β)

n

[ρmv2n

2+ ρsmvn(vs − vn) + ρE + P

]

+ (v(β)s − v(β)

n )ρs

[Λ +m

(vsvn − v2

n

2

)]+∑

β

∂U

∂rβ

(ρsvs + ρnvn)

− ia(ζ∗ − ζ)[Λ +m

(vsvn − v2

n

2

)]= −

∑β

∂

∂rβ

(v(β)s − v(β)

n )A. (3.18d)

We remind the reader that in these equations

a = a(ρ, θ, u), A = A(ρ, θ, u), u =(vs − vn)2

2, ρs =

1

mρ∂F (ρ, θ, u)

∂u

ρn = ρ− ρs; E = F (ρ, θ, u) − θ∂F (ρ, θ, u)

∂θ;

P = ρ2∂F (ρ, θ, u)

∂ρ, Λ = F (ρ, θ, u) + ρ

∂F (ρ, θ, u)

∂ρ,

ζ(t, r) = η(t, r)e−iχ(t,r) = η e−imΘ,

where Θ(t, r) is the velocity potential for superfluid flow:

v(α)s =

∂Θ

∂rα.

Equation (3.18d) can be simplified by introducing the entropy

S = −∂F∂θ

.

In fact, using the other equations of the group (3.18) we can transform thefirst term of (3.18d) as follows

∂

∂t

[ρE +

mρv2n

2+mρsvn(vs − vn)

]=

∂

∂t

[ρ(F − θ

∂F

∂θ

)− mρv2

n

2+mvn(ρsvs + ρnvn)

]

=∂ρ

∂t

(F + ρ

∂F

∂ρ

)− θ

∂

∂t

(ρ∂F

∂θ

)+mρs(v

(α)s − v(α)

n )∂(v

(α)s − v

(α)n )

∂t

− ∂ρ

∂t

mv2n

2− (ρs + ρn)mv(α)

n

∂v(α)n

∂t+∂

∂tmv(α)

n (ρsv(α)s + ρnv

(α)n )



=∂ρ

∂t

(Λ − mv2

n

2

)− θ

∂

∂t

(ρ∂F

∂θ

)+mρs(v

(α)s − v(α)

n )∂(v

(α)s − v

(α)n )

∂t

− ∂ρ

∂t

mv2n

2− (ρs + ρn)mv(α)

n

∂v(α)n

∂t+∂

∂tmv(α)

n (ρsv(α)s + ρnv

(α)n )

=∂ρ

∂t

(Λ − mv2

n

2

)− θ

∂

∂t

(ρ∂F

∂θ

)+mρs(v

(α)s − v(α)

n )∂(v

(α)s − v

(α)n )

∂t

− (ρs + ρn)mv(α)n

∂v(α)n

∂t+ (ρsv

(α)s + ρnv

(α)n )m

∂v(α)n

∂t

−mv(α)n

∂

∂rβ

[v(α)n (ρsv

(β)s + ρnv

(β)n ) + ρsv

(β)s (v(α)

s − v(α)n )] − v(α)

n

∂P

∂rα

− (ρs + ρn) v(α)n

∂U

∂rα

+ imv(α)n v(α)

s a(ζ∗ − ζ)

=∂ρ

∂t

(Λ − mv2

n

2

)− θ

∂

∂t

(ρ∂F

∂θ

)+mρs(v

(α)s − v(α)

n )∂v

(α)s

∂t

−mv(α)n v(α)

n

∂

∂rβ

(ρsv(β)s + ρnv

(β)n ) −mv(α)

n (ρsv(β)s + ρnv

(β)n )

∂v(α)n

∂rβ

−mv(α)n (v(β)

s − v(β)n )

∂ρsv(α)n

∂rβ

−mρsv(α)n v(β)

s

∂(v(α)s − v

(α)n )

∂rβ

− v(α)n

∂P

∂rα


∂U

∂rα

+ imv(α)n v(α)

s a(ζ∗ − ζ)

= −θ ∂∂t

(ρ∂F

∂θ

)−(Λ +

mv2n

2

)[ ∂

∂rβ(ρsv

(β)s + ρnv

(β)n ) − ia(ζ∗ − ζ)

]

+mρs(v(α)s − v(α)

n )∂v

(α)s

∂t−mρnv

(α)n v(β)

n

∂v(α)n

∂rβ

−mρsv(β)s v(α)

n

∂v(α)s

∂rβ

−mv(α)n (v(α)

s − v(α)n )

∂ρsv(β)n

∂rβ+ ima(ζ∗ − ζ)(v(α)

n v(α)s − v2

n)

− v(α)n

∂P

∂rα


∂U

∂rα

Also, the term∂

∂rβ

(v(β)n ρE) in (3.18d) can be rewritten

∂

∂rβ

(v(β)n ρE) =Λ

∂ρv(β)n

∂rβ

− P∂v(β)

n

∂rβ

− θ∂

∂rβ

(ρv(β)

n

∂F

∂θ

)


154 N. N. BOGOLUBOV

+mρsv(β)n (v(α)

s − v(α)n )

∂(v(α)s − v

(α)n )

∂rβ.

Now we rearrange the left-hand side of the (transformed) equation (3.18d),grouping together similar terms:

− θ∂

∂t

(ρ∂F

∂θ

)− θ

∂

∂rβ

(ρv(β)

n

∂F

∂θ

)

+ ρs(v(α)s − v(α)

n )[m∂v

(α)s

∂t+∂Λ

∂rα

+m(vsvn − v2

n

2

)+ U

]−[ρs(v

(α)s − v(α)

n )∂U

∂rα

+ (ρs + ρn)v(α)n

∂U

∂rα

− (ρsv(α)s + ρnv

(α)n )

∂U

∂rα

]

+ Λ[∂ρ∂t

+∂ρv

(β)n

∂rβ

+∂ρs(v

(β)s − v

(β)n )

∂rβ

− ia(ζ∗ − ζ)]

+ ia(ζ∗ − ζ)[Λ +m(v(β)

n v(β)s − v2

n) − Λ −m(v(β)

n v(β)s − v2

n

2

)+mv2

n

2

]− mv2

n

2

∂

∂rβ

[ρsv

(β)s + ρnv

(β)n − 2ρsv

(β)s − ρnv

(β)n − ρsv

(β)n + 2ρsv

(β)n + ρsv

(β)s

− ρsv(β)n

]−mv(α)n v(α)

s

∂

∂rβ

[ρsv

(β)s − ρsv

(β)n − ρs(v

(β)s − v(β)

n )]

−m∂v

(β)n

∂rβ

[ρnv

(α)n v(β)

n + ρnv(β)n (v(α)

s − v(α)n ) − ρsv

(β)n v(α)

n − ρnv(β)n v(α)

n

− ρsv(β)n v(α)

s + 2ρsv(β)n v(α)

n

]−m∂v

(β)s

∂rβ

[ρsv

(α)n v(β)

s + ρsv(β)n (v(α)

s − v(α)n )

− ρsv(α)n v(β)

n

]+∂(Pv

(β)n )

∂rβ

− v(β)n

∂P

∂rβ

− P∂v

(β)n

∂rβ

= mρsv(α)n v(β)

n

(∂v(α)n

∂rβ− ∂v

(β)n

∂rα

)+ θ

[∂(ρS)

∂t+

∂

∂rβ(ρv(β)

n S)]

= θ[∂(ρS)

∂t+

∂

∂rβ

(ρv(β)n S)

].

Thus we form the following equation for the entropy density

θ[∂(ρS)

∂t+∑

β

∂(ρv(β)n S)

∂rβ

]=∑

β

∂

∂rβ

[(v(β)

n − v(β)s )A(ρ, θ, u)

]. (3.19)



We can use the equations obtained above to prove that

A(ρ, θ, u) = 0. (3.20)

Consider the state of statistical equilibrium in the absence of an externalfield U , but with η a given function of r:

U = 0, ζ = ζ(r), θ = const, ρ = ρ(r), vn = 0, vs = vs(r).

Then we get from (3.19)

∑β

∂

∂rβ

v(β)s A

(ρ, θ,

v2s

2

)= 0, (3.21)

and from the continuity equation (3.18b)

∑β

∂

∂rβ(ρsv

(β)s ) = ia(ζ∗ − ζ). (3.22)

Moreover, we get from (3.18b)

∂

∂rα

Λ(ρ, θ,

v2s

2

)= 0. (3.23)

Remembering that

vs =∂Θ

∂r,

we see at once the we have three equations[(3.21)-(3.23)], for two unknownfunctions; this overdetermines the problem. This is the basis of ourassertion (3.20), as we shall now see.

Consider first the case when η, vs and the deviation of ρ from a constantvalue are vanishingly small. Then, neglecting terms of second order in thesequantities, we get

∑β

∂v(β)s

∂rβ

=ia

ρs

(ζ∗ − ζ); A(ρ, θ, 0)∑

β

∂v(β)s

∂rβ

= 0,

whence obviouslyA(ρ, θ, 0) = 0.


156 N. N. BOGOLUBOV

Now consider the more general case, when

vs(r) = w + δvs, ρ = ρ+ δρ(r); w = 0,

and δvs, δρ and ζ are vanishingly small. Then we get from (3.21) and (3.22)

A∑

β

∂δv(β)s

∂rβ

+∑

β

w(β)(∂A∂ρ

∂δρ

∂rβ

+∂A

∂u

∑γ

w(γ)∂δv(γ)s

∂rβ

)= 0,

ρs

∑β

∂δv(β)s

∂rβ+∑

β

w(β)(∂ρs

∂ρ

∂δρ

∂r+∂ρs

∂u

∑γ

w(γ)∂δv(γ)s

∂rβ

)= ia(ζ∗ − ζ).

(3.24)

Let us suppose

ζ(r) = eik r δG; δG = const.

Then obviously we can put

δvs = eik r δv + e−ik r δv∗,

δρ = eik r δΓ + e−ik r δΓ∗,

where δv and δΓ are infinitesimal constants.Let us choose the arbitrary vector k in such a way that

k w = 0.

With this choice of k we get

∑β

w(β)∂δρ

∂rβ

= 0,∑

β

w(β)∂δv(γ)s

∂rβ

= 0,

and therefore (3.24) takes the form

A∑

β

∂δv(β)s

∂rβ

= 0, ρs

∑β

∂δv(β)s

∂rβ

= ia(ζ∗ − ζ).

from which follows equation (3.20).We may now write in place of equation (3.19)

∂(ρS)

∂t+∑

β

∂(ρv(β)n S)

∂rβ= 0. (3.25)



Thus equations (3.18a)-(3.18c) and (3.25) constitute the hydrodynamicsequations for a superfluid liquid in our ideal-liquid approximation.

It should be remarked that for the case u = 0, η = 0 this system ofequations was first obtained by L. D. Landau [3] from phenomenologicalconsiderations.

4. Variational Equations and Green’s Functions

Let us now use equations (3.18), (3.25) to consider the case of infinitesimalderivation from the statistical equilibrium state of the liquid at rest. Put

ρ = ρ0 + δρ, vs = δvs, vn = δns, S = S0 + δS, U = δU, η = δη

thus making a transition from the hydrodynamic equations to the linearized“acoustic” equations. Since in this case η = ζ up to terms of second orderin the infinitesimals, the linearized equations take the form (we omit thesuperscript 0 on ρ and S)

∂δρ

∂t+ ρs

∑β

∂v(β)s

∂rβ+ ρn

∑β

∂v(β)n

∂rβ= i

√ρ0(η

∗ − η),

mρs∂v

(β)s

∂t+mρn

∂v(β)n

∂t= −∂δP

∂rα

− ρ∂U

∂rα

,

m∂v

(β)s

∂t= −∂(δΛ + U)

∂rα

,

ρ∂δS

∂t+ S

∂δρ

∂t+ ρS

∑β

∂v(β)n

∂rβ

= 0, (4.1)

where

√ρ0 = 〈ψ〉0,0 = 〈ψ†〉0,0,

δΛ =∂Λ

∂ρδρ+

∂Λ

∂θδθ = −Sδθ +

1

ρδP.

Before going on to examine the system of equations (4.1), we shallelucidate the connection between the various quantities involved and the


158 N. N. BOGOLUBOV

Green’s functions. The source term in the original Hamiltonian may beregarded as a variation (perturbation) of the Hamiltonian, i.e. we may writeH = H0 + δH , where

δH =

∫ψ(r, t)η∗(r, t) + ψ†(r, t)η(r, t) dr.

(Here and subsequently we put u = 0). Let us put

η(r, t) = e−iωt+εt+ik rηk + eiωt+εt−ik rη−k,

η∗(r, t) = e−iωt+εt+ik rη∗−k + eiωt+εt−ik rη∗k, (4.2)

where ε > 0, ε → 0. (This means we switch on δH adiabatically).Consider the variations

δφ = δ〈ψ(r, t)〉 = eiωt+εt δU (r) + eiωt+εt δB(r)

δφ∗ = δ〈ψ†(r, t)〉 = e−iωt+εt δB∗(r) + eiωt+εt δU ∗(r).

A general theorem on the variation of averages under the action of aperturbation switched on adiabatically (see e.g. reference [4]) tells us that if

we consider only the component of δH proportional to eik r, then

δU (r) = 2π

∫eik r ′ ψ(r);ψ(r′) ω+iε η

∗−k+ ψ(r);ψ†(r′) ω+iε ηk dr ′,

δB∗(r) = 2π

∫eik r ′ ψ†(r);ψ(r′) ω+iε η

∗−k+ ψ†(r);ψ†(r′) ω+iε ηk dr ′,

ψ(r);ψ(r′) E=1

2π

∫ aq; a−q E eiq(r−r ′) dq,

ψ†(r);ψ(r′) E=1

2π

∫ a†−q; a−q E eiq(r−r ′) dq,

ψ†(r);ψ†(r′) E=1

2π

∫ a†−q; a

†q E eiq(r−r ′) dq, . (4.2′)

Here E = ω + iε, and aq; a−q , etc., denote the Fourier components ofthe corresponding retarded Green’s functions. Using (4.2) and (4.2′) we find

δU (r) = 2π ak; a−k E η∗−k+ ak; a†k ηk eik r = δφk eik r,



δB∗(r) = 2π a†−k; ak E η∗−k+ ak; a†k ηk eik r = δφ∗

−k eik r.

In terms of the Fourier components δφk, δφ∗k the variations of φ, φ∗ can be

written in the form

δφ(t, r) = e−iωt+εt+ik r δφk + +eiωt+εt−ik r δφ−k,

δφ∗(t, r) = e−iωt+εt+ik r δφ∗−k + +eiωt+εt−ik r δφ∗

k.

The quantity δφ(t, r) does not appear explicitly in equations (4.1), but it isconnected with vs(t, r). Using the definition of vs, it is easy to obtain theexpression

v(α)s (t, r) =

i

2m√ρ0

(∂δφ∗(t, r)∂rα

− ∂δφ(t, r)

∂rα

).

We introduce the Fourier components of vs:

v(α)s (t, r) = e−iωt+εt+ik r v(α)

s (k) + eiωt+εt−ik r v(α)s (−k),

for which, from the expression for v(α)s and the definition of δφk, we have

v(α)s (k) =

π

m√ρ0

k(α) ak − a†−k, a−k E η∗−k+ ak − a†−k, a†k E ηk.

(4.3)Moreover, by definition:

a2(t, r) = φ∗(t, r)φ(t, r)

whence we get:

δa =1

2[δφ∗(t, r) + δφ(t, r)] =

∂a

∂ρδρ+

∂a

∂θδθ.

Then going over once again to Fourier components and using the definitionof δφk, we get

δa(t, r) = e−iωt+εt+ik r δak + eiωt+εt−ik r δa−k,

δak = π ak + a†−k, a−k E η∗−k+ ak + a†−k, a†k E ηk. (4.4)

Formulae (4.3) and (4.4) connect the hydrodynamic quantities obtainedfrom equation (4.1) with the Green’s functions. It should be noted thatsince the hydrodynamic equations are valid only for “slow” changes of the


160 N. N. BOGOLUBOV

hydrodynamic variables, the connection is asymptotic in nature. being trueonly for k 1/l and E 1/T , where l is the mean free path and T therelaxation time.

In order to find the solutions of the equations appropriate to this limitand so obtain asymptotic expressions for the Green’s functions, we need onlysubstitute equations (4.2) for η and η∗ in (4.1) (as in normal acoustic theory)and seek a solution proportional to ηk and η∗−k. Rewriting equation (4.1)(with δU = U = 0) as an equation for the Fourier components, we get:

−Eδρ(k) + ρs

∑β

k(β)v(β)s (k) + ρn

∑β

k(β)v(β)n (k) =

√ρ0(η

∗−k − ηk), (4.5a)

mE[ρsv(β)s (k) + ρnv

(β)n (k)] = k(α)

[∂P∂ρ

δρ(k) +∂P

∂θδθ(k)

], (4.5b)

mEv(α)s (k) = k(α)

[−Sδθ +

1

ρ

∂P

∂ρδρ(k) +

1

ρ

∂P

∂θδθ(k)

], (4.5c)

E[ρδS(k) + Sδρ(k)] = ρS∑

β

k(β) v(β)n (k), (4.5d)

δS(k) =∂S

∂θδθ +

∂S

∂ρδρ(k). (4.5e)

Let us start examining some limiting cases. Consider first the case E = 0.Then it follows from (4.5b) and (4.5c) that δρ(k) = δθ(k) = 0, and hencefrom (4.5d) that ∑

β

k(β) v(β)n (k) = 0.

Then we get from (4.5a)

ρs

∑β

k(β) v(β)s (k) =

√ρ0(η

∗−k − ηk),

or using (4.3)

πρsk2

mρ0

ak − a†−k, a−k E=0 η∗−k+ ak − a†−k, a

†k E=0 ηk = η∗−k − ηk.



Equating the coefficients of η∗−k and ηk on two sides of this equation, we get

ak − a†−k, a−k E=0=mρ0

πρs

1

k2,

ak − a†−k, a†k E=0= −mρ0

πρs

1

k2. (4.6)

Also we have

δa =∂a

∂ρδρ+

∂a

∂θδθ,

or, according to (4.4),

ak + a†−k, a−k E=0 η∗−k+ ak + a†−k, a

†k E=0 ηk = 0.

Hence,

ak; a−k E=0= − a†−k; a−k E=0,

ak; a†k E=0= − a†−k; a

†k E=0 . (4.7)

substituting these relations in (4.6), we find

ak; a−k E=0 − a†−k; ak E=0= 2 ak; a−k E=0=mρ0

πρs

1

k2

ak; a†k E=0 − a†−k; a

†k E=0= 2 ak; a

†k E=0= −mρ0

πρs

1

k2.

Thus we have obtained the “1/k2 theorem” [5] for the Green’s function, withthe coefficient explicit. It is interesting to note that the coefficient containsnot only ρs but also the actual density of the condensate particles ρ0.

Next, consider the special case θ = 0. Strictly speaking, thehydrodynamic equations can have only a formal meaning in this case, sincethe relaxation time as θ → 0 becomes very long. However, we shall considerthis case formally in order to see the results given by our formulae in thislimit. All expressions are now considerably simplified, since

ρs = ρ, ρn = 0, S = 0,∂S

∂ρ= 0,

∂S

∂θ= 0,

∂P

∂θ= 0,

δΛ =1

ρδP =

1

ρ

(∂P∂ρ

)θ=0

δρ.


162 N. N. BOGOLUBOV

Equations (4.5) give:

−Eδρ(k) + ρs

∑β

k(β)v(β)s (k) =

√ρ0(η

∗−k − ηk),

Emρv(α)s (k) = k(α)

(∂P∂ρ

)θ=0

δρ(k).

The second equation can be written:

δρ(k) =Eρ

k2c2

∑β

k(β)v(β)s (k), c2 =

1

m

(∂P∂ρ

).

Substituting this expression into the first equation, we get

∑β

k(β)v(β)s (k) =

k2√ρ0c2

ρ(E2 − c2k2)(ηk − η∗−k)

=πk2

m√ρ0

ak − a†−k, a−k E=0 η∗−k+ ak − a†−k, a

†k E=0 ηk.

Since δρ(k) contains not only first-order infinitesimals but also an extrasmall factor E, and we are interested in the asymptotic form of the Green’sfunctions for E, k → 0, equation (4.7) is still valid to lowest order, and so,equating the coefficients of ηk and η∗−k, we get:

ak; a−k ≈ − ρ0c2m

2πρ(E2 − c2k2),

ak; a†k ≈ ρ0c

2m

2πρ(E2 − c2k2).

Finally, consider the general case (θ = 0). From (4.5b) and (4.5c) wehave

mE[ρsv(α)s (k) + ρnv

(α)n (k)] = k(α)δP,

mE(ρs + ρn)v(α)s (k) = −k(α)ρSδθ(k) + k(α)δP.

Subtraction of the second equation from the first gives:

mEρn[v(α)n (k) − v(α)

s (k)] = ρSk(α)δθ(k). (4.8)



Equation (4.5a) can be rewritten as

−Eδρ(k) + ρ∑

β

k(β)v(β)s (k) + ρn

∑β

k(β)[v(β)n (k) − v(β)

s (k)] =√ρ0(η

∗−k − ηk).

(4.9)Using this equation to eliminate δρ(k) from (4.5d), we get:

EρδS(k) + Sρs

∑β

k(β)v(β)s (k) + Sρn

∑β

k(β)v(β)n (k)

− S√ρ0(η

∗−k − ηk) = S(ρs + ρn)

∑β

k(β)v(β)s (k).

Then, expressing δS(k) in terms of δθ(k) and δρ(k) [by equation (4.5e)] andagain eliminating δρ(k), we find

Eρ(∂S∂θ

)ρδθ(k) =

√ρ0(η

∗−k − ηk)

(S + ρ

(∂S∂ρ

)θ

)− ρ2

(∂S∂ρ

)θ

∑β

k(β)v(β)s (k) +

(Sρs − ρnρ

(∂S∂ρ

)θ

)∑β

k(β)[v(β)n (k) − v(β)

s (k)],

or, using (4.8)

Eδθ(k) = −ρ(∂S∂ρ

)θ(∂S

∂θ

)ρ

∑β

k(β)v(β)s +

Sρs − ρnρ(∂S∂ρ

)θ

ρ(∂S∂θ

)ρ

ρSk2

mEρnδθ(k)

+√ρ0

S + ρ(∂S∂ρ

)θ

ρ(∂S∂θ

)ρ

(η∗−k − ηk).

Therefore, we finally get

δθ(k) = −ρ(∂S∂ρ

)θmE

(∂S∂θ

)ρ

mE2 −

Sρs

ρn

− ρ(∂S∂ρ

)θ(∂S

∂θ

)ρ

∑β

k(β)v(β)s


164 N. N. BOGOLUBOV

+

√ρ0

(S + ρ

(∂S∂ρ

)θ

)mE

(∂S∂θ

)ρ

mE2 −

Sρs

ρn− ρ

(∂S∂ρ

)θ(∂S

∂θ

)ρ

(η∗−k − ηk). (4.10)

We can likewise express δρ(k) in terms of vs(k). From equation (4.9) wehave

δρ(k) =ρ

E

∑β

k(β)v(β)s (k) +

1

mE2ρSk2δθ(k) −

√ρ0

E(η∗−k − ηk).

Substituting (4.10) gives:

δρ(k) =ρ

E

∑β

k(β)v(β)s (k) −

ρ2k2S(∂S∂ρ

)θmE

∑β

k(β)v(β)s

mE2(∂S∂θ

)ρ

mE2 − Sk2

Sρs

ρn− ρ

(∂S∂ρ

)θ(∂S

∂θ

)ρ

+

√ρ0

E

Sk2(S + ρ

(∂S∂ρ

)θ

)mE

(∂S∂θ

)ρ

mE2 − Sk2

Sρs

ρn− ρ

(∂S∂ρ

)θ(∂S

∂θ

)ρ

− 1

(η∗−k − ηk).

(4.11)

Now, substituting the expression for δρ(k) and δθ(k) in the equation forv(α)(k) [equation (4.5c)], multiplying by k(α) and summing over α, we get:

mE∑

α

k(α)v(α)s (k) = −Sk2δθ(k) +

1

ρ

(∂P∂ρ

)θk2δρ(k) +

1

ρ

(∂P∂θ

)ρk2δθ(k)

=(1

ρ

(∂P∂θ

)ρ− S

) k2√ρ0

(S + ρ

(∂S∂ρ

)θ

)mE

ρ[mE2

(∂S∂θ

)ρ− Sk2

(Sρs

ρn

− ρ(∂S∂ρ

)θ

)] (η∗−k − ηk)



+1

ρ

(∂P∂ρ

)θk2

ρ

E−

ρ2Sk2(∂S∂ρ

)θ

E[mE2

(∂S∂θ

)ρ− Sk2

(Sρs

ρn− ρ

(∂S∂ρ

)θ

)]

×∑

α

k(α)v(α)s (k) +

1

ρ

(∂P∂ρ

)θk2

√ρ0

E

−

Sk2(S + ρ

(∂S∂ρ

)θ

)mE2

(∂S∂θ

)ρ− Sk2

(Sρs

ρn

− ρ(∂S∂ρ

)θ

) − 1

(η∗−k − ηk)

−(1

ρ

(∂P∂θ

)ρ− S

)k2

ρ(∂S∂ρ

)θmE

mE2(∂S∂θ

)ρ− Sk2

(Sρs

ρn

− ρ(∂S∂ρ

)θ

)×∑

α

k(α)v(α)s (k).

Solving this equation for∑

α k(α)v

(α)s (k), we get:

∑α

k(α)v(α)s (k) =

k2∆(k, E)√ρ0

mΩ(k, E)(η∗−k − ηk)

=πk2

m√ρ0

ak − a†−k, a−k E=0 η∗−k+ ak − a†−k, a

†k E=0 ηk (4.12)

where

Ω(k, E) = E4 − E2k2

m

S2 ρs

ρn

(∂S∂θ

)ρ

−(∂P∂θ

)ρ

(∂S∂ρ

)θ

1(∂S∂θ

)ρ

+(∂P∂θ

)ρ

]+k4

m2

(∂P∂ρ

)θS2 ρs

ρn

(∂S∂θ

)ρ

= E4 − E2k2

×[ 1

m

(∂P∂ρ

)θ+S2ρsθ

ρnmcv

]+k4

m

(∂P∂ρ

)θ

S2ρsθ

ρnmcv= (E2 − c21k

2)(E2 − c20k2).

(4.13)


166 N. N. BOGOLUBOV

Here cv = θ(∂S∂θ

)ρ, and the quantities c0, c1 are given by

c20,1 =1

2m

(∂P∂ρ

)θ+

1

2

S2ρsθ

ρnmcv

±√

1

4

[ 1

m

(∂P∂ρ

)θ+S2ρsθ

ρnmcv

]2

− 1

m

(∂P∂ρ

)θ

S2ρsθ

ρnmcv, (4.14)

(the upper sign referring to c0). The velocity c0 tends to the normal speedof sound both for ρs → 0 and for θ → 0. The velocity c1 is the speed of the“second-sound” vibration peculiar to a superfluid liquid and tends to zerofor ρs → 0. Finally, in equation (4.12) the quantity ∆(E, k) has the form

∆(E, k) =1

m

(∂P∂ρ

)θk2 S

2θ

ρncv− E2

[1

ρ

(∂P∂ρ

)θ−( ∂

∂ρρS)

θ

θ

ρcv

].

It is easy to show, as we did above for the case θ = 0, that δρ(k) andδθ(k) contain extra small factor of order k compared to vs(k). Hence, onceagain, equation (4.7) is satisfied to lowest order. Equating the coefficients ofηk and η∗−k on the right and left sides of (4.12), and using (4.7), we thereforefinally obtain the expressions:

ak; a−k ≈ ∆(E, k)ρ0

2πΩ(E, k), ak; a

†k ≈ −∆(E, k)ρ0

2πΩ(E, k). (4.15)

Obviously in the limits E → 0 or θ → 0 this reduces to the results obtainedabove.

From (4.15) and (4.13) we see that the Green’s functions have polescorresponding to two types of elementary excitation:

E = c0k, E = c1k.

In equations (4.15) the effects of damping are not taken into account;this is due to the fact that we have been considering only the ideal-liquidapproximation. It would be interesting to improve the asymptotic accuracyof (4.15) by making instead a “viscous-liquid approximation”; to do this weshould have to take into account in the equations of section 3 the terms “oforder µ” which we actually dropped. This problem is considerably simplifiedby the fact that to construct the Green’s functions we do not need the



complete hydrodynamic equations but only the linearized acoustic equations.The improved expressions for the Green’s function obtained in this waywould contain damping terms involving the viscosity, thermal conductivityand other kinetic coefficients.

In conclusion I should like to thank S. V. Iordanskii for considerableassistance in preparing these lectures for publication.

References

1. K. P. Gurov, “Quantum hydrodynamics,” Zh. Exper. i Theor. Fiz., 18,110 (1948); Zh. Exper. i Theor. Fiz., 20, 279 (1950).

2. S. V. Iordanskii, “Hydrodynamics of a rotating Bose system below thecondensation point,” Docl. Akad. Naul SSSR 153, 1, 74-77 (1963); Sov.Phys. ”Doklady”, English Transl.

3. L. D. Landau, Zh. Exper. i Theor. Fiz., 11, 592, (1941); Zh. Exper.i Theor. Fiz., 14, 112, (1944); English Transl. see Collected Papers ofL. D. Landau, ed. D. ter Haar, Gordon and Breach and Pergamon Press(1965).

4. D. N. Zubarev, “Double-time Green functions in statistical physics,”Uspekhi Fiz. Nauk, 21, 1, 71 (1960); Sov. Phys.–Usp. English Transl.,3, 320 (1960).

5. N. N. Bogoliubov,“Quasi-averages in statistical mechanics,” JINRpreprint, Dubna (1961).


CHAPTER 5

ON THE MODEL HAMILTONIAN OFSUPERCONDUCTIVITY

1. Statement of the Problem

The simplest model system considered in superconductivity theory is onedescribed by a hamiltonian which retains only interactions between particlesof opposite spin and momentum

H =∑

f

T (f)a†faf − 1

2V

∑f.f ′

λ(f)a†fa†−fa−f ′af ′ (1.1)

where f = (p, s), s± 1 and p is the momentum vector. For a given volumeV = L3, allowed values of p are given by:

px =2π

Lnx, py =

2π

Lny, pz =

2π

Lnz

where nx, ny, nz are integers. In (1.1) we also use the following notation:

−f = (−p,−s).

T (f) = p2

2m− µ, where µ is the chemical potential (µ > 0)

λ(f) =

J, for

∣∣∣ p2

2m− µ∣∣∣ ≤ ∆,

0, for∣∣∣ p2

2m− µ∣∣∣ > ∆.

For such a system, the BCS method [1] and the method of compensationof dangerous diagrams lead to identical results. Moreover, as was shown inreference [2], the Hamiltonian (1.1) is of great methodological interest in its

168



own right, since it provides on of the very few completely soluble problemsof statistical mechanics.

In reference [2] we showed that for this purpose we can obtain anexpression for the free energy which is exact in the limit V → ∞. The proofgiven there was along the following lines: The hamiltonian (1.1) was dividedin a particular way into two parts H0 and H1. The problem described by theHamiltonian H0 was solved exactly, and the effect of H1 was calculated byperturbation theory. It was proved that any given term of the correspondingseries expansion is asymptotically small in the limit V → ∞., which led to theconclusion that it is always legitimate to neglect the effect of H1 on passingto the limit of infinite volume. Needless to say, this kind of approach cannotpretend to mathematical rigour: however, it is worth pointing out that theproblems of statistical physics are often handled by far cruder methods. Forinstance, a very commonly used device consists in the selective summation ofvarious so-called “principal terms” of the perturbation series to the neglectof all the other terms, though the latter do not even tend to zero for V → ∞.

Some doubt was cast on the results of reference [2] when variousattempts to use the normal Feynman diagram technique (without taking

into account the “anomalous contractions” afa−f , a†fa†−f generated by

the canonical u − v transformation (see below)) failed to give the resultsanticipated. Furthermore, in reference [3] the summation of a certain classof Feynman diagrams led to a solution which is fundamentally different fromthe one obtained in references [1] and [2], and it was concluded that the latterwere invalid.

In view of this situation, we undertook an investigation [4] of the hierarchyof coupled equations for the Green’s functions which did not involve recourseto perturbation theory. In this work it was shown that the Green’s functionsfor the Hamiltonian H0 satisfies the whole chain of equations for the exactHamiltonian H = H0 + H1 to order 1/V . This tends to confirm the resultsof reference [3] and reveal the “ineffectiveness” of the additional term H1.

However, it is also possible to treat the problem form a purelymathematical standpoint. Once we have specified the Hamiltonian, sayin the form (1.1), we have a perfectly well-defined mathematical problem,which we also may solve rigorously, without any “physical assumptions”whatsoever. We need not to content ourselves with the knowledge thatthe approximate expressions satisfy the exact equations to order 1/V ; onthe contrary, we can actually evaluate the difference between the exact and


170 N. N. BOGOLUBOV

approximate expressions.With a view of the complete elucidation of the behavior of a system

with the Hamiltonian (1.1), we shall adopt in this work just such a purelymathematical standpoint. We shall investigate the hamiltonian (1.1) at zerotemperature and prove rigorously that the relative difference (E − E0)/E0

between the groundstate energies of H and H0, and also the differencebetween the corresponding Green’s functions, tends to zero for V → ∞;we shall set a bound on the rate of decrease in each case.

For the methodology reasons we shall find in convenient to consider arather more general Hamiltonian, which contains terms representing sourcesof creation and destruction of pairs;

H =∑

f

T (f)a†faf − ν∑

f

λ(f)

2(a−faf + a†fa

†−f )

− 1

2V

∑f,f ′

λ(f)λ(f ′)a†fa†−fa−f ′af ′ , (1.2)

where ν is a parameter on whose magnitude we place no restrictions; inparticular, ν may be equal to zero. Notice that the case ν < 0 need not beconsidered, since it can be reduced to the case ν > 0 by the trivial gaugetransformation

af → iaf , a†f → −ia†f .We emphasize that the only motive for considering the case ν > 0 is the lightit sheds on the situation in the physical case ν = 0.

For this investigation we shall not need all the specific properties of thefunctions λ(f), T (f) mentioned above; it will be quite sufficient for ourpurpose if they fulfil the following more general conditions:

1. The functions λ(f) and T (f) are real, piecewise continuous and obeythe symmetry conditions

λ(−f) = −λ(f); T (−f) = T (f)

2. λ(f) is uniformly bounded in all space, while

T (f) → ∞ for |f | → ∞3.

1

V

∑f

|λ(f)| ≤ const for V → ∞



4. limV →∞

1

2V

∑f

λ2(f)√λ2(f)x+ T 2(f)

> 1

for sufficiently small positive x.

We write the Hamiltonian (1.2) in the form

H = H0 + H1, (1.3)

where

H0 =∑

f

T (f)a†af − 1

2

∑f

λ(f)(ν + σ∗)a−faf (ν + σ)a†fa

†−f

+|σ|2V

2(1.4)

H1 = − 1

2V

(∑f

λ(f)a†fa†−f − V σ∗

)(∑f

λ(f)a−faf − V σ)

(1.5)

where σ is a complex number.Notice that if we fix σ by minimizing the ground-state energy of H0, and

then neglect H1, we arrive at the well-known approximate solution consideredin the work cited above [1,2,4]. Our present problem is to obtain bounds forthe difference of the corresponding Green’s functions. We shall prove thatthese differences vanish when we take the limitx V → ∞.

2. General Properties of the Hamiltonian

1. In this section we shall establish certain general properties of theHamiltonian H, (1.2). First, we consider the occupation number operatornf = a†faf . We shall prove that the quantities nf − n−f are constant ofmotion.

xIn recent years papers [7–12] have been published in which new methods weredeveloped to find asymptotically exact expressions for many-time correlation functionsand Green’s functions at arbitrary temperature θ. In addition, bounds were obtained forfree energy of system of BCS type. These bounds are exact in the limit V → ∞. On thebasis of analyzing and generalizing the results of [7–12], it became possible to formulatea new principle, the minimax principle [12], for a broad range of problems in statisticalphysics. (Remark added by the author in 1971.)


172 N. N. BOGOLUBOV

We have

a−faf (nf − n−f ) − (nf − n−f)a−faf = 0,

and also

a†−fa†f (nf − n−f ) − (nf − n−f)a

†−fa

†f = 0,

whence

H (nf − n−f ) − (nf − n−f)H = 0.

It follows thatd

dt(nf(t) − n−f (t)) = 0. (2.1)

2. We next show that the wave function φH corresponding to the minimumeigenvalue of the Hamiltonian H satisfy the equation

(nf − n−f )φH = 0 (2.2)

for arbitrary f .To prove this, we assume the contrary. Since the operator nf − n−f

commute with H and with one another, we can always choose φH to be aneigenfunction of all this operators:

nf − n−f =

1

0

−1.

We denote by K0, K+, K− respectively the classes of all those indices f forwhich

(nf − n−f)φH = 0 f ∈ K0

(nf − n−f − 1)φH = 0 f ∈ K+

(nf − n−f + 1)φH = 0 f ∈ K−.

Our hypothesis then reduces to the statement that the classes K+ and/orK1 are not empty and thaty

〈φ∗HH φH〉 ≤ 〈φ∗H φ〉 ≤

yWe use the notation 〈φ ∗ ψ〉 to denote the scalar product of the functions φ and ψ.



for arbitrary φ. We shall consider in particular functions φ satisfying theauxiliary conditions

(nf − n−f )φ = 0. (2.3)

Now we notice that if f ∈ K+, then

nf = 1; n−f = 0,

while if f ∈ K−, thennf = 0; n−f = 1.

Thus, we can write φH as a direct product

φH = φK0φK+φK−,

where

φK+ =∏

f∈K+

δ(nf − 1)δ(n−f); φK− =∏

f∈K−

δ(nf)δ(n−f − 1),

while φK0 is a function only of those nf for which f ∈ K0:

φK0 = F (. . . nf . . .); 〈φ†K0φK0〉 = 1 f ∈ K0.

Further, we notice that

a−fafδ(nf − 1)δ(n−f) = 0; a−fafδ(nf )δ(n−f − 1) = 0;

a†fa†−fδ(nf − 1)δ(n−f) = 0; a†fa

†−fδ(nf )δ(n−f − 1) = 0;

and hencea−fafφK+φK− = 0, a†fa

†−fφK+φK− = 0

if f ∈ K+ or f ∈ K−.Accordingly

HφH = ∑

f∈K+

T (f) +∑

f∈K−

T (f) +∑f∈K0

T (f)nf

− ν

2

∑f∈K0

λ(f)

2(a−faf + a†fa

†−f)

− 1

2V

∑f∈K0

∑f ′∈K0

λ(f)λ(f ′)a†fa†−fa−f ′af ′

φH,


174 N. N. BOGOLUBOV

and so it follows that

〈φ∗HHϕH〉 =

∑f∈K+

T (f) +∑

f∈K−

T (f)

+⟨φ∗

K0

∑f∈K0

T (f)nf − ν

2

∑f∈K0

λ(f)

2(a−faf + a†fa

†−f)

− 1

2V

∑f∈K0

∑f ′∈K0


φK0

⟩.

Let us now divide the set K+ +K− into two sets Q+ and Q−:

K+ +K− = Q+ +Q−

in such a way that Q+ includes all indices f from the set K+ +K− such thatT (f) ≥ 0, while Q− includes all those for which T (f) < 0. Because of thesymmetry property T (f) = T (−f) the indices f and −f always fall into Q+

and Q− as a pair. We can write

〈φ∗HHϕH〉 =

∑f∈Q+

|T (f)| +∑

f∈Q−

|T (f)|

+⟨φ∗

K0

∑f∈K0

T (f)nf − ν

2

∑f∈K0

λ(f)

2(a−faf + a†fa

†−f)

− 1

2V

∑f∈K0

∑f ′∈K0


φK0

⟩.

Let us now construct the function φ as a direct product by putting

φ = φK0φQ+φQ−,

where

φQ+ =∏

f∈Q+

δ(nf )δ(n−f); φQ− =∏

f∈Q−

δ(nf − 1)δ(n−f − 1).

(Here we the importance of the fact that f and −f always belong to Q+ orQ− as a pair). For such function we have

〈φ∗Hϕ〉 = − 2∑

f∈Q−

|T (f)|



+⟨φ∗

K0

∑f∈K0

T (f)nf − ν

2

∑f∈K0

λ(f)

2(a−faf + a†fa

†−f )

− 1

2V

∑f∈K0

∑f ′∈K0


.φK0

⟩

− 1

2V

∑f∈Q−

λ2(f).

It is obvious that〈φ∗

HHϕH〉 > 〈φ∗Hϕ〉.On the other hand, ψ by construction satisfies all the auxiliaryconditions (2.3), so that we arrive at a contradiction of our hypothesis. Thus,the statement (2.2) is proved.

A particular consequence of (2.2) is the fact that the total momentum ofthe state ψH is equal to zero:

∑f

f nfψH =1

2

∑f

f(nf − n−f)φH = 0. (2.4)

It is obvious from the above discussion that when attempting to findthe eigenfunction φH for the minimum eigenvalue of H we may alwaysrestrict ourselves to the class of functions φ which satisfy the auxiliaryconditions (2.3). For this special class of functions the hamiltonian H maybe expressed in terms of Pauli operators. Consider the operators

bf = a−faf , b†f = a†fa†−f .

Independently of the auxiliary condition we have

bfbf ′ = bf ′bf ; b†fb†f ′ = b†f ′b†f ; b2f = 0; (b†f)2 = 0;

bfb†−f − b†−fbf = 0; f = f.

Moreover, the auxiliary conditions imply that

b†fbf + bfb†f = nfn−f + (1 − nf)(1 − n−f) = 1

since nf and n−f are either both equal to zero or both equal to one. It

follows that, within the class of functions satisfying (2.3), the operators bf , b†f


176 N. N. BOGOLUBOV

constitute Pauli operators. When acting on this class of functions theHamiltonian has the form

H = 2∑

f>0

T (f)b†fbf − ν

2

∑f>0

(bf + b†f) −1

V

∑f>0,f ′>0

λ(f)λ(f ′)b†fbf ′. (2.5)

The restriction f > 0 is made in order to ensure that all the operators bfshall be independent (since bf = −b−f ). A Hamiltonian of this type wasconsidered in a previous paper by the author [5].

3. Upper Bound for the Minimum Eigenvalue of the Hamiltonian

We now consider the problem of finding an upper bound for the minimumeigenvalue of the Hamiltonian H (equation (1.2)). We start from therepresentation of H in the form (1.3), and denote by EH the minimumeigenvalue of H and by E0(σ) the minimum eigenvalue of H0 (equation (1.4))Since the operator H1 ≤ 0, the minimum eigenvalue of H0 cannot be smallerthan the minimum eigenvalue of H = H0 +H1, i.e.

E0(σ) ≥ EH (3.1)

for arbitrary σ. Thus, the set of minimum eigenvalues of theHamiltonians H0(σ) form an upper bound for EH , and the optimum boundis obtained by minimizing E0(σ) with respect to σ.

We shall now proceed to calculate the eigenvalues of the Hamiltonian H0.To perform the canonical transformation which diagonalizes the quadraticform H0 (1.4), we write down the identity

H0 =∑

f

√λ2(f)(ν + σ∗)(ν + σ) + T 2(f)(ufa

†f + v∗fa−f

)(ufaf + vfa†−f)

+1

2Vσ∗σ − 1

V

∑f

[√λ2(f)(ν + σ∗)(ν + σ) + T 2(f) − T (f)

],

(3.2)

where

uf =1√2

√1 +

T (f)√λ2(f)(ν + σ∗)(ν + σ) + T 2(f)

,

vf =−∈(f)√

2

√1 − T (f)√

λ2(f)(ν + σ∗)(ν + σ) + T 2(f)· σ + ν

|σ + ν| ,(3.3)



and where we have put

λ(f) = ∈(f)|λ(f)|, ∈(f) = signλ(f). (3.4)

Obviously:

u(−f) = u(f); v(−f) = −v(f); u2f + v2

f = 1. (3.5)

In general u is real and v complex. From (3.5) it follows that the operators

αf = ufaf + vfa†−f

α†f = ufa

†f + v∗fa−f

(3.6)

are fermion operators. Accordingly, we can rewrite the expression (3.2) forH0 as

H0 =∑

f

√λ2(f)(ν + σ∗)(ν + σ) + T 2(f)α†

fαf

+1

2Vσ∗σ − 1

V

∑f

[√λ2(f)(ν + σ∗)(ν + σ) + T 2(f) − T (f)

],

(3.7)

The minimum eigenvalue of H0 is obviously obtained by putting theoccupation number α†

fαf equal to zero. We then find the following expressionfor the ground state energy of the Hamiltonian

E0(σ) =1

2Vσ∗σ− 1

V

∑f

[√λ2(f)(ν + σ∗)(ν + σ) + T 2(f)−T (f)

]. (3.8)

To obtain the optimum upper bound for EH we must minimize E0(σ) withrespect to σ. For this purpose it is convenient to consider the cases ν = 0and ν > 0 separately.

(i) The case ν = 0

Writing x = σ∗σ > 0 we have

E0(σ) =1

2V F (x),


178 N. N. BOGOLUBOV

where

F (x) = x− 1

V

∑f

[√λ2(f)x+ T 2(f) − T (f)

].

In this case the minimization condition obviously determines only themodulus of σ, not its phase. We have

F ′(x) = 1 − 1

2V

∑f

λ2(f)√λ2(f)x+ T 2(f)

;

F ′′(x) =1

4V

∑f

λ4(f)

(√λ2(f)x+ T 2(f))3

.

Clearly F ′′(x) > 0 for 0 ≤ x ≤ ∞, so that F ′(x) can have at most one zeroin this interval. Taking into account the properties of the functions λ(f) andT (f) (see Section 1) we get

F ′(0) < 0; , F ′(∞) > 0

and so the interval 0 < x < ∞ contains a single solution of theequation F ′(x) = 0, and this solution defines the absolute minimum of thefunction F (x). Our final result is therefore:

V

2minF (x) ≥ EH(0 < x <∞). (3.9)

(ii) The case ν > 0

We put (ν + σ∗)(ν + σ) = x (so that, obviously, x > 0) and note theidentity

σ∗σ = x+ ν2 − ν(σ + ν + σ∗ + ν)

= (√x− ν)2 + 2ν√x− (σ + ν + σ∗ + ν).

The root is to be taken here and subsequently, as the positive root. Then wecan write

σ + ν =√x eiϕ; σ∗ + ν =

√x e−iϕ

andσ∗σ = (

√x− ν)2 + 2ν

√x(1 − cosϕ).



Therefore, we can express E0(σ) in the form

E0(σ) =V

2F (x) + V ν

√x(1 − cosϕ), (3.10)

where

F (x) = (√x− ν)2 − 1

V

∑f

[√λ2(f)x+ T 2(f) − T (f)

].

We then have

F ′(x) = 1 − ν√x− 1

2V

∑f

λ2(f)√λ2(f)x+ T 2(f)

;

F ′′(x) =ν

2x3/2+

1

4V

∑f

λ4(f)

(√λ2(f)x+ T 2(f))3

.

Since F ′′(x) > 0, clearly F ′(x) can have at most on e zero in the range0 ≤ x ≤ ∞. But,

F ′(0) = −∞, F ′(∞) = 1,

and therefore there exists a value x0 of x in the range 0 < x < ∞ such thatF ′(x0) = 0. This value defines the absolute minimum of the function F (x).

It is clear from (3.10) that the unique choice of σ which corresponds tothe absolute minimum of E0(σ) is

x = x0, ϕ = 0. (3.11)

Thus,σ + ν =

√x, σ =

√x− ν.

Hence, in the present case (ν > 0) both the amplitudes and the phase of σare fixed; in fact, σ must be real. Our final result, therefore, is

V

2minF (x) ≥ EH (0 < x <∞). (3.12)

It can be shown by the simple considerations of reference 2 that thecomplementary term H − H0 = H1 in equation (1.3) has no effect in thelimit to infinite volume. To prove this rigorously, however, we need not onlyan upper bound for EH but also a corresponding lower bound. To put inanother way, we should like to be able to eliminate completely the term(∑

f

λ(f)a†fa†−f − V σ∗

)(∑f

λ(f)a−faf − V σ).


180 N. N. BOGOLUBOV

Formally this could be achieved by treating σ not as a c-number but as theoperator

L =1

V

∑f

λ(f)a−faf .

However, if σ is an operator we cannot carry out a canonical transformationfrom the fermions operators a to the new fermion operators σ. In spite ofthis, we shall now try to generalize the identity (3.2) to this case; all that isnecessary is to determine the correct order of the operator. By this methodwe shall prove the theorem that the solution for H0 is also asymptoticallyexact for H in the limit of infinite volume.

4. Lower Bound for the Minimum Eigenvalue of the Hamiltonian

To derive the lower bound for the Hamiltonian (1.2) we first generalizethe identity (1.3) in such a way as to reduce the term H1 (equation (1.5))identically to zero. The way to do this is to take σ not as a c-number but asan operator, which we shall call L:

L =1

V

∑f

λ(f)a−faf . (4.1)

Instead of the c-numbers (ν + σ∗)(ν + σ) we introduce the operators:

K = (L+ ν)(L† + ν) + β2; K = (L† + ν)(L+ ν) + β2, (4.2)

where β is a constant.Consider now the following operators:

pf =1√2

√√Kλ2(f) + T 2(f) + T (f); pf = p†f

qf = −∈(f)√2

√√Kλ2(f) + T 2(f) − T (f) · 1√

K(L+ ν) (4.3)

Clearly

pfqf = −λ(f)

2(L+ ν) (4.4)

p2f =

1

2

√Kλ2(f) + T 2(f) + T (f)

(4.5)



q†fqf = (L† + ν)1

2K

√Kλ2(f) + T 2(f) − T (f)

(L+ ν). (4.6)

Taking into account the identity

ξ†F (ξξ†)ξ = ξ†ξF (ξξ†) (4.7)

which holds for any arbitrary operator ξ, we can rewrite equation (4.6) inthe form

q†fqf = (L† + ν)(L+ ν)1

2K

√Kλ2(f) + T 2(f) − T (f)

=

1

2

√Kλ2(f) + T 2(f) − T (f)

− β2

2K

√Kλ2(f) + T 2(f) − T (f)

.

(4.8)

With a view to the subsequent application if lemma II of the Appendix(equations (A.9), (A.10)) we write (4.8) as follows

q†fqf =1

2

√λ2(f)

(K +

2s

V

)+ T 2(f) − T (f)

− 1

2

√λ2(f)

(K +

2s

V

)+ T 2(f) −

√λ2(f)K + T 2(f)

− β2

2K

√Kλ2(f) + T 2(f) − T (f)

, (4.9)

where s forms an upper bound for the expression1

V

∑f

|λ(f)|2:

1

V

∑f

|λ(f)|2 ≤ s. (4.10)

Notice that the second term on the right-hand side of (4.9) is non-negative.In a similar way we can rewrite (4.5) as

p2f =

1

2

√λ2(f)

(K +

2s

V

)+ T 2(f) + T (f)

− 1

2

√λ2(f)

(K +

2s

V

)+ T 2(f) −

√Kλ2(f) + T 2(f)

. (4.11)


182 N. N. BOGOLUBOV

Now we consider the quantity

Ω =∑

f

(a†fpf + a−fq†f )(pfaf + qfa

†−f ). (4.12)

Using the fact that q†fqf = q†−fq−f and equation (4.4), we obtain

Ω =∑

f

a†fp2faf +

∑f

afq†fqfa

†f −

∑f

λ(f)

2

× (L† + ν)a−faf + a†fa†−f(L+ ν)

+R1 (4.13)

where

R1 =∑

f

λ(f)

2

(L†a−f + a−fL

†)af + a†f (a†−fL+ La†−f)

. (4.14)

Now observe that∑f

λ(f)

2

(L† + ν)a−faf + a†fa

†−f(L+ ν)

= V L†L+

V

2(νL+ νl†) (4.15)

and consequently

Ω +V

2L†L−

∑f

a†fp2faf +

∑f

afq†fqfa

†f

= −V2L†L+ ν(L+ L†) +R1. (4.16)

In other words, by virtue of (4.9) and (4.11) we have∑f

∑f

(a†fpf + a−fq†f )(pfaf + qfa

†−f)

+1

2

∑f

a†f√(

K +2s

V

)λ2(f) + T 2(f) −

√Kλ2(f) + T 2(f)

af

+1

2

∑f

af

√λ2(f)

(K +

2s

V

)+ T 2(f) −

√Kλ2(f) + T 2(f)

a†f

+1

2

∑f

af

β2

2K

√Kλ2(f) + T 2(f) − T (f)

a†f



− 1

2

∑f

a†f√

λ2(f)(K +

2s

V

)+ T 2(f) + T (f)

af

− 1

2

∑f

af

√λ2(f)

(K +

2s

V

)+ T 2(f) − T (f)

a†f +

V

2L†L

= −V2L†L+ ν(L+ L†) +R1. (4.17)

Let us introduce the notation:

∆1 =1

2

∑f

a†f√(

K +2s

V

)λ2 + T 2 −

√Kλ2 + T 2

af (4.18)

∆2 =1

2

∑f

af

√(K +

2s

V

)λ2 + T 2 −

√Kλ2 + T 2

a†f (4.19)

∆3 =1

2

∑f

afβ2

2K

√Kλ2 + T 2 − T

a†f . (4.20)

Then by virtue of lemma II (equations (A.9), (A.10)):

Ω ≥ 0, ∆1 ≥ 0; ∆2 ≥ 0; ∆3 ≥ 0. (4.21)

According to (4.17),

Ω + ∆1 + ∆2 + ∆3 − 1

2

∑f

a†f√(

K +2s

V

)λ2 + T 2 + T

af

− 1

2

∑f

af

√(K +

2s

V

)λ2 + T 2 − T

a†f +

V

2L†L

= −V2L†L+ ν(L+ L†) +R1. (4.22)

If we put

R2 =1

2

∑f

a†f[√(

K +2s

V

)λ2 + T 2

]af − af

√(K +

2s

V

)λ2 + T 2

(4.23)

R3 =1

2

∑f

af

[√(K +

2s

V

)λ2 + T 2

]a†f − a†f

√(K +

2s

V

)λ2 + T 2

(4.24)


184 N. N. BOGOLUBOV

then (4.22) becomes

Ω + ∆1 + ∆2+∆3 − R1 − R2 − R3 +V

2L†L

− 1

2

∑f

a†faf

√(K +

2s

V

)λ2 + T 2 + T

− 1

2

∑f

afa†f

√(K +

2s

V

)λ2 + T 2 − T

= −V2L†L+ ν(L+ L†) +R1. (4.25)

However

1

2

∑f

a†faf

√(K +

2s

V

)λ2 + T 2 + T

+1

2V∑

f

afa†f

√(K +

2s

V

)λ2 + T 2 − T

=1

2

∑f

√(K +

2s

V

)λ2 + T 2 − T

+∑

f

T (f)a†faf , (4.26)

and so

Ω + ∆1 + ∆2 + ∆3 − R1 −R2 − R3 +V

2(L†L− LL†)

+1

2

[LL† − 1

V

∑f

√(K +

2s

V

)λ2 + T 2 − T

]

=∑

f

T (f)a†faf − V

2

L†L+ ν(L† + L)

=∑

f

T (f)a†faf − ν∑

f

λ(f)

2(afaf + a†fa

†−f)

− 1

2V

∑f,f ′

λ(f)λ(f ′)a†fa†−fa−f ′af ′ = H (4.27)

Thus, we finally obtain

H =1

2VLL† − 1

V

∑f

[√(K +

2s

V

)λ2(f) + T 2(f) − T (f)

]



+ Ω + ∆1 + ∆2 + ∆3 − R1 −R2 − R3 +V

2(L†L− LL†). (4.28)

The expression (4.28) for the Hamiltonian (1.2) is simply an identity. Weshall treat the first term as a principal one; the terms R1, R2, R3 we shallprove to be an asymptotically small, and the terms Ω, ∆1, ∆2, ∆3 we shalldrop. Since they are positive (equation (4.21)) we obtain in this way a lowerbound for the eigenvalues of H .

It is easy to show that by virtue of (2.2)

−R1 +V

2(L†L− LL†) = − 1

V

∑f

λ2(f), (4.29)

where, according to(4.10),1

Vsumf |λ(f)|2 ≤ s. Further, in view of lemma IV

[(A.30) of the Appendix], we have

|R2| + |R3| ≤ c (4.30)

where

c =4

π

1

V

∑f

|λ(f)|21 +

|λ(f)|(

1V

∑ |λ(f)| + V)

2 1V

∑ |λ(f)|2 + V T 2(f)λ2(f)

∞∫

0

√t

(1 + t2)2dt. (4.31)

Hence, from (4.21) we have the following inequality for any normalizedfunction φ

〈φ∗Hφ〉 ≥ −(s + c)

+1

2V⟨φ∗(LL† − 1

V

∑f

[√(L+ ν)(L† + ν) + β2 +

2s

V

λ2 + T 2(f)

− T (f)])φ⟩.

However, s and c are independent of β; hence, taking the limit β → 0, wefind

〈φ∗Hφ〉 ≥ −(s + c)

+1

2V⟨φ∗(LL† − 1

V

∑f

[√(L+ ν)(L† + ν) +

2s

V

λ2 + T 2(f)

− T (f)])φ⟩. (4.32)


186 N. N. BOGOLUBOV

Now we have

LL† = (L+ ν)(L† + ν) − νL+ ν + L† + ν + ν2

LL† +2s

V=

(L+ ν)(L† + ν) +2s

V

− νL+ ν + L† + ν + ν2 (4.33)

Setting

(L+ ν)(L† + ν) +2s

V= X (4.34)

we can write

LL† +2s

V= (

√X − ν)2 + ν(2

√X − (L+ ν) − (L† + ν)). (4.35)

If, in the inequality of lemma I (A.1), (A.2), we put

ξ = L+ ν, ξ† = L† + ν

then we obtain

2

√(L+ ν)(L+ ν†) +

s

V− (L+ ν) − (L† + ν) ≥ 0, (4.36)

2√X − (L+ ν) − (L† + ν) ≥ 0. (4.37)

Therefore, defining a function F (x) as in section 3, i.e.

F (x) = (√x− ν)2 − 1

V

[√xλ2(f) + T 2(f) − T (f)

],

we can write formula (4.32) in the form

〈φ∗Hφ〉 ≥ − (2s+ c) +1

V〈φ∗F (X)φ〉

+ Vν

2〈φ∗2

√X − (L+ ν + L† + ν)φ〉

≥ − (2s+ c) +1

2〈φ∗F (X)φ〉, (4.38)

where X is now the operator defined by (4.34).



Let EH be the lowest eigenvalue of H and ψH is the correspondingeigenfunction; let EH0 be the lowest eigenvalue of H0, which, by (3.11) isgiven by

EH0 =V

2minF (x).

Suppose the absolute minimum of F (x) is obtained when

x = x0 = C2 (4.39)

Then we have

V

2F (C2) ≥ EH = 〈φ∗

HHφH〉 ≥ −(2s+ c) +1

2V 〈φ∗Hφ〉

≥ −(2s + c) +V

2F (C2). (4.40)

If we now take into account that the energy of the system must beproportional to the volume, we get as our final pair of bounds for theminimum eigenvalue of the Hamiltonian H (1.2):

0 ≤ EH0 −EH

V≤ 2s+ c

V. (4.41)

Now the quantities c [equation (4.31)] and s remain finite in the limit V →∞, according to the postulates of section 1. Hence the differencebetween the eigenvalues of the approximate Hamiltonian H0 and the exactHamiltonian H , divided by the volume of the system, decreases as 1/V inthe limit V → ∞. This result proves that the solution for the approximateHamiltonian H0(1.4) constitutes a solution for the exact Hamiltonian Hwhich is asymptotically exact in the limit of infinite volume.

We shall now show that it is asymptotically correct (i.e., correct toorder 1/V ) to treat the operator X defined by equation (4.34) as a c-number.Consider an arbitrary normalized function φ such that

〈φ∗Hφ〉 −EH ≤ c1 = const. (4.42)

Thus, using (4.38), (4.40) and (4.42), we have

〈φ∗(F (X) − F (C2))φ〉 + ν〈φ∗(2√X − (L+ ν + L† + ν)

)φ〉 ≤ l

Vl = 2(2s+ c+ c1). (4.43)


188 N. N. BOGOLUBOV

To proceed, we notice that both terms on the left-hand side of (4.43) arepositive. In fact, the second term is positive by virtue of lemma I of theAppendix (A.1), (A.2), and this implies that

〈φ∗(F (X) − F (C2))φ〉 ≤ 1

V, (4.44)

while on the other hand, there exists a value of ξ such that

F (X) − F (C2) =1

2F ′′(ξ)(X − C2)2. (4.45)

However, F ′′(x) is positive for all x

F ′′(x) =ν

2x3/2+

1

4

1

V

∑f

λ4(f)

(xλ2(f) + T 2(f))3/2;

1

2F ′′(ξ) ≥ α = const > 0. (4.46)

From equations (4.44)-(4.46) it follows that

〈φ∗|X − C2|2φ〉 ≤ 1

αV. (4.47)

Equation (4.47) shows that the operator X may with asymptotic accuracybe treated as a c-number.

Actually, in the case ν > 0 we can obtain rather more completeinformation about the expectation values of the operators L, L†. In fact,we shall prove that the mean square deviation of the operator ξ = L + νfrom the quantity C defined by (4.39) is asymptotically small in the limit ofinfinite volume.

We have the obvious inequality

(√X − C)2 =

(X − C2)2

(√X + C)

≤ 1

C2(X − C2)2. (4.48)

Hence, using (4.47), we get

〈φ∗(√X − C)2φ〉 ≤ 1

αC2V. (4.49)

Let us define the quantity C0 by

〈φ∗√Xφ〉 = C0. (4.50)



Then, from (4.49)

〈φ∗(√X − C0)

2φ〉 ≤ 〈φ∗(√X − C)2φ〉 ≤ 1

αC2V. (4.51)

Here the first inequality follows from the identity

〈φ∗(√X − C)2φ〉 = (C − C0)

2 + 〈φ∗(√X − C0)

2φ〉. (4.52)

From (4.51) we get the following inequality for the expectation value of theoperator X:

〈φ∗Xφ〉 − C20 ≤ 1

αC2V. (4.53)

Finally, from (4.51) and (4.52) follows an inequality for (C − C0)2, namely

(C − C0)2 ≤ 1

αC2V. (4.54)

Now putξ = L+ ν, ξ† = L† + ν. (4.55)

Then, for the mean square deviation of σ from C0 we have, by (4.34),

〈φ∗(C0 − ξ)(C0 − ξ†)φ〉≤ C2

0 + 〈φ∗Xφ〉 − C0〈φ∗(ξ + ξ†)φ〉. (4.56)

Using (4.53) and (4.54) we obtain

〈φ∗(C0 − ξ)(C0 − ξ†)φ〉 ≤ 2C20 +

l

αC2V− C0〈φ∗(ξ + ξ†)φ〉

= 〈φ†2√X − (ξ + ξ†)φ〉C0 +

l

αC2V≤ lC0

νV+

l

αC2V. (4.57)

Thus, we finally obtain the following bound for the fluctuations of ξ:

〈φ∗(C − ξ)(C − ξ†)φ〉 = 〈φ∗(C − C0 + C0 − ξ)(C − C0 + C0 − ξ†)φ〉≤ 2(C − C0)

2 + 2〈φ∗(C0 − ξ)(C0 − ξ†)φ〉≤ 2l

αC2V+

2lC0

νV+

2l

αC2V≤ const

V=

I

V, (I = const). (4.58)

Note that this bound is applicable only for ν > 0, since ν appears in thedenominator of the right-hand side of the inequality (4.58).


190 N. N. BOGOLUBOV

A few comments on the results obtained here are appropriate at thisstage. Suppose ν = 0. Then, as we saw above, in any state with energyasymptotically close to the ground state EH the operator L†L is equal, withasymptotic accuracy, to the c-number C2. However, these states possess nosimilar properties with respect to the operators L,L† themselves, as we shallnow see.

Consider a state φH with the ground state energy EH ; in generaldegeneracy can occur, so that we will have not just one φH but a linearmanifold φH of possible states with the same (minimum) energy. Sincein the case under consideration (ν = 0) the operator N =

∑f

a†faf , which

represents the total number of particles in the system, commutes exactlywith the Hamiltonian H , we can always choose from this manifold φH afunction φ′

H for which N takes some definite value N0. Then

〈φ∗′HLφ

′H〉 = 0; 〈φ∗′HL†φ′

H〉 = 0.

Consequently, L cannot have even an asymptotically well-defined value inthe state φ′

H , since if did L†L would be (asymptotically) equal to zero in thisstate rather than to the finite quantity C2.

Consider now the manifold φ of states with energies asymptoticallyclose to EH . Since L, L† commute approximately with H , we might expectthat we can choose from φ a function φ for which L†, L take asymptoticallywell-defined values.

Such is indeed the case. For instance, one function with the requiredproperties is φ′

H0, the state corresponding to the minimum eigenvalue of H0.

Indeed, as we saw, φ′H0

is defined by the relations

αfφ = 0; where αf = ufaf − vfα†−f

uf =1√2

√1 +

T (f)√λ2(f)C2 + T 2(f)

uf =∈(f)√

2

√1 − T (f)√

λ2(f)C2 + T 2(f).

We can express L in terms of the fermion operators αf , α†f ; the result is:

L =1

V

∑f

λ(f)u2

fα−faf − v2fα

†fα

†−f − 2ufvfα

†faf

+

1

V

∑f

ufvfλ(f).



However,1

V

∑f

ufvf =1

2V

∑f

|λ(f)|2C√λ2(f)C2 + T 2(f)

= C,

and therefore,

〈ψ∗H0

(L† − C)(L− C)φH0〉 ≤const

V;

〈ψ∗H0

(L− C)(L† − C)φH0〉 ≤const

V.

Thus, for φH0, L and L† are asymptotically equal to C. This is preciselythe reason for the success of the approximation method which replaces Hmwhich conserves the particle number N exactly, by H0, for which N is nolonger an exact constant of motion.

We can now see that it is also possible to formulate the approximationmethod in such a way as not to break the law of conservation of particlenumber. To do so we introduce, instead of the Fermi operators αf , theoperators

αf = ufaf − vfL

|C| a†−f ,

which obey Fermi commutation relations with asymptotic accuracy. Then αf

decreases and α†f increases the particle number N by unity. These operators

are analogous to the operators

bf =a†f√N0

αf ,

which were introduced in the theory of superfluidity [6] to eliminate thecondensate. Indeed, there is a strong analogy in general between the Bose-condensate operators a0, a

†0 and the operators L, L† in the present case.

As soon as we include in H a term containing sources of pairs (i.e., putν > 0) then L, L† immediately take asymptotically well-defined values foreigenstates of H with energies near EH . We can see here an analogy with thecase of ferromagnetism in an isotropic medium; in the absence of the externalmagnetic field the direction of the axis of magnetization is not well-defined,but as soon as we introduce an arbitrary weak field acting in a given directionthe magnetization vector immediately orient itself in that direction.


192 N. N. BOGOLUBOV

Finally, we point out that the relations

L†L ∼ C2 (ν = 0)

ν + L ∼ C; ν + L† ∼ C (ν > 0)

enable us to prove that the correlation expectation values

〈φ∗H . . . afl(tl) . . . a

†fj(tj) . . . φH〉

for the Hamiltonian H are also asymptotically equal to the correspondingexpectation values for H0. For ν > 0 this is true for all averages of thetype indicated; for ν = 0 it is of course true only for those in which thenumbers of creation and annihilation operators are equal, i.e. for averagesof operators which conserve particle number. We shall now proceed to provethis statement.

5. Green’s Functions (Case ν > 0)

In this section we turn to the problem of finding asymptotic limits for theGreen’d functions and correlations functions in the case ν > 0. The existenceof these limits implies that in the limit of infinite volume the solution ofthe equations of motion constructed for the Green’s functions from theHamiltonian H0, (1.4), will differ by an asymptotically small amount fromthe corresponding solutions for the full model Hamiltonian H , (1.2).

Consider the equation of motion for the operators af , a†f . From (1.2)

and (4.1) we obtain

idaf

dt= T (f)af − λ(f)a†−f(ν + L),

ida†fdt

= −T (f)a†f + λ(f)(ν + L†)a−f , (5.1)

and therefore

ida−f

dt= T (f)a−f − λ(f)a†f(ν + L),

ida†−f

dt= −T (f)a†−f + λ(f)(ν + L†)af . (5.2)



We put [cf. (3.3) and (3.11)]

uf =1√2

√1 +

T (f)√C2λ2(f) + T 2(f)

vf =∈(f)√

2

√1 − T (f)√

C2λ2(f) + T 2(f)(5.3)

where C is a number given by (4.39), and introduce new fermion operators

α†f = ufa

†f + vfa−f . (5.4)

We then have

idα†

f

dt= uf i

da†fdt

+ vf ida−f

dt

= uf

−T (f)a†f + λ(f)(ν + L†)a−f

+ vf

T (f)a−f + λ(f)a†f(ν + L)

= −a†f

T (f)uf − λ(f)vf(ν + L)

+λ(f)(ν + L†)af + T (f)vfa−f

= −a†fT (f)uf − λ(f)vfC

+λ(f)Caf + T (f)vfa−f +Rf ,

where

Rf = R(1)f +R

(2)f

R(1)f = ufλ(f)(L† + ν − C)a−f

R(2)f = vfλ(f)a†f(L+ ν − C). (5.5)

Now we introduce the following identities

T (f)uf − λ(f)vfC =[√

C2λ2(f) + T 2(f)]uf

T (f)vf + λ(f)ufC = −[√

C2λ2(f) + T 2(f)]vf . (5.6)

It follows that

idα†

f

dt+[√

C2λ2(f) + T 2(f)]α†

f = Rf (5.7)

and also

idαf

dt−[√

C2λ2(f) + T 2(f)]α†

f = −R†f . (5.8)


194 N. N. BOGOLUBOV

The next step is to obtain bounds for various quantities connected withR and R†. We have

〈φ∗HRfR

†fφH〉 ≤ 2〈φ∗

HR(1)f R

(1)†f φH〉 + 2〈φ∗

HR(2)f R

(2)†f φH〉

= 2u2fλ

2(f)〈φ∗H(L† + ν − C)a−fa

†−f(L+ ν − C)φH〉

+ 2v2fλ

2(f)〈φ∗Ha

†f (L+ ν − C)(L† + ν − C)afφH〉.

However, since |a−fa†−f |2 ≤ 1, it follows that

〈φ∗H(L† + ν − C)a−fa

†−f (L+ ν − C)φH〉

≤ 〈φ∗H(L† + ν − C)(L+ ν − C)φH〉

and also, using equation (A.18), that

〈φ∗Ha

†f (L+ ν − C)(L† + ν − C)afφH〉≤ 2s

V+ 〈φ∗

Ha†f (L

† + ν − C)(L+ ν − C)afφH〉

=2s

V+ 〈φ∗

H(L† + ν − C)a†faf (L+ ν − C)φH〉

≤ 2s

V+ 〈φ∗

H(L† + ν − C)a(L+ ν − C)φH〉.

Thus,

〈φ∗HRfR

†fφH〉 ≤ 2λ2(f)〈φ∗

H(L† + ν − C)(L+ ν − C)φH〉+ 2λ2(f)v2

f

2s

V. (5.9)

By an entirely analogous procedure we obtain

〈φ∗HR

†fRfφH〉 ≤ 2λ2(f)〈φ∗

H(L+ ν − C)(L† + ν − C)φH〉+ 2λ2(f)u2

f

2s

V. (5.10)

Now, we proved above [cf. (4.58)] that

〈φ∗H(L† + ν − C)(L+ ν − C)φH〉 ≤ I

V

〈φ∗H(L+ ν − C)(L† + ν − C)φH〉 ≤ I

V



Hence, introducing the constant

γ = 2(I + 2s), (5.11)

we can write

〈φ∗HRfR

†fφH〉 ≤ γ

V|λ(f)|2, 〈φ∗

HR†fRfφH〉 ≤ γ

V|λ(f)|2. (5.12)

We can actually state a more general set of inequalities. Consider any setof operators Af , each of which is a linear combination of operators af and

a†−f ,

Af = pfaf + qfa†−f (5.13)

with bounded coefficients:

|pf |2 + |qf |2 ≤ const. (5.14)

We shall prove that

|〈φ∗HAf1 . . . Afl

RfAfl+1. . . AfmR

†fAfm+1 . . . φH〉| ≤ const

I

|〈φ∗HAf1 . . . Afl

R†fAfl+1

. . . AfmRfAfm+1 . . . φH〉| ≤ const

I. (5.15)

Proof. We notice first of all that

Laf − afL = 0; L†a†f − a†fL† = 0;

|La†f − a†fL| ≤2|λ(f)|V

; |L†af − afL†| ≤ 2|λ(f)|

V.

Thus, for example,

〈φ∗HAf1 . . . (L+ ν − c)Afj

. . . (L† + ν − c)Afi. . . φH〉

= Z + 〈φ∗H(L+ ν − c)Af1 . . . Afn(L† + ν − c)φH〉,

where

|Z| ≤ const

V.

Therefore,

〈φ∗HAf1 . . . (L+ ν − c)Afj

. . . (L† + ν − c)Afi. . . φH〉


196 N. N. BOGOLUBOV

≤ const

V+ |Af1| . . . |Afn|〈φ∗

H(L+ ν − c)(L† + ν − c)φH〉 ≤ const

V.

(5.16)

Similarly we can prove that

〈φ∗HAf1 . . . (L

† + ν − c)Afj. . . (L+ ν − c)Afi

. . . φH〉 ≤ const

V. (5.17)

Moreover, we have:

〈φ∗HAf1 . . . (L+ ν − c)Afj

. . . (L+ ν − c)Afi. . . φH〉

≤ const

V+ 〈φ∗

H(L+ ν − c)Af1 . . . Afn(L+ ν − c)φH〉

≤ const

V+√〈φ∗

H(L+ ν − c)Af1 . . . AfnA†fnA†

f1(L† + ν − c)φH〉

×√

〈φ∗H(L+ ν − c)(L+ ν − c)φH〉

≤ const

V+ |Af1| . . . |Afn|

√〈φ∗

H(L+ ν − c)(L† + ν − c)φH〉

×√

〈φ∗H(L+ ν − c)(L+ ν − c)φH〉

≤ const

V(5.18)

and it is easily shown that in a similar way

〈φ∗HAf1 . . . (L

† + ν − c)Afj. . . (L† + ν − c)Afi

. . . φH〉 ≤ const

V. (5.19)

From (5.16)-(5.19) it follows that the inequalities (5.15) are satisfied.We can now set about finding limits for the correlation functions.

Using (5.7), we obtain

id

dt〈φ∗

Hα†f(t)αfφH〉 = −

√c2λ2(f) + T 2(f)〈φ∗

Hα†f(t)αfφH〉

+ 〈φ∗HRf (t)αfφH〉. (5.20)

Here, and subsequently, we write

αf (0) = αf ; a†f(0) = a†f .

Since the equation

idJ(t)

dt= −ΩJ(t) +R(t)



has the solution

J(t) = J(0) eiΩt + eiΩt

t∫

0

e−iΩtR(t) dt,

we may write

〈φ∗Hα

†f(t)αfφH〉 =ei

√c2λ2(f)+T 2(f) t〈φ∗

Hα†fαfφH〉 + ei

√c2λ2(f)+T 2(f) t

×t∫

0

ei√

c2λ2(f)+T 2(f) t 〈φ∗Hα

†f(t)αfφH〉 dt. (5.21)

On the other hand, since φH is the eigenfunction of H corresponding to itsminimum eigenvalue, the usual spectral representation gives

〈φ∗Hα

†f (t)αfφH〉 =

t∫

0

Jf (ν) e−iνt dν, (5.22)

where

Jf ≥ 0;

t∫

0

Jf (ν) dν ≤ 1. (5.23)

Let us define a function

h(t) =

2∫

0

ω2(2 − ω)2 e−iωt dω. (5.24)

Obviously this function is regular on the whole of the real axis. Integratingby parts, we easily see that for |t| → ∞, h(t) decreases in such a way that

|h(t)| ≤ const

|t|3 . (5.25)

Thus, the integral∞∫

−∞

|th(t)| dt (5.26)


198 N. N. BOGOLUBOV

is finite.We now put √

c2λ2(f) + T 2(f) = Ω (5.27)

and note that

h(Ωt) =1

Ω

2Ω∫

0

ν2(2Ω − ν)2 e−iνt dν. (5.28)

It is clear form the above that∞∫

−∞

h(Ωt) e−iνt dt = 0, for ν ≥ 0 (5.29)

and therefore that (5.22) implies

∞∫

−∞

〈φ∗Hα

†f (t)αfφH〉h(Ωt) dt = 0. (5.30)

Thus, it follows from (5.21) that

〈φ∗Hα

†fαfφH〉

∞∫

−∞

eiΩth(Ωt) dt

−∞∫

−∞

h(Ωt)eiΩt( t∫

0

e−iΩt′〈φ∗HRf (t

′)αfφH〉 dt′)dt. (5.31)

However,∞∫

−∞

eiΩth(Ωt) dt =2π

Ω. (5.32)

Therefore

〈φ∗Hα

†fαfφH〉 ≤ Ω

2π

∞∫

−∞

|h(Ωt)| t∫

0

|〈φ∗HRf(t

′)αfφH〉| dt′dt. (5.33)

Using (5.12), we have

〈φ∗HRfαfφH〉 ≤

√|〈φ∗

HR†fRfφH〉| |〈φ∗

Ha†fafφH〉|



≤( γV

)1/2

|λ(f)| |〈φ∗Ha

†fafφH〉|1/2. (5.34)

and consequently

〈φ∗Hα

†fαfφH〉

≤ Ω

2π

∞∫

−∞

|h(Ωt)| |t|dt( γV

)1/2

|〈φ∗Ha

†fafφH〉|1/2

≤ 1

2πΩ

∞∫

−∞

|h(τ)τ | dτ( γV

)1/2

|〈φ∗Ha

†fafφH〉|1/2|λ(f)|.

Thus,

〈φ∗Hα

†fαfφH〉 ≤ |λ(f)|2

2π(C2|λ(f)|2 + T 2(f)

) γV

( ∞∫

−∞

|h(τ)τ | dτ)2

. (5.35)

From this formula we can immediately derive a number of inequalities.Using Schwartz’s inequality and the fact |a†faf | ≤ 1, we obtain from (5.35)

|〈φ∗Hα

†f1. . . α†

fsαgl

. . . αg1φH〉|≤√〈φ∗

Hα†f1. . . α†

fsαfs . . . αf1φH〉〈φ∗

Hα†g1 . . . α

†glαgl

. . . αg1φH〉≤√

〈φ∗Hα

†f1αf1φ〉〈φ∗

Hα†g1αg1φ〉 ≤

const

V. (5.36)

We also have

|〈φ∗Hαf1 . . . αfsφH〉|

≤√

〈φ∗Hαf1 . . . αfs−1α

†fs−1

. . . α†f1φH〉〈φ∗

Hα†fsαfsφH〉

≤√

〈φ∗Hα

†fsαfsφH〉 ≤ const√

V(5.37)

and

|〈φ∗Hα

†f1. . . α†

fsφH〉| ≤

√〈φ∗

Hα†f1αf1φH〉 ≤ const√

V. (5.38)

We may now compare the expectation values

〈φ∗HUf1 . . .UfsφH〉


200 N. N. BOGOLUBOV

(where Uf may stand for af or a†f) with the corresponding values calculatedby using the Hamiltonian H0 with ν + σ set equal to C. For convenience weshall denote the two kind of expectation values by

〈Uf1 . . .Ufs〉H , and 〈Uf1 . . .Ufs〉H0

respectively. We wish to establish bounds for the differences

〈Uf1 . . .Ufs〉H − 〈Uf1 . . .Ufs〉H0. (5.39)

It is appropriate at this point to outline the calculation of the quantities〈Uf1 . . .Ufs〉H0 . We use the transformation

a†f = ufα†f − vfα−f

af = ufαf − vfα†−f

and then reduce the product Uf1 . . .Ufs to a sum of normal products (i.e.,products in which all the α†s precede all the αs). Since all terms of the type

〈α† . . . α†〉H0 ; 〈α . . . α〉H0; 〈α† . . . α〉H0 (5.40)

are equal to zero, we obtain in this way an expression for 〈Uf1 . . .Ufs〉H0.We can apply the same procedure to calculate the quantities

〈Uf1 . . .Ufs〉H . Obviously the difference (5.39) is entirely due to termsproportional to expectation values of the form

〈α† . . . α†〉H ; 〈α . . . α〉H ; 〈α† . . . α〉H, (5.41)

which, in general, unlike the terms (5.40), are not equal to zero. However, wehave certain inequalities for the terms (5.41), namely formulae (5.36)-(5.38).We are therefore led to the result

|〈Uf1 . . .Ufs〉H − 〈Uf1 . . .Ufs〉H0| ≤const√V. (5.42)

We next turn to the double-time correlation functions. We shall provethat the absolute magnitudes of all differences of the type

〈Bf1(t) . . . BflUfm(τ) . . .Ufn(τ)〉H − 〈Bf1(t) . . . Bfl

Ufm(τ) . . .Ufn(τ)〉H0

(5.43)



(where Uf and Bf may denote either af or a†f ) can be at most quantities of

order 1/√V .

We first notice that while

〈α†f1

(t) . . . αfj(t)Ufm(τ) . . .Ufn(τ)〉H0 = 0, (5.44)

we also have

|〈α†f1

(t) . . . αfj(t)Ufm(τ) . . .Ufn(τ)〉H |

≤√

〈α†f1

(t)αf1(t)〉H〈w†w〉H =√〈α†

f1αf1〉H〈w†w〉H , (5.45)

wherew = . . . αfj

(t)Ufm(τ) . . .Ufn(τ).

Therefore

|〈α†f1

(t) . . . αfj(t)Ufm(τ) . . .Ufn(τ)〉H | ≤

√〈α†

f1αf1〉H ≤ const√

V. (5.46)

It follows that we need only prove that differences of the type

〈af1(t) . . . afl(t)Ufm(τ) . . .Ufn(τ)〉H − 〈af1(t) . . .Ufn(τ)〉H0

have absolute magnitude of order equal to or less than 1/√V .

Let us use the notation

〈af1(t) . . . afl(t)Ufm(τ) . . .Ufn(τ)〉H = Γ(t− τ). (5.47)

From (5.8) we have

iΓ(t− τ)

dt− Ω(f1) + . . .+ Ω(fl)

Γ(t− τ) = ∆(t− τ), (5.48)

whereΩ(f) =

√C2λ2(f) + T 2(f)

and

∆(t− τ) = ∆1(t− τ) + . . .+ ∆l(t− τ)

∆1(t− τ) = −〈R†f1

(t)αf2(t) . . . αfl(t)Ufm(τ) . . .Ufn(τ)〉H

. . .


202 N. N. BOGOLUBOV

∆l(t− τ) = −〈αf1(t) . . . αfl−1(t)R†

fl(t)Ufm(τ) . . .Ufn(τ)〉H .

However

|∆s(t− τ)| ≤√

〈αf1(t) . . . αfs−1(t)R†fs

(t) . . . αfl(t)α†

fl(t) . . . Rfs(t) . . . α

†1(t)〉H

×√

〈2U †fn

(τ) . . .U †fm

(τ)Ufm(τ) . . .Ufn(τ)〉=√

〈αf1 . . . αfs−1R†fs. . . αfl

α†fl. . . Rfs . . . α

†1〉H〈U †

fn. . .Ufn〉

≤√

〈αf1 . . . αfs−1R†fs. . . αfl

α†fl. . . Rfs . . . α

†1〉H

and therefore, using (5.15),

|∆s(t− τ)| ≤ const√V. (5.49)

Consequently

|∆(t− τ)| ≤ s√V

where s = const. (5.50)

From (5.48) we have [cf. (5.21)]

Γ(t− τ) = Γ(0) e−iΩ(f1)+...+Ω(fl)(t−τ)

+ exp[−iΩ(f1) + . . .+ Ω(fl)(t− τ)

] t−τ∫

0

eiΩ(f1)+...+Ω(fl)ω∆(ω) dω.

(5.51)

Hence, using (5.50),∣∣∣Γ(t− τ) − Γ(0) e−iΩ(f1)+...+Ω(fl)(t−τ)∣∣∣ ≤ s√

V|t− τ |. (5.52)

We also have

〈αf1(t) . . . αfl(t) . . .Ufm(τ) . . .Ufn(τ)〉H0

= e−iΩ(f1)+...+Ω(fl)(t−τ)〈αf1 . . . αfl. . .Ufm . . .Ufn〉H0. (5.53)

From (5.52) and (5.53) its follows that

D ≡|〈αf1(t) . . . αfl(t)Ufm(τ) . . .Ufn(τ)〉H



− 〈αf1(t) . . . αfl(t)Ufm(τ) . . .Ufn(τ)〉H0 |

≤ s√V|t− τ | + |〈αf1 . . . αfl

Ufm . . .Ufn〉H − 〈αf1 . . . αflUfm . . .Ufn〉H0|.

The second term on the right-hand side is the difference of two equal-timeexpectation values. As we have proved above, [cf. (5.42)], such differencesare all of order 1/

√V . Thus, we have succeeded in proving the following

inequality for the double-time expectation value

|〈Bf1(t) . . . BflUfm(τ) . . .Ufn(τ)〉H − 〈Bf1(t) . . . Bfl

Ufm(τ) . . .Ufn(τ)〉H0|≤ G1√

V|t− τ | + G2√

V; G1, G2 = const. (5.54)

These inequalities can be generalized to the case of the multiple-timeexpectation values,

〈Ps(ts)Ps−1(ts−1) . . . P1(t1)〉Pj(t) = U (j)

1 (t) . . .U (j)l (t) (5.55)

where U(j)

s (t) may, as usual, denote αf(t) or α†f (t). In fact, we shall prove

that

|〈Ps(ts) . . . P1(t1)〉H−〈Ps(ts) . . . P1(t1)〉H0|≤ (Ks|ts − ts−1| + . . .+K2|t2 − t1| +Qs√

V(5.56)

where

Kj = const, Qs = const. (5.57)

The proof is easily given by induction. We shall assume the relation (5.56)true for the (s−1)-time averages and prove it for the s-time ones. Reasoningas in the double-time case, we see that it will be sufficient to prove (5.56) forthe average of the type

Ps(t) = αf1(t) . . . αfl(t).

For such cases we have

〈Ps(ts)Ps−1(ts−1) . . . P1(t1)〉H0


204 N. N. BOGOLUBOV

= exp−i(Ωf1 + . . .+ Ωfl

)(ts − ts−1)〈Ps(ts−1)Ps−1(ts−1) . . . P1(t1)〉H0 .

(5.58)

On the other hand, from (5.8) and (5.15), and argument leading to (5.52),we see that

|〈Ps(ts)Ps−1(ts−1) . . . P1(t1)〉H − exp−i(Ωf1 + . . .+ Ωfl

)(ts − ts−1)

× 〈Ps(ts−1)Ps−1(ts−1) . . . P1(t1)〉H | ≤ K(s)1 |ts − ts−1|√

V, (5.59)

where K(s)1 = const. Thus,

|〈Ps(ts) . . . P1(t1)〉H − 〈Ps(ts) . . . P1(t1)〉H|

≤K(s)1 |ts − ts−1|√

V+ |〈Ps(ts−1)Ps−1(ts−1) . . . P1(t1)〉H

− 〈Ps(ts−1)Ps−1(ts−1) . . . P1(t1)〉H0| (5.60)

But the second term on the right-hand side is the difference of two (s−1)-timecorrelation functions, for which, by hypothesis, the required inequalities hasbeen established. Thus they are also true for the s-time expectation values.

Thus, the use of H0 gives an asymptotically exact expression for allcorrelation functions of the type

〈Ps(ts) . . . P1(t1)〉.

As a consequence, the same is true for the Green’s functions constructedfrom operators of this type.

Note: We could have sharpened the above inequalities, replacingconst√V

byconst

Veverywhere, had we chosen to replace C in the definition of uf , vf ,

and H0 by the quantity

C1 = 〈L+ ν〉H = 〈L† + ν〉H . (5.61)

Since we have [cf. (4.58)]

(C − C1)2 ≤ const

V,



all inequalities of the type (5.12), (5.15), (5.35) remain valid; but we nowalso have the following useful relations:

|〈Af1 . . . Rf . . . Afn〉| ≤const

V

|〈Af1 . . . R†f . . . Afn〉| ≤

const

V. (5.62)

To prove them it is sufficient to expand the expressions

〈Af1 . . . Rf . . . Afn〉; 〈Af1 . . . R†f . . . Afn〉, (5.63)

by expressing all the a’s and a†’s in terms of α’s and α†’s. Then we canrepresent the expressions (5.63) as sums of terms of the type

〈α† . . . α〉H〈(L+ ν − C1) . . . α〉H ; 〈α† . . . (L+ ν − C1)〉H〈(L† + ν − C1) . . . α〉H; 〈α† . . . (L† + ν − C1)〉Hconst〈L+ ν − C1〉H ≡ 0; const〈L† + ν − C1〉H ≡ 0 (5.64)

and “commutation” terms of order 1/V . (The last two terms in (5.64) arezero in virtue of (5.61).) Applying to (5.64) the inequality

|〈AB〉| ≤√

|〈AA†〉|√|〈B†B〉|,

and also (5.35), we see that all these terms will be of order 1/V , so that (5.62)is proved.

We shall now use these additional relations. Consider expression of thetype

〈α†f1. . . α†

fn〉,

which is obviously independent of t. For this reason we have

d

dt〈α†

f1. . . α†

fn〉H =

⟨ ddtα†

f1. . . α†

fn

⟩H

+ . . .+⟨α†

f1. . .

d

dtα†

fn

⟩H

= 0. (5.65)

Consequently, we get from (5.7)

(Ω(f1)+ . . .+ . . .Ω(fn))〈α†f1. . . α†

fn〉H = 〈Rf1 . . . α

†fn〉H + . . .+ 〈α†

f1. . . Rfn〉H .

(5.66)


206 N. N. BOGOLUBOV

According to (5.62), we have

|〈Rf1 . . . α†fn〉H + . . .+ 〈α†

f1. . . Rfn〉H | ≤

D

V; D = const. (5.67)

Therefore

|〈α†f1. . . α†

fn〉| ≤ D

V (Ω(f1) + . . .+ . . .Ω(fn)). (5.68)

Hence, by taking the complex conjugate, we also have

|〈αf1 . . . αfn〉| ≤D

V (Ω(f1) + . . .+ . . .Ω(fn)). (5.69)

Using our new inequalities (5.68) and (5.69) in place of the old ones (5.37)and (5.38) [but keeping (5.36)] we can prove the following relation,

|〈Uf1 . . .Ufs〉H − 〈Uf1 . . .Ufs〉H0 | ≤const

V(5.70)

which replaces (5.42).In a similar way we can sharpen the inequalities for all the correlation

functions of the type considered above. Rather than give a general proof, weshall merely find a bound for the difference

〈αf1(t) . . . αfl(t)α†

g1(τ) . . . α†

gr(τ)〉H − 〈αf1(t) . . . α

†gr

(τ)〉H0 . (5.71)

DefiningΓH,H0(t− τ) = 〈αf1(t) . . . α

†gr

(τ)〉H,H0 , (5.72)

we have

idΓH(t− τ)

dt= (Ω(f1) + . . .+ Ω(fl))ΓH(t− τ) + ∆H(t− τ), (5.73)

where

∆H(t− τ) = −∑

j

〈αf1(t) . . . R†fj

(t) . . . αfl(t)α†

g1(τ) . . . α†

gr(τ)〉H . (5.74)

Differentiating (5.74) with respect to t, we find

id∆H(t− τ)

dt= −(Ω(g1) + . . .+ Ω(gr))∆H(t− τ) + ζH(t− τ), (5.75)



where

ζ(t− τ) = −∑j,s

〈αf1(t) . . . R†fj

(t) . . . αfl(t)α†

g1(τ) . . . Rfs(τ) . . . α

†gr

(τ)〉H .

(5.76)However, from (5.15), we have

|ζ(t− τ)| ≤ Q

V, where Q = const. (5.77)

Thus, from (5.75) we obtain in the usual way [cf., e.g., (5.48)-(5.52)]

∣∣∆H(t− τ) − ∆H(0) expi[Ω(g1) + . . .+ Ω(gr)](t− τ)∣∣ ≤ Q

V|t− τ |. (5.78)

From (5.62) and (5.74), we obtain

|∆H(0)| ≤ Q1

V; Q1 = const. (5.79)

Therefore,

|∆H(t− τ)| ≤ Q1 +Q|t− τ |V

. (5.80)

Substituting this inequality in (5.73), we find∣∣ΓH(t− τ) − ΓH(0) expi[Ω(f1) + . . .+ Ω(Fl)](t− τ)∣∣≤ Q1|t− τ | +Q|t− τ |2 1

2

V(5.81)

On the other hand we have

ΓH0(t− τ) = ΓH0(0) expi[Ω(f1) + . . .+ Ω(Fl)](t− τ), (5.82)

so that

|ΓH(t− τ) − ΓH0(t− τ)| ≤ |ΓH(0) − ΓH0(0)|

+Q1|t− τ | +Q|t− τ |2 1

2

V. (5.83)

By (5.70),

|ΓH(0) − ΓH0(0)| = 〈αf1 . . . αflα†

g1. . . α†

gr〉H


208 N. N. BOGOLUBOV

− 〈αf1 . . . αflα†

g1. . . α†

gr〉H0 ≤

Q2

V; Q2 = const., (5.84)

and so, finally

|〈αf1(t) . . . αfl(t)α†

g1(τ) . . .α†

gr(τ)〉H − 〈αf1(t) . . . α

†gr

(τ)〉H0|

≤ Q2 +Q1|t− τ | +Q|t− τ |2 12

V. (5.85)

By proceeding further along these lines we could easily replace quantitiesof order 1/

√V by quantities of order 1/V in all inequalities obtained in this

section.

6. Green’s Function (Case ν = 0)

In the preceding section we derived all the necessary limits for the Green’sfunctions in the case ν > 0. Since some of the inequalities used in that section[e.g., (4.58)] are meaningless when ν = 0, the results cannot be carried overdirectly to this case, which therefore requires special consideration.

Since now the operators L and L† do not take asymptotically well-defined values in the lowest energy eigenstate φH , we shall work with fermionoperators defined rather differently from those used earlier,

αf = ufaf + vfa†−f

L

C, (6.1)

where

uf =1√2

√1 +

T (f)√C2λ2(f) + T 2(f)

vf = −∈(f)√2

√1 − T (f)√

C2λ2(f) + T 2(f). (6.2)

These operators do not obey Fermi commutation relation exactly; however,they do obey them with asymptotic accuracy.

To obtain the required limits we shall first have to establish a number ofinequalities. Consider first of all the expressions∑

Ω(f)α†fαf ,



whereΩ(f) =

√C2λ2(f) + T 2(f). (6.2′)

Substituting (6.1), we have

∑Ω(f)α†

fαf =∑

Ω(f)ufa

†f + vf

L†

Ca−f

ufaf + vfa

†−f

L

C

=∑

Ω(f)u2

fa†faf + v2

f

L†

Ca−fa

†−f

L†

C+ ufvf

L†

Ca−faf + ufvfa

†fa

†−f

L†

C

.

However, we have

∑Ω(f)v2

f

L†

Ca−fa

†−f

L†

C= −

∑Ω(f)v2

f

L†

Ca†faf

L†

C+∑

Ω(f)v2f

L†LC2

= −∑

Ω(f)v2fa

†f

L†

Caf

L†

C+∑

Ω(f)v2f

L†LC2

.

Moreover, since

ufvf = −Cλ(f)

2Ω(f),

we also have

−∑

Ω(f)ufvf

L†

Ca−faf + ufvfa

†fa

†−f

L†

C

= V L†L.

Consequently,

∑Ω(f)α†

fα =∑

f

Ω(f)u2

fa†faf − v2

fa†f

L†LC2

af

+∑

Ω(f)v2f

L†LC2

− V L†L

=∑

f

Ω(f)(u2f − v2

f )a†faf −

∑f

Ω(f)v2fa

†f

L†L− C2

C2af

+∑

Ω(f)v2f

L†LC2

− V L†L

hold. Since,Ω(f)(u2

f − v2f) = T (f)

we have

H =∑

T (f)a†faf − VL†L2


210 N. N. BOGOLUBOV

=∑

Ω(f)α†fαf + V

L†L2

−∑

Ω(f)v2f

L†LC2

+∑

Ω(f)v2fa

†f

L†L− C2

C2af .

As a result, we obtain

H =∑

Ω(f)α†faf +

V

2

C2 − 2

V

∑Ω(f)v2

f

+V (L†L− C2)

2

−∑

Ω(f)v2f

L†L− C2

C2+∑

Ω(f)v4f

L†L− C2

C2

+∑

Ω(f)v2f

a†fL†L− C2

C2af − v2

f

L†L− C2

C2

.

On the other hand

C2V

2−∑

Ω(f)v2f +

∑Ω(f)v4

f = C2V

2−∑

Ω(f)u2fv

2f

=V

2

C2 − 1

2VC2∑ λ2(f)√

C2λ2(f) + T 2(f)

=V

2C2F ′(C2);

where

F (C2) = C2 − 2

V

∑Ω(f)v2

f .

Hence

H =∑

Ω(f)α†fαf − w +

V

2(L†L− C2)F ′(C2) + F (C2), (6.3)

where

w = −∑

Ω(f)v2f

a†fL†L− C2

C2af − v2

f

L†L− C2

C2

. (6.4)

By definition [cf. (3.8) and (4.39)], C2 is a root of the equation

F ′(x) = 0.

Moreover, we have〈φ∗

HHφH〉 ≤ F (C2).

It follows that〈φ∗

H

∑Ω(f)α†

fαfφH〉 ≤ 〈φ∗HwφH〉. (6.5)



We now set about finding an inequality for the expectation value of w.We recall the definition (6.1) of the operators α:

α†f = ufa

†f + vf

L†

Ca−f ,

α−f = −vfa†f

L

C+ ufa−f .

From this results

u2fα

†f − vf

L†

Cα−f = u2

fa†f + v2

f

L†

Ca†fL

C= a†f

(u2

f + v2f

L†LC2

).

We put

η†f = a†fv2f

C2 − L†LC2

= v2f

C2 − L†LC2

a†f +2λ(f)L†

V C2a−f

ηf = v2f

C2 − L†LC2 f

= v2faf

C2 − L†LC2

+2λ(f)

V C2a†−fL; (6.6)

then

a†f = ufα† − vf

L†

Cα−f + η†f

af = ufα− vfα†−f

L

C+ ηf . (6.7)

Let us now go back to (6.4) and write

w = w1 + w2 + w3;

w1 =∑

Ω(f)v2fufα

†f

C2 − L†LC2

af

=∑

Ω(f)v2fufα

†faf

C2 − L†LC2

+∑

Ω(f)v2fufα

†f

2λ(f)

V C2a†−fL,

w2 =∑

Ω(f)v2fη

†f

C2 − L†LC2

af

=∑

Ω(f)v2fη

†faf

C2 − L†LC2

+∑

Ω(f)v2fη

†f

2λ(f)

V C2a†−fL,


212 N. N. BOGOLUBOV

w3 =∑

Ω(f)v2f

−vf

L†

Cα−f

C2 − L†LC2

af − v2f

C2 − L†LC2

= −

∑Ω(f)v3

f

L†

C

L†LC2

α−f − α−fL†LC2

af

+∑

Ω(f)v2f

−vf

L†

C

(C2 − L†LC2

)(α−faf + afα−f)

− v2f

C2 − L†LC2

+∑

Ω(f)v3f

L†

C

(C2 − L†LC2

)afα−f .

We can now find bounds for w1, w2, and w3. For w1 we use theinequality (4.47) proved above, which we write in the form⟨

φ∗H

(C2 − L†LC2

)φH

⟩≤ G

V⟨φ∗

H

(C2 − LL†

C2

)φH

⟩≤ G

V(6.8)

where G = const. We then have∣∣〈φ∗Hw1φH〉

∣∣ ≤∑Ω(f)v2fuf

∣∣∣⟨φ∗Hα

†faf

(C2 − L†LC2

)φH

⟩∣∣∣+∑

Ω(f)v2fuf

∣∣〈φ∗Hα

†fa

†fLφH〉

∣∣2|λ(f)|V C2

≤∑

Ω(f)v2fuf

√〈φ∗

Hα†fafa

†fαf 〉

√⟨φ∗H(C2 − L†L

C2

)2

φH

⟩+∑

Ω(f)v2fuf

√〈φ∗

Hα†αfφH〉〈φ∗

HL†a−fa

†−fLφH〉2|λ(f)|

V C2

≤∑

Ω(f)v2fuf

(GV

)1/2√〈φ∗

Hα†αfφH〉

+1

V

∑Ω(f)v2

fuf2|λ(f)|V C2

|L|〈φ∗Hα

†αfφH〉

≤√

〈φ∗H

∑Ω(f)α†αfφH〉

√G

V

∑f

Ω(f)v4fu

2f

+2|L|C2

√V

√1

V

∑f

Ω(f)v4fu

2f |λ2(f)|

.

Consequently, we obtain

|〈φ∗Hw1φH〉| ≤ R1

√〈φ∗

H

∑Ω(f)α†αfφH〉; R1 = conts.



In an entirely analogous way we can show that

|〈φ∗Hw2φH〉| ≤ R2, where R2 = conts.

We now come to w3. We notice that

α−faf + afα−f =(−vfa

†f

L

C+ ufa−f

)af

+ af

(−vfa

†f

L

C+ ufa−f

)= −vf

C(a†fLaf + afa

†fL)

= −vf

C(a†faf + afa

†f)L = −vf

CL.

Therefore [cf. the expression for w3]:

∆ ≡∑

f

Ω(f)v2f

−vf

L†

C

(C2 − L†LC2

)(α−faf + afα−f) − v2

f

C2 + L†LC2

=∑

f

Ω(f)v2f

v2

f

L†

C

(C2 − L†LC2

)LC

− v2f

C2 + L†LC2

=∑

f

Ω(f)v4f

L†

C2

(LL† + L†LC2

)L−

∑f

Ω(f)v4f

(C2 − L†LC2

)2

and further [cf. (A.18)]

〈φ†H∆φH〉 ≤

∑f

Ω(f)v4f

⟨φ∗

H

L†

C

(LL† − L†LC2

)LCφH

⟩

=2

V 2C2

∑f,f ′

Ω(f)v4fλ

2(f ′)⟨φ∗

H

L†

C(1 − a†f ′af ′ − a†−f ′a−f ′)

L

CφH

⟩

≤ 2|L|2C4

1

V

∑f

Ω(f)v4f

1

V

∑f ′λ2(f ′) ≤ const.

We also have

∑f

Ω(f)|vf |3⟨φ∗

H

L†

C

(L†LC2

α−f − α−fL†LC2

)afφH

⟩≤ const

∑f

Ω(f)|vf |3⟨φ∗

H

L†

C

(C2 − L†LC2

)afα−fφH

⟩


214 N. N. BOGOLUBOV

≤ R3

√∑f

Ω(f)〈φ∗Hα

†fαfφH〉.

Thus, collecting the expressions for w1, w2, and w3, we find:

〈φ∗HwφH〉 ≤ γ1

√〈φ∗

H

∑f

Ω(f)α†fαfφH〉 + γ2,

whereγ1 = const, γ2 = const.

Substituting this inequality in (6.5), we obtain

〈φ∗H

∑f

Ω(f)α†fαfφH〉 ≤ γ1

√〈φ∗

H

∑f

Ω(f)α†fαfφH〉 + γ2.

If we put

x = 〈φ∗H

∑f

Ω(f)α†fαfφH〉,

then

x2 − γ1x ≤ γ2,(x− γ1

2

)2

≤ γ2 +γ2

1

4,

and so

x ≤ γ1

2+

√γ2 +

γ21

4.

Thus, ⟨φH

1

V

∑f

Ω(f)α†fαfφH

⟩≤ R

V, (6.9)

where

R =(γ1

2+

√γ2 +

γ21

4

)2

= const.

We may now turn to the equation of motion. For ν = 0 we obtain fromequations (5.1), (5.2),

idaf

dt= T (f)af − λ(f)a†−fL

ida−f

dt= T (f)a−f − λ(f)a†fL. (6.10)



Therefore

idL

dt=

1

V

∑λ(f)T (f)a−f + λ(f)a†fLaf

+1

V

∑λ(f)a−fT (f)af + λ(f)a†−fL

=2

V

∑λ(f)T (f)a−faf +

1

V

∑λ2(f)(a†faf + a−fa

†−f)L

=2

V


1

V

∑λ2(f)(a†faf + afa

†f )L

=2

V


1

V

∑λ2(f)(2a†faf − 1)L.

We now notice that

−2λ(f)T (f)ufvfL

C+ λ2(f)(2v2

f − 1)L

= λ2(f)T (f)

Ω(f)L+ λ2(f)

(1 − T (f)

Ω(f)− 1)L = 0.

Consequently,

idL

dt= D1 +D2

D1 =2

V

∑λ(f)T (f)

a−faf + ufvf

L

C

D2 =

2

V

∑λ2(f)(a†faf − v2

f )L. (6.11)

Now, from (6.7), we have

a−faf + ufvfL

C=(ufα−f + vfα

†f

L

C

)(ufαf − vfα

†−f

L

C

)+ η−faf + a−fηf − η−fηf + ufvf

L

C

=u2fα−fαf − v2

fα†f

L

Cα†−f

L

C− ufvfα−fα

†f

L

C

+ ufvfα†f

L

Cαf + η−faf + a−fηf − η−fηf + ufvf

L

C

=u2fα−fαf − ufvf(α−fα

†−f + α†

−fα−f − 1)L

C


216 N. N. BOGOLUBOV

− v2fα

†f

L

Cα†−f

L

C+ ufvf

(α†

f

L

Cαf + α†

−fα−fL

C

)+ η−faf + a−fηf − η−fηf . (6.12)

We also have

α−fα†−f + α†

−fα−f − 1 =(−vfa

†f

L

C+ ufa−f

)(−vf

L†

Caf + ufa

†−f

)+(−vf

L†

Caf + ufa

†−f

)(−vfa

†f

L

C+ ufa−f

)− 1

= v2fa

†f

LL†

C2af + u2

fa−fa†−f − ufvfa

†f

L

Ca†−f − ufvfa−f

L†

Caf

+ v2f

L†

Cafa

†f

L

C+ u2

fa†−fa−f − ufvfa

†−fa

†f

L

C− ufvf

L†

Cafa−f − 1

= v2fa

†f

LL† − L†LC2

af + v2f

L†LC2

+ u2f − 1 − ufvfa

†f

(LCa†−f − a†−f

L

C

)− ufvf

(a−f

L†

C− L†

Ca−f

)af

and therefore

α−fα†−f + α†

−fα−f − 1 =2

V 2

∑g

v2fa

†f

λ2(g)

C2(1 − a†gag − a†−ga−g)af

+ v2f

L†L− C2

C2+ ufvfa

†faf

2λ(f)

V+ ufvfa

†faf

2λ(f)

V. (6.13)

As a result we obtain

〈φ∗HD1D

†1φH〉 =

2

V

∑f

λ(f)T (f)u2f〈φ∗

Hα−fαfD†1φH〉

− 2

V

∑f

λ(f)T (f)⟨φ∗

Hα†f

v2

f

L

Cα†−f

L

C− ufvf

(LCαf + αf

L

C

)D†

1φH

⟩

− 4

V 3

∑f, g

λ(f)T (f)ufv3f

λ2(g)

C2

⟨φ∗

Ha†f(1 − a†gag − a†−ga−g)af

L

CD†

1φH

⟩

+2

V

∑f

λ(f)T (f)〈φ∗H(η−faf − ηfa−f + [a−fηf + ηfa−f ] − η−fηf)D

†1φH〉

− 2

V

∑f

λ(f)T (f)⟨φ∗

H

v2

f

L†L− C2

C2+ 2ufvfa

†faf

2λ(f)

V

LCD†

1φH

⟩.




〈φ∗Hα−fαfD

†1φH〉 =〈φ∗

Hα−fD†1αfφH〉

+ 〈φ∗Hα−f(αfD

†1 −D†

1αf)φH〉,we can now use (6.8) and (6.9) to establish that

〈φ∗HD1D

†1φH〉 ≤ Γ1

V, Γ1 = const. (6.14)

In the same way we obtain

〈φ∗HD

†1D1φH〉 ≤ Γ2

V, Γ2 = const. (6.15)

We now turn to the expression D2. We have

a†faf − v2f = a†fηf + η†faf − η†fηf

+(ufα

†f − vf

L†

Cα−f

)(ufαf − vfα

†−f

L

C

)− v2

f

=u2fα

†fαf + v2

f

L†

Cα−fα

†−f

L

C− v2

f − ufvfα†−fα

†−f

L

C

− ufvfL†

Cα−fαf + a†fηf + η†faf + η†fηf

=u2fα

†fαf + v2

f

L†

C(α−fα

†−f + α†

−fα−f − 1)L

C− v2

f

C2 − L†LC2

− v2f

L†

Cα−fα

†−f

L

C− ufvfα

†fα

†−f

L

C− ufvf

L

Cα−fαf

+ α†fηf + η†faf − η†fηf . (6.16)

From this relation and the inequalities (6.8) and (6.9), we can show that

〈φ∗HD2D

†2φH〉 ≤ Γ3

V, Γ3 = const

〈φ∗HD

†2D2φH〉 ≤ Γ3

V. (6.17)

It follows from (6.11) that

⟨φ∗

H

(dLdt

)†dLdtφH

⟩≤ Γ

V; Γ = const


218 N. N. BOGOLUBOV

⟨φ∗H

dL

dt

(dLdt

)†φH

⟩≤ Γ

V. (6.18)

Let us go back again to the equations of motion (6.10). Using (6.1)and (6.2), we have

idα†

f

dt=i

d

dt

(ufa

†f + vf

L†

Ca−f

)= uf i

da†fdt

+ vfL†

Cida−f

dt

+ vf idL†

dt

a−f

C= uf−T (f)a†f + λ(f)L†a−f

+ vf

L†

CT (f)a−f + λ(f)a†fL + vf i

dL†

dt

a−f

C

= − a†fT (f)uf − λ(f)vf

L†LC

+L†

Cufλ(f)C + T (f)vfa−f

+ vf idL†

dt

a−f

C

= − a†fT (f)uf − λ(f)vfC +L†

Cufλ(f)C + T (f)vfa−f

− a†fλ(f)vfC2 − L†L

C+ vf i

dL†

dt

a−f

C.

However, [cf. (5.6)]

ufλ(f)C + T (f)vf = Ω(f)vf

T (f)uf − λ(f)vfC = Ω(f)uf (6.19)

and so

idα†

f

dt+ Ω(f)α†

f = Rf , (6.20)

where

Rf = −a†fλ(f)vfC2 − L†L

C+ vf (D

†1 +D†

2)a−f

C.

We now have

〈φ†HR

†fRfφ〉 ≤2

⟨φ†

H

C2 − L†LC

afa†f

C2 − L†LC

φH

⟩λ2(f)v2

f

+ 2⟨φ∗

H

a†−f

C(D1 +D2)(D

†1 +D†

2)a−f

CφH

⟩v2

f

≤2λ2(f)v2f

⟨φ∗

H

(C2 − L†L)2

C2φH

⟩



+ 2v2f

⟨φ∗

H

a†−f

C(D1 +D2)(D

†1 +D†

2)a−f

C

− (D1 +D2)a†−fa−f

C2(D†

1 +D†2)φH

⟩+ 2〈φ∗

H(D1 +D2)(D†1 +D†

2)φH〉v2

f

C2

and also

〈φ†HRfR

†fφ〉 ≤ 2

⟨φ∗

Ha†f

(C2 − L†L)

C2

)2

afφH

⟩λ2(f)v2

f +2v2

f

C2

× 〈φ∗H(D†

1 +D†2)(D1 +D2)φH

⟩= 2λ2(f)v2

f

⟨φ∗

H

a†f(C2 − L†L

C

)×(C2 − L†L

C

)af −

(C2 − L†LC

)a†faf

(C2 − L†LC

)φH

⟩+ 2λ2(f)v2

f

⟨φ∗

H

(C2 − L†LC2

)2

φH

⟩+

2v2f

C2〈φ∗

H(D†1 +D†

2)(D1 +D2)φH

⟩.

Hence, it follows that

〈φ†HRfR

†fφ〉 ≤ v2

f

S

V

〈φ†HR

†fRfφ〉 ≤ v2

f

S

V, where S = const. (6.21)

Once we have established equation (6.20) and the inequality (6.21), wecan repeat word for word the arguments used in the preceding section inconnection with the case ν > 0. We now obtain

〈φ∗Hα

†HαHφH〉 ≤ S

V

v2f

2πΩ2(f)

( ∞∫

−∞

|h(τ)τ | dτ)2

. (6.21′)

This inequality is a considerable improvement on (6.9); the latter merelyshowed that the average with respect to f of the sum of quantities〈φ∗

Hα†HαHφH〉 is of order 1/V , whereas (6.21′) shows this quantity is itself of

order 1/V for each value of f separately.


220 N. N. BOGOLUBOV

From (6.21′) we can immediately obtain limits for the single-timeaverages. Let Uf represent either af or a†f , and consider those operatorsof the form

Uf1Uf2 . . .UfK,

which conserve particle number. We shall prove that

∣∣〈Uf1Uf2 . . .UfK〉H − 〈Uf1Uf2 . . .UfK

〉H0

∣∣ ≤ const√V. (6.22)

We begin by noticing that φH and φH0 satisfy the condition (2.3)

(a†faf − a†−fa−f )φ = 0.

Therefore,〈Uf1Uf2 . . .UfK

〉may be written as a sum of terms of the type

〈. . . a†faf . . . a†ga

†−g . . . a−hah . . .〉,

where the indices ±f, ±g, ±h, . . . are all different. The number of indices gmust of course be equal to the number of indices h. Now it is obvious that

〈. . . a†faf . . . a†ga

†−g . . . a−hah . . .〉H0 =

∏f

v2f

∏g

(−ugvg)∏h

(−uhvh),

so that we need to only prove that

∣∣〈. . . a†faf . . . a†ga

†−g . . . a−hah . . .〉H

∣∣−∏f

v2f

∏g

(−ugvg)∏h

(−uhvh) ≤ const√V.

(6.23)For the results proved above [see (6.12), (6.13), (6.16)] we have

a−hah + uhvh

L

C= u2

hα−hαh − uhvh

2

V 2

∑f

v2ha

†h

λ2(f)

C2(1 − a†faf − a†−fa−f )

× ah + v2h

L†L− C2

C2+ uhvha

†hah

4λ(f)

V

LC

− v2hα

†h

L

Ca†−h

L

C

+ uhvh

(α†

h

L

Cαh + α†

−hα−hL

C

)+ η−hah + a−hηh − η−hηh (6.24)



and

a†faf − v2f = u2

fα†fαf − v2

f

L†

Cα†−fα−f

L

C− ufvfα

†fα

†−f

L

C

− ufvfL†

Cα−fαf + v2

f

L†

C

2

V 2

∑g

v2fa

†f

λ2(g)

C2(1 − a†gag − a†−ga−g)af

+ v2f

L†L− C2

C2+ ufvf

4λ(g)

Va†faf

+ a†fηf + η†faf − η†fηf . (6.25)

We now transfer each operator α† to the left-hand end of the term in whichit occurs, each α to the right-hand end and each operator (L†L − C2)/C2

(occurring for instance, in η or η†) to the one end or the other - exactly whichend is unimportant in this case. Since all the indices f, g are different thecommutators arising from these permutations will be quantities of order 1/V .Then, invoking once again the inequality

|〈AB〉| ≤√〈AA†〉〈B†B〉

we see that as soon as an operator α† find itself adjacent to the left-hand end,an α to the right-hand end or an (L†L − C2)/C2 to either, we immediatelyhave a quantity of order of magnitude 1/

√V at most. Consequently∣∣∣〈. . . a†faf . . . a

†ga

†−g . . . a−hah . . .〉H

−∏

f

v2f

⟨. . . (−ugvg)

L†

C(−uhvh)

L

C. . .⟩

H

∣∣∣ ≤ const√V. (6.26)

However, the number of indices g is equal to number of indices h, and thequantities L and L† may be permuted, to within an error of order 1/V .Therefore ⟨

. . . (−ugvg)L†

C(−uhvh)

L

C. . .⟩

H

differs from ∏g

ugvg

∏h

uhvh

⟨(L†LC2

)l⟩H

by terms of order 1/V . On the other hand the expression

⟨(L†LC2

)l⟩H


222 N. N. BOGOLUBOV

differs form unity by terms of order 1/√V at most. This therefore concludes

the proof of (6.3) and also of (6.22).We now turn to the double-time correlation functions. We shall prove the

following general inequality:∣∣〈Bf1(t) . . .Bfl(t); Ug1(τ) . . .Ugk

(τ)〉H− 〈Bf1(t) . . .Bfl

(t); Ug1(τ) . . .Ugk(τ)〉H0

∣∣ ≤ K(t− τ) +K1√V

K = const, K1 = const (6.27)

where the operators Bf , Ug may represents either a or a†. As always in thiscontext, we assume that the operator

Bf1 . . .Ugk

conserves particle number.By virtue of the supplementary condition (2.3) obeyed by φH and φH0,

rearrangement of these operators in the “correct” order allows us to reducethe averages being considered to a sum of terms of the type

〈 . . . a†f(t)af (t) . . . a†g(t)a

†−g(t) . . . a−h(t)ah(t) . . .

. . . a†k(t) . . . af(t) . . . a†g′(τ)af ′(τ) . . . a†g′(τ)a

†−g′(τ) . . .

. . . a−h′(τ)ah′(τ) . . . ak′(τ) . . . a†q′(τ) . . .〉, (6.28)

where the number of operators α and α† is the same, and the indices ±f ,±g, ±h, ±k, ±q are all different form one another, as are the indices ±f ′,±g′, ±h′, ±k′, ±q′. It follows that it is sufficient to prove (6.27) for averagesof the type (6.28).

We start by using for the pairs a†a, a†a†, aa formulae (6.24) and (6.25),and for single operators a, a† formulae (6.7). Next we transfer all operatorsα†(t) and L†(t)L(t)−C2 to the left end and all operators α(τ) and L†(τ)L(τ)−C2 to the right end of terms in which they occur. Since, as we noted above,all the indices are different and since we permute in this way only operatorswith the same time argument, all the commutators arising from this processwill be of order 1/V . Again, as soon as either an α†(t) or an L†(t)L(t) − C2

finds itself adjacent to the left-hand end, or an α(τ) and L†(τ)L(τ) − C2 tothe right-hand end, we immediately get a quantity of order 1/V at worst. It



therefore only remains to prove that an inequality of the type (6.23) holdsfor averages of the form

Γ(t− τ)

= 〈αf1(t) . . . αfl(t)Lk(t)L†q(t)L†q1(τ)Lk1(τ)α†

g1(τ) . . . α†

gr(τ)〉. (6.29)

We now use the equation of motion (6.11) and the inequalities givenby (6.18)-(6.21). We find

i∂ΓH(t− τ)

∂t− Ω(f1) + . . .+ Ω(fl)ΓH(t− τ) = ∆(t− τ)

with

|∆(t− τ)| ≤ G√V, where G = const.

Hence, since

ΓH(t− τ) = e−iΩ(f1)+...+Ω(fl)(t−τ)ΓH(0)

+ e−iΩ(f1)+...+Ω(fl)(t−τ)

t−τ∫

0

e−iΩ(f1)+...+Ω(fl)z∆(z) dz

we obtain

∣∣ΓH(t− τ) − e−iΩ(f1)+...+Ω(fl)(t−τ)ΓH(0)∣∣ ≤ G|t− τ |√

V. (6.30)

On the other hand

ΓH0(t− τ) = e−iΩ(f1)+...+Ω(fl)(t−τ)ΓH0(0) (6.31)

and therefore

ΓH0(t− τ) = 〈αf1(t) . . . αfl(t)α†

g1(τ) . . . α†

gr(τ)〉H0 C

k+q+q1+k1. (6.32)

Thus, we have∣∣ΓH(t− τ) − ΓH0(t− τ)∣∣

≤ ∣∣ΓH(0) − ΓH0(0)∣∣+ G|t− τ |√

V


224 N. N. BOGOLUBOV

=∣∣〈αf1 . . . αfl

Lk(L†)q+q1Lk1α†q1. . . α†

qr〉H

− Ck+k1+q+q1〈αf1 . . . αflα†

q1. . . α†

qr〉H0

∣∣ + G|t− τ |√V

. (6.33)

Suppose that two of indices, f1, . . . , fl, in (6.33) coincide. Then, that theexpression

α2f =(ufaf + vfa

†−f

L

C

)(ufaf + vfa

†−f

L

C

)=v2

fa†−f

L

Ca†−f

L

C+ ufvf

a†−f

L

Caf + afa

†−f

L

C

=v2

fa†−f

C(La†−f − a†−fL)L− ufvf (a†−faf + afa

†−f )

L

C

= − 2v2

fλ(f)

C2Va†−fafL (6.34)

is of order 1/V , we see that 〈. . .〉H will be of the same order, while thecorresponding average with respect to H0 is simply equal to zero. Thesame result of course also applies in the case where one or more pairs ofthe indices g1, . . . , gr are identical.

Next suppose that there is one (or more) index fj among the f ’s whichdoes not occur among the g’s. Then we can transfer αfj

to the right-hand endin the expression 〈. . .〉H , obtaining in the process commutators of order 1/V .We easily see that in this case the average with respect to H will be atmost of order 1/

√V . Again, the corresponding average with respect to H0 is

rigorously equal to zero. The situation is the same if there is one (or more)index among g’s which does not occur among the f ’s.

Thus, it only remains to consider the cases for which

1. All f1, . . . , fl are different, and

2. The set g1, . . . , gr is identical with the set f1, . . . , fl (the order ofenumeration being in general different).

Now we rearrange the operators on the right-hand side of (6.13) inthe “correct” order; that is we replace α†

g1. . . α†

grby α†

f1. . . α†

fl. This

rearrangement can of course be performed exactly for H0; for H it introducesan error which, as usual, is asymptotically small. Next we notice that sincethe operators inside the pointed brackets conserve particle number, and the



number of α’s and α†’s are equal, k + k1 must be equal to q + q1. We cantherefore write our expression as

〈αf1 . . . αflLk(L†)k+k1Lk1α†

fl. . . α†

f1〉

make the replacement Lk(L†)k+k1Lk1 → (L†L)k+k1 and transfer this factor tothe right-hand end; this process introduces an error of order 1/V . Finally,we notice that∣∣〈αf1 . . . αfl

α†fl. . . α†

f1(L†L)k+k1〉H

− 〈αf1 . . . αflα†

fl. . . α†

f1〉H0C

2(k+k1)∣∣ ≤ const√

V. (6.35)

Thus, we obtain from (6.33)

∣∣ΓH(t− τ) − ΓH0(t− τ)∣∣ ≤ G|t− τ |√

V+

K√V

+ C2(k+k1)∣∣〈αf1 . . . αfl

α†fl. . . α†

f1〉H − 〈αf1 . . . αfl

α†fl. . . α†

f1〉H0

∣∣. (6.36)

However, since all the f ’s are different,

〈αf1 . . . αflα†

fl. . . α†

f1〉H0

= 〈αf1α†f1〉H0〈αf2α

†f2〉H0 . . . 〈αfl

α†fl〉H0 = 1.

In the average with respect to H such a decomposition can also be made –not, of course, exactly in this case, but with an error of order 1/V . Thisconclude our proof of (6.27).

As in the case ν > 0, we could obtain analogous asymptotic limits for themultiple-time correlation function; we shall not do so here. The interestedreader is now in a position to carry out all the relevant calculations himself,along the patterns developed above. As in the case ν > 0 the inequalitiescan be sharpened from const/

√V to const/V by replacing the constant C in

the Hamiltonian H0 by the quantity

C1 =√〈L†L〉H

which, generally speaking, differs from C by a term of order 1/√V . We shall

not give the proof of this statement here.


226 N. N. BOGOLUBOV

Appendix A.

In this appendix we prove various relations used in the text.z The operatorsconsidered are assumed to be completely continuous, since all operatorsoccurring in the text are of this type.

Lemma I.Let the operator ξ satisfies the condition

|ξξ† − ξ†ξ| ≤ 2s

V, (A.1)

where s is a number, and let ε equal either +1 or −1. Then the followinginequality holds,

2

√ξ†ξ +

s

V− ∈(ξ + ξ†) ≥ 0. (A.2)

Proof. Assume the contrary; then there exists a normed function ϕ suchthat

2

√ξ†ξ +

s

V−∈(ξ + ξ†)

ϕ = −ρϕ,

where ρ > 0. Then we have

(2

√ξ†ξ +

s

V+ ρ)ϕ = ∈(ξ + ξ†)ϕ. (A.3)

Now we use the fact that ifAϕ = Bϕ and if the operatorsA, B are Hermitian,then

〈ϕ∗A2ϕ〉 = 〈ϕ∗B2ϕ〉. (A.4)

From (A.4) and (A.1) we have

⟨ϕ∗(2

√ξ†ξ +

s

V+ ρ)2

ϕ⟩

= 〈ϕ∗(ξ + ξ†)2ϕ〉= 2〈ϕ∗(ξξ† + ξ†ξ)ϕ〉 − 〈ϕ∗(ξ† − ξ)(ξ − ξ†)ϕ〉≤ 2〈ϕ∗(ξξ† + ξ†ξ)ϕ〉 ≤ 2

⟨ϕ∗(ξ†ξ +

2s

V+ ξ†ξ

)ϕ⟩

≤ 4⟨ϕ∗(ξ†ξ +

s

V

)ϕ⟩, (A.5)

zWe shall denote the norm of a functions as follows: ‖ϕ‖ =√〈ϕ∗ϕ〉; and the norm of

an operator U by |U | = sup‖U ϕ‖, where ‖ϕ‖ = 1.



which cannot be satisfied for ρ > 0. Thus, (A.2) is proved.Corollary. Interchanging ξ and ξ†, we also have

2

√ξξ† +

s

V− ∈(ξ + ξ†) ≥ 0. (A.6)

Similarly we can prove the inequalities:

2

√ξξ† +

s

V+ i∈(ξ − ξ†) ≥ 0. (A.7)

2

√ξ†ξ +

s

V+ i∈(ξ − ξ†) ≥ 0. (A.8)

Lemma II.Let ξ satisfy the condition

|ξξ† − ξ†ξ| ≤ 2s

V. (A.9)

Then √ξξ† +

2s

V+ A2 −

√ξ†ξ + A2 ≥ 0, (A.10)

where A is a real c-number.Proof. Assume the contrary; then there exists a normed function ϕ such

that √ξξ† +

2s

V+ A2 −

√ξ†ξ + A2

ϕ = −ρϕ. (A.11)

Hence √ξξ† +

2s

V+ A2 + ρ

ϕ =

√ξ†ξ + A2ϕ. (A.12)

Using (A.4) we therefore get

⟨ϕ∗(√

ξξ† +2s

V+ A2 + ρ

)2

ϕ⟩

= 〈ϕ∗(ξ†ξ + A2)ϕ〉

≤⟨φ∗(ξξ† +

2s

V+ A2

)ϕ⟩, (A.13)

which cannot be satisfied for ρ > 0.


228 N. N. BOGOLUBOV

Corollary. Interchanging the operators ξ and ξ†, we get√ξ†ξ +

2s

V+ A2 −

√ξξ† + A2 ≥ 0. (A.14)

Also, if α and λ are real c-numbers, we have√λ2(ξξ† +

2s

V+ α2

)+ A2 −

√λ(ξ†ξ + α2) + A2 ≥ 0, (A.15)

√λ2(ξ†ξ +

2s

V+ α2

)+ A2 −

√λ(ξξ† + α2) + A2 ≥ 0. (A.16)

Note to Lemma II.Put

ξ =1

V

∑f

λ(f)a−faf + ν ≡ L+ ν. (A.17)

Then,

ξξ† − ξ†ξ =2

V 2

∑f

λ2(f)(1 − a†faf − a†−fa−f ). (A.18)

Suppose λ(f) satisfies the condition

1

V

∑f

λ2(f) ≤ s.

Then

|ξξ† − ξ†ξ| ≤ 2s

V,

and so √λ2(f)

(L+ ν)(L† + ν) + α2 +

2s

V

+ T 2(f)

−√λ2(f)(L+ ν)(L† + ν) + α2 + T 2(f) > 0. (A.19)

Lemma III (generalization of lemma II).Again let

|ξξ† − ξ†ξ| ≤ 2s

V.



Consider operators U , U † with norm

|U | ≤ 1, |U †| ≤ 1,

such that

|U ξ†ξU † − ξ†U U †ξ| ≤ 2l

V. (A.20)

Then

2

√ξξ† +

s+ l

V−∈(ξU † − U ξdag) ≥ 0, (A.21)

where ∈ is equal either to +1 or to −1.Proof. Assume the contrary. Then there exists a normed function ϕ such

that

2

√ξξ† +

s+ l

V−∈(ξU † − U ξdag)

ϕ = −ρϕ, ρ > 0. (A.22)

Hence, (2

√ξξ† +

s+ l

V+ ρ)

= ∈(ξU † − U ξ†)ϕ. (A.23)

According to 2-A.4, it follows that

⟨ϕ∗(2

√ξξ† +

s+ l

V+ ρ)2

ϕ⟩

= 〈ϕ∗(ξU † − U ξ†)2ϕ〉= 2〈ϕ∗ξU †U ξ† + U ξ†ξU ϕ〉 − 〈ϕ∗(ξU † − U ξ)(U ξ† − ξU †)ϕ〉≤ 2〈ϕ∗ξU †U ξ† + U ξ†ξU ϕ〉. (A.24)

However, since by hypothesis |U | ≤ 1, |U †| ≤ 1, we have |U †U | ≤, andconsequently

〈ϕ∗ξU †U ξ†〉 ≤ 〈ϕ†ξξ†ϕ〉. (A.25)

From (A.20) and (A.25) we obtain

〈ϕ∗U ξ†ξU †ϕ〉 = 〈ϕ∗ξ†U U †ξϕ〉 + 〈ϕ∗U ξ†ξU † − ξ†U U †ξϕ〉≤ 〈ϕ∗ξ†U U †ξϕ〉 +

2l

V≤ 〈ϕ∗ξ†ξϕ〉 +

2l

V

≤ 〈ϕ∗ξξ†ϕ〉 +2(l + s)

V=⟨ϕ∗(ξξ† +

2(l + s)

V

)ϕ⟩. (A.26)


230 N. N. BOGOLUBOV

Thus, using (A.24), we can write

⟨ϕ∗(2

√ξξ† +

s+ l

V+ ρ)2

ϕ⟩≤ 4⟨ϕ∗(ξξ† +

(l + s)

V

)ϕ⟩. (A.27)

However, such an inequality is impossible for ρ > 0, which proves therelation (A.21)

Note to Lemma III.

Put

ξ = L+ ν; U = ag.

Then

|U ξ†ξU † − ξ†U U †ξ| = |ag(L† + ν)(L+ ν)a†g − (L† + ν)aga

†g(L+ ν)|

= |ag(L† + ν)(L+ ν)a†g − (L† + ν)ag(L+ ν)a†g

+ (L† + ν)ag(L+ ν)a†g − (L† + ν)aga†g(L+ ν)|

≤ (|L| + ν)|La†g − a†gL| + |agL† − L†ag|

≤ (|L| + ν)4

V|λ(g)|,

where [cf. the definition (A.17)]

|L| ≤ 1

V

∑f

|λ(f)|,

since |af | ≥ 1. Hence, by (A.21),

2

√(L+ ν)(L† + ν) +

1

Vs+ (|L| + ν)2|λ(g)|

− ∈(L+ ν)a†g + ag(L† + ν) ≥ 0. (A.28)

Putting U = iag, we also obtain

2

√(L+ ν)(L† + ν) +

1

Vs+ (|L| + ν)2|λ(g)|

+ i∈(L+ ν)a†g − ag(L† + ν) ≥ 0. (A.29)



Lemma IV.

Let β be a real c-number; let α2 = β2 +2s

Vand let ν ≥ 0. Then

√(L+ ν)(L† + ν) + α2

λ2(f) + T 2(f) af

− af

√(L+ ν)(L† + ν) + α2

λ2(f) + T 2(f) ≤ const

V. (A.30)

The same inequality holds with af replaced by a†f .Proof. Consider an arbitrary normed function ϕ and form the expression

〈ϕ∗√(Q+ α2)λ2(f) + T 2(f)(af + a†f ) − (af + a†f )

×√

(Q+ α2)λ2(f) + T 2(f)ϕ〉 = E , (A.31)

whereQ = (L+ ν)(L† + ν).

To examine the expression (A.31) we use the following identity,

√z −√

z0 =1

π

∞∫

0

1

z0 + ω− 1

z + ω

√ω dω,

where z0 is an arbitrary positive number. We also observe that if A and Bare operators,

− 1

AB +B

1

A=

1

A(AB −BA)

1

A.

Thus, we have

E =1

π

∞∫

0

⟨ϕ∗ λ2(f)

(Q+ α2)λ2(f) + T 2(f) + ωQ(af + a†f) − (af + a†f )Q

× 1

(Q+ α2)λ2(f) + T 2(f) + ωϕ⟩√

ω dω.

However,

Qaf − afQ = (L+ ν)L†af − afL†,

L† =1

V

∑f

λ(f)a†fa†−f , L†af − afL

† = − 2

Vλ(f)a†−f ,


232 N. N. BOGOLUBOV

and therefore

Q(af + a†f ) − (af + a†f )Q

= − 2

Vλ(f)(L+ ν)a†−f +

2

Vλ(f)a−f(L

† + ν).

Thus, we find

|E | =∣∣∣Ei

∣∣∣ =2|λ(f)|3

π

×∣∣∣∣∣∞∫

0

⟨ϕ∗ 1

(Q+ α2)λ2(f) + T 2(f) + ω

(L+ ν)a†−f − af(L† + ν)

i

× 1

(Q+ α2)λ2(f) + T 2(f) + ωϕ⟩√

ω dω

∣∣∣∣∣.Using (A.29) and changing the variable of integration, we obtain

|E | ≤ 4|λ(f)|3πV

∞∫

0

⟨ϕ∗

√Q+

1

V(s+ 2|λ(f)|)(|L| + ν)(

Q+ α2 +T 2(f)

λ2(f)+ τ)2

ϕ

⟩√τ dτ

However, by definition, α2 = β2 + 2sV

, therefore

√Q+

1

V(s+ 2|λ(f)|)(|L| + ν)√

Q+ α2 +T 2(f)

λ2(f)+

2|λ(f)|(|L|+ ν)

V

=

√Q+ α2 +

T 2(f)

λ2(f)

√√√√√1 +2|λ(f)|(|L|+ ν)

V Q+ V α2 + VT 2(f)

λ2(f)

<

√Q+ α2 +

T 2(f)

λ2(f)

√√√√√1 +|λ(f)|(|L| + ν)

s +1

2VT 2(f)

λ2(f)



<

1 +

|λ(f)|(|L| + ν)

2s+ VT 2(f)

λ2(f)

√Q+ α2 +

T 2(f)

λ2(f).

Put

Λ = Q+ α2 +T 2(f)

λ2(f)≥ α2.

Then

|E | ≤ 4|λ(f)|3πV

1 +

|λ(f)|(|L| + ν)

2s+ VT 2(f)

λ2(f)

∞∫

0

⟨ϕ∗

√Λ

(Λ + τ)2ϕ⟩√

τ dτ.

We now expand the function ϕ in terms of the eigenfunctions of theoperator Λ:

ϕ =∑

CΛϕΛ;∑

|CΛ|2 = 1.

We then obtain

∞∫

0

⟨ϕ∗

√Λ

(Λ + τ)2ϕ⟩√

τ dτ =∑

Λ

|CΛ|2∞∫

0

√Λτ dτ

(Λ + τ)2

=∑

Λ

|CΛ|2∞∫

0

√t dτ

(1 + t)2=

∞∫

0

√t dτ

(1 + t)2.

Thus, for an arbitrary normed function, ϕ,

|E | =∣∣∣⟨ϕ∗

[√(Q+ α2)λ2 + T 2; af + a†fi

]ϕ⟩∣∣∣ ≤ Sf ,

where

Sf =4|λ(f)|3πV

1 +

|λ(f)|(∑ |λ(f)| 1

V+ ν)

21

V

∑ |λ(f)|2 + VT 2(f)

λ2(f)

∞∫

0

√t dτ

(1 + t)2.


234 N. N. BOGOLUBOV

However, the operator

[√(Q+ α2)λ2 + T 2; af + a†fi

]is Hermitian and consequently,∣∣[√(Q+ α2)λ2 + T 2; af + a†f ]

∣∣ ≤ Sf .

In an entirely analogous way we can show that∣∣[√(Q+ α2)λ2 + T 2; af − a†f ]∣∣ ≤ Sf .

Since,|U | + |B| ≥ |U + B|,

it follows that ∣∣[√(Q+ α2)λ2 + T 2; af ]∣∣ ≤ Sf .

Since we also have Sf ≤ const/V , this proves the required lemma. (Thesecond half of the lemma follows from the obvious fact that |U | ≤ Sf thenalso |U †| ≤ Sf .)

Appendix B.

The Principle of Extinction of Correlations

In our lectures ”The principle of extinction of correlations and the quasi-averages method” we formulated a principle of extinction of correlationsbetween particles for system in a state of statistical equilibrium. Theprinciple may be formulated as follows:

If Us(xs, ts) represents either the field operator ψ(xs, ts) or its adjointψ†(xs, ts), then the correlation functions

〈U1(x1, t1) . . .Us(xs, ts) . . .Un(xn, tn)〉, (B.1)

may be decomposed into the product of correlation functions

〈U1(x1, t1) . . .Us−1(xs−1, ts−1s)〉〈Us+1(xs+1, ts+1) . . .Un(xn, tn)〉 (B.2)

provided the set of points x1, . . . , xs is infinitely distant from the set of pointsxs+1, . . . , xn (the times t1, . . . , ts, . . . , tn are assumed fixed). In cases where



the number of creation and annihilation operators inside the brackets is notequal, the averages 〈. . .〉 must be understood as quasi-averages.

The system described by our model Hamiltonian is one of the few forwhich the principle of extinction of correlation may be demonstrated bydirect calculation. Below we prove this on the basis of the asymptotic limitsderived in the text.

Consider the ”vacuum expectation values” of the field operators in thecoordinate representation,

ψ−(t, x) =1√V

∑(f<0)

af(t) ei(fx)

ψ+(t, x) =1√V

∑(f>0)

a†f(t) e−i(fx). (B.3)

Here, f represents both momentum and spin indices (k, σ); the sums f < 0,f > 0 mean summation over all f with σ fixed (σ = ±), and (fx) = (k · r).For instance, we have,

〈ψσ1(t, x)ψ†σ2

(t, x′)〉H0 =1

V

∑(f>0)

|uf |2 eif(x−x′) δ(σ1 − σ2)

= 1

V

∑(f>0)

eif(x−x′) − 1

V

∑(f>0)

|vf |2 eif(x−x′)δ(σ1 − σ2) (B.4)

where uf and vf are the coefficients of the canonical transformation.Obviously the term

1

V

∑(f>0)

|vf |2 eif(x−x′)

goes over in the limit V → ∞ to the integral

1

(2π)3

∫|vf |2 eif(x−x′) dk.

This integral is absolutely convergent, since

∫|v2

f |2 dk =1

2

∫ √T 2(f) + λ2(f)C2 − T (f)2

T 2(f) + λ2(f)C2dk <∞.


236 N. N. BOGOLUBOV

As for the expression1

V

∑(f>0)

|uf |2 eif(x−x′)

we may similarly say that in the limit V → ∞ it goes over to the delta-function

1

(2π)3

∫eif(x−x′) dk.

However, we must of course understand the words ”limit” and ”convergenceof a function” in a rather different sense; in fact, that appropriate to thetheory of generalized functions. We shall digress for a moment to recall themeaning of the relation

fV (x1, . . . , xe) −−−→V →∞

f(x1, . . . , xe) (B.5)

or equivalentlyf(x1, . . . , xe) = lim

V →∞fV (x1, . . . , xe)

in that theory.Consider the class C(q, r) (where q and r are positive integers) of

continuous and infinitely differentiable functions h(x1, . . . , xe) such that forthe entire space Ee of the point (x1, . . . , xe) the following relations are fulfilled

|x1| + . . .+ |xe|α|h(x1, . . . , xe)| ≤ conts

α = 0, 1, . . . , r

|x1| + . . .+ |xe|α∣∣∣ ∂s1+...+seh

∂xs11 . . . ∂xse

e

∣∣∣ ≤ const

α = 0, 1, . . . , r; s1 + . . .+ se = 0, 1, . . . , q.

Then, if we can find positive numbers q, r such that for every function h ofthe class C(q, r) we have

∫h(x1, . . . , xe)fV (x1, . . . , xe) dx1 . . . dxe →∫

h(x1, . . . , xe)f(x1, . . . , xe) dx1 . . . dxe

we shall say that the generalized limit relation (B.5) is fulfilled. As wesaw above, the averages of products of ψ(t, x) and ψ†(t, x) may contain



generalized functions; therefore we must understand the correspondingasymptotic relations (for the limit V → ∞) in the sense described above.

Consider the expression

〈ψσ1(t1, x1)ψ†σ2

(t2, x2)〉 =1

V

∑(f>0)

〈af(t1)a†f (t2)〉 eif(x−x′) δ(σ1 − σ2).

We have ∫h(x1 − x2)〈ψσ1(t1, x1)ψ

†σ2

(t2, x2)〉 dx1

=1

V

∑(f>0)

〈af(t1)a†f(t2)〉h(f)δ(σ1 − σ2),

where

h(f) =

∫h(x) ei(f ·x) dx.

By an appropriate choice of the indices q, r of the class C(q, r) to whichh(x) belongs, we can arrange that h(x) shall decrease faster than any desiredpower of |f |−1 in the limit |f | → ∞. For present purpose we need only ensurethat

1

V

∑f

|h(f)| ≤ K = const.

Then, noticing that according to (6.36)

⟨af (t1)a

†f(t2)〉H − 〈af (t1)a

†f (t2)〉H0

∣∣ ≤ s1|t1 − t2| + s2√V

,

where s1, s2 = const, we obtain∣∣∣∫ h(x1 − x2)〈ψσ1(t1, x1)ψ

†σ2

(t2, x2)〉H − 〈ψσ1(t1, x1)ψ†σ2

(t2, x2)〉H0

dx1

∣∣∣≤ 1

V

∑f

⟨af(t1)a

†f (t2)〉H − 〈af(t1)a

†f(t2)〉H0

∣∣ |h(f)|

≤ ks1|t1 − t2| + s2

V−−−→V →∞

0.

Accordingly the following generalized limit relation holds,

〈ψσ1(t1, x1)ψ†σ2

(t2, x2)〉H − 〈ψσ1(t1, x1)ψ†σ2

(t2, x2)〉H0 → 0. (B.6)


238 N. N. BOGOLUBOV

We can see by direct calculations that

〈ψσ1(t1, x1)ψ†σ2

(t2, x2)〉H0 =1

V

∑(f>0)

|uf |2 e−iΩ(f)(t1−t2)+if(x1−x2) δ(σ1 − σ2)

and hence, it is also true in the generalized sense that

〈ψσ1(t1, x1)ψ†σ2

(t2, x2)〉H−

∫|uf |2 exp−iΩ(f)(t1 − t2) + if(x1 − x2) dkσ(σ1 − σ2) −−−→

V →∞0. (B.7)

From (B.6) and (B.7) we finally get

limV →∞

〈ψσ1(t1, x1)ψ†σ2

(t2, x2)〉H

=

∫|uf |2 exp−iΩ(f)(t1 − t2) + if(x1 − x2) dkδ(σ1 − σ2)

= ∆(t1 − t2, x1 − x2) − F (t1 − t2, x1 − x2)δ(σ1 − σ2), (B.8)

where

∆(t, x) =

∫e−Ω(f)t+ifx dk

F (t, x) =

∫|vf |2 e−Ω(f)t+ifx dk. (B.9)

In an entirely analogous way we obtainaa

limV →∞

〈ψ†σ2

(t2, x2)ψσ1(t1, x1)〉H = F (t2 − t1, x1 − x2)δ(σ1 − σ2). (B.10)

Now consider the two-particle expressions

〈ψ(t1, x1)ψ(t2, x2)ψ†(t′2, x

′2)ψ

†(t′1, x′1)〉.

We have

〈ψ(t1, x1)ψ(t2, x2)ψ†(t′2, x

′2)ψ

†(t′1, x′1)〉

=1

V 2

∑〈af1(t1)af2(t2)a

†g2

(t′2)a†g1

(t′1)〉aaThis relation is also true in the generalized sense, owing to the absolute convergence

of the integral defining F (t, x).



× expif1x1 + if2x2 − ig2x′2 − ig1x

′1. (B.11)

Since the total momentum is conserved, and is equal to zero for φH (andφH0) we see that the expressions

〈af1(t1)af2(t2)a†g2

(t′2)a†g1

(t′1)〉 (B.12)

can be different form zero only if

f1 + f2 = g2 + g1. (B.13)

We now recall that by (2.1) and (2.2) the quantity nf(t) − n−f (t) (where

nf = a†faf) is a constant of the motion and that φH0 (and φH) satisfy theadditional condition

(nf − n−f )φ = 0.

Finally, we notice that

(nf − n−f )ah = ah(nf − n−f) + δ(f − h) + δ(f + h).

As a result we have, for arbitrary f ,

〈af1(t1)af2(t2)a†g2

(t′2)a†g1

(t′1)〉= 〈1 + nf − n−faf1(t1)af2(t2)a

†g2

(t′2)a†g1

(t′1)〉= 〈1 + nf (t1) − n−f (t1)af1(t1)af2(t2)a

†g2

(t′2)a†g1

(t′1)〉= 〈af1(t1)1 + nf(t1) − n−f(t1) − δ(f − f1) + δ(f + f1)

× af2(t2)a†g2

(t′2)a†g1

(t′1)〉= 〈af1(t1)1 + nf(t2) − n−f(t1) − δ(f − f1) + δ(f + f1)

× af2(t2)a†g2

(t′2)a†g1

(t′1)〉= 〈af1(t1)af2(t2)1 + nf (t2) − n−f (t1) − δ(f − f1) + δ(f + f1)

− δ(f − f2) + δ(f + f2)a†g2(t′2)a

†g1

(t′1)〉 = . . .

= 〈af1(t1)af2(t2)a†g2

(t′2)a†g1

(t′1)

× 1 + nf(t2) − n−f(t1) − δ(f − f1) + δ(f + f1) + δ(f + f2)

− δ(f − f2) + δ(f − g2) − δ(f + g2) + δ(f − g1) − δ(f + g1)〉= 1 − δ(f − f1) + δ(f + f1) − δ(f − f2) + δ(f + f2)

+ δ(f − g2) − δ(f + g2) + δ(f − g1) − δ(f + g1)


240 N. N. BOGOLUBOV

× 〈af1(t1)af2(t2)a†g2

(t′2)a†g1

(t′1)〉.This identity shows that the quantities (B.12) can be different from zero

if, for arbitrary f , the following relation is satisfied,

−δ(f − f1) + δ(f + f1) − δ(f − f2) + δ(f + f2)

+ δ(f − g2) − δ(f + g2) + δ(f − g1) − δ(f + g1) = 0.

This relation can be fulfilled simultaneously with (B.13) only in the followingcases:

f1 + f2 = 0; g1 + g2 = 0 (B.14)

f1 = g1; f2 = g2 (B.15)

f1 = g2; f2 = g1. (B.16)

Moreover, in the cases (B.15) and (B.16) we can always assume that g1 = g2,since

a†g(t′2)a

†g(t

′1)φH = 0. (B.17)

This last relation follows from the fact that

(ng − n−g)a†g(t

′2)a

†g(t

′1)φH = a†g(t

′2)a

†g(t

′1)(ng − n−g + 2)φH =

= 2a†g(t′2)a

†g(t

′1)φH .

Since the only possible eigenstates of ng − n−g are ±1 and 0, this relationcan only be fulfilled by the satisfaction (B.17).

Thus, we can reduce (B.11) to the form

〈ψ(t1, x1)ψ(t2, x2)ψ†(t′2, x

′2)ψ

†(t′1, x′1)〉

=∑f, g

1

V 2〈a−f (t1)af (t2)a

†g(t

′2)a

†−g(t

′1)〉 expif(x2 − x1) − ig(x′2 − x′1)

+∑f, g“f =g

f+g =0

”

1

V 2〈af (t1)ag(t2)a

†g(t

′2)a

†f (t

′1)〉 expif(x1 − x′2) + ig(x2 − x′2)

+∑f, g“f =g

f+g =0

”

1

V 2〈af (t1)ag(t2)a

†f (t

′2)a

†g(t

′1)〉 expif(x1 − x′2) + ig(x2 − x′1).

(B.18)



Now we turn to the limit V → ∞. We consider the class C(q, r) of functionsh(x, y) and fix q and r so that

1

V 2

∑f,g

|h(f, g)| ≤ const

where

h(f, g) =

∫h(x, y) ei(fx+gy) dx dy.

Since for fixed t1, t2, t′2, t

′1 we have [cf. (5.56)]

〈af(t1)ag(t2)a†f(t

′2)a

†g(t

′1)〉H

− 〈af(t1)ag(t2)a†f(t

′2)a

†g(t

′1)〉H0 ≤

const√V,

it follows that∣∣∣∫ h(x, y)ΓH(t1, t2, t′2, t

′1|x, y) − ΓH0(t1, t2, t

′2, t

′1|x, y)

∣∣∣≤ const√

V−−−→V →∞

0,

where

ΓH(t1, t2, t′2, t

′1|x, y)

=1

V 2

∑f, g“f =g

f+g =0

”〈af(t1)ag(t2)a

†f(t

′2)a

†g(t

′1)〉 ei(fx+gy).

Thus, we obtain the generalized limit relations

ΓH(t1, t2, t′2, t

′1, x1 − x′2, x2 − x′1)

− ΓH0(t1, t2, t′2, t

′1, x1 − x′2, x2 − x′1) −−−→

V →∞0.

However, a direct calculation, as in the case (B.4), shows that

ΓH0(t1, t2, t′2, t

′1, x1 − x′2, x2 − x′1)

= − 1

V 2

∑f, g“f =g

f+g =0

”|uf |2|ug|2e−iΩ(f)(t1−t′2)−iΩ(g)(t2−t′1) ei(f(x1−x′

2)+g(x2−x′1))


242 N. N. BOGOLUBOV

→ −∆(t1 − t′2, x1 − x′2) − F (t1 − t′2, x1 − x′2)× ∆(t2 − t′1, x2 − x′1) − F (t2 − t′1, x2 − x′1)δ(σ1 − σ′

2)δ(σ2 − σ′1)

(B.19)

where Ω(f) is defined by (6.2′) and ∆(t, x) and F (t, x) by (B.9).

Consequently

limV →∞

ΓH(t1, t2, t′2, t

′1, x1 − x′2, x2 − x′1)

= −∆(t1 − t′2, x1 − x′2) − F (t1 − t′2, x1 − x′2)× ∆(t2 − t′1, x2 − x′1) − F (t2 − t′1, x2 − x′1)δ(σ1 − σ′

2)δ(σ2 − σ′1).

We can deal with the other terms on the right-hand side of (B.18) in anexactly similar way. Now let us put

Φσ(t, x) = −∫ufvf e−iΩ(f)t−ifx dk

=

∫cλ(f)

2Ω(f)e−iΩ(f)t−ifx dk. (B.20)

Then, we can write the generalized limit relation in the form

limV →∞

〈ψσ1(t1, x1)ψσ2(t2, x2)ψ†σ′2(t′2, x

′2)ψ

†σ′1(t′1, x

′1)〉

= Φσ2(t1 − t2, x1 − x2)Φσ′2(t′2 − t′1, x

′2 − x′1)δ(σ1 + σ2)δ(σ

′1 + σ′

2)

+ δ(σ1 − σ′1)δ(σ2 − σ′

2)∆(t1 − t′1, x1 − x′1) − F (t1 − t′1, x1 − x′1)× ∆(t2 − t′2, x2 − x′2) − F (t2 − t′2, x2 − x′2)− δ(σ1 − σ′

2)δ(σ2 − σ′1)∆(t1 − t′2, x1 − x′2) − F (t1 − t′2, x1 − x′2)

× ∆(t2 − t′1, x2 − x′1) − F (t2 − t′1, x2 − x′1). (B.21)

By an entirely analogous procedure we can obtain formulae for other productsof the field operators ψ · ψ†.

We shall the example of (B.21) to illustrate the principle of extinction ofcorrelations. We need only observe that

F (t, x) → 0, |x| → ∞Φ(t, x) → 0, |x| → ∞ (B.22)



andbb

∆(t, x) → 0, |x| → ∞. (B.23)

Let us fix the times t1, t2, t′2, t

′1 and spatial differences

x1 − x′1, x2 − x′2

at some finite values. Now we let the remaining spatial differences

x1 − x2, x′1 − x′2, x1 − x′2, x2 − x′1

tend to infinity. Then the two-point function

limV →∞

〈ψσ1(t1, x1)ψσ2(t2, x2)ψ†σ′2(t′2, x

′2)ψ

†σ′1(t′1, x

′1)〉H (B.24)

will decompose into the product

∆(t1 − t′1, x1 − x′1) − F (t1 − t′1, x1 − x′1)× ∆(t2 − t′2, x2 − x′2) − F (t2 − t′2, x2 − x′2)δ(σ1 − σ′

1)δ(σ2 − σ′2)

which, by (B.8), is equal to

limV →∞

〈ψσ1(t1, x1)ψ†σ′1(t′1, x

′1)〉H lim

V →∞〈ψσ2(t2, x2)ψ

†σ′2(t′2, x

′2)〉H . (B.25)

Now we consider a second aspect of the extinction of correlations. Againwe fix the times t1, t2, t

′2, t

′1, and this time also the spatial differences

x1 − x2, x′1 − x′2.

Then we let the remaining spatial differences

x1 − x′1, x2 − x′2, x1 − x′2, x2 − x′1

tend to infinity. Then the function (B.24) decomposes into the product

Φ(t1 − t2, x1 − x2)Φ(t′2 − t′1, x′2 − x′1)Φσ2Φσ′

2δ(σ1 + σ2)δ(σ

′1 + σ′

2). (B.26)

For ν > 0,

Φσ(t1 − t2, x1 − x2) = limV →∞

〈ψ−σ(t1, x1)ψσ(t2, x2)〉HbbThe function ∆(t, x) is itself generalized; (B.23) is of course also true in the generalized

sense.


244 N. N. BOGOLUBOV

Φσ(t′2 − t′1, x′2 − x′1) = lim

V →∞〈ψ−σ(t′2, x

′2)ψσ(t′1, x

′1)〉H (B.27)

so that (B.24) decomposes into the product of averages

limV →∞

〈ψσ1(t1, x1)ψ†σ2

(t2, x2)〉H limV →∞

〈ψσ′2(t′2, x

′2)ψ

†σ′1(t′1, x

′1)〉H . (B.28)

The two relations (B.25) and (B.28) are the expressions of the principle ofextinction of correlations for the two-particle average considered.

For the case ν = 0,

〈ψ(t1, x1)ψ(t2, x2)〉H = 0,

and the relation (B.27) is no longer true. However, in this case we canintroduce the “quasi-averages”

〈ψσ1(t1, x1)ψσ2(t2, x2)〉H = limν>0ν→0

limV →∞

〈ψσ1(t1, x1)ψσ2(t2, x2)〉

= Φσ2(t1 − t2, x1 − x2)δ(σ1 + σ2) (B.29)

and replace the product of averages in (B.28) by a product of quasi-averages.Thus the relations obtained above illustrate the general principle of

extinction of correlations.

References

1. J. Bardeen, L. Cooper and J. Schrieffer, Phys. Rev., 108, 1175 (1957).

2. N. N. Bogoliubov, D. N. Zubarev, and Yu. A. Tserkovnikov, Docl. Akad.Nauk SSSR, 117, 788 (1957); Sov. Phys. “Doklady”, English Transl.,2, 535, (1957).

3. R. E. Prange, Bull. Am. Phys. Soc., 4, 225 (1959).

4. N. N. Bogoliubov, D. N. Zubarev, and Yu. A. Tserkovnikov, Zh. Exper.i Teor. Fiz., 39, 120 (1960); Sov. Phys. JETP, English Transl., 12, 88(1961).

5. N. N. Bogoliubov, Zh. Exper. i Teor. Fiz., 34, 73 (1958); Sov. Phys.JETF, English Transl., 7, 51 (1958).

6. N. N. Bogoliubov, Izv. Akad. Nauk SSSR Ser. Fiz. 11, 77 (1947);translation in D. Pines, The Many-Body Problem, Benjamin, New Work(1961).

7. N. N. Bogoliubov, Jr. Preprint JINR P4-4184. Dubna, 1968.



8. N. N. Bogoliubov, Jr. Preprint JINR P4-4175. Dubna, 1968.

9. N. N. Bogoliubov, Jr. Preprint ITPh-67-1. Kiev, 1967.

10. N. N. Bogoliubov, Jr. Preprint ITPh-68-65. Kiev, 1968.

11. N. N. Bogoliubov, Jr. Preprint ITPh-68-67. Kive, 1968.

12. N. N. Bogoliubov, Jr. Yad. Phys. 10, 425 (1969) [Sov. J. Nucl. Phys.10, 243 (1970).

******

In connection with this work I should like to express sincere gratitude toD. N. Zubarev, S. V. Tyablikov, Yu. A. Tserkovnikov, and E. N. Yakovlev.


ii



PART II

247


ii



CHAPTER 6

MODEL HAMILTONIANS WITH FERMIONINTERACTION

In many-body theory most problems of physical interest are rathercomplicated and usually insoluble. Model systems permitting a mathematicaltreatment of these problems are therefore acquiring considerable interest.

Unfortunately, however, in concrete problems in many-body theorythere is usually no adequate correspondence between a real system and itsmathematical models; one must be content with a model whose propertiesdiffer substantially from those of the real system and in solving problems onemust use approximate methods lacking the necessary mathematical rigour.

Of considerable interest in this connection is the study of those few modelswhich have some resemblance to real systems yet admit exact solution;fundamental properties of many-body can be established in this way. Systemsof non-interacting particles can be taken as examples of systems which canbe solved exactly. Although, of course, this model seems rather trivial, it isused as a starting point in most problems in many-body theory. In the theoryof metals one can often leave the mutual interaction of the valence electronout of consideration. In the shell model of nuclei in its simplest form onecan explain many general properties of nuclear spectra without introducinginteraction between the particles into the treatment.

One of the most important problems in statistical physics is the study ofexactly soluble cases. The fact is that this study makes an essential contri-bution to our understanding of the extremely complex problems of statisticalphysics and, in particular, serves as a basis for the approximate methods usedin this field. Up to the present time, the class exactly soluble dynamicalmodel systems has consisted mainly of one- and two-dimensional systems.

In this study we shall concentrate on the treatment of certain modelsystems of a general type which can be solved exactly, e.g. models

249


250 N. N. BOGOLUBOV, Jr

with four-fermion pair interaction which have as their origin the BCSmodel problems and are applicable in the theory of superconductivity; thedetermination of asymptotically exact solutions for these models has beeninvestigated by Bogolyubov, Zubarev and Tserkovnikov [1,2]. In these papersan approximation procedure was formulated in which ideas of a methodbased on the introduction of “approximating (trial) Hamiltonians” werepropounded and reasons were given for believing that the solution obtainedwas asymptotically exact on passing to the usual statistical mechanicallimit V → ∞.

1. General Treatment of the Problem. Some Preliminary Results

We begin here with the problem of the asymptotic calculation of quasi-averages based on works [13, 14, 17].

For a natural approach to the proper formulations of these we recall aseries of results which we established earlier for a hamiltonian of the form

H = T − 2V gJJ† (1)

in which

T =∑

f

T (f)a†faf ; T (f) =p2

2m− µ

J =1

2V

∑f

λ(f)a†fa†−f (2)

and af and a†f are Fermi amplitudes, m, µ and g are positive constants, and

f = (p, σ)

where σ is the spin index, which takes the values ±12. The function

λ(f) = λ(p, σ)

occurring in (2) is a real and continuous in the spherical layer∣∣∣∣ p2

2m− µ

∣∣∣∣ ≤ ∆

(where ∆ is a certain positive constant) and equal to zero outside it; it alsopossesses the property of antisymmetry

λ(−f) = −λ(f); − f = (−p,−σ).


MODEL HAMILTONIANS 251

We note finally that in summation “over f” the components pα (α = 1, 2, 3)of the vector p take the values 2πnα/L, while nα runs over all integers(−∞, ∞). L3 = V , where V is the volume of the system, which belowwill tend to ∞.

To introduce the quasi-averages we add to the Hamiltonian H terms with“pair sources”, e.g.

−νV (J + J†)

where ν is a positive constant.Thus, the Hamiltonian under consideration will be

Γ = T − 2V gJJ† −−νV (J + J†). (3)

The quasi-averages for the Hamiltonian H are introduced as the limits, asV → ∞, of the usual averages for the Hamiltonian Γ, with the sequentialpassage to the limit

≺ . . . H= limν→0ν>0

limV →∞

〈. . .〉Γ.

We have shown [7]cc that the simplest binary-type correlation averages

〈a†f(t)af (tau)〉Γ, 〈a†f (t)a†−f (τ)〉Γ, 〈a−f(t)af (tau)〉Γare asymptotically (V → ∞) close to the corresponding averages taken forthe “trial Hamiltonian”:dd

Γa(C) = T − 2V g(CJ† + C∗J − C∗C) − νV (J + J†)

= T − 2V g(C +

ν

2g

)J † +

(C∗ +

ν

2g

)J

+ 2V gC∗C.

(There is, however, another approach, viz. to deal with infinite volume formthe start. Such a situation was studied in a paper by Petrina (1970).) Thequantity C accuring in Γa(C) is determined from the condition that the freeenergy has an absolute minimum

fΓa(C) = min

ccWe note that this techniques has also been found to be useful in the study of exactlysolvable quasi-spin models (see, for example, [9, 33] and [34]).

ddIn this formulas we should have written 2V gC∗C · 1, where 1 is the unit operator;however, since this will not lead to misunderstandings anywhere, below we shall not writeout the unit operator explicitly.



in the whole complex C-plane.Since Γa(C) is a quadratic form in the Fermi amplitudes, this Hamiltonian

can be diagonalized by means of a u− v transformation:

af = u(f)αf − v(f)α†f , (4)

where αf and α†f are new Fermi amplitudes, and

u(f) =1

2

√1 +

T (f)

E(f),

v(f) = −λ(f)

(C +

ν

2g

)√

2∣∣∣λ(f)

(C +

ν

2g

)∣∣∣√

1 − T (f)

E(f),

E(f) =

√T 2(f) + 4λ2(f)g2

∣∣∣C +ν

2g

∣∣∣2.In the new Fermi amplitudes the Hamiltonian will take the form

Γa(C) =

2gC∗C − 1

2V

∑f

(E(f) − T (f)

)V +

∑f

E(f)α†fαf (5)

so that the free energy per unit volume calculated on the basis of thisHamiltonian will be

fΓa(C) = 2gC∗C − 1

2V

∑f

E(f) − T (f)

− θ

V

∑f

ln(1 + e−E(f)/θ). (6)

It is clear form this that the absolute minimum of fΓa(C) as a function ofthe complex variable is found at a real value of C, and at a value such that

C +ν

2g> 0.

This minimazing value of C depends general on V and ν:

C = C(V, ν).



In our above-mentioned paper (Bogolyubov, Jr., 1967) it was shown that

C(V, ν) → C(ν) (7)

asV → ∞ (ν(fixed) > 0)

andC(ν) −→

ν→0(ν>0)

C(0).

Here C = C(ν) realizes the absolute minimum of the asymptotic expression

f∞Γa(C) = limV →∞

Γa(C)

= 2gC∗C − 1

2(2π)3

∫dfE(f) − T (f)

− θ

(2π)3

∫df ln(1 + e−E(f)/θ).

As throughout the book, the “integral over f” denotes integration p andsummation over σ: ∫

df(. . .) =∑

σ

∫dp(. . .).

The value C(0) ≥ 0 is chosen as the number giving the absolute minimum ofthe function

f∞Ha(C) = 2gC2 − 1

2(2π)3

∫dfE(f)− T (f)

− θ

(2π)3

∫df ln(1 + e−E(f)/θ),

in whichE(f) =

√T 2(f) + 4g2C2λ2(f).

As we have noted already, we have proved in our work that the differenceof the binary averages constructed on the basis of the model (Γ) and trial(Γa) Hamiltonians tends to zero as V → ∞ for any fixed value of ν > 0.

On the other hand, in view of (4) and (5), these averages are calculatedeasily for Γa. For example, we have

〈a†f(t)af (τ)〉Γa = u2(f)eiE(f)(t−τ) e−E(f)/θ

1 + e−E(f)/θ



+ v2(f)e−iE(f)(t−τ) 1

1 + e−E(f)/θ,

〈a†f(t)a†−f (τ)〉Γa = u(f)v(f)

eiE(f)(t−τ) e−E(f)/θ

1 + e−E(f)/θ

+ e−iE(f)(t−τ) 1

1 + e−E(f)/θ

, (8)

As can be seen, the right-hand sides here are defined for all p, and not onlyfor the quasi-discrete values

p =(2πn1

L,

2πn2

L,

2πn3

L

),

which are the only values for which the amplitudes af and a†f , and therebythe left-hand sides of the expressions (8), are defined.

Moreover, the right-hand sides of (8) as functions of f depend on V onlythrough the quantity C = C(V, ν). The passage to the limit V → ∞ thereforereduces, because of (7), to replacing C(V, ν) by C(ν) in these functions. Thesubsequent passage to the limit ν → 0 (ν > 0) corresponds to replacementof C(ν) by C(0).

Thus, if we put ν = 0, C = C(0) in the functions of f occurring inthe right-hand sides of (8), these functions will represent the correspondingquasi-averages for the Hamiltonian H .

We note that the most complicated of our proofs was that establishingthe relation

〈. . .〉Γ − 〈. . .〉Γa → 0 (9)

as V → ∞ for binary expressions of the type indicated above. To provethem, we had to show first that

〈(J − C(V, ν))(J† − C(V, ν))〉Γ → 0 (10)

as V → ∞ and then establish the asymptotic relations (9). We mustemphasize that the problem of investigating situations with more complicatedaverages was not solved in the papers cited [4, 6, 7]. Moreover, the proofof the properties (10) and (7) was based on the specific features of theHamiltonian (1) and could not be extended to model Hamiltonians of moregeneral form.

In our work [4,6–8], we have constructed a new method which enables usto extend the above-mentioned results to the case of many-time averages of



Fermi amplitudes or field functions, and, therefore, for model Hamiltoniansof more complicated structure. If we wished to apply this method to theinvestigation of the Hamiltonian Γ it would be appropriate to start from therepresentation:

Γ = Γa(C(ν)) − 2gV (J − C(V, ν))(J† − C(V, ν))

= T − 1

2

∑f

Λ(f)a†fa†−f + a−faf + 2gV C2

− 2gV (J − C(V, ν))(J† − C(V, ν)), (11)

whereΛ(f) = 2gλ(f)

(C(ν) +

ν

2g

). (12)

In this case, in view of (7) and (10)

〈(J − C(V, ν))(J† − C(V, ν))〉Γ ≤ εV → 0. (13)

However, we shall not study only this Hamiltonian here.As will be shown under much wider conditions in the following chapters,

model Hamiltonians with properly chosen source terms can also be reducedto a form similar to (11).

We turn now to consider the situation when

Γ = Γa +H1, (14)

where

Γa =∑

f

T (f)a†faf − 1

2

∑f

Λ∗(f)a−faf + Λ(f)a†fa

†−f

+ K ,

where

K = const, T (f) =p2

2m− µ,

andH1 = −V

∑α

Gα(Jα − Cα)(J†α − C∗

α), (15)

where

Jα =1

2V

∑f

λα(f)a†fa†−f .



Here the summation over f runs over the above-mentioned quasi-discrete set,which we shall call the set φV . We shall examine this model system under thefollowing conditions, which we shall call conditions I (which were formulatedin [13, 14]).

1. The functions λα(f) and Λ(f) are defined and bounded in the wholespace φ of the points f = (p, σ).

2. The series ∑α

|Gα||λα(f)|2 = P (f)

converges uniformly in φ and the function P (f) represented by it satisfiesthe inequalities:

P (f) ≤M1 = const,

1

V

∑f

P (f) ≤M2 = const.

3. The inequality ⟨∑α

|Gα|(Jα − Cα)(J†α − C∗

α)⟩

Γ≤ εV

is satisfied, whereεV → 0 as V → ∞.

4. The function Λ(f) and the constants Cα satisfy the inequalities

1

V

∑f

|Λ(f)|2 ≤MΛ,

∑α

|Gα||Cα|2 ≤MC .

5. The functions λα(f) and Λ(f) are antisymmetric with respect toreflection:ee

λα(−f) = −λα(f), Λ(−f) = −Λ(f).

eeThis last condition is not, essentially, a restrictive one. Any sum of the form∑f

F (f)a†fa†−f ;

∑f

F (f)a−faf



Here α takes integer values. (In general, the sums over α imply aninfinite number of terms; in the next sections, however, we considerproblems in which α takes a finite number of values.)

With these conditions we prove a series of theorems on the asymptoticcloseness of averages taken over the Hamiltonians Γ and Γa respectively.

Apart from conditions I when considering field functions

ψσ(r, t) =1√V

∑p

apσ(t)ei(p·r),

ψ†σ(r, t) =

1√V

∑p

a†pσ(t)e−i(p·r)

(here the sum runs over the set of quasi-discrete p) we shall now have toimpose the following additional conditions, which we will call conditions I′:

1. The functionsΛ(f) = Λ(p, σ) (σ = ±1

2)

are defined and bounded in the whole space E of points p and areindependent of V .

2. The discontinuities of these functions form a set of measure zero in thespace E (see [31]).

These conditions I′ mean that as a result of the passage to the limit(V → ∞), we go over form the sums to Riemann integrals. In fact, theintegrability in the Riemann sense of some bounded functions is ensured bythe fact that the set of its discontinuities is of a measure zero.

2. Calculation of the Free Energy for Model System withAttraction

In the following we shall study dynamical systems which correspond toattraction of fermions. We shall begin by calculating the free energy for

can always be reduced to the form

∑f

F (f) − F (−f)2

a†fa†−f ;

∑f

F (f) − F (−f)2

a−faf ,

in which the coefficient function [F (f) − F (−f)]/2 is already antisymmetric with respectto the reflection: f → −f .



model Hamiltonians with four-fermion interactions. This problem, as we haveshown [5], is of great interest in the study of model problems in the problemof superconductivity and serves as an example of an exact calculation ofthe free energy for model systems of the BCS type [34, 35].ff The resultsand upper bounds obtained here also constitute a proof of the Theorem 1formulated in this section.

We shall start from the Hamiltonian

H = T − 2V∑1≤αs

JαJ†α. (16)

If we take the following Fermi-operator expressions for the operators Tand Jα:

T =∑

f

T (f)a†faf , Jα =1

2V

∑f

λα(f)a†fa†−f , (17)

we obtain the usual BCS Hamiltonian

H =∑

f

T (f)a†faf − 1

2V

∑f,f ′

J (f, f ′)a†fa†−fa−f ′af ′ . (18)

In fact it is not necessary for our discussion that the operators T and Jα havethe explicit form (17)

It is sufficient to impose the following general conditions:

|Jα| ≤M1, |TJα − JαT | ≤M2,

|J†αJβ − JβJ

†α| ≤

M3

V, |JαJβ − JβJα| ≤ M3

V, (19)

where M1, M2 andM3 are constants as V → ∞ and the symbol | . . . | denotesthe norm of the indicated operators. We assume also that the free energyper unit volume for the Hamiltonian H = T is bounded by a constant andthat the number of terms s in the sum (16) is fixed.

We thus start from the Hamiltonian (16) with the condition (19). Wetake the trial Hamiltonian to have the usual form

H0 = T − 2V∑

1≤α≤s

(CαJ†α + C∗

αJα) + 2V∑

1≤α≤s

|Cα|2. (20)

ffThis treatment also solves a number of problems raised by Wentzel [35] concerningthe asymptotically exact calculation of the free energy and can also be applied to certainquasi-spin Hamiltonians (cf. [12, 15] and [32]).



Here the Cα are complex constants determined from the condition that thefunction

fH0 = − 1

Vθ ln Tr e−H0/θ (21)

have its absolute minimum value in the domain of all the complexvariables (C1, . . . , Cs). We shall denote this complex set of points (C1, . . . , Cs)by Es. By making use of the minimizing values of C, we calculate the freeenergy per unit volume for the trail Hamiltonian:

fH0(C) = minEs

fH0(C). (22)

We also take the corresponding free energy for the Hamiltonian (16):

fH = − 1

Vθ ln Tr e−H/θ. (23)

We shall prove that the difference FH0 − fh tends to zero as V → ∞.For this it is convenient to consider fist the auxiliary problem with theHamiltonian

Γ = H − V∑

1≤α≤s

(ναJα + ν∗αJ†α), (24)

where ν1, . . . , νs are arbitrary non-zero complex parameters. In this problem,the corresponding trial Hamiltonian has the form

Γ0 = H0 − V∑

1≤α≤s

(ναJα + ν∗αJ†α). (25)

The complex quantities C = (C1, . . . , Cs) occurring here are also determinedfrom the condition for the absolute minimum of the function

fΓ0(C) = − 1

Vθ ln Tr e−Γ0/θ. (26)

We shall obtain an upper bound for the difference fΓ0 −fγ , and show thisdifference to be asymptotically small as V → ∞. Here

fΓ0 = minEs

fΓ0(C)

and fΓ is the free energy per unit volume for the Hamiltonian Γ. Althoughthis bound for fΓ0 − fγ will be found for |να| 0, it turns out that it is uniform



with respect to να → 0, so that we can then pass to the limit να = 0(1 ≤ α ≤ s). We thereby obtain a bound for fH0 − fH , proving it to beasymptotically small as V → ∞.

We therefore begin by treating the trial Hamiltonian Γ0. It is not difficultto show that the problem of the absolute minimum of the function (26) has asolution and that this absolute minimum is realized for finite values Ck −C0

k

(1 ≤ k ≤ s). This can be seen by using the inequalities∑α

|Cα|2 + (|Cα| + 2M1)2 − 4M2

1 s+ γ + fT ≥ fΓ0(C)

≥∑

α

|Cα|2 + (|Cα| − 2M1)2 − 4M2

1 s− γ + fT

≥∑

α

|Cα|2 − 4M21 s+ γ + fT ,

C = (C1, . . . , Cs), γ = 2M1

∑α

|να| (1 ≤ α ≤ s). (27)

Thus, the function fΓ0(C) has an absolute minimum at some point C0 =(C0

1 , . . . , C0s ).

Because the function fΓ0(C) is continuously differentiable, at thepoint C = C0 we have

∂fΓ0(C)

∂Cα

= 0 (1 ≤ α ≤ s),

i.e. an equation for Cα:

Cα = 〈Jα〉Γ0 =Tr Jα e−Γ0/θ

Tr e−Γ0/θ

Taking into account the conditions (19) on the operators Jα, we obtain |Cα| ≤M1 = const.

We turn now to the derivation of inequalities limiting the difference inthe free energies per unit volume fΓ0 − fΓ in terms of averages of

A

V= −2

∑1≤α≤s

(Jα − Cα)(J†α − C∗

α). (28)

For this we note that Γ = Γ0+A and introduce the intermediate Hamiltonian

Γt = Γ0 + tA ,



which for t = 0 coincides with the trial Hamiltonian (25) and for t = 1coincides with the original Γ (24). The constants C = (C1, . . . , Cs) occurringin Γt are assumed to be fixed and to be independent of the parameter t.

Let us consider the partition functiongg and free energy for theintermediate Hamiltonian Γt:

Qt = Tr e−Γt/θ, ft(C1, . . . , Cs) = − θ

VlnQt, Qt = e−V ft/θ. (29)

Differentiating the equality (29) twice with respect to t using operatordifferentiation, we arrive at the formula

−Vθ

∂2ft

∂t2+V 2

θ2

(∂ft

∂t

)=

1

θ2Qt

1∫

0

TrA e−(Γt/θ)τA e−(Γt/θ)(1−τ) dτ.


∂ft

∂t=

1

V

TrA e−Γt/θ

Tr e−Γt/θ=

1

V〈A 〉t,

we find

−∂2ft

∂t2=

1

θV

1

Qt

1∫

0

TrA e−(Γt/θ)τA e−(Γt/θ)(1−τ) dτ − 〈A 〉2

=1

θV Qt

1∫

0

TrB e−(Γt/θ)τB e−(Γt/θ)(1−τ) dτ ;

B = A − 〈A 〉.

Going over to a matrix representation in which the Hamiltonian Γt isdiagonal, we have

−∂2ft

∂t2=

1

θV Qt

1∫

0

dτ∑n,m

BnmBmn e−(Etm−Et

n)/θτ−Etn/θ

ggMathematical questions concerning the existence and analytical properties of partitionfunctions have been considered by the author in [11]. The theorem proved there has beenfurther generalized by H. D. Maison [29].



=1

θV Qt

1∫

0

dτ∑n,m

|Bnm|2 e−(Etm−Et

n)/θτ−Etn/θ ≥ 0.

Hence it follows, in particular, that ∂2t /∂t

2 ≤ 0, and therefore ∂ft/∂t =〈A 〉t/V decreases with increase of the parameter t. Furthermore, takinginto account that fΓ does not depend on C, we have

fΓ0(C) − fΓ = −1∫

0

∂ft

∂tdt =

1∫

0

〈A 〉tV

dt ≥ 0.

Since this relation is true for all C = (C1, . . . , Cs), we also have

minEs

fΓ0 ≥ fΓ, fΓ0 ≥ fΓ.

We shall integrate the inequality 〈A 〉Γt ≥ 〈A 〉Γ (0 ≤ t ≤ 1).Substituting the expression (28) for A , we convince ourselves that theinequality

fΓ0(C) − fΓ ≤ 2∑

1≤α≤s

〈(Jα − Cα)(J†α − C∗α)〉Γ

is true for any C = (C1, . . . , Cs). We put here Cα = 〈Jα〉Γ (0 ≤ t ≤ 1) andnote that

fΓ0 = minEs

fΓ0(C) ≤ fΓ0(〈J〉Γ).

Thus,

fΓ0 − fΓ ≤ fΓ0(〈J〉Γ) − fΓ ≤ 2∑

1≤α≤s

⟨(Jα − 〈Jα〉Γ)(J†

α − 〈J†α〉Γ)

⟩Γ

and, finally,

0 ≤ fΓ0 − fΓ ≤ 2∑

1≤α≤s

⟨(Jα − 〈Jα〉Γ)(J†

α − 〈J†α〉Γ)

⟩Γ, hh (30)

where, as always,fΓ0 = min

EsfΓ0(C). (31)

hhFor convenience, in the reminder of this section we shall omit the subscript in statisticalaverages over the Hamiltonian Γ, i.e. we shall write 〈. . .〉 ≡ 〈. . .〉Γ.



Let us recall our main problem. We want to show that the differencefΓ0 − fΓ is asymptotically small as V → ∞. It follows from (30) that weshall have solved our problem if we can demonstrate the asymptotic smallnessof the average on the right-hand side of (30).

Taking into account the main idea of a paper by the author (Bogolyubov,Jr., 1966a), we express this right-hand side in terms of ∂2f/∂ν∗α∂α.Differentiating, we have

−1

θ

∂2f

∂ν∗α∂α

=V

θ2

1∫

0

Tr(D(α) e−(τ/θ)Γ (D(α))† e−(1−τ)/θΓ)

dτ

Tr e−Γ/θ,

whereD(α) = Jα − 〈Jα〉, (1 ≤ α ≤ s).

Going over to the matrix representation in which Γ is diagonal, we find

−1

θ

∂2f

∂ν∗α∂α=

V

Qθ2

∑n,m

1∫

0

D(α)nm e−(τ/θ)Em (D(α)

mn)† e−(1−τ)/θEm dτ

=V

Qθ2

∑n,m

∣∣D(α)nm

∣∣2 1∫

0

e−(τ/θ)Em−(1−τ)/θEm dτ

=V

Qθ2

∑n,m

∣∣D(α)nm

∣∣2 e−Em/θ − e−En/θ

En − Em≥ 0.

Using Holders inequality, we have the following bound:

V

Q

∑n,m

∣∣D(α)nm

∣∣2∣∣e−Em/θ − e−En/θ∣∣

≤(− ∂2f

∂ν∗α∂α

)3/2(VQ

∑n,m

∣∣D(α)nm

∣∣2|En −Em|2(e−Em/θ − e−En/θ

))1/3

.

We carry out the simple transformations:

V

Q

∑n,m

∣∣D(α)nm

∣∣2|En − Em|2(e−Em/θ − e−En/θ

)

=V

QTr e−Γ/θ

(ΓD(α) −D(α)Γ

)((D(α))†Γ − Γ(D(α))†

)



+((D(α))†Γ − Γ(D(α))†

)((D(α))†Γ − Γ(D(α))†

)=V

⟨(ΓJα − JαΓ)(ΓJα − JαΓ)† + (ΓJα − JαΓ)†(ΓJα − JαΓ)

⟩≤2VM 2,

whereM = M2 + 4M1M3s+ 2M3

∑1≤α≤s

|να|.

Hence we obtain

V

Q

∑n,m

∣∣D(α)nm

∣∣2∣∣e−Em/θ − e−En/θ∣∣ ≤ (

− ∂2f

∂ν∗α∂α

)2/3

(2VM 2)1/3.

Furthermore,

V

Q

∑n,m

∣∣D(α)nm

∣∣2e−En/θ ≤ θV

Q

∑n,m

∣∣D(α)nm

∣∣2(En − Em)

(e−Em/θ − e−En/θ)

+V

Q

∑n,m

∣∣D(α)nm

∣∣2∣∣e−Em/θ − e−En/θ∣∣,

where

V

Q

∑n,m

∣∣D(α)nm

∣∣2e−En/θ =V

QTrD(α)(D(α))† e−Γ/θ = V 〈D(α)(D(α))†〉

= V⟨(Jα − 〈Jα〉)(J†

α − 〈J †α〉)

⟩.

Thus, we finally obtain⟨(Jα − 〈Jα〉)(J†

α − 〈J†α〉)

⟩≤

(− ∂2f

∂ν∗α∂α

) θV

+(2M 2)1/3

V 2/3

(− ∂2f

∂ν∗α∂α

)2/3

.

Substituting this inequality into (30), we find

0 ≤ fΓ0−fΓ ≤ 2θ

V

∑1≤α≤s

(− ∂2f

∂ν∗α∂α

)

+2

V 2/3(2M 2)1/3

∑1≤α≤s

(− ∂2f

∂ν∗α∂α

)2/3

. (32)



Hence we can see that our problem would be solved if we could show thatthe second derivatives

∣∣∂2f/∂ν∗α∂α

∣∣ are bounded by a constant as V → ∞.Unfortunately, we are unable to prove such a statement. We must start fromthe boundedness of the first derivatives

∣∣∂f/∂να

∣∣ ≤M1 (1 ≤ α ≤ s).Because of this, we develop a method in which it will not be necessary

to use the boundedness of the second derivatives and by means of which wecan demonstrate the asymptotic smallness of the difference

a = fΓ0 − fΓ.

For the following, it will be more convenient to transform to the polarvariables rα, ϕα:

rα = rα(να, ν∗α), ϕα = ϕα(να, ν

∗α), (1 ≤ α ≤ s)

in inequality (32). Accordingly,

f(ν1, ν∗1 , . . . , νs, ν

∗s ) → f(r1, ϕ1, . . . , rs, ϕs).

Then∂2f

∂ν∗α∂α

=1

4

1

rα

∂

∂rα

(rα∂f

∂rα

)+∂2f

∂r2α

1

r2α

. (33)

We now make use of the inequality∣∣a(r1, . . . , rs;ϕ1, . . . , ϕs)∣∣ ≤ ∣∣a(r1, . . . , rs;ϕ1, . . . , ϕs)

− a(ξ1, . . . , ξs; η1, . . . , ηs)∣∣ +

∣∣a(ξ1, . . . , ξs; η1, . . . , ηs)∣∣

≤∑

1≤α≤s

∣∣∣ ∂a∂rα

∣∣∣max

|rα − ξα| +∑

1≤α≤s

∣∣∣ ∂a∂ϕα

∣∣∣max

|ϕα − ηα|

+∣∣a(ξ1, . . . , ξs; η1, . . . , ηs)

∣∣, (34)

we take

rα + l ≤ ξα ≤ rα + 2l, ϕα ≤ ηα ≤ ϕα + δα, δα =l

rα

, (1 ≤ α ≤ s),

so that

a(ξ1, . . . , ξs; η1, . . . , ηs)



=

r1+2l∫r1+l

. . .rs+2l∫rs+l

dr1 . . . drsϕ1+δ1∫

ϕ1

. . .ϕs+δs∫

ϕs

dϕ1 . . . ϕs a(r1, . . . , rs;ϕ1, . . . , ϕs)∏

1≤α≤srα

(1/2)3∏

1≤α≤s

[(rα + 2l)2 − (rα + l)2

]δα

(35)

We note that ∣∣∣ ∂f∂rα

∣∣∣ ≤ 2M1,∣∣∣ ∂f∂ϕα

∣∣∣ ≤ 2M1rα

and

∣∣∣ ∂a∂rα

∣∣∣ ≤ 4M1,∣∣∣ ∂a∂ϕα

∣∣∣ ≤ 4M1rα. (36)

Therefore, the first two terms in inequality (34) can be bounded as follows:

∑1≤α≤s

∣∣∣ ∂a∂rα

∣∣∣max

|rα − ξα| +∑

1≤α≤s

∣∣∣ ∂a∂ϕα

∣∣∣max

|ϕα − ηα|

≤ 4M1s · 2l + 4M1ls = 12M1sl. (37)

Starting from formulae (32) and (35), we find a bound for theexpression a(ξ1, . . . , ξs; η1, . . . , ηs). We put (33) into the right-hand side ofthe inequality (32). We then multiply (32) by the product r1r2 . . . rs andintegrate it over all values of the variables r1, . . . , rs, ϕ1, . . . , ϕs within thefollowing limits:

rα + l ≤ rα ≤ rα + 2l, ϕα ≤ ϕα ≤ ϕα + δα, δα =l

rα

, (1 ≤ α ≤ s).

We then obtain

0 ≤∫. . .

∫a(r1, . . . , rs, ϕ1, . . . , ϕs)r1r2 . . . rs dr1 . . . drsdϕ1 . . . ϕs

≤ θ

2V

∫. . .

∫F1r2r3 . . . rs + F2r1r3 . . . rs

+ . . .+ Fsr1r2 . . . rs−1dr1 . . . drsdϕ1 . . . ϕs

+M 2/3

V 2/3

∫. . .

∫F 2/3

1 r1/31 r2r3 . . . rs + F

2/32 r1r

1/32 r3 . . . rs



+ . . .+ F 2/3s r1r2 . . . rs−1r

1/3s dr1 . . . drsdϕ1 . . . ϕs. (38)

Here

Fα =∂

∂rα

(rα∂(−f)

∂rα

)+

1

rα

∂

∂ϕα

(∂(−f)

∂ϕα

)≥ 0, (1 ≤ α ≤ s).

By considering the separate terms of the part of (38) containing thefactor θ/2V , we see that bounds can be found for these by integratingsuccessively over rα and ϕα in each of them (1 ≤ α ≤ s) and using theinequalities (36) to find bounds for the resulting first derivatives ∂f/∂ϕα

and ∂f/∂rα; for all the terms of this sum, we then find

θ

2V

∑1≤β≤s

2M1(δβ + 2)

l · 2s−1δβ

∏1≤α≤s

(rα + 2l)2 − (rα + l)2δα. (39)

By applying Holder’s inequality and using analogous reasoning for all termsin the sum containing the factor M 2/3/V 2/3, we obtain

M 2/3

V 2/321/3

∑1≤β≤s

(2M1(δβ + 2)

)2/3

l2/3 · 2s−1δ2/3β

∏1≤α≤s

(rα + 2l)2 − (rα + l)2δα. (40)

Using the formulae (34)-(40), we now obtain a bound for fΓ0 − fΓ =a(r1, . . . , rs;ϕ1, . . . , ϕs):

0 ≤ fΓ0 − fΓ ≤ 12M1ls+2θM1

V

∑1≤β≤s

(δβ + 2)

lδβ

+M 2/3

V 2/3(4M1)

2/3∑

1≤β≤s

(δβ + 2)2/3

l2/3δ2/3β

. (41)

On the other hand, we note that δβ = l/rβ and choose R in such a way,

R ≥ |ν1|, . . . , |νs|,

that R ≥ rβ (β = 1, . . . , s) and δβ ≥ δ = l/R. Then, using the obviousinequality (l + 2R)2/3 ≤ l2/3 + (2R)2/3, we find from (41)

0 ≤ fΓ0 − fΓ ≤ 12M1ls+2θM1

V l2s(l + 2R) +

M 2/3

V 2/3s

(4M1)2/3

l2/3



+M 2/3

V 2/3s

(4M1)2/3

l2/322/3R2/3. (42)

We now choose l, which is an arbitrary positive quantity, such that

12M1l =M 2/3(4M1)

2/3

V 2/3l2/3.

Then

l =P

V 2/5, P =

M 2/5

22/533/5M3/51

= const.

Putting this expression for l into the inequality (42), we find

0 ≤ fΓ0 − fΓ ≤ 24M1sP

V 2/5+

2θM1s

V 3/5P+

4θM1sR

V 1/5P 2

+M 2/3

V 2/15P 4/3s (4M1)

2/3 22/3R2/3 for |να| < R (1 ≤ α ≤ s).

Hence it is clear that the difference fΓ0 − fΓ vanishes as V → ∞.We note that in the above bound, we can take the limit να = 0 (1 ≤ α ≤

s) and finally prove the statement we made earlier about the asymptoticsmallness of the difference fΓ0 − fΓ:

0 ≤ fΓ0 − fΓ ≤ 24M1sP

V 2/5+

2θM1s

V 3/5P, (43)

where P is a simple combination of the original constants M1, M2, and M3.It is also clear that the above bound is uniform as θ → 0, and, therefore, theinequality (43) is valid for θ ≥ 0.

We have thus proved the following theorem:

Theorem 1. Let the Hamiltonian of the system be

H = T − V∑

1≤α≤s

gαJαJ†α (44)

and let the operators T and Jα in (44) satisfy the following conditions:

T = T †, |Jα| ≤M1,

|TJα − JαT | ≤M2, |JαJβ − JβJα| ≤ M3

V,



|J†αJβ − JβJ

†α| ≤

M3

V; M1, M2, M2 = const. (45)

In addition, let the free energy per unit volume, calculated for theHamiltonian T , be bounded by a constant:

|f(T )| ≤ M0 = const. (46)

We construct the trail Hamiltonian

H(C) = T − 2V∑

α

gα(CαJ†α + C∗

αJα − CαC∗α), (47)

where C = (C1, . . . , Cs) and C1, . . . , Cs are complex numbers. Then thefollowing inequalities are valid:ii

0 ≤ min(C)

fH(C) − f(H) ≤ ε( 1

V

), (48)

where ε(1/V ) → 0 (as V → ∞) uniformly with respect to θ in theinterval (0 ≤ θ ≤ θ0) where θ0 is an arbitrary fixed temperature.

3. Further Properties of the Expressions for the Free Energy

Having formulated Theorem 1, we shall now study the question of theexistence of the limit

limV →∞

f(H).

We shall assume that, in addition to the conditions of Theorem 1, thefollowing condition is fulfilled. For any complex C1, . . . , Cs the limit

limV →∞

fH(C)ii(1) We shall denote the free energy per unit volume for some Hamiltonian A by f(A),

or, if we wish to emphasize its dependence on the volume, by fV (A).(2) By min

(C)we shall always mean the absolute minimum of the function f(C) in the

space of all points C.(3) ε(1/V ) is given by (cf.(43))

ε( 1V

)=

24M1sP√qV 2/5

+2θM1s√gV 3/5P

,

where P is a constant and g > 0 is the smallest of the gα.



exists. We shall denote this limit by

f∞H(C).We putjj

FV (C) = fH(C) − 2∑

α

gαCαC∗α

and note that∂FV (C)

∂Cα

= −2gα〈J †α〉H(C)

∂FV (C)

∂C∗α

= −2gα〈Jα〉H(C).

Therefore, from (1) we have

∣∣∣∂FV (C)

∂Cα

∣∣∣ ≤ 2gαM1,∣∣∣∂FV (C)

∂C∗α

∣∣∣ ≤ 2gαM1,

whence ∣∣FV (C ′) − FV (C ′′)∣∣ ≤ 4M1

∑1≤α≤s

gα|C ′α − C ′′

α|. (49)

Thus, the set of functions

FV (C) (V → ∞)

is uniformly continuous.Since we have the convergence

FV (C) → F∞(C) = f∞H(C) − 2∑

α

gαCαC∗α (V → ∞)

at each point C, we see that this convergence will be uniform on the setM (R) of points C defined by the inequalities

|C1| ≤ R1, . . . , |Cs| ≤ R,

for any fixed value of R.

jjWe stress that the function FV (C) need not be treated from the standpoint of thetheory of functions of a complex variables. It is not difficult to see that, in essence, thefunction FV (C) is a function of a real variables and that these can be taken as the realand imaginary parts of the variables Cα.



Therefore,∣∣fV H(C) − f∞H(C)∣∣ =∣∣FV (C) − F∞(C)

∣∣≤ ηV (R) → 0, (V → ∞) (50)

for C ∈ M (R).On the other hand, it follows from (49) that∣∣FV (C1, C2, . . . , Cs) − FV (0, C2, . . . , Cs)

∣∣ ≤ 4M1g1|C1|.

Hence, we have

fH(C1, C2, . . . , Cs) − fH(0, C2, . . . , Cs)= FV (C1, C2, . . . , Cs) − FV (0, C2, . . . , Cs) + 2g1|C1|2≥ −4g1M1M1|C1| + 2g1|C1|2. (51)

We denote the lower bound of FH(C) in the space of the points C by

inf(C)

fH(C).

Obviously,fH(C1, C2, . . . , Cs) ≥ inf

(C)fH(C)

and, therefore, it follows from(51) that

fH(C) − inf(C)

fH(C) ≥ 2g1|C1|(|C1| − 2M1).

Replacing C1 by Cα (α = 1, 2, . . . , s) in the above discussion, we find also

fH(C) − inf(C)

fH(C) ≥ 2gα|Cα|(|Cα| − 2Mα), α = 1, 2, . . . , s.

Hence it is clear that if |Cα| > 2M1 for at least one α, then

fH(C) > inf(C)

fH(C).

Therefore, the lower bound of fH(C) on the set M (2M1) is equal to thelower bound of this function on the whole space of points C. Since FH(C)is continuous and the set M (2M1) is bounded and closed, this lower bound



is attained on M (2M1), i.e. an absolute minimum of the function underconsideration exists and is realized at certain points:kk

C = C(V ) ∈ M (2M1).

On the other hand, taking (49) into account and passing to the limit V → ∞,we find

|F∞(C ′) − F∞(C ′′)| ≤ 4M1

∑1≤α≤s

gα|C ′α − C ′′

α|.

Hence, repeating exactly the above treatment, we see that the function

f∞H(C) = F∞(C) + 2∑

α

gαCαC∗α

also has an absolute minimum in the space of all the points C, which isrealized at certain points

C = C ∈ M (2M1).

From (50), we have now:

f∞H(C(V )) − fH(C(V )) ≤ ηV (2M1),

fH(C) − f∞H(C) ≤ ηV (2M1).

But, by definition of the absolute minimum,

f∞H(C(V )) ≥ f∞H(C),fH(C) ≥ fH(C(V )).

Consequently

f∞H(C) − fH(C(V )) ≤ ηV (2M1),

fH(C(V )) − f∞H(C) ≤ ηV (2M1),

or|f∞H(C) − min

(C)fH(C)| ≤ δV ,

whereδV = ηV (2M1) → 0, (V → ∞).

kkGenerally speaking, the point C(V ) of the absolute minimum is not unique.



Taking Theorem 1, we finally obtain

−δV ≤ f∞H(C) − f(H) ≤ ε( 1

V

)+ δV .

Thus, we have now proved the following theorem:

Theorem 2. If the conditions of Theorem 1 are fulfilled, and if for anycomplex values of C1, . . . , Cs the limit

f∞H(C) = limV →∞

fH(C)

exist, then:

1. This limit function has an absolute minimum is the space of all points C,which is realized at certain points

C = C ∈ M (2M1).

2. The inequalitiesf∞H(C) = lim

V →∞fH(C)

are valid, where

ε( 1

V

)→ 0, δV → 0 as V → ∞,

δV = max(C∈M (2M1))

∣∣fH(C) − f∞H(C)∣∣.4. Construction of Asymptotic Relations for the Free Energy

We shall now make a special study of those cases when the operators T andJα in the Hamiltonian (44) have the form (17)

As can easily be shown, the conditions of Theorem 1 will be fulfilled insuch cases if

(a)1

V

∑p

|T (p)λα(p, σ)| ≤ Q0,

(b)1

V

∑p

|λα(p, σ)| ≤ Q1,



(c)1

V

∑p

|λα(p, σ)|2 ≤ Q2. (52)

Here, α = 1, . . . , s; σ = ±1/2, Q0, Q1, Q2 = const. Then, for example, inthe inequalities (1) we can put

M1 = Q1, M3 = Q2, M2 = 2Q0.

Here, let the functions λα(p, σ) satisfy, in addition to the inequalities (52),the following conditions:

|λα(p, σ)| ≤ Q, Q = conts. (53)

The set of the discontinuities of the functions λα(p, σ) is a set

of measure zero in the space E. (54)

We shall show that in this situation the conditions of Theorem 2 are alsofulfilled.

Before proceeding to this problem, we note that the inequalities (52)and (53) are not independent. In fact, (52c) follows form theinequalities (52b) and (52a). Also, (52b) follows from (52a) and (53). Thus,all the inequalities imposed here on the λα are fulfilled if the inequalities (52a)and (53) are true.

We note further that (52a) and (53) hold if λα satisfy the inequalities

|λα(p, σ)| ≤ K

(p2 + a)3, K , a = const. (55)

We turn now to the question of the fulfillment of the conditions ofTheorem 2. Since, in the situation being studied, the conditions ofTheorem 1 are fulfilled, we need only show that for any fixed complexquantities C1, . . . , Cs the limit

f∞H(C) = lim(V →∞)

fV H(C) (56)

exists. For this, we write the operator form of the trial Hamiltonian asfollows:

H(C) =∑

f

T (f)a†faf − 1

2

∑f

Λ(f)a†fa

†−f + Λ∗(f)a−faf



+ 2V∑

α

gαC∗αCα, (57)

where

λ(f) = 2∑

α

gαC∗αλα(f). (58)

Going over to the Fermi amplitudes αf , α†f , which are related to the old af , a

†f

by the transformation

af = u(f)αf − v(f)α†−f ,

a†−f = u(f)α†−f + v∗(f)αf ,

with

u(f) =1√2

√1 +

T (f)

E(f), v(f) = − Λ(f)√

2|Λ(f)|

√1 − T (f)

E(f),

we diagonalize the form (57) and obtain

H(C) =∑

f

E(f)α†fαf + V

2∑

α

gαC∗αCα − 1

V

∑f

[E(f) − T (f)], (59)

where

E(f) =√T 2(f) + |Λ(f)|2.

Hence, we find

fV H(C) = − θ

VlnTr e−H(C)/θ = 2

∑α

gαC∗αCα

− 1

C

∑f

[E(f) − T (f)] − θ

V

∑f

ln(1 + e−E(f)/θ),

or, separating the indices p and σ in the summation,

fV H(C) = 2∑

α

gαC∗αCα −

∑σ

1

2V

∑p

E(p, σ) −

( p2

2m− µ)

− θ∑

σ

1

V

∑p

ln(1 + e−E(p,σ)/θ). (60)



On the other hand, it is not difficult to see that if we have some boundedfunction F (p), defined everywhere on the space E, whose discontinuities forma set of measure zero, then

1

V

∑p∈Sr

F (p) → 1

(2π)3

∫

Sr

F (p) dp (61)

for any sphere Sr with arbitrary fixed radius r. In fact, such a function willbe Riemann-integrable in the region Sr. For the summation points

p =(2πn1

L,

2πn2

L,

2πn3

L

)we have

∆px∆py∆pz =(2π

L

)=

(2π)3

V,

so that(2π)3

V

∑p∈Sr

F (p)

will be the Riemann sum for the integral∫Sr

F (p) dp. We note also that if

1

V

∑p

|F (p)| ≤ A = const,

then1

V

∑p∈Sr

|F (p)| ≤ A.

Hence, passing to the limit V → ∞, we have

1

(2π)3

∫

Sr

F (p) dp ≤ A.

Because of the arbitrariness of the radius r, we see that F (p) is an absolutelyintegrable function in the whole space, such that

1

(2π)3

∫F (p) dp ≤ A.



For a given function F (p), let the following inequality be valid:

1

V

∑p∈E−Sr

|F (p)| = ηr,

where (E−Sr) denotes the set of the points of E lying outside the sphere Sr,does not depend on V , and

ηr → 0, (r → ∞).

Then, obviously

1

V

∑p

F (p) → 1

(2π)3

∫F (p) dp. (62)

In fact, we shall fix an arbitrary small number ε > 0 and choose r = r0 suchthat

ηr0 ≤ε

4.

In view of (61), we can find a number V0 such that, for V ≥ V0, the inequality∣∣∣ ∑p∈Sr0

F (p) − 1

(2π)3

∫

Sr0

F (p) dp∣∣∣ ≤ ε

2

holds. We have, therefore∣∣∣∑p

F (p) − 1

(2π)3

∫F (p) dp

∣∣∣≤

∣∣∣ ∑p∈Sr0

F (p) − 1

(2π)3

∫

Sr0

F (p) dp∣∣∣

+∑

p∈E−Sr0

|F (p)| − 1

(2π)3

∫

E−Sr0

|F (p)| dp

≤ ε

2+ε

4+ε

4= ε

for any V ≥ V0, and this establishes the validity of (62).After these trivial remarks, we turn to the expression (60). It is clear

from (58) and the conditions imposed above on the λα that

|Λ(p, σ)| ≤ Λ0 = const,



1

V

∑p

|Λ(p, σ)| ≤ Λ1 = const.

We see that the discontinuities of the function Λ(p, σ), are consequently alsoof the function

E(p, σ) −( p2

2m− µ

)=

√( p2

2m− µ

)2

+ |Λ(p, σ)|2 −( p2

2m− µ

)form a set of measure zero in the space E. Further, for

p2 ≥ 4mµ,

we havep2

2m− µ ≥ p2

4m,

0 ≤ E(p, σ) −( p2

2m− µ

)≤ |Λ(p, σ)|2

2p2

4m

=2m

p2|Λ(p, σ)|2.

Hence,1

V

∑p∈E−Sr

(p2≥4mµ)

E(p, σ) −

( p2

2m− µ

)≤ 2m

p2Λ1.

Thus, taking into account the remarks made above, we see that

1

V

∑p

E(p, σ) −

( p2

2m− µ

)

→ 1

(2π)3

∫E(p, σ) −

( p2

2m− µ

)dp.

Further, we have

ln1 + e−E(p,σ)/θ < e−E(p,σ)/θ ≤ const e−p2/2mθ.

Since this function decreases sufficiently rapidly as p → ∞ and itsdiscontinuities form a set of measure zero, we also obtain

1

V

∑p

ln1 + e−E(p,σ)/θ → 1

(2π)3

∫ln1 + e−E(p,σ)/θ dp.



This also proves the validity of the property (56). We have here

f∞H(C) = 2∑

α

gαCαC∗α − 1

2

∑σ

1

(2π)3

∫E(p, σ) −

( p2

2m− µ

)dp

− θ∑

σ

1

(2π)3

∫ln1 + e−E(p,σ)/θ dp,

or, more compactly,

f∞H(C) = 2∑

α

gαCαC∗α − 1

2(2π)3

∫E(f) − T (f) df

− θ

(2π)3

∫ln1 + e−E(f)/θ df. (63)

Here, the integration∫(. . .) df implies the operation

∑σ

∫(. . .) dp.

Thus, in the case under investigation, if the conditions (52a), (53) and (54)are fulfilled, the condition of Theorem 2 are satisfied. As was noted above,in the proof of this theorem, the convergence

fV H(C) − f∞H(C) → 0, (V → ∞) (64)

is uniform on any bounded set of points C.

5. On the Uniform Convergence with Respect to θ of the FreeEnergy Function and on the Bounds for the Quantities δV

We shall show that, in the above case, the convergence (64) is also uniformwith respect to θ in the interval (0 < θ ≤ θ0), where θ0 is any fixedtemperature. It can be seen that this property will be established once wehave shown that∣∣∣ ∂

∂θ

(fV H(C) − f∞H(C))∣∣∣ ≤ X = const (0 < θ ≤ θ0) (65)

and this will be our aim now. We have

∂

∂θfV H(C) = − 1

V

∑f

ln1 + e−E(f)/θ − 1

θV

∑f

E(f)e−E(f)/θ

1 + e−E(f)/θ.



SinceE

θe−E/2θ ≤ 2

e,

we can write∣∣∣ ∂∂θ

fV H(C)∣∣∣ ≤ 1

V

∑f

e−E(f)/θ +2

e

1

V

∑f

e−E(f)/θ

≤(1 +

2

e

)∑f

e−E(f)/θ0 .

Completely analogously, we find

∣∣∣ ∂∂θ

f∞H(C)∣∣∣ ≤ (

1 +2

e

) 1

(2π)3

∫e−E(f)/θ0 df

and, therefore

∣∣∣ ∂∂θ

(fV H(C) − f∞H(C))∣∣∣≤

(1 +

2

e

) 1

V

∑f

e−E(f)/θ +1

(2π)3

∫e−E(f)/θ0 df

.

In view of the rapid falling off of e−E(f)/2θ0 as |p| → ∞, the integral

∫e−E(f)/2θ0 df

has a finite value, and

1

V

∑f

e−E(f)/θ → 1

(2π)3

∫e−E(f)/θ0 df.

Inequality (65) is thus established, and the uniformity of the convergence (64)with respect to θ in the interval (0 < θ ≤ θ0) is thereby also proved.Therefore, in the case under consideration, in Theorem 2 the relation

δV → 0 (V → ∞)

holds uniformly with respect to θ in this interval.



Thus, putting

ε( 1

V

)+ δV = δV ,

we can formulate paragraph 2 of Theorem 2 in the form

|f∞H(C) − f(H)| ≤ δV , (66)

δV → 0 (V → ∞) uniformly with respect to θ in the interval (0 < θ ≤ θ0).An explicit expression for ε(1/V ) has been obtained. (See footnote on

page 269.) It would also not be difficult to obtain an explicit expression forthe bound δV of the difference

fV H(C) − f∞H(C) (67)

if we impose on λα(f) the appropriate conditions of smoothness and fallingof as |p| → ∞. In fact, as we have seen, (67) is the difference between aRiemann sum and the corresponding integral,so that here we can make useof well-known technique form the theory of the approximate calculations ofthree dimensional integrals.

Thus, we can show, for example, that if the functions λα of the point (p)are continuous and differentiable everywhere, with the possible exception ofcertain sufficiently smooth discontinuity surfaces, and go to zero, togetherwith ∂λα/∂p, sufficiently rapidly as p→ ∞, then

δV ≤ const

L=

const

V 1/3.

In the case when λα are everywhere continuous, possess derivatives of secondorder with respect to (p), and, as p → ∞, go to zero along with theirderivatives of up to and including second order, we can obtain the strongbound:

δV ≤ const

V 2/3.

6. Properties of Partial Derivatives of the Free Energy Function.Theorem 3

We shall now study the partial derivatives of the function

f∞H(C) = 2∑

α

gαC∗αCα



− 1

2(2π)3/2

∫E(f) − T (f) + 2θ ln(1 + e−E(f)/θ) df (68)

with respect to the variables C1, . . . , Cs, C∗1 , . . . , C

∗s . We have

U =∂

∂Cα

E(f) − T (f) + 2θ ln(1 + e−E(f)/θ)

=

1 − 2

1 + eE(f)/θ

∂E(f)

∂Cα

= 2tanh[E(f)/2θ]

E(f)

∑β

gβC∗βλβ(f)

gαλ

∗α(f).

But, by virtue of (53) and the inequality

0 ≤ tanh x

x≤ 1,

we see that U is a bounded function of (p) in E:

|U | ≤ 1

θ

∑β

|Cβ|Q2gα.

It is clear also that U is a continuous and differentiable function of C in thewhole space of points (C). On the other hand, since | tanhx| ≤ 1, we alsohave

|U | ≤ 2

E(f)

∑β

gαgβ|Cβ| · |λβ(f)λ∗α(f)|

≤ 1∣∣∣ p2

2m− µ

∣∣∣∑

β

gαgβ|Cβ| · |λβ(f)λ∗α(f)|

Hence, it is not difficult to see that U(p) is absolutely integrable in E, and∣∣∣∣∫U dp −

∫

Sr

U dp

∣∣∣∣ =

∣∣∣∣∫

E−Sr

U dp

∣∣∣∣ ≤∫

E−Sr

|U | dp

≤ 4mgα

r2

∑β

gβ|Cβ|√∫

|λβ(f)|2 df∫|λα(f)|2 df

for r2 ≥ 4mµ. Consequently∫Sr

U dp →∫U dp (r → ∞)



uniformly with respect to C on any bounded set of points (C).Thus, the expression (68) can be differentiated with respect to Cα (or C∗

α)under the integral sign, and the corresponding derivatives

∂f∞H(C)

∂Cα

= 2gαC∗α − gα

(2π)3

∫tanh[E(f)/2θ]

E(f)

∑β

gβC∗βλβ(f)

gαλ

∗α(f) df,

∂f∞H(C)

∂C∗α

= 2gαCα − gα

(2π)3

∫tanh[E(f)/2θ]

E(f)

∑β

gβCβλ∗β(f)

gαλα(f) df

(69)

will be continuous functions of C in the whole space of points (C).It is not difficult to see that an analogous treatment is valid for partial

derivatives of f∞H(C) of any order with respect to the variables C1, . . . , Cs,C∗

1 , . . . , C∗s .

In fact, on further differentiation of the expression for U , in addition tothe factor tanh[E(f)/2θ]/E(f), the expressions

1

E

∂

∂E

(tanh[E/2θ]

E

)E=E(f)

,

1

E

∂

∂E

( 1

E

∂

∂E

tanh[E/2θ]

E

)E=E(f)

,

also appear. These are bounded functions of E (since tanh[E/2θ]/E can,for small E, be expanded in a Taylor series in even powers of E), and, asE → ∞, fall off like

1

E2∼ const

p4,

1

E3∼ const

p6, . . . .

Moreover, on differentiation of U with respect to the variables C, there appearfurther polynomials in Cα, C

∗α and λα(f), which also fail to invalidate the

above arguments.Returning to the expressions (69) for the first derivatives, we see that,

since they are continuous functions of C at the points C = Cat which theabsolute minimum of the function f∞H(C) occurs, we have

2gαC∗α − gα

(2π)3

∫tanh[E(f)/2θ]

E(f)

∑β

gβC∗βλβ(f)

gαλ

∗α(f) df = 0,



2gαCα − gα

(2π)3

∫tanh[E(f)/2θ]

E(f)

∑β

gβCβλ∗β(f)

gαλα(f) df = 0. (70)

Thus, summarizing the results just obtained, we see that the followingtheorem holds:

Theorem 3. If in the Hamiltonian (44) the operators T and Jα have theform (17) and the functions λα(f) satisfy the conditions (52a), (53) and(54) then:

1.|fV H(C) − f∞H(C)| ≤ δV (71)

for|Cα| ≤ 2M1, α = 1, . . . , s

where δV → 0 uniformly with respect to θ in the interval (0 < θ < θ0).

Here, f∞H(C) is given by the expression (68) and possess continuouspartial derivatives of all orders with respect to the variables C1, . . . , Cs,C∗

1 , . . . , C∗s for all complex values of these variables.

This function has an absolute minimum in the space of all the points (C),which is realized at certain points C = C:

min(C)

f∞H(C) = f∞H(C),

satisfying the equation (70)

2. The inequality|fV H(C) − f∞H(C)| ≤ δV (72)

holds, where δV = (ε(1/V ) + δV ) → ∞ uniformly with respect to θ inthe interval (0 < θ < θ0).

We shall now add a rider to this theorem.

7. Rider to Theorem 3 and Construction of an AuxiliaryInequality

The point C = C at which the function f∞H(C) attains an absoluteminimum is, in general, not unique. However, in the particular case when



the absolute minimum is realized at the point C = 0, the uniqueness propertyholds.

In other words, if

min(C)

f∞H(C) = f∞H(0),

thenf∞H(C) > f∞H(0)

for C = 0 (i.e. for C such that at last one of the components Cα = 0). Toestablish this property of the free energy (68) taken for the trial Hamiltonian,we shall assume that the opposite is true.

Then there exists a point C = 0 such that

f∞H(C) = f∞H(0). (73)

We putC =

√τ C, τ > 0,

and consider the function

φ(τ) = f∞H(√τ C).

Then, because of (73),φ(1) = φ(0). (74)

Making use of the expression (68) and differentiating, we find

dφ(τ)

dτ= 2

∑α

gα|Cα|2 − 1

(2π)3

∫tanh[E(f)/2θ]

E(f)

∣∣∣∑β

gβCβλ∗β(f)

∣∣∣2 df,d2φ(τ)

dτ 2=

4

(2π)3

∫eE/θ

(1 + eE/θ)

sinh(E/θ) − E/θ

E3

∣∣∣∑β

gβCβλ∗β(f)

∣∣∣4 df.Since

sinh(E/θ) −E/θ

E3> 0,

d2φ(τ)/dτ2 can go to zero only if, for all f ,∣∣∣∑β

gβCβλ∗β(f)

∣∣∣ = 0



identically. But in this case,∫tanh[E(f)/2θ]

E(f)

∣∣∣∑β

gβCβλ∗β(f)

∣∣∣2 d = 0

also, so thatdφ(τ)

dτ= 2

∑α

gα|Cα|2 > 0, τ ≥ 0.

But this inequality contradicts (74). Consequently,

d2φ(τ)

dτ 2> 0. (75)

On the other hand, since C = 0 gives the absolute minimum of f∞H(C),we have

φ(τ) ≥ φ(0), τ > 0.

Therefore (dφ(τ)/dτ)τ=0 cannot be negative:(dφ(τ)

dτ

)τ=0

≥ 0.

Hence, it follows from (75) that

dφ(τ)

dτ> 0 for τ > 0

and, consequently,φ(1) > φ(0),

which again contradicts (74). Our rider is thus proved.To conclude this section, which contains preliminary results relating to

the properties of the free energies fV (H), fV H(C) and f∞H(C), weshall prove one more equality, which we shall use frequently in the followingdiscussions.

We shall consider systems defined by a Hamiltonian which dependslinearly on some parameter τ :

Hτ = Γ0 + τΓ1.

We shall formally define the expression

fV (Hτ) = − θ

Vln Tr e−Hτ /θ,



which we shall call the free energy per unit volume for the model system Hτ .Differentiating this expression, we have

d

dτfV (Hτ ) =

1

V

Tr Γ1 e−Hτ /θ

Tr e−Hτ /θ=

1

V〈Γ1〉Hτ (76)

and

d2fV (Hτ)

dτ 2= −1

θ

1∫

0

TrΓ1 e−(Hτ /θ)ξΓ1 e−(Hτ /θ)(1−ξ) dξTr e−Hτ /θ

whereΓ1 = Γ1 − 〈Γ1〉Hτ .

But, as we have shown in section 1 of this chapter

d2fV (Hτ)

dτ 2≤ 0, (77)

in view of whichdfV (Hτ )

dτ

τ=1

≤ dfV (Hτ)

dτ≤

dfV (Hτ )

dτ

τ=0

(0 ≤ τ ≤ 1)

and, therefore, for the difference

fV (Γ0 + Γ1) − fV (Γ0) =

1∫

0

d

dτfV (Hτ) dτ

we obtain the inequality

dfV (Hτ)

dτ

τ=1

≤ fV (Γ0 + Γ1) − fV (Γ0) ≤dfV (Hτ )

dτ

τ=0

.

Thus, on the basis of (76) we have established the following importantinequality

1

V〈Γ1〉Γ0+Γ1 ≤ fV (Γ0 + Γ1) − fV (Γ0) ≤ 1

V〈Γ1〉Γ0. (78)

We shall make use of these inequalities later when we specify concrete modeland trial systems and choose the source terms in an appropriate way.



8. On the Difficulties of Introducing Quasi-Averages

We shall now study the question of the determination of quasi-averages.Let A be some operator of the type for which the limit Theorems wereformulated in [13, 14], e.g. a product of Fermi amplitudes, field functions orsimilar operators. Then the quasi-average

≺ A H

of such an operator will be defined, for the Hamiltonian (44) underconsideration, as the limit

≺ A H= limν→0

(lim

V →∞〈A 〉Γ

)(79)

of an ordinary average〈A 〉Γ

taken over a Hamiltonian Γ obtained from H by adding “source terms” to it:

Γ = H − V∑

α

(ναJ†α + ν∗αJ

†)

= T − 2V∑

α

gαJαJ†α − V

∑α

(ναJ†α + ν∗αJ). (80)

We now wish to call attention to certain difficulties associated with thedefinition (79). Thus, in the definition given, there is no indication in whichregion the parameters ν must lie, or how they must tend to zero in order toensure convergence in the definition (79).

We shall show that, even in the simplest cases, if |ν| tends to zeroarbitrary, the limit lim

η→0may not exist.

We shall take, as an example, the Hamiltonian (11)

H = T − 2V gJJ†,

which we have examined in a number of papers;ll the basic results of thesewere summarized briefly in [13, 14].

We recall that here T and J are given by the formulae (2), and thefunction λ(f) satisfies all the conditions imposed in §1 (conditions I and I′).

llSee, for example, [4, 6] and [7].



For Γ, we took a Hamiltonian with real positive ν:

Γ = Γν = T − 2V gJJ† − νV (J + J†), (ν > 0). (81)

As we have shown, Γ reduces to the form (11)

Γ = Γa − 2V g(J − C(ν)

)(J † − C(ν)

),

Γa = T − 1

2

∑f

Λ(f)a†fa†−f + a−faf + 2gV C2,

Λ(f) = 2gλ(f)C(ν) +

ν

2g

.

Hence,

C(ν) +ν

2g> 0

and the quantity C = C(ν) realizes the absolute minimum of thefunction f∞Γ(C):

min(C)

f∞Γ(C) = f∞Γ(C(ν))

Moreover (see (13)),

〈(J − C(ν))(J† − C(ν)

)〉Γ ≤ εV → 0, (V → ∞). (82)

It can be seen that the Hamiltonian belongs to the class (14), and, by virtueof the inequality (82) and the conditions imposed on λ(f), the conditions Iand I′ of §1 are fulfilled.

Because of this, we can make use of the above-mentioned limit theoremsand establish the existence of limit of the type

limV →∞

〈A 〉Γ = limV →∞

〈A 〉Γa .

We note further that, as has already been pointed out (see (7)),

C(ν) → C(0) = C (ν > 0, ν → 0). (83)

On the other hand, the expression

limV →∞

〈A 〉Γa



can be expanded in explicit form using the rules of Bloch and de Dominics,and it can be proved in a completely elementary way that the passage tothe limit ν → 0 (ν > 0) can be made and reduces simply to replacing C(ν)by C in this expression, i.e. to replacing the averaging over Γa by averagingover H(C).

In this way we can establish the existence of the quasi-averages

≺ A H= lim“ν→0,ν>0

” limV →∞


〈A 〉H(C). (84)

With the definition (79), and for the case when

C = 0, (85)

we shall now examine how the situation changes when we go over to complexvalues of ν and, in place of the Hamiltonian (81), we take

Γν, ν∗ = T − 2V gJJ† − V (νJ† + ν∗J). (86)

Here, we shall put ν = |ν|eiϕ and note that Γν, ν∗ can be reduced to the formΓ = Γ|ν| (i.e. to the Hamiltonian (81) with |ν| in place of ν) by means of thegauge transformation

af → af ei(ϕ/2), a†f → a†f e−i(ϕ/2).

Thus, we obtain, for example,

〈a†f(t)a†−f (τ)〉Γν, ν∗ = e−iϕ〈a†f(t)a†−f (τ)〉Γ|ν|

=ν∗

|ν| 〈a†f (t)a†−f(τ)〉Γ|ν| . (87)

The limitlim|ν|→0

limV →∞

〈a†f (t)a†−f(τ)〉Γ|ν| (88)

obviously exists and is given by the formula (8) in which C replaces inthe expression for u(f), v(f) and E(f). Then, in the case (85) underconsideration, the expression (88) does not go identically to zero.

Consequently, although

limV →∞

〈a†f(t)a†−f(τ)〉Γν, ν∗



always exists for |ν| > 0, the limit

limν→0

limV →∞

〈a†f(t)a†−f(τ)〉Γν, ν∗ (89)

does not, for the trivial reason that the ratio ν/|ν| does not tend to any limitas ν → 0.

The limit (89) exists only when ν tends to zero in such a way that theratio ν/|ν| is finite.

In the general case (80), with the passage to the limit ν → 0 thesituation is found, naturally, to be even more complicated. Apart from gaugeinvariance (due to the gauge group), other groups of transformations can alsooccur, e.h. the rotation group.

We shall now direct our attention to a difficulty which is specific for s > 1.

We take the Hamiltonian

H = T − 2V gJ1J†1 − 2V gJ2J

†2 ,

Γ = H − V ν1(J1 + J †1) + ν2(J2 + j†2).

Here, we take ν1 and ν2 to be real and positive.

We put here

J1 =J√2, J2 = − J√

2,

where the operators J and T have the same form as in the Hamiltonian (1),and (81).

In the given case, H will, in this way, be the same Hamiltonian (1) thatwe have just considered.

We take ν1 = ν2. Then the source will drop out completely:

Γ = H

and, since the operator H conserves the number of particles, we have,identically,

〈a†fa†−f 〉Γ = 0.

It can be seen that in such a situation we cannot define a quasi-averagecorrectly at all.



9. A New Method of Introducing Quasi-Averages

In order to avoid difficulties of the above type, we propose that ν be takenproportional to C with positive proportionality coefficients:

νa = rα Cα, rα > 0, α = 1, 2, . . . , s. (90)

In such a case, we shall consider the trial Hamiltonian

Γα = T − 2V∑

α

gαCαJ†α + C∗

αJα

− V∑

α

rαCαJ†α + C∗

αJα + const. (91)

Here we shall not write out the constant term, since it affect neither thecalculations of the averages 〈. . .〉Γα nor the equation of motion. It can beseen that we have obtained a trial Hamiltonian for H with the transformedparameters

gα → gα +rα

2,

Hr = T − 2V∑

α

(gα +

rα

2

)JαJ

†α.

In order that Γa from (91) be a trial Hamiltonian not for Hr but for theoriginal H , we must replace gα by gα − rα/2 in the expression (80) for Γ (inwhich ν is taken in accordance with (90)), thereby obtaining,

Γ = T − 2V∑

α

(gα +

rα

2

)JαJ

†α − V

∑α

rαJ†αCα + JαC

∗α.

It is clear that here, apart from adding “sources”, we have performed a“renormalization” of the parameters gα.

We can add any constant term to this expression for Γ since it will notaffect either the average 〈. . .〉Γ or the equation of motion.

As such a constant term, we shall take

V∑

α

rαC∗αCα.

Then the Hamiltonian Γ will be represented by the form

Γ = T − 2V∑

α

(gα +

rα

2

)JαJ

†α − V

∑α

rαJ†αCα + JαC

∗α



+ V∑

α

rαC∗αCα = H + V

∑α

rα(Jα − Cα)(J†α − C∗

α). (92)

We emphasize that here, as always, C denotes the point at which the absoluteminimum of the function f∞H(C) (see (63)) is attained. For notationalconvenience, in (92) we put

rα = 2ταgα, where τα > 0.

Thus, we shall be concerned with a Hamiltonian having the form

Γ = H + 2V∑

α

ταgα(Jα − Cα)(J†α − C∗

α) = T − 2v∑

α

gαJαJ†α

+ 2V∑

α


α)

τα > 0, α = 1, 2, . . . , s. (93)

We shall show that, with the above choice of Γ, no difficulties will now arisein the definition of the quasi-averages

A H= limτ→0

limV →∞

〈A 〉Γ, τα > 0, α = 1, . . . , s.

For this, we note first of all that, for

τ1 = 1, . . . , τs = 1,

from (93) we shall have:

Γ = T − 2V∑

α

gα(J †αCα + JαC

∗α) + 2V

∑α

gαC∗αCα = H(C)

(cf. also formula (99)). Since

(Jα − Cα)(J†α − C∗

α) ≥ 0,

we see that

H(C) − Γ ≥ 0 and Γ −H ≥ 0 for 0 < τα < 1. (94)

Consequently, the inequalities

fV H(C) ≥ fV (Γ) ≥ fV (H) (for 0 < τα < 1) (95)



are valid. However,

0 ≤ fV H(C)−fV (H) ≤ ∣∣f∞H(C)−fV (H)∣∣+ ∣∣f∞H(C)−fV H(C)∣∣.

Therefore, on the basis of Theorem 3, we obtain

0 ≤ fV H(C) − fV (H) ≤ δV + δV .

Hence, taking (95) into account, we find

0 ≤ fV (Γ) − fV (H) ≤ δV + δV ,

0 ≤ fV H(C) − fV (Γ) ≤ δV + δV . (96)

We make use now of inequality (78) and substitute into it

Γ0 = H, Γ1 = Γ −H = 2V∑

α


α).

Then, form the first of the inequalities (96), we obtain

2∑

α

ταgα〈(Jα − Cα)(J†α − C∗

α)〉Γ ≤ δV + δV . (97)

We have thus proved the following theorem:

Theorem 4. Let the conditions of Theorem 3 be fulfilled, and let Γ berepresented by the expression (93) in which

0 < τα < 1, α = 1, 2, . . . , s.

Then the following inequalities hold:

0 ≤ fV (Γ) − fV (H) ≤ δV + δV → 0 as V → ∞,∑α

gα〈(Jα − Cα)(J†α − C∗

α)〉Γ ≤ δV + δV

2τ0→ 0 as V → ∞, (98)

where τ0 is the smallest of the quantities τ1, τ2, . . . , τs.



RIDER TO THEOREM 4

We shall consider the more general case when, in the expression

H = T − 2V∑

α

gαJαJ†α,

the operators T and Jα are not represented by (17) and satisfy only theconditions of Theorem 2.

Then, replacing Theorem 3 by Theorem 2 in the arguments carriedthrough above, we see that Theorem 4 remains true.

We note, further, that in the case when the operators have the specificform (17) and the conditions of Theorem (3) are fulfilled, the Theorem 4proved above enables us to transform the Hamiltonian Γ directly to theform (14), (15), and the conditions I and I′ of §1 are found to be fulfilled.

In fact, we have

H = T − 2V∑

α

gα(J†αCα + JαC

∗α) + 2V

∑α

gαC∗αCα

− 2V∑

α

gα(Jα − Cα)(J†α − C∗

α),

and, therefore

Γ = H(C) − 2V∑

α

gα(1 − τα)(Jα − Cα)(J†α − C∗

α), (99)

where

H(C) = T − 2V∑

α

gα(J†αCα + JαC

∗α) + 2V

∑α

gαC∗αCα.

It can be seen that this Hamiltonian has the form of the Hamiltonian(14), (15) in which we put

Γα = H(C),

Λ(f) = 2∑

α

gαλα(f)C∗α,

K = 2V∑

α

gαC∗αCα,

Gα = 2gα(1 − τα) > 0, Cα = Cα. (100)



By virtue of Theorem 4, the following inequality is fulfilled:∑α

Gα〈(Jα − Cα)(J†α − C∗

α)〉Γ ≤ εV , (101)

where

εV =δV + δVτ0

(V → ∞)

uniformly with respect to the temperature θ in any interval of the form(0 < θ ≤ θ0).

The validity of paragraph 3 of condition I is thereby also established.The remaining paragraphs of conditions I and I′ follow trivially form the

inequalities (52) and (53), the condition (54), the fact that s is finite in thesum over α, and the fact that the quantities C are independent of V .

We can therefore make use of all the limit theorems proved in [13, 14].Since Γa = H∞(C), we write the theorems on the existence of the limits

limV →∞


〈A 〉Γa

in the formlim

V →∞〈A 〉Γ = lim

V →∞〈A 〉H(C).

But H(C) is independent of the parameters τ in the case under considerationwhen

0 < τα < 1, α = 1, 2, . . . , s. (102)

Therefore, the expressionlim

V →∞〈A 〉Γ

is also independent of the τ lying in the region (102)Consequently, when all the τ1, . . . , τs tend to zero while remaining

positive, we have, trivially

limτ→0

limV →∞


〈A 〉H(C).

We can define the quasi-average in this situation by the relations

≺ A H= limV →∞


〈A 〉H(C) (103)

in which the τ can take any values from the range (102).



We emphasize again that the most important point in our arguments wasthe establishment of the inequality (97), based on the inequality (96).

It can be seen that from the inequality (96) there follows the asymptoticrelation

limV →∞

fV (Γ) = limV →∞

fV (H). (104)

10. The Question of the Choice of Sign for the Source-Terms

We note, in passing, that the above asymptotic relation ceases, in general,to be true for negative values of τ . In fact, for example,

τα = −ωα; ωα > 0 (α = 1, 2, . . . , s). (105)

ThenΓ = Γω = H − 2V

∑α

ωαgα(Jα − Cα)(J†α − C∗

α).

We make use of the inequality (78), substituting in it

Γ0 = Γ, Γ1 = 2V∑

ωαgα(Jα − Cα)(J†α − C∗

α).

Then in (78) the Hamiltonian will be

Γ0 + Γ1 = H

andfV (H) − fV (Γω) ≥ 2

∑ωαgα〈(Jα − Cα)(J†

α − C∗α)〉H . (106)

But, obviously,〈a†fa†−f〉H = 〈a−faf〉H = 0,

and therefore〈Jα〉H = 0, 〈J†

α〉H = 0.

We have, consequently

〈(Jα − Cα)(J†α − C∗

α)〉H = 〈JαJ†α〉H + |Cα|2,

whence, by virtue of (106), we find:

fV (H) − fV (Γω) ≥ 2∑

ωαgα|Cα|2,



and, passing to the limit,

limV →∞

fV (H) − limV →∞

fV (Γω) ≥ 2∑

ωαgα|Cα|2. (107)

Thus, if C = 0, the equality (104) is not true for negative τ (105).It was to take this fact into account that we required that the

proportionality coefficients rα in our choice (90) of parameters να

characterizing the sources included in the Hamiltonian be positive.

11. The Construction of Upper-Bound Inequalitiesin the Case C = 0

We give here a separate treatment of the case when C = 0, i.e. when

C1 = C2 = . . . = Cs = 0. (108)

In this case,

H(C) = T,

and, therefore,

fv(H) − fV (T ) → 0 as V → ∞.

Thus, the interaction terms

−2V∑

1≤α≤s

gαJαJ†α

of the Hamiltonian H are asymptotically (as V → ∞) ineffective in thecalculation of the free energy.

Further, we have (compare with (93))

Γ = Γτ = H + 2V∑

α

gαταJαJ†α = T − 2V

∑α

gα(1 − τα)JαJ†α, (109)

and, in view of our earlier proofs, we can write an upper bound for acorrelation average constructed on the basis of this Hamiltonian:

⟨∑α

gαJαJ†α

⟩Γ≤ δV + δV

2τ0→ 0 as V → ∞. (110)



We shall show that, in the given case (108), we have also⟨∑α

gαJαJ†α

⟩Γ≤ ζV → 0 (V → ∞). (111)

For this, we shall take the Hamiltonian

Hω = T − 2V (1 + ω)∑

α

gαJαJ†α (1 > ω > 0), (112)

and formulate a trial Hamiltonian

Hω(C) = T − 2V (1 + ω)∑

α

gα(CαJα + C∗αJ

†α)

+ 2V (1 + ω)∑

α

gαC∗αCα. (113)

We denote by C(ω) the point C giving the absolute minimum of thefunction f∞H∞(C). If for any positive ω, however small,

C(ω) = 0, (114)

the proof of the relation (111) is trivial. We need only replace H by Hω inthe equality (110) and for τα take

τα =ω

1 + ω

in the Hamiltonian Γτ . Then the Hamiltonian Γ in (110) coincides with H .It remains, therefore, to consider the case when (114) is not true for somepositive value of ω, however small.

We note that the value C = C(ω) must satisfy the equations

∂f∞Hω(C)∂C∗

α

= 0 (0 ≤ α ≤ s);

i.e. from (69),

C(ω)α =

1

2(2π)3

∫tanh[Eω(f)/2θ]

Eω(f)

∑β

(1 + ω)gβC(ω)β λ∗β(f)

λα(f) df,



where

E2ω(f) =

( p2

2m− µ

)2

+ 4(1 + ω)2∣∣∣∑

β

gβC(ω)β λ∗

β(f)∣∣∣2.

Hence, it follows that

|C(ω)α | ≤ 1

4(2π)3

∫|λα(f)| df ≤ Q1

2. (115)

We now put

(1 + ω)Cα = Xα, (116)

and note that

f∞Hω(C) = f∞H(X) − 2ω

1 + ω

∑α

gα|Xα|2. (117)

Thus, for

X (ω)α = (1 + ω)C(ω)

α ,

the expressions on the right-hand side of (117) attains an absolute minimum.Therefore

f∞H(X(ω)) − 2ω

1 + ω

∑α

gα|X(ω)α |2 ≤ f∞H(0). (118)

On the other hand,

f∞H(0) = min(X)

f∞H(X),

in view of which,

f∞H(X(ω) ≥ f∞H(0).Hence,

0 ≤ f∞H(X(ω) − f∞H(0) ≤ 2ω

1 + ω

∑α

gα|X(ω)α |2. (119)

We shall show now that

X(ω) → 0 (ω → 0). (120)



In fact, we assume the opposite to be the case. Then, since, by virtue of (115),X(ω) is bounded:

|X(ω)α | ≤ (1 + ω)

Q1

2≤ Q1,

we can always choose a sequence of positive ω′ → 0 such that

X (ω′) → X,

withX = 0. (121)

Putting ω = ω′ in (119) and passing to the limit, we find

f∞H(X) = f∞H(0). (122)

But, as we have seen, if the point C = 0 gives the absolute minimum of thefunction f∞H(C), no other points realizing the absolute minimum of thisfunction exists; in view of this, (122) and (121) are inconsistent and we havearrived at a contradiction.

Thus, the relation (120) is proved. Noting that

f∞H(0) = f∞(T ),

form (117) and (119) we obtain

−2ω

1 + ω

∑α

gα|X (ω)α |2 ≤ f∞Hω(C(ω)) − f∞(T ) ≤ 0;

i.e.;0 ≤ f∞(T ) − f∞Hω(C(ω)) ≤ ωξ(ω), (123)

where

ξ(ω) =2

1 + ω

∑α

gα|X (ω)α |2 → 0 as ω → 0. (124)

We now invoke Theorem 3. Since, in the case under consideration, H(C) =H(0) = T , we can write

|fV (H) − f∞(T )| ≤ δV → 0, (V → ∞).

For the Hamiltonian Hω, we also have

|fV (Hω) − f∞Hω(C(ω))| ≤ δV (ω) → 0, as V → ∞.



Here, δV (ω) denotes δV for Hω. On the other hand,

0 ≤ fV (H) − fV (Hω) = fV (H) − f∞(T ) + f∞(T )

− f∞Hω(C(ω)) + f∞Hω(C(ω)) − fV (Hω),

and, therefore,

0 ≤ fV (H) − fV (Hω) ≤ δV + δV (ω) + ωξ(ω). (125)

We shall now make use of the inequality (78); we substitute in it

Γ0 = Hω, Γ1 = 2V ω∑

α

gαJαJ†α,

Γ0 + Γ1 = H.

Then, from (125) we obtain

⟨∑α

gαJαJ†α

⟩H≤ δV + δV (ω)

2ω+

1

2ξ(ω). (126)

However, this inequality is valid for any value of ω in the interval (0 < ω < 1),and its left-hand side is entirely independent of ω. Consequently, the left-hand side of (126) will not exceed the lower bound of the right-hand side inthe given interval: ⟨∑

α

gαJαJ†α

⟩H≤ ζV ,

ζV = inf(0<ω<1)

δV + δV (ω)

2ω+

1

2ξ(ω)

.

It only remains for us to show that ζV goes to zero as V → ∞.We shall fix an arbitrary small number . On the basis of (124), we can,

in the interval (0 < ω < 1) under consideration, fix a number ω0 such that

ξ(ω0) ≤ .

We see then that

ζV ≤ δV + δV (ω)

2ω+

2.

But, since ω0 is fixed, we have

δV + δV (ω)

2ω→ 0 as V → ∞.



We can find a value V0 such that

δV + δV (ω)

2ω≤

2for V ≥ V0.

Thus,ζV ≤ for V ≥ V0;

i.e., ζV → 0 as V → ∞, and the relation (111) is proved.It is now clear that H is of the type (1.14), (1.15), with

Γa = T, Λ(f) = 0, Cα = 0, Gα = 2gα.

By virtue of (111), paragraph 3 of condition I is fulfilled; the remainingparagraphs of conditions I and I′ are trivial in the given case.

We shall, therefore, make use of the limit theorems proved in [13,14]. Wesee that, for the operators A with which these theorems are concerned, wecan write

limV →∞

〈A 〉H = limV →∞

〈A 〉T .Applying (103), we see that, in the case under investigation (C = 0), we have

≺ A H= limV →∞

〈A 〉H = limV →∞

〈A 〉T .

Thus, the quasi-averages and the usual averages of the operators consideredabove are asymptotically equal to the corresponding averages taken over theHamiltonian T . Moreover, the interaction terms in H turn out to have noeffect here.

Some applications for Approximating Hamiltonian Method (AHM) werealso done in [3,5,6,16,18–28,30]. It is also possible to consider Hamiltonianswith coupling constants of different signs and formulate on this basis minimaxprinciple for the Hamiltonian

H = T0 + 2V∑

(1≤α≤r)

gαJαJ†α − 2V

∑(r+1≤α≤r+s)

gαJαJ†α,

T0 =∑

f

( p2

2m− µ

)a†faf , Jα =

1

2V

∑f

λα(f)a†fa†−f ,

λα(−f) = −λα(f), gα > 0.

For more details, the reader is referred to the works by the author [10, 16].



References

1. Bogolyubov, N. N., Zubarev, D. N., and Tserkovnikov, Yu. A. Docl.Acad. Nauk. SSSR 117, 788 (1957) [Sov. Phys. Doclady 2, 535 (1957)].

2. Bogolyubov, N. N., Zubarev, D. N., and Tserkovnikov, Yu. A. Zh. Eksp.Teor. Phys. 39, 120 (1960) [Sov. Phys. JETP 12, 88 (1960)].

3. Bogolyubov, N.N., Jr. Ukrainsk. Matematichesk. Journal 17, 3 (1965)[in Russian].

4. Bogolyubov, N. N., Jr. Vestn. Mosc. Univ. Fiz. Atsron. 21(1), 94(1966) [Moscow Univ. Phys. Bull. 21(1), 67 (1966)].

5. Bogolyubov, N. N., Jr. Physica 32, 933 (1966).

6. Bogolyubov, N. N., Jr. Docl. Acad. Nauk. SSSR 168, 766 (1966) [Sov.Phys. Doclady 11, 482 (1966)].

7. Bogolyubov, N. N., Jr. ITPK (Institute for Theoretical Physics, Kiev)Preprint 67-1 (1967).

8. Bogolyubov, N. N., Jr. JINRD Preprint P4-4184 (1968).

9. Bogolyubov, N. N., Jr. Docl. Acad. Nauk. SSSR 182, 797 (1968) [Sov.Phys. Doclady 13, 1111 (1969)].

10. Bogoliubov, N. N., Jr. Yad. Fiz. 10, 425 (1969) [Sov. J. Nucl. Phys.10, 243 (1970)].

11. Bogoliubov, N. N., Jr. Physica 41, 601 (1969).

12. Bogolyubov, N. N., Jr. and Shumovski, A. S. Int. J. Pure Appl. Phys.8, 121 (1970).

13. Bogolyubov, N. N., Jr. Theoret. and Math. Phys. 4, 929 (1970).


15. Bogolyubov, N. N., Jr. and Shumovski, A. S. Phys. Lett. 35A, 380(1971).

16. Bogolyubov, N. N., Jr. A Method for Studying Model Hamiltonians,Pergamon Press (1972).


18. Bogolyubov, N. N., Jr. and Plechko V.N. Physica 82A, (163) (1975);Docl. Acad. Nauk. SSSR 288, 1061 (1976).

19. Bogolyubov, N. N., Jr. and Plechko V.N., Repnikov N. F. Theoret. andMath. Phys. 24, 886 (1975).

20. Bogolyubov, N. N., Jr., Shumovski A.S. Tr. Mat. Inst., Akad. NaukSSSR 136 351 (1975) (in Russian).



21. Bogolyubov, N. N., Jr., Brankov J.G., Zagrebnov V.A., Kurbatov A.M.,Tonchev N.S. The Approximating Hamiltonian Method in StatisticalPhysics. Publ. House Bulg. Acad. Sci., Sofia, 1981 [in Russian].

22. Bogolyubov, N. N., Jr., Plechko V.N., Shumovski A. S. Fiz. Elem.Chastits At. Yadra 14, 1443 (1983) [in Russian].

23. Bogolyubov, N. N., Jr., Brankov J.G., Zagrebnov V.A., Kurbatov A.M.,Tonchev N.S. Russian Math. Surveys 39, 1 (1984).

24. Bogolyubov, N. N., Jr. and Bogolyubova, E. N. Ukrainian Journal ofPhysics 45, 4 (2000).

25. Bogolubov, N.N., Jr., Bogolubova, E.N. and Kruchinin, S.P. Calculationof Correlation Functions for Superconductivity Models in New Trendsin Superconductivity, NATO Science Series 67, 277 (2002).

26. Brankov, J.G., Danchev, D. M., Tonchev N.S. Theory of CriticalPhenomena in Finite-Size Systems: Scaling and Quantum Effects, Seriesin Modern Condensed Matter Physics v. 9, World Scientific, Singapore,2000.

27. Brankov, J.G. and Tonchev, N.S. Cond. Matter Physics 14, 13003(2011)

28. Brankov J.G. and Tonchev N.S. Phys. Rev. E 85, 031115 (2012).

29. Maison, H.D. Preprint Th. 1299-CERN, Geneva, (1971).

30. Plechko V. N. Theoret. and Math. Phys. 28 677 (1976).

31. Shilov, G. E. and Gurevich, V. L. Integral, Measure and Derivative.A Unified Approach, Nauka, Moscow, 1964 (Translation published byPrentice-Hall, Inc., N. Y., 1966)

32. Shumovski, A. S. ITPK Preprint 71-57E, 71-56P (1971).

33. Thirring, W. and Wehrl, A. Commun. Math. Phys. 4, 303 (1967).

34. Thirring, W. Preprint, Institute of Theoretical Physics, University ofVienna (1968).

35. Wentzel, G. Helv. Phys. Acta. 33, 859 (1960).

June 30, 2014 14:0 Quantum Statistical Mechanics b1891-refs page 306


Additional References

I. GENERAL REFERENCES

1. N. N. Bogoliubov, Collection of Scientific Papers, In 12 vol.ed. A.D.Sukhanov (Moscow, Nauka, 2005-2009).

2. N. N. Bogoliubov and N. N. Bogoliubov, Jr., Introduction to QuantumStatistical Mechanics, 2nd ed. (World Scientific, Singapore, 2009).

3. N. N. Bogoliubov, Jr., B. I. Sadovnikov, A. S. Schumovsky,Mathematical Methods for Statistical Mechanics of Model Systems,(Nauka, Moscow, 1989) [in Russian].

4. N. N. Bogoliubov Jr. and D. P. Sankovich, N. N. Bogoliubov andStatistical Mechanics, Usp. Mat. Nauk. 49, 21 (1994).

5. D. Ya. Petrina, Mathematical Foundations of Quantum StatisticalMechanics, (Kluwer Academic Publ., Dordrecht, 1995).

6. P.A. Martin and F. Rothen. Many-body Problems and Quantum FieldTheory, (Springer, Berlin, 2004).

7. D. V. Shirkov, 60 Years of Broken Symmetries in Quantum Physics(from the Bogoliubov theory of superfluidity to the Standard Model), Usp.Fiz. Nauk 179, 581 (2009).

8. A.L. Kuzemsky, Bogoliubov’s Vision: Quasiaverages and BrokenSymmetry to Quantum Protectorate and Emergence, Int. J. Mod. Phys.B24, 835-935 (2010).

II. BOGOLIUBOV TRANSFORMATIONS

1. W. Witschel, On the General Linear (Bogoliubov) Transformation forBosons, Z.Physik B21, 313 (1975)

2. S. N. M. Ruijsenaars, On Bogoliubov Transformations. The GeneralCase. Annals of Physics 116, 105 (1978).

3. S. N. M. Ruijsenaars, Integrable Quantum Field Theories and BogoliubovTransformations, Annals of Physics 132, 328 (1981).

4. Nguyen Ba An, A Step-by-step Bogoliubov Transformation Method forDiagonalising a Kind of non-Hermitian Effective Hamiltonian, J. Phys.C: Solid State Phys. 21, L1209 (1988).



5. F. J. W. Hanle and A. Klein, Generalization of the QuantizedBogoliubov-Valatin Transformation, Phys.Lett. B 229, 1 (1989).

6. P. C. W. Fung, C. C. Lam and P. Y. Kwok, U-Matrix Theory,Bogoliubov Transformation and BCS Trial Wave Function, Science inChina 32, 1072 (1989).

7. W.-S. Liu and X.-P. Li, Time-dependent formulation of theBogoliubov transformation and time-evolution operators for time-dependent quantum oscillators, Europhys. Lett. 58, 639 (2002).

8. J.-W. van Holten and K. Scharnhorst, Nonlinear BogolyubovValatintransformations and quaternions, J. Phys. A: Math. Gen. 38 10245(2005).

9. L. Bruneaua and J. Dereziski, Bogoliubov Hamiltonians and one-parameter groups of Bogoliubov transformations, J. Math. Phys. 48,022101 (2007).

10. A. I.Vdovin, A. A. Dzhioev, Thermal Bogoliubov Transformation inNuclear Structure Theory, Physics of Particles and Nuclei 41, 2093(2010).

11. K. Scharnhorst and J.-W. van Holten, Nonlinear Bogolyubov-Valatintransformations: Two modes, Annals of Physics 326, 2868 (2011).

12. J. Katriel, A Nonlinear Bogoliubov Transformation, Phys. Lett. A 307,1 (2003).

13. K.Takayanagi, Utilizing Group Property of Bogoliubov Transformation,Nucl. Phys. A 808, 17 (2008).

14. S. Caracciolo, F. Palumbo and G. Viola, Bogoliubov Transformationsand Fermion Condensates in Lattice Field Theories, Annals of Physics324, 584 (2009).

III. QUASIAVERAGES

1. N. N. Bogoliubov, Jr., Method of Calculating Quasiaverages, J. Math.Phys. 14, 79 (1973).

2. N. N. Bogoliubov and N. N. Bogoliubov, Jr., Introduction to QuantumStatistical Mechanics, 2nd ed. (World Scientific, Singapore, 2009).

3. D. Ya. Petrina, Mathematical Foundations of Quantum StatisticalMechanics, (Kluwer Academic Publ., Dordrecht, 1995).



4. S. V. Peletminskii, A. I. Sokolovskii, Flux Operators of PhysicalVariables and the Method of Quasiaverages, Theor. Math. Phys. 18,121 (1974).

5. V. I. Vozyakov, On an Application of the Method of Quasiaverages inthe Theory of Quantum Crystals, Theor. Math. Phys. 39, 129 (1979).

6. D. V. Peregoudov, Effective Potentials and Bogoliubov’s Quasiaverages,Theor. Math. Phys. 113, 149 (1997).

7. N. N. Bogoliubov, Jr., D. A. Demyanenko, M. Yu. Kovalevsky, N.N. Chekanova, Quasiaverages and Classification of Equilibrium Statesof Condensed Media with Spontaneously Broken Symmetry, Physics ofAtomic Nuclei 72, 761 (2009).

8. A. L. Kuzemsky, Symmetry Breaking, Quantum Protectorate andQuasiaverages in Condensed Matter Physics, Physics of Particles andNuclei 41, 1031 (2010).

9. A. L. Kuzemsky, Quasiaverages, Symmetry Breaking and IrreducibleGreen Functions Method, Condensed Matter Physics(http://www.icmp.lviv.ua/journal), 13, N4, p.43001: 1-20 (2010).

IV. BOGOLIUBOV’S INEQUALITIES

1. A. Ishihara, The Gibbs-Bogoliubov inequality, J. Phys. A 2, 539 (1968).

2. N. D. Mermin, Some Applications of Bogoliubov’s Inequality inEquilibrium Statistical Mechanics, J. Phys. Soc. Japan 26 Supplement,203 (1969).

3. S. Okubo. Some General Inequalities in Quantum Statistical Mechanics,J. Math. Phys. 12, 1123 (1971)

4. B. I. Sadovnikov, V. K. Fedyanin, N. N. Bogoliubov’s Inequalities inSystems of Many Interacting Particles with Broken Symmetry, Theor.Math. Phys. 16, 368 (1973).

5. J. C. Garrison, J. Wong, Bogoliubov Inequalities for Infinite Systems,Commun. Math. Phys. 26, 1 (1972).

6. L. Pitaevskii, S. Stringari, Uncertainty Principle, QuantumFluctuations, and Broken Symmetries, J. Low Temp. Phys. 85, 377(1991).



7. A. V. Soldatov, Generalization of the Peierls-Bogoliubov Inequalityby Means of a Quantum-Mechanical Variational Principle, Physics ofParticles and Nuclei 31, 138 (2000).

8. A. L. Kuzemsky, Bogoliubov’s Vision: Quasiaverages and BrokenSymmetry to Quantum Protectorate and Emergence. Int. J. Mod. Phys.B 24, 835 (2010).

V. SUPERFLUIDITY, BOSE GAS

1. V.V. Tolmachev, Theory of Bose Gas. (Izd-vo Mosk. Un-ta, Moscow,1969) [in Russian].

2. V. A. Zagrebnov, Bogoliubov’s Theory of a Weakly Imperfect BoseGas and its Modern Development, in: N. N. Bogoliubov, Collectionof Scientific Papers In 12 vol. ed. A. D. Sukhanov, vol.8. (Moscow,Nauka, 2007) p.576.

3. E.H. Lieb, The Bose Gas: a Subtle Many-Body Problem, Theproceedings of the XIII International Congress of Mathematical Physics,London, July 18-24, 2000. arXiv:math-ph/0009009v1

4. V. A. Zagrebnov and J.-B. Bru, The Bogoliubov Model of WeaklyImperfect Bose Gas, Phys. Rep., 350, 291 (2001).

5. M. Corgini and D. P. Sankovich, Study of a Non-Iinteracting Boson Gas,Int. J. Mod. Phys. B 16, 497 (2002).

6. V.I. Yukalov, Self-consistent Theory of Bose-condensed Systems. Phys.Lett., A 359, 712 (2006).

7. V. I. Yukalov and H. Kleinert, Gapless Hartree-Fock-BogoliubovApproximation for Bose Gases, Phys. Rev., A 73, 063612 (2006).

8. N. N. Bogoliubov, Jr. and D. P. Sankovich, Bogoliubov’s Approximationfor Bosons, Ukr. J. Phys., 55, 104 (2010).

9. D. P. Sankovich, Bogoliubov’s Theory of Superfluidity, Revisited, Int. J.Mod. Phys. B 24, 5327 (2010).

10. V. I. Yukalov, Basics of Bose-Einstein Condensation. Physics ofParticles and Nuclei, 42, 460 (2011).

11. L. Pitaevskii, S. Stringari, Bose-Einstein Condensation, (OxfordUniversity Press, Oxford, 2003).

12. C. J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases,(Cambridge University Press, Cambridge, 2002).



13. A. Griffin, T. Nikuni, E. Zaremba, Bose-Condensed Gases at FiniteTemperatures, (Cambridge University Press, Cambridge, 2009)

14. A. Griffin, New light on the intriguing history of superfluidity in liquid4He. J. Phys.: Condens. Matter 21, 164220 (2009).

15. A. Verbeure, Many-Body Boson Systems: Half a Century Later,(Springer, Berlin, 2011).

16. C. W. Gardiner, Particle-number-conserving Bogoliubov method whichdemonstrates the validity of the time-dependent Gross-Pitaevskiiequation for a highly condensed Bose gas, Phys. Rev., A 56, 1414 (1997).

17. A. Brunello, F. Dalfovo, L. Pitaevskii, S. Stringari, How to Measure theBogoliubov Quasiparticle Amplitudes in a Trapped Condensate, Phys.Rev. Lett. 85, 4422 (2000).

18. J. M. Vogels, K. Xu, C. Raman, J. R. Abo-Shaeer, W. Ketterle,Experimental Observation of the Bogoliubov Transformation for a Bose-Einstein Condensed Gas, Phys. Rev. Lett. 88, 060402 (2002).

19. M. W. J. Romans and H. T. C. Stoof, Bogoliubov theory of Feshbachmolecules in the BEC-BCS crossover, Phys. Rev., A 74, 053618 (2006).

20. E. H. Lieb, R. Seiringer, J. Yngvason, Bose-Einstein Condensation andSpontaneous Symmetry Breaking, Rep. Math. Phys. 59, 389 (2007).

21. P. Shygorin and A. Svidzynsky, Modified Bogolyubovs Derivation of theTwo-Fluid Hydrodynamics, Ukr. J. Phys. 55, 109 (2010).

22. A. A. Svidzinsky and M. O. Scully, Condensation of N bosons:Microscopic approach to fluctuations in an interacting Bose gas, Phys.Rev., A 82, 063630 (2010).

VI. SUPERCONDUCTIVITY

1. N. N. Bogoliubov, To the Question of Model Hamiltonian inSuperconductivity Theory, Sov. J. Part. Nucl. 1, 1 (1971).

2. V. V. Tolmachev, Theory of Fermi Gas. (Izd-vo Mosk. Un-ta, Moscow,1973) [in Russian].

3. C. T. Chen-Tsai, On the Bogoliubov-Zubarev-Tserkovnikov Method inthe Theory of Superconductivity, Chinese J. Phys. 3, 22 (1965).

4. D. H. Kobe, Green’s Functions for Fermion Systems with PairingCorrelations, Ann. Phys. 20, 279 (1962).



5. D. H. Kobe, Green’s Functions Derivation of the Method of ApproximateSecond Quantization, Ann. Phys. 25, 121 (1963).

6. D. H. Kobe, Self-Energy for the Bogoliubov Quasiparticle. Ann. Phys.28, 400 (1964).

7. D. H. Kobe, Best Energy Criterion and the Principle of Compensationof Dangerous Diagrams, Ann. Phys. 40, 395 (1966).

8. D. H. Kobe, Derivation of the Principle of Compensation of DangerousDiagrams, J. Math. Phys. 8, 1200 (1967).

9. G. A. Raggio and R.F. Werner, The Gibbs Variational Principle forGeneral BCS-type Models, Europhys. Lett. 9, 633 (1989).

10. D. Waxman, Fredholm Determinant for a Bogoliubov Hamiltonian.Phys. Rev. Lett. 72, 570 (1994).

11. N. N. Bogoliubov, V. A. Moskalenko, On Question of the Existence ofSuperconductivity in the Hubbard Model, Theor. Math. Phys. 86, 16(1991).

12. V.V. Tolmachev, Superconducting BoseEinstein condensates of Cooperpairs interacting with electrons, Phys. Lett. A 266, 400 (2000).

13. H. Matsui, T. Sato, T. Takahashi, S.-C. Wang, H.-B. Yang, H. Ding, T.Fujii, T. Watanabe, A. Matsuda, BCS-Like Bogoliubov Quasiparticlesin High-Tc Superconductors Observed by Angle-Resolved PhotoemissionSpectroscopy, Phys. Rev. Lett. 90, 217002 (2003).

14. M. de Llano and V. V. Tolmachev, A Generalized BoseEinsteinCondensation Theory of Superconductivity Inspired by Bogolyubov. Ukr.J. Phys. 55, 79 (2010).

15. J. Batle, M. Casas, M. Fortes, F. J. Sevilla, M. A. Solis, M. de Llano,O. Rojo and V. V. Tolmachev, BCS and BEC Finally Unified: aBrief Review, in: Condensed Matter Theories, volume 18 (Nova SciencePublishers, Inc., 2003) p.111.


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