PHYSICAL REVIEW D 72, 067301 (2005)

Preclassical solutions of the vacuum Bianchi I loop quantum cosmology

Ghanashyam Date*The Institute of Mathematical Sciences, CIT Campus, Chennai-600 113, India

(Received 13 May 2005; published 12 September 2005)

*Electronic

1550-7998=20

Loop quantization of diagonalized Bianchi class A models, leads to a partial difference equation as theHamiltonian constraint at the quantum level. A criterion for testing a viable semiclassical limit has beenformulated in terms of existence of the so-called preclassical solutions. We demonstrate the existence ofpreclassical solutions of the quantum equation for the vacuum Bianchi I model. All these solutions avoidthe classical singularity at vanishing volume.

DOI: 10.1103/PhysRevD.72.067301 PACS numbers: 98.80.Qc, 04.60.Pp, 98.80.Bp

Loop Quantum Gravity (LQG) is the leading candidatefor a manifestly background independent approach to con-structing a quantum theory of gravity [1]. This approach isparticularly well suited in the context where the Einsteintheory indicates occurrence of singularities entailinghighly dynamical geometries with extreme curvatures.The methods employed in this approach can be adaptedand tested in the simpler context of cosmological models.Quantizing the cosmological models along the lines ofLQG has lead to the development of Loop QuantumCosmology (LQC) [2].

One of the crucial simplification available in LQC is theexistence of the triad representation and knowledge ofcomplete spectrum of the volume operator so crucial forquantization of the Hamiltonian constraint. The fact thatthe holonomies of the connection are well defined opera-tors but not the connection it self, is directly responsible forthe two main features of LQC: (a) the quantumHamiltonian constraint leads to a difference equation [3]and (b) inverses of triad components have bounded spectra[4,5]. Both these features lead to the absence of ‘‘singular-ities’’ in the quantum theory [6–9].

While loop quantization of cosmological models is wellspecified, one also needs to check if the quantum dynamicsadmits solutions (semiclassical states) which can approxi-mate the classical description. A natural way to recoverclassical behavior would be in terms of the expectationvalues of suitable observables in the semiclassical states.LQC (and LQG) being constrained systems, makes such arecovery of classical behavior, more complicated. Thesolutions of the Hamiltonian constraint, the only relevantone for LQC, are typically distributional and one needs toequip the space of solutions with a new (physical) innerproduct and also identify suitable (Dirac) observables.Addressing these aspects is at a preliminary stage [10–13].

Current understanding of the semiclassical limit of LQCis centered around the notion of preclassicality. The ar-ticulation of this notion has undergone a few changes and itis useful to note them. Originally proposed in [14], pre-classicality was thought in terms of obtaining the contin-

address: shyam@imsc.res.in

05=72(6)=067301(4)$23.00 067301

uum Wheeler-De Witt equation from the fundamentaldifference equation by a limit in which the Barbero-Immirzi parameter is taken to zero. Subsequently, theidea of preclassical limit was replaced by the idea of apreclassical approximation [15]. To obtain (modified)Einstein dynamics from the Hamilton-Jacobi equation pro-vided by a WKB approximation [16–18], it was shown tobe sufficient to have approximate preclassical solution(s)with a finite domain of validity. The property of localstability is needed in the construction of such solutions[15] and is satisfied by difference equations of LQC [9].Recently, the methods of analyzing the asymptotic behav-ior of exact solutions of the difference equations have beendeveloped [19,20] which provide the sharpest yet formu-lation of preclassicality of a solution [19]. Briefly, thecriterion is that asymptotically, small scale oscillations inthe solution be suppressed.

Preclassical solutions so identified are known to exist forisotropic models [6,14,20] and some of the LRS models[19]. However, for the anisotropic, vacuum Bianchi Imodel, preclassical solutions were shown not to exist[21]. While this result is true for solutions that pass throughvanishing volume, there are more possibilities which per-mit existence of preclassical solutions. Since the works in[8,9], the so-called Bohr quantization has been developed[7] which is crucial for the existence of preclassical solu-tions and we incorporate it in the brief summary of thequantum theory given below.

The kinematical Hilbert space is spanned by orthonor-malized vectors labeled as j�1; �2; �3i; �I 2 R. Theseare properly normalized eigenvectors of the triad operatorspI with eigenvalues 1

2�‘2P�I, where � is the Barbero-

Immirzi parameter and ‘2P

:� 8�G@ :� �@. The volumeoperator is also diagonal in these labels with eigenvaluesV� ~�� given by �12�‘

2P�

3=2���������������������j�1�2�3j

p. Here we have used

the vector notation to denote the triple ��1; �2; �3�.Imposing the Hamiltonian constraint operator on generalvectors of the form jsi �

P~�s� ~��j ~�i leads to the funda-

mental difference equation for the wave function s� ~��.Here the sum is over countable subsets of R3. The wavefunction s� ~�� has to be invariant under simultaneous re-versal of signs of a pair of �I ’s and this corresponds to the

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BRIEF REPORTS PHYSICAL REVIEW D 72, 067301 (2005)

residual freedom of reversing the sign of any two of thetriad components [8].

In the present context of vacuum Bianchi I model, theHamiltonian constraint leads to the equation [9],X

~�12

A12� ~�; ~�12�s� ~�; ~�12� � cyclic � 0; where (1)

~�12 � ��1; �2; �01; �

02� with each of the �� taking values�1;

s� ~�; ~�12� � s��1 ��0�1 ��0�01; �2 ��0�2 ��0�02; �3�; �0 is an order 1 parameter and,

A12� ~�; ~�12� � V� ~�; ~�12�d��3���1�2 � �01�02�

d��� :��j1��0��1j1=2 � j1��0��1j1=2 � � 0

0 � � 0

(2)

Note that while the Eq. (1) is defined for all�I 2 R, it isactually a difference equation since only the coefficientss� ~��, differing in steps of �2�0 are constrained by theequation. One can make this explicit by setting �I: �2�0�I � 2�0NI; NI 2 Z and S ~N� ~�� :� s� ~��. Clearly,there are infinitely many ‘‘sectors’’ labeled by ~� with �I 2�0; 1�. Only those sectors for which at least one of the �I iszero, one will encounter zero volume.

It is convenient to absorb the factors of volume eigen-values into the wave functions by defining t� ~�� :�V� ~��s� ~�� so that the Eq. (1) becomes,X

~�12

d��3���1�2 � �01�02�t� ~�; ~�12� � cyclic � 0 (3)

This preserves the gauge invariance condition on the s� ~��.Furthermore, due to the explicit volume eigenvalues,t�1;�2;�3

� 0 if any of the �I’s equal zero.Making a product ansatz for the wave function and

introducing the difference operator �,

t� ~�� :� z1��1�z2��2�z3��3�; (4)

�zI��I� :� fzI��I � 2�0� � zI��I � 2�0�g; (5)

the difference Eq. (3) can be written as,

d��3��z1��1��z2��2�z3��3� � cyclic � 0: (6)

The gauge invariance conditions then translate tozI���I� � �zI��I�;8I, where � � �1. The vanishingof t� ~�� when any of the �I � 0, translates into the condi-tion: zI�0� � 0;8I.

An exact solution of the partial difference equation canbe obtained by setting,

�zI��I� � �Id��I�zI��I�; 8I; 8�I 2 R (7)

where �I are some constants which have to satisfy,

�1�2 � �2�3 � �3�1 � 0 ��XI

�I

�2�XI

�2I : (8)

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Thus we obtain a class of exact solutions of the partialdifference Eq. (3) from those of three, ordinary differenceEqs. (7) with parameters �I satisfying (8).

The original partial difference Eq. (1) is linear and withreal coefficients, AIJ’s. So, without loss of generality, wecan assume the wave function, s� ~�� to be real. Since for theproduct ansatz, all zI are independent, these must be real aswell which requires that the �I’s be real.

If all �I are zero, then zI’s are constants and so is t� ~��.These cannot satisfy the condition zI�0� � 0 without mak-ing t� ~�� � 0 identically. The nontrivial solutions thencannot pass through zero volume and must belong to thesectors with �I � 0. This shows that in the sectors with�I � 0;8I, there is an exact preclassical solution to (1)namely t�2�0 ~�� 2�0

~N� � constant �� 0�;8 ~N 2 Z3.If at least one �I � 0, then we can always take out a

common factor from all �I and ensurePI�I � 1.

Equivalently, a common scaling of �I can be absorbedby a common inverse scaling of the d��� functions, whichamounts to a scaling of the volume which cancels out in(1). The class of solutions that is being constructed can thusbe parameterized exactly in the same manner as the clas-sical Kasner solution. In particular either two of the �’s arezero or exactly one is negative while other two are positive.From now on we will restrict to 0< j�Ij< 1.

To explore preclassicality of the separable solution, letus focus on (7), suppressing the label I. Introducing thenotation: � :� 2�0�� 2�0n; n 2 Z; � 2 �0; 1� andz��� :� Z���n , (7) can be written as:

Z�n�2 � Z�n � �d��; n� 1�Z�n�1; 8n 2 Z

d��; n� :� j1�1

2��� n�j1=2 � j1�

1

2��� n�j1=2

(9)

We have infinitely many decoupled sectors, labeled by �and for each of these we have a second order, ordinarydifference equation. Because of linearity, only one condi-tion is enough to determine a solution. Only for the sector� � 0 (� is integer multiple of 2�0), the condition z�0� �0 is relevant and it fixes the solution completely. The gaugeinvariance condition translates into the identification:

Z1����n�1� � �Z�n; 8n 2 Z; � � �1 (10)

which restricts the Z’s in the same sector, only for � � 0; 12 .

For all other sectors the gauge invariance condition relatestwo different sectors. Under the identification implied by(10), the equation satisfied by Z�n goes over to the equationsatisfied by Z1��

n automatically with the same value of �.Therefore the solutions in the sector �1� �� can all beobtained from solutions in the sector � via (10). For � � 1

2 ,

the gauge invariance condition requires Z1=2�1 � �Z1=2

0which fixes the solution completely. For the two sectors,� � 0; 1

2 , preclassicality of the solution is not optional.From now on the superscript � is suppressed.

Consider (9). Defining the ratios un�1 :� Zn�1=Zn, onecan see that ��un�2 � �d��; n� 1��un�1 � 1Zn � 0.

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There are several possibilities now (n 0 for definite-ness). If Zn remains nonzero for all n, then un ! �1 asn! 1. The un ! �1 is referred to as a sequence withsign oscillations. Since un�2 � �d��; n� 1� and un�1,must have the same sign, it is clear that if u1 > 0 and �>0, then un > 0 for all n. For �< 0 however, un maybecome negative for some n0 and then stay negative sub-sequently. Whether this could happen depends on the valueof u1. Thus it is conceivable that for some positive valuesof u1, one could have a sequence without sign oscillations.If Zn converges to zero, then junj converges to a value� 1,once again allowing for un ! �1. The sequence un couldalso converge to 0 in which case there could be oscillationsabout 0, but these are suppressed. By preclassicality wemean either absence or suppression of sign oscillations.

To identify preclassical sequences, we employ the gen-erating function technique [19]. The function d��� beingan algebraic function poses some difficulties which can behandled in exactly the same way as in [19]. Basically, oneseparates out the large n part of d��; n� and solves theequation perturbatively, Zn � an �

P1k�1 Z

kn. The leading

order term, an satisfies the equation with d��; n� replacedby �2�n� ����1. Let us focus on n 0 so that the absolutesigns can be removed. Setting � � 4 the equation defin-ing the an sequence is,

an�2 �2

n� 1� �an�1 � an � 0; n 0: (11)

To account for nonintegral �, we define a generatingfunction F�x� :�

P1n�0 anx

n�� and the function G�x� :�x�1�F�x� � a0x��, which satisfies a differential equationequivalent to the difference Eq. (11),

ddx��1� x2�G� a1x�� � 2G� a0��� 1�x� � 0 (12)

For nonzero �, the second term is singular at x � 0implying that G�x� will not be analytic at x � 0.However, it is integrable so that G�x� is continuous at x �0. The series representation requires it to vanish at x � 0.

The Eq. (12) can be easily integrated to give,

G�x� � �1� x��1�1� x���1

�c0 �

Z x�1� t1� t

�

� fa0�1� ��t� � a1�t

��1g

�(13)

Notice that given any a0; a1, the solution to (11) iscompletely determined and so should be G�x�. The abovesolution for G�x� contains an indefinite integral and anarbitrary constant of integration, c0. We must choose c0

and convert the indefinite integral to a definite one suchthat G�x� corresponds to the sequence specified by thegiven a0; a1. The only value of G�x� we know withouthaving to know the full sequence is G�0� � 0.Furthermore both the integrands in the integrals in (13)are integrable at x � 0. Thus it is possible to imposeG�0� � 0 which then requires c0 � 0. The generating

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function is then obtained as,

�1� x�G�x� � �1� x��1�1� x���a0I��; ; x�

� a1I��� 1; ; x�; (14)

I��; ; x� :� �1� ��Z x

0

�1� t1� t

�t� (15)

One may already note that the integrals are finite at bothx � �1 but the prefactor is not. Thus singularities of �1�x�G�x� are controlled by the prefactor which is indepen-dent of �. Apart from the � dependence of the integrals,one does not expect qualitative behavior of G�x� to beaffected by �. Furthermore, since there are two free pa-rameters (a0; a1) which specify the sequence and only oneof these is relevant one due to the linearity, one can at themost impose only one condition capturing preclassicality,to get a nontrivial solution.

Now consider the behavior of G�x� as x! �1. For therange of 0< jj< 1, both the integrals exist (and arepositive), but the prefactor diverges. A divergence inG�x� at x � �1 implies unsuppressed sign oscillationswhich are to be avoided for preclassical sequences [19].Clearly, to avoid this singularity in G�x�, the integrals mustadd up to zero which determines a1 in terms of a0. For x ��1, one has,

I��; ;�1� � ��1��I��;�; 1�; (16)

I��;�; 1� � �1� ��B�1� �; 1� �

� F��; �� 1;�� �� 2;�1�; (17)

where, B and F are the Beta function and the hypergeo-metric functions [22]. The condition of no singularity atx � �1 gives,

a0I��;�; 1� � a1I��� 1;�; 1�: (18)

This determines the sequence uniquely modulo a trivial,overall scaling. The sequence satisfying (18) has sup-pressed sign oscillations. Its convergence properties aredetermined by the x! 1 behavior of �1� x�G�x�. For x �1 the integrals again exist but now the prefactor divergesfor > 0 and vanishes for < 0 and so does the sequencefang. However, �1� x�G�x� is integrable at x � 1 whichimplies that the asymptotic behavior of an is bounded by n.Making a power law ansatz for asymptotic an, one can seefrom (11) (and indeed from (9) as well) that an n. Since�I come with both signs, both behaviors must be admis-sible. For � 0, one gets a1 � a0 implying an � a08n 0 which is obviously preclassical.

As anticipated, these results are exactly analogous tothose obtained in [19]. Indeed, for � � 0, Eqs. (11)–(13)go over to the equations of [19]. Now the boundary con-dition is G�0� � a1 which gives c0 � a1 and definite in-tegral has the lower limit as 0. Demanding nonsingularityofG��1�, determines a1 exactly as in [19]. The behavior atx � 1 is similar to that for the nonzero � case.

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BRIEF REPORTS PHYSICAL REVIEW D 72, 067301 (2005)

Thus, there certainly exist sectors such that for eachchoice of the separation constant �; 0< j�j< 1, one canselect a unique solution of (11) which is preclassical.These solutions of course have to be improved by comput-ing the corrections Zkn [19]. The asymptotic power lawbehavior of an will continue to hold also for Zn.

All these statements hold for n 0. Having determinedZ�0 ; Z

�1 , by preclassicality, Zn<0 can be determined by the

exact Eq. (9). Whether Z�n is preclassical also for n < 0,can be inferred by testing for preclassicality of Z1��

n forpositive n, using (10).

As noted earlier, the sectors � � 0; 12 already have a

unique solution due to the conditions Z00 � 0 and Z1=2

�1 �

�Z1=20 respectively. If these conditions are imposed on the

leading preclassical sequence fang, then clearly there areno nontrivial solutions in the � � 0 sector. For � � 1=2sector, numerically, the gauge invariance condition and thepreclassicality condition seem to hold only for � � 0 with� � 1. For other sectors, preclassicality is the only condi-tion imposed and solutions can be constructed.

The full wave function is the product of the three se-quences and apart from an overall constant factor, is com-pletely determined. One can build more general (andnonseparable) solutions by taking complex linear combi-nations with coefficients being functions of ~�. Clearly, acombination involving a diverging and a vanishing solutionwill be a diverging one and still without sign oscillations.Since such a solution involves a distribution of ~�, theseparameters themselves would not be identified with theclassical Kasner parameters, I (say) satisfying

PII �

1 �PI

2I . Rather, one would imagine constructing linear

combinations which ‘‘peak’’ in some suitable sense,

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around three triad values pI0 and a Kasner parameter ~0.(Since the reduced phase space of the vacuum Bianchi Imodel is 4 � 6� 2 dimensional, one needs four parame-ters to specify a classical solution and these could beconveniently taken as three initial triad values and aKasner parameter value.) If such a construction can becarried out, then one would be able to claim that thequantum theory has ‘‘sufficient number of semiclassicalsolutions’’ as expected from the classical theory.

We note that since solutions of the Hamiltonian con-straint are expected to be distributional in general, kine-matical normalizability of the preclassical (or otherwise)solutions is not directly mandated. The requirement ofpreclassicality for both signs of n is an open issue. Foran alternative treatment of separable solutions, see [23].

In summary, we have shown that in every sector �I �

0; 12 , there exist a one parameter family of preclassical

solutions. For �I � 0, the solution is in fact exact, possiblycorresponding to the Minkowski space-time. All thesesolutions skip the vanishing volume eigenvalues. By con-trast, in the �I � 0 sector, there are no preclassical solu-tions [21]. The richness of the loop quantization,manifested by the infinitely many sectors, is crucial forthis result; an observation also made in [11] in the isotropiccontext. We have heuristically indicated how these familiescan be used to see if loop quantization does admit ‘‘enoughsemiclassical states’’. The exact, nonsingular solutions ofthe effective dynamics of vacuum Bianchi I models givenin [24] also exhibit a similar feature of avoiding vanishingvolume which motivated this work.

I would like to thank Martin Bojowald for helpful re-marks and Golam Hossain for discussions.

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