new vista on excited states. contents monte carlo hamiltonian: effective hamiltonian in low energy...

New Vista On Excited States New Vista On Excited States

ContentsContents

• Monte Carlo Hamiltonian:• Effective Hamiltonian in low energy • /temperature window

• - Spectrum of excited states• - Wave functions• - Thermodynamical functions • - Klein-Gordon model• - Scalar φ^4 theory• - Gauge theory

• Summary

Critical review of Lagrangian vs Critical review of Lagrangian vs Hamiltonian LGT Hamiltonian LGT

• Lagrangian LGT: • Standard approach- very sucessfull. • Compute vacuum-to-vacuum transition

amplitudes• Limitation: Excited states spectrum, • Wave functions

• Hamiltonian LGT:• Advantage: Allows in principle for

computation of excited states spectra and wave functions.

• BIG PROBLEM: To find a set of basis states which are physically relevant!

• History of Hamilton LGT: - Basis states constructed from

mathematical principles (like Hermite, Laguerre, Legendre fct in

QM). BAD IDEA IN LGT!

- Basis constructed via perturbation theory: Examples: Tamm-Dancoff, Discrete Light

Cone Field Theory, …. BIASED CHOICE!

STOCHASTIC BASISSTOCHASTIC BASIS

• 2 Principles: - Randomness: To construct states which sample a

HUGH space random sampling is best.- Guidance by physics: Let physics tell us which

states are important. Lesson: Use Monte Carlo with importance

sampling! Result: Stochastic basis states. Analogy in Lagrangian LGT to eqilibrium

configurations of path integrals guided by exp[-S].

Construction of BasisConstruction of Basis

…

t

T

0 X

4X

fiX

2

T 3X 5X2X1X 6X

…

7X.. . . . . .

.

.inX

Box FunctionsBox Functions

Monte Carlo HamiltonianMonte Carlo Hamiltonian

M ijT =< x i∣e−HT / ℏ∣x j i , j∈1,2, .. . , N

H. Jirari, H. Kröger, X.Q. Luo, K.J.M. Moriarty, Phys. Lett. A258 (1999) 6.C.Q. Huang, H. Kröger,X.Q. Luo, K.J.M. Moriarty, Phys.Lett. A299 (2002) 483.

Transition amplitudes between position states.

Compute via path integral. Express as ratio of path integrals. Split action: S =S_0 + S_V

M ijT =M0 ijT

∫ [dx ]exp[−SV ]exp[−S0]∣xj,oxi,T

∫ [dx ]exp[−S0 ]∣xj,0xi,T

=M0 ij T exp[−SV ]¿

¿

Diagonalize matrix

M T =U D T U

Uik=< x i∣E

eff k¿

¿Dk T =exp[−Eeff kT /ℏ ]

Spectrum of energies and wave funtions

Effective Hamiltonian

H eff =∑ k∣Eeff k ⟩ E

eff k ⟨ Eeff k∣

Many-body systems – Quantum field theory:Essential: Stochastic basis: Draw nodes x_i from probability distribution derived from physics – action.

Path integral. Take x_i as position of paths generated by Monte Calo with importance sampling at a fixed time slice.

P y =∫ [dx ]exp [−S ]∣0

y

∫ dy∫ [dx ]exp [−S ]∣0y

Thermodynamical functions:

Definition: Z β =Tr [exp−βH ] ,

U β =−∂ logZ∂ β

U β =N s

2a t

1N t

⟨ ∂∂at

S⟩Lattice:

Monte Carlo Hamiltonian: Z eff β =∑

n=1

N

exp[−βEeff n ] ,

U eff β =−1Zeff β

∑n=1

N

Eeff

nexp[−βEeff

n ]

Klein Gordon ModelKlein Gordon ModelX.Q.Luo, H. Jirari, H. Kröger, K.J.M. Moriarty, Non-perturbative Methods and Lattice QCD, World Scientific Singapore (2001), p.100.

Energy spectrumEnergy spectrum

Free energy beta x F

Average energy U

Specific heat C/k_B

Scalar ModelScalar Model

C.Q. Huang, H. Kröger,X.Q. Luo, K.J.M. Moriarty

Phys.Lett. A299 (2002) 483.

Energy spectrumEnergy spectrum

Free energy FFree energy F

Average energy UAverage energy U

Entropy SEntropy S

Specific heat CSpecific heat C

LLatticeattice gauge theory gauge theory

Principle: Physical states have to be gauge invariant!

Construct stochastic basis of gauge invariant states.

∣U ⟩≡∣U 12 ,U 23, . .. ⟩

∣U inv ⟩=ZN∫ dg1dg2dg3 .. .∣g1U 12g2−1 , g2U 23g3

−1 ,. . . ⟩

Abelian U(1) gauge group. Abelian U(1) gauge group. Analogy: Q.M. – Gauge theoryAnalogy: Q.M. – Gauge theory

l = number of links = index of irreducible representation.

[ P , X ]=ℏ/i⇔ [ E , U ]=− U⟨p∣x ⟩=exp ipx /2πℏ⇔⟨l∣U ⟩=U l

Fourier Theorem – Peter Weyl Theorem

∑l=0,±1,±2, . ..

∣l ⟩ ⟨ l∣=1, ⟨ l'∣l⟩=δl ' , l

∑l=0,±1,±2, . ..

⟨U '∣l ⟩ ⟨ l∣U ⟩=δ U '−U

∫ dU ∣U ⟩ ⟨U ∣=1, ⟨U '∣U ⟩=δ U '−U

∫dU ⟨ l'∣U ⟩ ⟨U∣l ⟩=δl ' , l

∫ dU ∣U ⟩ ⟨U ∣=1, ⟨U '∣U ⟩=δ U '−U

Transition amplitude between Transition amplitude between Bargmann statesBargmann states

⟨U12fi ,U23

fi ,U43fi ,U14

fi ∣exp [−HelecT /ℏ ]∣U12in ,U23

in ,U43in ,U14

in ⟩

¿ ∏ij=12,23 ,43,14 { ∑

nij=0,±1,±2, . .exp[−g2ℏT

2anij2 ]cos [nijα ij

fi−aijin ]}

Transition amplitude between Transition amplitude between gauge invariant statesgauge invariant states

inv ⟨U12fi ,U23

fi ,U 43fi ,U14

fi ∣exp [−HelecT /ℏ ]∣U12in ,U23

in ,U 43in ,U14

in ⟩inv

¿ 12π 4

∫0

2π

dβ1.. .∫0

2π

dβ4

∏ij=12,23,43,14{ ∑

nij=0,±1,±2, ..exp[−g2ℏ T

2anij2 ]cos [nijα ij

fi−aijinβi−β j ]}

Result:Result:

• Gauss’ law at any vertex i:

∑jnij=0

inv ⟨U12fi ,U23

fi ,U 43fi ,U14

fi ∣exp [−HelecT /ℏ ]∣U12in ,U23

in ,U 43in ,U14

in ⟩inv=

∏ { ∑nplaq=0,±1,±2, ..

exp[−g2ℏT2a

4nplaq2 ]cos [nplaq θplaq

fi −θplaqin ]}

θplaq=α 12α 23α 34α 41Plaquette angle:

Electric Hamiltonian…Electric Hamiltonian…Lattice results versus Lattice results versus

analytical results analytical results

Energy Spectrum 2x2Energy Spectrum 2x2

Energy Spectrum 3x3Energy Spectrum 3x3

Energy Spectrum 4x4Energy Spectrum 4x4

Energy Spectrum 10x10Energy Spectrum 10x10

4x44x4

Scaling of energy levels 2x2Scaling of energy levels 2x2

Scaling of wave functions 2x2Scaling of wave functions 2x2

Scaling of excited states: energy - Scaling of excited states: energy - wave fct. 2x2wave fct. 2x2

Scaling of exited states: energy - Scaling of exited states: energy - wave fct. 2x2wave fct. 2x2

Energy scaling: 3x3, a_s=1Energy scaling: 3x3, a_s=1

Energy scaling: 3x3, a_s=0.05Energy scaling: 3x3, a_s=0.05

Energy scaling: 6x6Energy scaling: 6x6

Wave fct scaling: 6x6Wave fct scaling: 6x6

Wave fct scaling: ground state + 1st Wave fct scaling: ground state + 1st excited state: 6x6excited state: 6x6

Wave fct scaling. 2Wave fct scaling. 2ndnd excited state: excited state: 6x6 6x6

Wave fct scaling: 3Wave fct scaling: 3rdrd excited state: excited state: 6x66x6

Spectrum: 8x8Spectrum: 8x8

Spectrum -Degeneracy: 8x8Spectrum -Degeneracy: 8x8

Spectrum - Error estimate: 8x8Spectrum - Error estimate: 8x8

Energy scaling: 8x8Energy scaling: 8x8

Thermodynamics: Average energy Thermodynamics: Average energy U: 2x2U: 2x2

Free energy FFree energy F

Entropy SEntropy S

Specific heat CSpecific heat C

Including Magnetic Term…Including Magnetic Term…

Comparison of electric and...Comparison of electric and...

... full Hamiltonian: 2x2, a_s=a_t=1... full Hamiltonian: 2x2, a_s=a_t=1

a_s=1, a_t=0.05a_s=1, a_t=0.05

Application of Monte Carlo Hamiltonian- Spectrum of excited states- Wave functions- Hadronic structure functions (x_B, Q^2) in

QCD - S-matrix, scattering and decay amplitudes.

IV. OutlookIV. Outlook