Lorentz symmetry in QFTon quantum Bianchi I space-time

Andrea Dapor,1,* Jerzy Lewandowski,1,† and Yaser Tavakoli2,‡

1Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, ul. Hoza 69, 00-681 Warsaw, Poland2Departamento de Fısica, Universidade da Beira Interior, Rua Marques d’Avila e Bolama, 6200 Covilha, Portugal

(Received 11 July 2012; published 5 September 2012)

We develop the quantum theory of a scalar field on loop quantum cosmology Bianchi I geometry. In

particular, by considering only the single modes of the field, the evolution equation is derived from the

quantum scalar constraint: it is shown that the same equation can be obtained from quantum field theory

on a ‘‘classical’’ effective geometry. We then study the dependence of this effective space-time on the

wave vector of the modes (which could in principle generate a deformation in local Lorentz symmetry), by

analyzing the dispersion relation for propagation of the test field on the resulting geometry. We show that

when we disregard the backreaction no Lorentz violation is present, despite the effective metric being

different than the classical Bianchi I. Furthermore, a preliminary analysis of the correction due to

inclusion of backreaction is briefly discussed in the context of Born-Oppenheimer approximation.

DOI: 10.1103/PhysRevD.86.064013 PACS numbers: 04.60.�m, 04.60.Pp, 98.80.Qc

I. INTRODUCTION

Quantum field theory (QFT) in curved space-time is atheory wherein matter is treated quantum-mechanically,but gravity is treated classically in agreement with generalrelativity [1,2]. Despite its classical treatment of gravity,QFT on curved space-time has provided a good approxi-mate description in circumstances where the quantumeffects of gravity do not play a dominant role. On the otherhand, during the cosmological era arbitrarily close to theclassical singularity, such effects cannot be neglectedand the theory is no longer valid. This suggests that thebackground classical space-time in the Planck regimenear the singularity has to be replaced by a quantumbackground [3].

Loop quantum cosmology (LQC) [4,5] has become, inthe last few years, an interesting candidate for a quantumdescription of the early Universe at the Planck scale,providing a number of concrete results. This approach,that comprises a family of symmetry-reduced cosmologi-cal models within the framework of loop quantum grav-ity (LQG) [6–8], was first fully applied to the spatiallyk ¼ 0 Friedman-Robertson-Walker (FRW) cosmologycoupled to a massless scalar field (that also serves asan internal time parameter [9–11]). It was shown numeri-cally that the big bang singularity is replaced by aquantum bounce, and that the energy density is boundedby a critical density �cr � 0:41�Pl of the order of thePlanck density (see also Refs. [12–16]). Furthermore,extensions of this quantization method were successfullyexposed for k ¼ 0 models with or without cosmologicalconstant [17,18], k ¼ 1 closed models [19,20], k ¼ �1open models [21], and for the simplest anisotropic

cosmology (namely, Bianchi I [22], Bianchi II [23], andBianchi IX [24] models).Discrete approaches to quantum gravity lead to a break-

down of the usual structure of space-time at around thePlanck scale, with possible violations of Lorentz symme-try. This can have phenomenological implications such as adeformation of the dispersion relations for propagatingparticles (modes of a matter field) [25–27]. These modifi-cations would change the speed of propagation, introduc-ing delays/advances for particles of different energies.These can be in principle detected in experimental testssuch as observation of gamma ray bursts [25].Of course, to study such effects in the framework of

LQC, one needs to couple a quantum field to a LQC space-time, in other words, to develop QFT on LQC geometry.From a field-theoretic perspective, LQC provides a familyof well-defined quantum geometries on which quantummatter can propagate. The first step in the construction ofQFT on such quantum geometries has been developed inRef. [28], for a real massless scalar field on k ¼ 0 FRWquantum space-time. Therein, a mode decomposition ofthe quantum fields was given, so that the field could beregarded as a collection of decoupled harmonic oscillators.This allows one to study each quantummode separately, byignoring the infinite number of degrees of freedom of thefield. In that work, it was shown that each mode of the field‘‘sees’’ the underlying quantum geometry as an effectiveclassical one. In principle, such effective geometry coulddepend on the wave vector of the mode, thereby producingan (apparent) Lorentz violation: different quanta wouldsimply propagate on different space-times. It turns outthat in the FRW case this effective metric is the same forall modes, and thus no Lorentz violation is present. Apossible explanation of this lies in the fact that FRWspace-time is conformally flat and, therefore, masslessfields (such as the one used in Ref. [28]) do not distinguishit from Minkowski space-time. If this is the case, a more

*adapor@fuw.edu.pl†jerzy.lewandowski@fuw.edu.pl‡tavakoli@ubi.pt

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significant test would be to consider more complicatedspace-times such as the Bianchi type-I model.

The purpose of this paper is to generalize the model ofQFT on FRW cosmological space-time to that of theBianchi I model and analyze the issue of Lorentz symmetryin this context. In light of the Belinskii-Khalatnikov-Lifshitz conjecture [29,30], such space-times are themost interesting ones since they are more likely to repre-sent the cosmological dynamics in the Planck-scale regimewhen the matter source (without any symmetry assump-tion) is a massless scalar field [31]. On the other hand, asalready stated, it is hoped that anisotropy will lead to awave vector-dependent effective geometry, thus producinga modification in the particles’ dispersion relation.

The paper is organized as follows. In Sec. II we brieflyreview the LQC-like quantization of Bianchi-I geometry.In Sec. III, we consider a test scalar field on this back-ground Bianchi-I space-time: in Sec. III A we summarizethe essential features of QFT on a classical Bianchi Ibackground geometry; in Sec. III B we develop QFT onquantum Bianchi-I space-time. In Sec. IV the dynamics ofQFT on classical and quantum backgrounds are comparedand an effective classical geometry is shown to emerge. InSec. V we investigate the possibility that field quanta ofdifferent energies could in fact probe different effectivegeometries, i.e., different aspects of the underlying quan-tum geometry, possibly producing Lorentz violation. Inparticular, we compare the propagation of the test fieldon classical and effective geometries, derive the modifieddispersion relation, and study the status of causality.In Sec. VI we discuss the results and conclusions.Appendix A complements our discussion on mode decom-positions of the test field, while Appendix B probes theconsistency of our findings under a different choice ofphysical time.

II. BACKGROUND QUANTUM GEOMETRY

In this section we present a brief summary of LQC ofBianchi-I space-time.

A. Classical Bianchi-I space-time

Let us consider the background space-time manifold Mto be topologically M ¼ R� T3, where T3 is the 3-torus[whose coordinates xi range in ð0; 1Þ]. As is standard in theliterature on Bianchi models, we will restrict ourselves tothe diagonal Bianchi-I metric

g��dx�dx� ¼ �dt2 þX3

i¼1

a2i ðtÞðdxiÞ2: (1)

This choice of coordinates is not the most general: we havechosen the lapse function for the time coordinate t asNt ¼ 1, and the shift vector Na

t ¼ 0. In LQG the canonicalpair consists of the gravitational SUð2Þ connection Aa

i andthe densitized triad Ei

a, which for metric (1) are given by

Aia ¼ ci�i

a; Eai ¼ pi�

ai ; (2)

where ci, pi are constants. Therefore, the phase space of

gravity, �gr, is coordinated by the canonical variables

ðci; piÞ, satisfying the canonical Poisson algebra

fci; pjg ¼ 8�G��ij; (3)

with �, the Immirzi parameter. The relations between thephase space variables pi and the scale factors ai are

p1¼a2a3; p2¼a3a1; p3¼a1a2: (4)

The matter source for this model will be a masslessscalar field T, which will also serve as the relationaltime. Therefore, it is convenient to work with a harmonictime function �, i.e., satisfying h� ¼ 0. Since � is homo-geneous, the spatial part of this equation vanishes, andhence we get d2�=dt2 ¼ 0. Integrating this over the fidu-cial cell V , we get

Zd4x

d2�

dt2¼

Zdt

d2�

dt2

ZVd3x ¼ d�

dtV ¼ const; (5)

where V is the volume of the fiducial cell V , which isgiven in terms of the gravitational variables as

V ¼ ja1a2a3j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

q: (6)

Choosing const ¼ 1 in Eq. (5), we get dt=d� ¼ V. Thisobject allows us to change the space-time coordinates fromðt; xiÞ to ð�; xiÞ: the corresponding lapse function, N�, canbe obtained using the relation N�d� ¼ Ntdt,

N� ¼ Nt

dt

d�¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

q: (7)

In terms of the new coordinates ð�; xiÞ, the Bianchi-I metric(1) then becomes

g��dx�dx�¼�N2

�d�2þX3

i¼1

a2i ð�ÞðdxiÞ2

¼jp1p2p3j��d�2þX3

i¼1

ðdxiÞ2p2i

�: (8)

General relativity is a constrained theory, and inAshtekar variables (2) the constraints comprise so-calledGauss constraint, vector constraint, and scalar constraint.However, since we have restricted ourselves to diagonalhomogeneous metrics and fixed the internal gauge, theGauss and the vector constraints are identically satisfied.The only nontrivial constraint (i.e., the homogeneous partof the scalar constraint) can be expressed by restricting theintegration in the full theory to the fiducial cell V ,

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Cgr¼ZVd3xN�Cgr

¼� 1

8�G�2

ZVd3x

N�

jp1p2p3j�ðp1p2c1c2þp2p3c2c3þp3p1c3c1Þ

¼� 1

8�G�2ðp1p2c1c2þp2p3c2c3þp3p1c3c1Þ: (9)

On the other hand, the massless scalar field T and itsconjugate momentum PT coordinate the phase space ofmatter, denoted by �T . The energy density for the masslessscalar field is given by �T ¼ P2

T=2V2, so the contribution

of T to the scalar constraint can be obtained as

CT ¼ZVd3xN�CT ¼ P2

T

2; (10)

where we used CT ¼ ffiffiffiq

p�T .

Putting together the two contributions (9) and (10), weobtain the total scalar constraint for the system (namely,the scalar constraint of the geometry, denoted by Cgeo) as

Cgeo ¼ Cgr þ CT: (11)

Physical states of the classical geometry lie on the con-straint surface defined on �geo ¼ �T � �gr by the equation

Cgeo ¼ 0. The �-evolution of any phase space function f is

obtained by the Poisson brackets,

df

d�¼ ff; Cgeog: (12)

In particular, the �-evolutions of the scalar field T and itsconjugate momentum PT read

dT

d�¼ fT; Cgeog ¼ PT; (13)

dPT

d�¼ fPT; Cgeog ¼ 0: (14)

From Eq. (14) it is seen that the conjugate momentum PT

of the scalar field T is a constant of motion (which ischosen here to be positive), and hence Eq. (13) describesa linear evolution of the scalar field T with respect to �,

T ¼ PT� (15)

for a vanishing integration constant. This equation impliesthat the phase space variable T is a good time parameter for�-evolution of the system. In other words, although T doesnot have the physical dimensions of time, it is a goodevolution parameter; in LQC literature it is known as therelational time. Using the relation NTdT ¼ N�d� in (13),we get an expression for the lapse function NT ,

NT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

pPT

: (16)

Thus, we can change coordinates from ð�; xiÞ to the physi-cal ðT; xiÞ one. Then, the metric (8) becomes

g��dx�dx�¼jp1p2p3j

�� 1

P2T

dT2þX3i¼1

ðdxiÞ2p2i

�: (17)

B. Quantum Bianchi-I space-time

The previous construction was preparatory for loopquantization, which we sketch now. The kinematicalHilbert space of the Bianchi-I model, H kin, is given asthe tensor product of the Hilbert space H gr of the gravi-

tational sector and the Hilbert space H T. More precisely,the states in H gr can be labeled by three real numbers,

which are arranged in a vector ~� [22]. The gravitationalpart of Cgeo is turned into a difference operator acting

on H gr.

On the other hand, the scalar field sector is quantizedaccording to the Schrodinger picture: the Hilbert space isH T ¼ L2ðR; dTÞ, while the dynamical variables T and PT

are promoted to operators on it,

T ¼ T; PT ¼ �iℏ@

@T: (18)

Therefore, the kinematical Hilbert space of geometry isgiven by H kin ¼ H T �H gr. The scalar constraint op-

erator Cgeo is well-defined on H kin and is given by

C geo ¼ � 1

2ðℏ2@2T � IÞ � 1

2ðI ��Þ; (19)

where � is a difference operator on H gr defined in

Ref. [22].Physical states of the quantum geometry are those

�oðT; ~�Þ 2 H kin lying in the kernel of Cgeo. In other

words, they are solutions to equation

� ℏ2@2T�oðT; ~�Þ ¼ ��oðT; ~�Þ: (20)

One can further restrict solutions of the Eq. (20) to thespace spanned by the ‘‘positive frequency’’ solutions, i.e.,the solutions to

�iℏ@T�oðT; ~�Þ ¼ffiffiffiffiffiffiffiffij�j

p�oðT; ~�Þ ¼: Ho�oðT; ~�Þ: (21)

Wewill take this as our physical Hilbert space of geometry,H o

phys, endowed with scalar product

h�oj�0oi ¼

X�1;�2;�3

�oðTo; ~�Þ�0oðTo; ~�Þ; (22)

where To is any ‘‘instant’’ of internal time T.

III. THE TEST QUANTUM FIELD

This section is divided into two parts. In the first section,for the convenience of the reader, we present a briefsummary of the essential features of QFT on a classicalBianchi-I background space-time. In the second, we

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consider the case of a quantum Bianchi-I background, aspresented in the previous section.

A. Quantum field theory on classicalBianchi-I background

We consider the background space-time to be theclassical Bianchi-I model (of Sec. IIA), equipped with thecoordinates ðx0; xjÞ, in which xj 2 T3, with x0 2 R being ageneric time coordinate. In these coordinates, the Bianchi-Ibackground space-time is described by the metric

g��dx�dx� ¼ �N2

x0ðx0Þdx20 þX3i¼1

a2i ðx0ÞðdxiÞ2: (23)

Let us consider a real (inhomogeneous) test scalarfield1 ðx0; ~xÞ on this background space-time, whoseLagrangian is

L ¼ 1

2ðg��@�@��m22Þ: (24)

Performing the Legendre transformation, one gets thecanonically conjugate momentum for the test field ,denoted by �, on a x0 ¼ const slice. Then, for the pair

ð;�Þ, the classical solutions of the equation of motion

[coming from (24)] can be expanded in Fourier modes

ðx0; ~xÞ ¼ 1

ð2�Þ3=2X~k2L

~kðx0Þei ~k� ~x;

�ðx0; ~xÞ ¼ 1

ð2�Þ3=2X~k2L

�~kðx0Þei ~k� ~x;(25)

where ðk1; k2; k3Þ 2 ð2�ZÞ3 span a three-dimensional lat-tice L (see Appendix A).

Using the Fourier transform (25) in Poisson algebraof and �, it follows that the Fourier coefficients ~k

and �~k satisfy f~k; � ~k0 g ¼ �~k;� ~k0 and the reality conditions

~k ¼ �� ~k and �~k ¼ �� ~k. The Hamiltonian follows from

(24) as

Hðx0Þ ¼ 1

2

Zd3x

Nx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3jp

��2

þX3i¼1

ðpi@iÞ2

þ jp1p2p3jm22

�

¼ Nx0

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

p X~k2L

���~k� ~k þ

�X3i¼1

ðpikiÞ2

þ jp1p2p3jm2

��~k~k

�: (26)

In Appendix A it is shown that, by a wise choice ofvariables, it is possible to rewrite (26) as the Hamiltonian

for a collection of decoupled harmonic oscillators.Specifically, we obtain

Hðx0Þ :¼X~k2L

H~kðx0Þ ¼Nx0

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

p

� X~k2L

�p2

~kþ

�X3i¼1

ðpikiÞ2 þ jp1p2p3jm2

�q2~k

�;

(27)

where q ~k and p~k are the two conjugate variables associated

with the ~k modes. Notice that, for each mode ~k, the termH~kðx0Þ in Eq. (27) is nothing but the Hamiltonian of a

single harmonic oscillator with a time-dependent mass/frequency.In this paper we will focus on the dispersion relation of

the quantum field (see Sec. V), for which studying a singlemode q ~k is sufficient. On the other hand, the quantization

of the full system would require to take into accountall modes. In particular, renormalization of the UV limitis a crucial element of the definition of the QFT; withoutit, expressions such as Eq. (27) are entirely formal.Considering such an infinite-dimensional system is not atall straightforward, and leads to the whole topic of QFT incurved space-time. However, as long as one is interestedonly in a single mode (or a finite set), quantization is on theline of the quantum harmonic oscillator: the Hilbert spaceis H ~k ¼ L2ðR; dq ~kÞ, and the dynamical variables q ~k and

p~k are promoted to operators on it, q ~kc ðq ~kÞ ¼ q ~kc ðq ~kÞand p ~kc ðq ~kÞ ¼ �iℏ@=@q ~kc ðq ~kÞ. Time evolution (with

respect to generic x0) is generated by the time-dependent

Hamiltonian operator H ~kðx0Þ via the Schrodinger equationiℏ@x0c ðx0;q ~kÞ¼ H ~kðx0Þc ðx0;q ~kÞ

¼ Nx0ðx0Þ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1ðx0Þp2ðx0Þp3ðx0Þjp

�p2

~kþ�X3i¼1

ðpikiÞ2

þjp1p2p3jm2

�q2~k

�c ðx0;q ~kÞ: (28)

This concludes the study of QFT on classical Bianchi-Ibackground. However, to consider QFT on quantumBianchi-I background, it is necessary to fix the choice oftime in accordance with the previous section. In the x0 ¼ �

case, it is N� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

p, and (28) reduces to

iℏ@�c ð�; q ~kÞ ¼ H�; ~kð�Þc ð�; q ~kÞ

¼ 1

2½p2

~kþ!2

�; ~kq2~k�c ðx0; q ~kÞ; (29)

where we defined the (time-dependent) frequency

!2�; ~k

:¼ X3i¼1

ðpikiÞ2 þ jp1p2p3jm2 (30)

for any ~k mode.

1This is to be considered a test field, so that it does not induceany backreaction on the geometry, which can then be thought ofas background geometry.

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B. Quantum field theory on quantumBianchi-I background

We consider a system of general relativity coupledto a homogeneous massless scalar field T, togetherwith a massive and, in general, inhomogeneous testfield , propagating on the homogeneous background.The scalar constraint on the full phase space of thegravitational field, scalar field T and test field , isexpressed as

C ¼ Cgr þ CT þ C: (31)

Let us fix the time coordinate � with the lapse func-

tion N� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

pfor the whole system. Then, the

scalar constraint becomes

C� ¼ Cgeoð�Þ þHð�Þ; (32)

where Cgeo is given in Eq. (11) and Hð�Þ is obtained

from (27),

Hð�Þ ¼X~k2L

H�; ~k ¼1

2

X~k2L

½p2~kþ!2

�; ~kq2~k�: (33)

To pass to the quantum theory, let us focus just on the

single mode ~k 2 L. The total kinematical Hilbert space for

the system is given by the tensor productH ð ~kÞkin ¼ H geo �

L2ðR; dq ~kÞ. Therefore, from (32) the scalar constraint for

the QFT (of the single mode ~k) on the quantum Bianchi-Ibackground is given by

C�; ~k:¼ Cgeo þ H�; ~k

¼�ℏ2

2ð@2T � Igr � I ~kÞ �

1

2ðIT ��� I ~kÞ þ ðIT � H�; ~kÞ:

(34)

The Hamiltonian operator H�; ~k in Eq. (34) is given by

H�; ~k ¼1

2

�p2

~kþ

�X3i¼1

p2i k

2i þ jp1p2p3jm2

�q2~k

�; (35)

acting on the kinematical Hilbert space H gr �H ~k.

Notice that, although pi are operators, they commutewith each other and with matter operators p ~k and q ~k

and so does the ‘‘frequency operator’’ (in roundbrackets).

Physical states, �ðT; ~�; q ~kÞ, are those � 2 H ð ~kÞkin

that satisfy

�ℏ2@2T�ðT; ~�; q ~kÞ ¼ ½H2o � 2H�; ~k��ðT; ~�; q ~kÞ; (36)

where Ho ¼ ffiffiffiffiffiffiffiffij�jpas above. Since the operator H2

o �2H�; ~k on the right-hand side of (36) is symmetric on

H ð ~kÞkin, we can restrict to its self-adjoint extension on a

suitable domain.2 On the physical Hilbert space, this op-

erator is identified with P2T , which is classically a positive

Dirac observable; thus, we can consider just the positive

part of the spectrum of H2o � 2H�; ~k.

Remark. Consider the m ¼ 0 isotropic case for definite-

ness. Classically, for large volumes v� p3=2 it is H2o � v2

and H�; ~k � v4=3k2, so the classical inequality H2o �

2H�; ~k � 0 can always be satisfied for k � 1 provided

that v is large enough. Quantum-mechanically, one usesthe boundedness of operator � by �crv

2 (a property inde-pendent of the matter content). From the Einstein equation

it follows that H�; ~k ¼ H2o � P2

T , and hence H�; ~k itself is

bounded in a similar way. Since H�; ~k is the Hamiltonian of

a harmonic oscillator with frequency proportional to v2=3k,the ground state of this operator has eigenvalue propor-

tional to v2=3k=2. Because of boundedness, such frequencymust be less then or equal to �crv

2 for all k. This sets ak-dependent lower bound to the volume

v ��

k

2�cr

�3=4 ¼: vmin; (37)

which semiclassical states of geometry achieve at thebounce. We thus see that k can be arbitrarily large: thegeometry will ‘‘react’’ by concentrating on large volumes.From these considerations, it follows that the subspace

of the kinematical Hilbert space where H2o � 2H�; ~k is

positive-definite is nonempty for all ~k: restricting to sucha subspace, we are able to define the square root of operator

H2o � 2H�; ~k. Therefore, physical states are solutions to the

Schrodinger-like equation

�iℏ@T�ðT; ~�; q ~kÞ ¼ ½H2o � 2H�; ~k�1=2�ðT; ~�; q ~kÞ: (38)

The space spanned by these solutions is the physical Hilbert

space H ð ~kÞphys of the theory (for the chosen mode ~k). The

scalar product on the physical Hilbert space is simply

h�j�0i ¼ X�1;�2;�3

Z 1

�1dq ~k�ðTo; ~�; q ~kÞ�0ðTo; ~�; q ~kÞ: (39)

If the test field approximation holds, the backreaction ofthe field is neglected on the homogeneous backgroundgeometry. Therefore, the theory is physically relevant only

when we treat 2H�; ~k as a perturbation to H2o. Using this

assumption, we obtain from (38) the following approxima-tion (for a discussion, see Ref. [28]):

2Here we are assuming that such extension exists. In case thisis not true for the operator �, one should seek an alternativequantization, such that H2

o � 2H�; ~k admits a self-adjoint exten-sion. In fact, what follows does not rely on the details ofoperator Ho.

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�iℏ@T�ðT; ~�; q ~kÞ ¼ ½Ho � H�1

2o H�; ~kH

�12

o ��ðT; ~�; q ~kÞ¼: ½Ho � HT; ~k��ðT; ~�; q ~kÞ: (40)

In the last step we introduced the operator HT; ~k which

implements the Hamiltonian that generates evolution inrelational time T. More precisely, at the classical level,the Hamiltonian that generates evolution in T can be foundby considering the lapse functionNT ¼ P�1

T N� in Eq. (27):it is HT; ~k¼H�; ~kNT=N�¼H�; ~kP

�1T H�; ~kH

�1o (the last step

holding thanks to the test field approximation, which at

physical level allows us to put PT ¼ðH2o�2H�; ~kÞ1=2Ho).

At the quantum level, the operator HT; ~k in Eq. (40) is a par-

ticular quantum realization of the classical function HT; ~k.

IV. EFFECTIVE BIANCHI-I GEOMETRY

In the previous section, we have constructed QFTon a Bianchi-I quantum background geometry, obtainingEq. (40) as the quantum counterpart of Eq. (29) for QFTonclassical space-time. In this section, by a comparison be-tween these two dynamical equations, we will discuss therelation between the aspects of QFT in classical and inquantum geometry. To do this, we shall now construct aclassical geometry limit of the QFT on quantum Bianchi-Ispace-time.

At the quantum-geometry level, Eq. (40) provides a

quantum evolution equation for the state �ðT; ~�; q ~kÞ,depending on (the ~kth mode of) the test field and the

quantum geometry (encoded in ~�). On the other hand, onthe classical geometry, c ðT; q ~kÞ given in Eq. (29) is the

evolution of the state of the test field on the time-dependent classical background geometry. To comparethe two frameworks, we shall consider the test fieldapproximation, that is, we disregard the backreaction ofmatter on geometry. Thus, we can write the state of thesystem as a tensor product of the state of geometry andthe state of matter, as the two are disentangled,

�ðT; ~�; q ~kÞ ¼ �oðT; ~�Þ � c ðT; q ~kÞ; (41)

where the geometry evolves through Ho, i.e.,

�iℏ@T�o ¼ Ho�o [see Eq. (21)]; in other words, it is

�oðT; ~�Þ ¼ eiTHo=ℏ�oð0; ~�Þ. Plugging this in Eq. (40) and

projecting both sides on �oðT; ~�Þ, one finds an equationfor the matter only,

iℏ@Tc ðT;q ~kÞ¼1

2

�hH�1

o ip2~kþhH�1

2o

�X3i¼1

p2i ðTÞk2i

þjp1ðTÞp2ðTÞp3ðTÞjm2

�H

�12

o iq2~k�c ðT;q ~kÞ;

(42)

where we moved the T-dependence from �o to thegravitational operators (i.e., we describe the geometry

sector in Heisenberg picture), so that hAðTÞi denotes theexpectation value on the quantum state of geometry

�oð0; ~�Þ of gravitational operatorAðTÞ ¼ e�iTHo=ℏAeiTHo=ℏ: (43)

Equation (42) is an evolution equation for (the ~kth modeof) a quantum field on an ‘‘effective’’ geometry. Let usassume that this geometry is described by a metric �g�� of

the form

�g��dx�dx� ¼ � �N2ðTÞdT2 þ j �p1 �p2 �p3j

X3i¼1

ðdxiÞ2�p2i

: (44)

Then, by setting x0 ¼ T in Eq. (28) we obtain

iℏ@Tc ðT; q ~kÞ ¼ H ~kðTÞc ðT; q ~kÞ

¼ �NTðTÞ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij �p1 �p2 �p3jp

�p2

~kþ

�X3i¼1

ð �pikiÞ2

þ j �p1 �p2 �p3jm2

�q2~k

�c ðT; q ~kÞ; (45)

which is a classical time-dependent evolution equation for

(the ~kth mode of) the test field with respect to T. Now, bycomparing (45) and (42), we find the following relations:

�NðTÞ ¼ hH�1o i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij �p1 �p2 �p3j

q; (46)

�NðTÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij �p1 �p2 �p3jp �p2

i ¼ hH�12

o p2i ðTÞH�1

2o i; (47)

�NðTÞm2 ¼ m2 hH�1

2o jp1ðTÞp2ðTÞp3ðTÞjH�1

2o iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij �p1 �p2 �p3j

p : (48)

These relations hold only if we ignore the quantum fluc-tuations of the geometry and assume that the underlyingquantum geometry state �o is peaked on an effective

trajectory (with expectation values �PT and �pi of Ho andpi, respectively). In cosmological applications, the semi-classical states�o of geometry have very small dispersionsalong the entire effective trajectory, thus these assumptionsare justified.Let us consider now the system of Eqs. (46)–(48). There

are five equations with four variables, �NT and �pi, andhence there exists no general solutions.3 However, consid-ering a massless test field, the last equation is identically

3This of course does not mean that massive scalar fields cannotexist on quantum geometry. It simply suggests that there is noeffective geometry of the form (44) that reproduces the equationof motion of such a field on the quantum geometry. Moreover,note that (46) and (48) are classically equivalent, so we canexpect that what follows also applies to the massive case, as longas we disregard quantum fluctuations (that is to say, we consideronly semiclassical states of geometry). Nevertheless, for defi-niteness we will set m ¼ 0 in the following.

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verified, and the remaining equations reduce to a solvablesystem of four equations with four variables. It turns outthat there exists a unique solution for �NT and �pi given by

�NðTÞ ¼ hH�1o i1=4

�Y3i¼1

hH�12

o p2i ðTÞH�1

2o i

�14; (49)

�p i ¼�hH�1=2

o p2i ðTÞH�1=2

o ihH�1

o i�1

2: (50)

These relations determine the effective space-time (44) interms of expectation values of the gravitational operatorson the quantum geometry state �o. It should be noticedthat the effective metric components (49) and (50) do not

depend on ~k, which indicates that all modes of the field‘‘probe’’ the same effective background geometry, or theirenergy, independently. For this reason we expect that noLorentz violation is present, and indeed it can be shownthat there is no quantum gravity effect on the dispersionrelation for the field : we will study this issue in thenext section.

To conclude, it is worth mentioning that the effectivegeometry defined in (49) and (50) comes from the m ¼ 0

choice, or more precisely the m k ~kk limit. In this limit,Eq. (48) can be disregarded, and one finds the presentedresult. On the other hand, one may consider the opposite

limit, namely,m � k ~kk. In this case, the three equations in(47) are to be disregarded, and the system does not have aunique solution. However, if one assumes that both m and

k ~kk are large, then it is Eq. (46) that drops from the system,and one has again a unique solution,

�N T ¼� Q

ihH�1

2o p2

i H�1

2o i

hH�12

o jp1p2p3jH�12

o i

�12; (51)

�p i ¼ hH�12

o jp1p2p3jH�12

o i� hH�1

2o p2

i H�1

2o iQ

jhH�1

2o p2

j H�1

2o i

�12: (52)

As already pointed out in footnote 3, the effective metricdefined in (51) and (52) is classically equivalent to the onedefined in (49) and (50). Nevertheless, if fluctuations of thesemiclassical state of geometry could consistently be takeninto account, it would seem that two different behaviors areto be expected from light and heavy particles, respectively.

V. DISPERSION RELATIONS ANDLORENTZ SYMMETRY

A prediction of many approaches to quantum gravitycomes from the study of in vacuo ‘‘dispersion relation.’’This is the relation between the frequency ! and the wave

vector ~k of a mode of a field in vacuo. In QFT, thedispersion relation links the momentum and the energy

of quanta, and thus for fundamental fields its form isdictated by Lorentz symmetry (namely, by the mass-shellconstraint).Deformed dispersion relations are a rather natural pos-

sibility in quantum gravity [25–27], requiring a modifica-tion of Lorentz symmetry, which is then said to be‘‘broken’’ by quantum gravity effects. If that is the case,Lorentz invariance is only an approximate symmetry of thelow-energy world. Hence, quantum gravity effects couldbe observed by relying on modified dispersion relationsfor the matter (such as photons or neutrinos) travelingon a quantum gravitational background. More precisely,Planck-scale effects are expected to be negligible in stan-dard circumstances, but observations of highly energeticparticles traveling long distances may set bounds on vio-lation of (standard) Lorentz symmetry. Some of theseproperties may be experimentally detectable in satellitefacilities (e.g., Gamma-ray Large Area Space Telescope[32] or Alpha Magnetic Spectrometer [33]), using asprobes light from distant astrophysical sources, such asgamma ray bursts (GRBs).We analyze the form of the effective geometry obtained

in the previous section, addressing the question on whethera Lorentz violation is produced in our model. To do this, letus first consider the Hamiltonian of the test field on theclassical background given by Eq. (27). This equationdetermines the evolution of modes q ~k in time coordinate

x0. More precisely, for each mode ~k 2 L, the x0-evolutionof each pair of variables ðq ~k; p ~kÞ is given by the Hamilton

equations

dq ~k

dx0¼ fq ~k; H ~kg;

dp ~k

dx0¼ fp~k;H ~kg; (53)

where H~k is defined by Eq. (A6). Using Eq. (53) one can

get the wave equation for each mode q ~k (see Appendix A).

For the case x0 ¼ �, the lapse function isN� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

p,

and we get the wave equation for each q ~k [see Eq. (A22)] as

d2q ~k

d�2þ!2

�; ~kq ~k ¼ 0; (54)

where !�; ~k is the dispersion relation for any mode ~k and is

given by Eq. (30). In particular, for massless field we have

!2�; ~k

¼ X3i¼1

p2i k

2i : (55)

Let us consider now a cosmological observer, whose

4-velocity is u� ¼ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1=g00p

; 0; 0; 0Þ. This observerdefines a local orthonormal frame fe�a g as follows. First,the 4-velocity fixes the 0th basis vector as e�0 ¼ u� then

we are free to orient the three spatial basis vectors aswe wish—provided that they are all normalized andorthogonal to e

�0 . An obvious choice in our case is

e�1 ¼ ð0; ffiffiffiffiffiffiffiffiffiffiffiffi1=g11

p; 0; 0Þ, e�2 ¼ ð0; 0; ffiffiffiffiffiffiffiffiffiffiffiffi

1=g22p

; 0Þ, and e�3 ¼ð0; 0; 0; ffiffiffiffiffiffiffiffiffiffiffiffi

1=g33p Þ. We can write this basis compactly as

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e�a ¼ ��a

1ffiffiffiffiffiffiffiffiffiffijgaajp ; (56)

where there is no sum over a. Now, if k� ¼ ð!k; k1; k2; k3Þis the wave 4-vector of the quantum field, the 4-vector seenby the cosmological observer is given by its projection onthe orthonormal basis

Ka ¼ k�e�a ¼

�!kffiffiffiffiffiffiffiffiffiffiffi�g00

p ;k1ffiffiffiffiffiffiffig11

p ;k2ffiffiffiffiffiffiffig22

p ;k3ffiffiffiffiffiffiffig33

p�: (57)

The observed 3-velocity of the mode is then

Vi ¼ dK0

dKi

¼ffiffiffiffiffiffiffiffiffiffiffiffi� giig00

sd!k

dki; (58)

and the norm of this vector, i.e., kVk2 :¼ ijViVj,

is simply

kVk2 ¼ Xi

ðViÞ2 ¼ �Xi

giig00

�d!k

dki

�2: (59)

This norm corresponds to the speed of the particle asmeasured by the cosmological observer. In the case athand (i.e., for a massless scalar field on classicalBianchi-I space-time), it is

kVk2 ¼ Xi

1

p2i

�kip

2i

!�; ~k

�2 ¼ 1: (60)

As expected, the observed speed of massless particlescoincides with the speed of light; this confirms the localLorentz symmetry.

We can similarly study the case of a massless scalar fieldon the effective geometry. The corresponding wave equa-tion (54) for the effective geometry (44) can be furtherwritten as (see Appendix A)

d2Q ~k

dT2þ�2

T; ~kQ ~k ¼ 0; (61)

where Q ~k, denotes the modified modes q ~k, and is defined

in Eq. (A19) as

Q ~k:¼ q ~kffiffiffiffiffiffiffiffiffiffiffiffiffi

hH�1o i

q : (62)

Since Eq. (61) has the form of a harmonic oscillator, thequantity �T; ~kðTÞ encodes the (modified) dispersion rela-

tion of the test field on the effective geometry (44). Themodified dispersion relation �T; ~kðTÞ then is given by

�2T; ~k

ðTÞ ¼��!2~k� �2

4� 1

2

d�

dT

�; (63)

where � and �!~k, are defined by Eqs. (A17) and (A18),

and reads

� ¼ �d lnhH�1o i

dT; (64)

�! 2~k¼ hH�1

o i2�X3i¼1

ð �pikiÞ2 þ j �p1 �p2 �p3jm2

�: (65)

It should be noted that, in principle,�T; ~k in Eq. (63) can be

imaginary, since in the m ¼ 0 case one can make �!2~k

arbitrarily small by choosing small wave vectors ~k.However, two facts should be taken into account: first,the T3 topology of space introduces an IR cutoff; second,

if the test field approximation holds, then H�1o commutes

with the total Hamiltonian, and can then be consideredconstant in T, so that � vanishes.4

Substituting Eqs. (64) and (65) in Eq. (63), for a mass-less test field, we get

�2T; ~k

ðTÞ ¼�hH�1

o iXi

k2i hH�12

o p2i ðTÞH�1

2o i � 1

4

�d lnhH�1

o idT

�2

þ 1

2

d2 lnhH�1o i

dT2

�: (66)

As already pointed out, hH�10 i is independent of time T,

and hence Eq. (66) reduces to

�2T; ~k

ðTÞ ¼ hH�1o iX

i

k2i hH�12

o p2i ðTÞH�1

2o i: (67)

Using Eq. (59) we thus obtain the 3-velocity of modespropagating on the effective geometry as

kVk2 ¼ �Xi

�gii�g00

�d�T; ~k

dki

�2

¼ 1

�2T; ~k

Xi

k2i hH�1o ihH�1

2o p2

i ðTÞH�12

o i ¼ 1: (68)

This equation confirms our expectation that no Lorentzviolation is presented in our model herein.

VI. CONCLUSIONS AND DISCUSSION

In this work we considered an improved dynamics LQCof Bianchi-I setting [22] among the anisotropic class ofmodels for a background cosmological quantum space-time, coupled with a massless scalar field. We developedthe QFT of the test field on this background quantumgeometry by considering a mode decomposition of suchtest field. It was shown that the QFTon this quantum space-time can be seen as a QFT on classical curved space-time;this background geometry is described by a classical ef-fective metric, which can be thought of as emergent fromthe underlying quantum space-time once a mode of the

4As for the ~k ¼ 0 mode of the massless field (for which onehas�T; ~k ¼ 0), note that it is not a harmonic oscillator, but rathera spatially constant scalar field. We can think of it as thehomogeneous part of , and use it instead of T (indeed, theKlein-Gordon equation for q0 reads @2q0=@T

2 ¼ 0, whose so-lution is linear in T).

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field is coupled to it. It should be emphasized that this‘‘effective metric’’ is of a different nature than the usualsemiclassical metric of LQC effective dynamics [10,11],since it arises from the peculiar procedure we devisedinvolving the presence of a matter field. Despite highexpectations, this effective geometry does not depend onany specific chosen mode of the test field, and thereforethere is no Lorentz violation. To investigate this in moredetails, we analyzed the dispersion relation for the matterfield propagating on the background space-time; we haveshown that it defines a speed in vacuowhich coincides withthe speed of light.

A possible explanation for this could be that we havedisregarded the backreaction of the matter on the quantumgeometry, i.e., we have been using the test field approxi-mation (41). It is possible to lift this approximation, byincluding the backreaction via a Born-Oppenheimer (B-O)scheme. However, there are two caveats to this procedure:

(i) First, if the backreaction of the mode on geometry isto be accounted, then one should, in principle, takeinto account the backreaction of everything else, aswell. Indeed, there is no reason to believe that thebackreaction of a specific mode of the specific fieldhas to be stronger than the others. Thus, by onlyconsidering this case, we are in essence restrictingfrom QFT to quantum mechanics of a single har-monic oscillator. The physical relevance of thismodel can then be questioned.

(ii) In order to apply the B-O scheme, the ‘‘perturba-

tion’’ HT; ~k in Eq. (40) must act on the geometry

sector at most as a multiplication. However, this is

not the case since HT; ~k involves Ho. To circumvent

this problem, we might choose a different relationaltime: for the choice ~T ¼ T=PT the right-hand side

of the Eq (40) becomes H2o=2� H�; ~k, whereas the

perturbation H�; ~k acts only by multiplication on the

gravitational part. Now, this choice of time is un-common in LQC, and someone might think that theresult we got at the ‘‘test-field order’’ does not holdfor ~T. In Appendix B we will show that this is notthe case: even by starting with ~T, when disregardingthe backreaction, no Lorentz violation is present.

We henceforth sketch the B-O procedure, bearing inmind these two remarks mentioned above, for the simpli-fied FRW case (where pi ¼ p ¼ a2).

Following the ideas of B-O approximation, by denotingthe gravitational degrees of freedom as ‘‘heavy’’ and thematter degrees of freedom as ‘‘light,’’ one finds that thestate of the system has the form

�ð ~T; �; q ~kÞ ¼ �oð ~T; �Þ � c ð ~T; q ~kÞ þ ��ð ~T; �; q ~kÞ;(69)

where the correction is a nonsimple tensor product of thetwo sectors,

�� ¼ X�;i

f�i’o� � i: (70)

Here, ’o� is the geometry-eigenstate of �=2 with eigen-

value Eo�, and i is the matter-eigenstate of H�; ~k with

eigenvalue �iðpÞ. The explicit form of the coefficientsf�i is given by

f�i :¼X���

c�bih’o

�j�iðpÞj’o�i

Eo� � Eo

�

; (71)

where c� and bi are the coefficients of the expansions of�o on f’o

�g, and of c on f ig, respectively (so they are

fixed by the 0th order). Note that H�; ~k acts on matter as the

Hamiltonian of a harmonic oscillator with square fre-quency k2p2, so its eigenvalues are linear in k, and thusf�i � k.From Eq. (69) it is hard to extract a dynamical equation

for the matter only, since it is entangled with the geometry.Thus, we ‘‘force’’ the state to be again of the disentangledform: � ¼ ½�o þ ��1� � c , where ��1 ¼

P�;if�i’

o�.

Plugging this new state into the constraint equation

� iℏ@ ~T� ¼�1

2�� H�; ~k

��; (72)

and projecting on �o (we may assume the correction ��1

to be orthogonal to �o), one finds a Shrodinger-like equa-tion for c only,

iℏ@ ~Tc ¼ 1

2

��ℏ2ð1þ �Þ d2

dq2~kþ k2hp2ið1þ �Þq2~k

�c :

(73)

Here, as before, hAi :¼ h�ojAð ~TÞj�oi, and � is really anoperator acting on matter; notice that if we assume c to be(at least approximately) an eigenstate of �, then we canthink of � as a number.5 Since � is proportional to f�i, itsdependence on k is linear, and thus we can write Eq. (73) as

iℏ@ ~Tc ¼ 1

2

��ℏ2ð1þ �kÞ d2

dq2~kþ k2hp2ið1þ �kÞq2~k

�c ;

(74)

where we introduced � to make explicit the linear-dependence of � on k. The number � has dimensions oflength (inverse of k), and is to be computed from � once

the eigenequations for � and H�; ~k have been solved.

Comparing Eq. (74) with Eq. (45) in the same wayas done for the 0th order (i.e., in the test field approxima-tion case), one is led to an effective metric that indeeddepends on k,

5Formally, � is given by

� ¼ h�ojH�; ~kj�oi�1h�ojH�; ~kj��1i:

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�g��dx�dx� ¼ �ð1þ �kÞ2hp2i3=2d ~T2 þ

ffiffiffiffiffiffiffiffiffihp2i

qðdx2

þ dy2 þ dz2Þ: (75)

Moreover, by carrying out the dispersion relation analysisfor Eq. (74) in the metric (75), it is easy to check that thespeed of the quanta with energy proportional to k asmeasured by a cosmological observer is at first-order in �,

kVk ¼ 1þ 1

2�k: (76)

Thus, even though the above computation involved numer-ous approximations and assumptions, a prediction ofLorentz violation is obtained, where the speed of particlesdeviates linearly from the standard value c ¼ 1. RecentGRB observations [34] have probed linear deviations up toenergies of order 107J, observing no Lorentz violation.This value is just 2 orders of magnitude away from thePlanck scale (EPl 109J), at which quantum gravity issupposed to become important. This would correspond inour case to � ‘Pl. Hence, there is the concrete possibilitythat the next generation of GRB observations be able tocheck our ‘‘prediction.’’ Of course, if the prediction isdisproved, this does not mean that LQG (or LQC) is wrong:we have used many approximations to get to Eq. (76), andmany of them could not be completely justified in thegravitational context. The result (76) should thus be re-garded as an indication, rather than a real prediction (es-pecially in light of the fact that we are considering aquantum mechanical model, not the complete QFT, asexplained in the first caveat above).

Another source of doubt comes from the followingconsideration: on one hand, we know that backreactionof matter on classical geometry does not induce Lorentzviolation; on the other, from the very definition of theproblem we have that � accounts for both the classicaland the quantum part of the backreaction (formally, � ¼�C þ �Q). Thus, it would be more appropriate if only �Q

appeared in (76). A possible way out of this problem lies inthe observation that, while Eq. (74) is obtained from thefirst-order in the backreaction on quantum geometry,Eq. (45) comes from an effective metric �g�� of the non-

backreacted Bianchi-I type. Thus, the comparison of thetwo [which leads to (75) and ultimately to (76)] is notcompletely justified. It would be more consistent to com-pare (74) with the dynamical equation of a harmonic

oscillator on an effective metric �gð~kÞ

�� obtained as follows:

take the harmonic oscillator (with ~k-dependent frequency)on the FRW ‘‘vacuum’’ g��, and evaluate its energy-

momentum tensor hT��i; plug this into the Einstein equa-

tion, and solve for the metric. This produces a classical

metric gð~kÞ

�� which ‘‘knows’’ about the matter (at first-order), i.e., it takes into account the first-order modificationto classical geometry due to the presence of the harmonicoscillator. One can then consider a generic effective metric

�gð~kÞ

�� of the kinematical family of gð~kÞ

��, and find the dynami-cal equation for a harmonic oscillator on such an effectivegeometry. This is the equation to be compared with (74) inorder to single out the ‘‘quantum part’’ of the backreaction,�Q. Despite being an extremely complicate procedure,

there is a change that the form of (76) will not be alteredmuch (possibly, only by the replacement of �with �Q), and

in the worst-case scenario a functionally more complicate~k-dependence should appear in kVk, so that Lorentz sym-metry is in fact broken.It should be noted that we never used the explicit form of

Ho: it follows that this analysis can be applied to allquantum cosmologies which, absent matter, describe thestate of geometry by a wave function�o evolving through

Ho. Also, we would like to remark that this Lorentz devia-tion does not come from the discrete nature of the quantumspace-time: if we describe quantum gravity on a fixedgraph, then it is not a surprise that Lorentz symmetry isbroken, because from the point of view of matter propaga-tion only certain modes are normal with respect to thegiven space-time ‘‘crystal’’ [35]. On the contrary, we neverinvoked any feature of the Planck-scale space-time. Ouronly request is that it makes sense to consider only alimited number of gravitational degrees of freedom (inFRW case, just the scale factor). While this is of course astrong approximation, it does not involve any assertion onthe microscopic discreteness of space-time.Aside from this last indication of Lorentz violation at

first-order in backreaction, we still have much to say. Weproved that at 0th order no symmetry breaking is present inthe Bianchi-I case, thus extending the result of Ref. [28].Also, while the search for symmetry breaking was ourdriving force, we exposed the first steps toward quantumfield theory on quantum Bianchi-I space-times. This is aresult useful and interesting in its own right, and worthrefining (in particular, more work should be done to be ableto consider an infinite number of modes). It is to be saidthat developments in this direction are expected from thestudy of quantum perturbations [36], which is an area ofgrowing interest in the LQC community.These are the first nontrivial findings about Lorentz

invariance in LQC, and a first step to understanding it infull LQG. Previous attempts to study Lorentz invariance inLQG have been done, but none of them attacked theproblem from the indirect perspective we adopted, namely,via an effective ‘‘classical’’ metric that reproduces thesame results as the semiclassical approximation of theunderlying quantum theory of geometry. We would liketo focus the attention on the issue of Lorentz symmetry,which has not been clarified yet in the complete theory. Ourresult proves that no violation is present in Bianchi-I case(at least when backreaction can be disregarded), and alinear deviation could be present once the backreaction istaken into account. However, such findings do not solve theproblem. In this search, the tools of ‘‘effective metric’’ and

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dispersion relation analysis can be used for more realisticcases of matter coupling (such as the electromagneticfield). We hope this work will trigger the question ofLorentz invariance in LQG, as this approach to quantumgravity does not yet give a definitive statement. Such astatement could be in principle used to falsify the theory,especially considering the great effort on the observationalside [25–27,34]. We would like to know where LQG standsin this context. Of course more work is needed, and wehope to start a discussion in this direction within the LQG/LQC community.

ACKNOWLEDGMENTS

We are grateful to Kristina Giesel for the enlighteningdiscussion about Born-Oppenheimer approximation, andto Abhay Ashtekar for his comments about its applicabil-ity. The authors would also like to thank the unknownreferee for useful remarks and suggestions. This workwas partially supported by the Grant No. 182/N-QGG/2008/0 (PMN) of Polish Ministerstwo Nauki iSzkolnictwa Wyzszego. Y. T. thanks the ESF sponsorednetwork ‘‘Quantum Geometry and Quantum Gravity’’ fora short visit grant to collaborate with theWarsaw group. Heis supported by the Portuguese Agency Fundacao para aCiencia e Tecnologia through the fellowship SFRH/BD/43709/2008. He also would like to thank the warm hospi-tality of the Department of Theoretical Physics of theUniversity of Warsaw, where the idea for this work wasinitiated.

APPENDIX A: MODE DECOMPOSITION

Let us consider a decomposition of ~k and �~k as

~k¼1ffiffiffi2

p ðð1Þ~kþ ið2Þ

~kÞ; � ~k¼

1ffiffiffi2

p ð�ð1Þ~kþ i�ð2Þ

~kÞ: (A1)

Since the reality conditions are satisfied, not all variables in(A1) are independent. In particular, we have

ð1Þ� ~k

¼ð1Þ~k; ð2Þ

� ~k¼�ð2Þ

~k; �ð1Þ

� ~k¼�ð1Þ

~k; �ð2Þ

� ~k¼��ð2Þ

~k:

(A2)

Because there exist relations between the ‘‘positive’’ and

‘‘negative’’ modes ~k and � ~k, one can split the lattice Linto positive and negative parts,

Lþ ¼ f ~k: k3 > 0g [ f ~k: k3 ¼ 0; k2 > 0g [ f ~k: k3¼ k2 ¼ 0; k1 > 0g;

L� ¼ f ~k: k3 < 0g [ f ~k: k3 ¼ 0; k2 < 0g [ f ~k: k3¼ k2 ¼ 0; k1 < 0g: (A3)

Observe that, if ~k 2 Lþ, then � ~k 2 L�. Using this fact,we are now able to split the summation in the Hamiltonian(26) into its positive and negative parts as H ¼ Hþ

þ

H� . InH

� , we change the dummy index ~k into� ~k, so that

we can convert its sum fromP

� ~k2L�to

P~k2Lþ

, the same

as the sum in Hþ . Thus, we get

Hðx0Þ¼1

2

X~k2Lþ

Nx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3jp

���~k� ~kþ ��� ~k�� ~k

þ�X3i¼1

ðpikiÞ2þjp1p2p3jm2

�ð �~k~kþ �� ~k� ~kÞ

�

¼1

4

X~k2Lþ

X�¼1;2

Nx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3jp

�ð�ð�Þ

~kÞ2þð�ð�Þ

� ~kÞ2

þ�X3i¼1

ðpikiÞ2þjp1p2p3jm2

��ðð�Þ

~kÞ2þðð�Þ

� ~kÞ2��

¼1

2

X~k2Lþ

X�¼1;2

Nx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3jp

�ð�ð�Þ

~kÞ2

þ�X3i¼1

ðpikiÞ2þjp1p2p3jm2

�ðð�Þ

~kÞ2�; (A4)

in which we have used Eq. (A1) in the second step, andEq. (A2) in the third step. In the final expression all themodes are independent, so this is really a collection ofindependent harmonic oscillators. We can do even better if

we think of � as the ‘‘sign’’ of ~k. In this way we cancombine the sums

P~k2Lþ

andP

�¼1;2 into a single one:P~k2L. In terms of these variables, the Hamiltonian (A4)

has the form

Hðx0Þ ¼X~k2L

H~kðx0Þ; (A5)

where H~kðx0Þ is given by

H~k ¼Nx0

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

p�p2

~kþ

�X3i¼1

ðpikiÞ2 þ jp1p2p3jm2

�q2~k

�;

(A6)

and we defined the following real variables:

q ~k ¼8><>:ð1Þ

~kif ~k 2 Lþ

ð2Þ� ~k

if ~k 2 L�; p ~k ¼

8><>:�ð1Þ

~kif ~k 2 Lþ

�ð2Þ� ~k

if ~k 2 L�:

(A7)

Using the relation f~k; � ~k0 g ¼ �~k;� ~k0 together with

Eq. (A1), it then follows:

fð1Þ~k; �ð1Þ

� ~k0g ¼ �~k ~k0 ; fð2Þ

~k; �ð2Þ

� ~k0g ¼ ��~k ~k0 : (A8)

So that we can compute fq ~k; � ~k0 g as follows: for all ~k, ~k0 2Lþ we get

fq ~k; � ~k0 g ¼ fð1Þ~k; �ð1Þ

~k0g ¼ fð1Þ

~k; �ð1Þ

� ~k0g ¼ �~k ~k0 ; (A9)

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for all ~k 2 Lþ, and ~k0 2 L� we have

fq ~k; � ~k0 g ¼ fð1Þ~k; �ð2Þ

� ~k0g ¼ 0; (A10)

for all ~k 2 L�, and ~k0 2 Lþ we have

fq ~k; � ~k0 g ¼ fð2Þ� ~k; �ð1Þ

~k0g ¼ 0; (A11)

and finally, for all ~k, ~k0 2 L� we get

fq ~k;� ~k0 g¼ fð2Þ� ~k;�ð2Þ

� ~k0g¼�fð2Þ

~k;�ð2Þ

� ~k0g¼�~k ~k0 : (A12)

We then conclude that q ~k and p~k are canonically

conjugates

fq ~k; p ~k0 g ¼ �~k ~k0 : (A13)

Therefore, each pair ðq ~k; p ~kÞ describes a harmonic oscil-

lator, decoupled from all others and evolving according toH~kðx0Þ defined in Eq. (A6).

From this knowledge, we can go back and reconstructthe equation of motion for q ~k. From Hamilton equations

dq ~k

dx0¼ fq ~k; H ~kg ¼

Nx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3jp p~k

dp ~k

dx0¼ fp~k; H ~kg

¼ � Nx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3jp �

�X3i¼1

ðpikiÞ2 þ jp1p2p3jm2

�q ~k

(A14)

one gets

d2q ~k

dx20¼ d

dx0

�Nx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

p�p~k þ

Nx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3jp dp~k

dx0

¼ d

dx0ln

�Nx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

p�dq ~k

dx0� N2

x0

jp1p2p3j

��X3i¼1

ðpikiÞ2 þ jp1p2p3jm2

�q ~k: (A15)

For simplicity, we rewrite Eq. (A15) as

d2q ~k

dx20þ �

dq~k

dx0þ!2

~kq ~k ¼ 0; (A16)

where the time-dependent parameters�ðx0Þ and!~kðx0Þ aregiven by

� ¼ � d

dx0ln

�Nx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

p�; (A17)

!2~k¼ N2

x0

jp1p2p3j�X3i¼1

ðpikiÞ2 þ jp1p2p3jm2

�: (A18)

In order to get the dispersion relation for mode ~k, it isbetter to write Eq. (A16) in its normal form. To do this, letus consider a new variable Q ~k,

Q ~k:¼ q ~k exp

�1

2

Z x0�ðx00Þdx00

�: (A19)

With this new variable, Eq. (A16) can be rewritten as

d2Q ~k

dx20þ�2

~kQ ~k ¼ 0; (A20)

where �2~kis given by

�2~k¼

�!2

~k� �2

4� 1

2

d�

dx0

�: (A21)

In the case of harmonic time x0 ¼ �, � ¼ 0 (since

Nx0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

p), and Eq. (A20) reduces to

d2q ~k

d�2þ!2

�; ~kq ~k ¼ 0; (A22)

with

!2�; ~k

¼ X3i¼1

ðpikiÞ2 þ jp1p2p3jm2: (A23)

APPENDIX B: A DIFFERENT CHOICE OF TIME

It could be argued that the main result of our work(namely, the independence of the effective metric on thefield mode) is due to the approximations taken herein(cf., Sec. IV). For this reason, here summarize such ap-proximations as(i) the expansion of the square root in Eq. (38) to get

Eq. (40).

(ii) the test field approximation �ðT; ~�; q ~kÞ ¼�oðT; ~�Þ � c ðT; q ~kÞ, that is, the assumption that

backreaction can be disregarded.

As discussed in Sec. VI, a possible way to lift the secondapproximation is to make use of B-O scheme. However, todo so one needs to choose a different relational time. If thisseems awkward at first glance, one should remember thattime in general relativity is simply a gauge parameter.Moreover, with the choice

~T :¼ T

PT

; (B1)

we find again the same result as above, thus making it morerobust. Incidentally, this choice also lifts the first approxi-mation, that is, the square root in Eq. (38) disappears.The new relational time parameter ~T is again a good

parameter for the gauge flow of C. Indeed, the conjugatemomentum of ~T is P ~T ¼ P2

T=2, so we have

CT ¼ P2T

2¼ P ~T; (B2)

and hence d ~T=d� ¼ 1, and dP ~T=d� ¼ 0, from which itfollows that in fact

~T ¼ �: (B3)

ANDREA DAPOR, JERZY LEWANDOWSKI, AND YASER TAVAKOLI PHYSICAL REVIEW D 86, 064013 (2012)

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The lapse function of ~T is then

N ~T ¼ N� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijp1p2p3j

q; (B4)

so that, instead of Eq. (17) we have

g��dx�dx� ¼ jp1p2p3j

��d ~T2 þX3

i¼1

ðdxiÞ2p2i

�: (B5)

Quantization of the geometrical part is then on the lines ofthe T-case: one promotes ~T and P ~T to operators but thistime, in Cgeo, only a first derivative in ~T appears. Physical

states are thus those �oð ~T; ~�Þ 2 H T �H gr that verify

the Schrodinger-like equation

� iℏ@ ~T�oð ~T; ~�Þ ¼ 1

2��oð ~T; ~�Þ: (B6)

For the QFT on quantum geometry, we consider the full

Hamiltonian constraint for mode ~k of the matter field and harmonic time �,

C �; ~k ¼ Cgeo � I ~k þ I ~T � H�; ~k; (B7)

where H�; ~k acts on both gravitational and matter degrees

of freedom, and is explicitly given by (for masslessscalar field)

H �; ~k ¼1

2

�p2

~kþ

�X3i¼1

p2i k

2i

�q2~k

�: (B8)

Physical states are hence those �ð ~T; ~�; q ~kÞ that solve the

Schrodinger-like equation

� iℏ@ ~T�ð ~T; ~�;q ~kÞ¼1

2½��2H�; ~k��ð ~T; ~�;q ~kÞ

¼1

2½H2

0�2H�; ~k��ð ~T; ~�;q ~kÞ: (B9)

Note the difference with the T-case: here we do not haveany square root on the right-hand side. Since classically

H�; ~k ¼ H ~T; ~k, one can write H�; ~k ¼ H ~T; ~k. Approximating

again the state as the disentangled tensor product of

geometry and matter (where now �o obeys �iℏ@ ~T�o ¼12 H

2o�o), and projecting on �o, we find a Schrodinger-

like equation for c ðT; q ~kÞ,

iℏ@ ~Tc ¼1

2

��ℏ2 @2

@q2~kþ�X3i¼1

hp2i ð ~TÞik2i

�q2~k

�c : (B10)

By the comparison with the equation obtained for c inthe case of an effective classical geometry �g��, (45), we

read off the effective terms

�N ~Tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij �p1 �p2 �p3jp ¼ 1; �p2

i ¼ hp2i ð ~TÞi: (B11)

Therefore, the effective geometry probed by the mode ~kof the field is obtained as

�g��dx�dx� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijhp2

1ð ~TÞihp22ð ~TÞihp2

3ð ~TÞijq

���d ~T2 þX3

i¼1

ðdxiÞ2hp2

i ð ~TÞi�: (B12)

Once again, �g�� does not depend on the mode ~k: even

with this choice of time, which allows to lift the squareroot approximation and to push the analysis to theinclusion of backreaction, we see no violation ofLorentz symmetry at the test field order.

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