absence of singularities in loop quantum cosmology

7/30/2019 Absence of Singularities in Loop Quantum Cosmology

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arXiv:g

r-qc/0102069v114Feb2001

Absence of Singularity in Loop Quantum Cosmology

Martin Bojowald

Center for Gravitational Physics and Geometry, The Pennsylvania State University,

104 Davey Lab, University Park, PA 16802, USA

It is shown that the cosmological singularity in isotropic minisuperspaces is naturally removedby quantum geometry. Already at the kinematical level, this is indicated by the fact that the inversescale factor is represented by a bounded operator even though the classical quantity diverges at theinitial singularity. The full demonstation comes from an analysis of quantum dynamics. Because ofquantum geometry, the quantum evolution occurs in discrete time steps and does not break downwhen the volume becomes zero. Instead, space-time can be extended to a branch preceding theclassical singularity independently of the matter coupled to the model. For large volume the correctsemiclassical behavior is obtained.

CGPG01/21gr-qc/0102069

On a macroscopic scale, the gravitational field is suc-cessfully described by general relativity, which is experi-mentally well tested in the weak field regime. However,this classical theory must break down in certain situa-tions where it predicts singularities, i.e. boundaries of

space-time which can be reached by observers in finiteproper time, but beyond which an extension of the space-time manifold is impossible [1]. An outstanding exam-ple is the big-bang singularity appearing in cosmologi-cal models. At this point curvature diverges whence theclassical theory completely breaks down and has to bereplaced by a quantum theory of gravity. However, upto now there is no complete quantum theory of gravity,and so the problem has been approached by first carry-ing out a symmetry reduction (by requiring isotropy andhomogeneity) and then quantizing the resulting minisu-perspace models which have only a finite number of de-grees of freedom [2,3]. In the context of these models, as

yet, there is no definitive resolution of the status of theinitial singularity. Furthermore, generally the methodsused in this analysis can easily miss some key features ofthe full theory. Indeed, while it has been speculated fora long time that quantum gravity may lead to a discretestructureof space and time which could cure classical sin-gularities, it has not been possible to embody this ideain standard quantum cosmological models.

By now, there are promising candidates for a quan-tum theory of gravity. The results reported in this letterare obtained in the framework of quantum geometry [4]which does predict a discrete geometry because, e.g., thespectra of geometric operators as area and volume arediscrete [57]. Although temporal observables have notbeen included in the full theory, it is clear that the space-time structure is very different from that used in generalrelativity. But this difference can be important only atvery short scales or in high curvature regimes like theone close to the classical singularity. This leads to thebasic question raised here: What happens to the classicalcosmological singularity in quantum geometry?

The first step in our approach is the construction of

isotropic states in full quantum geometry; we first quan-tize and then carry out a symmetry reduction. This,however, is not a straightforward problem because thediscrete structure of space, represented by a graph (spinnetwork) embedded in space, necessarily breaks any con-

tinuous symmetry. But symmetric states can be definedas generalized states of quantum geometry [8] which canbe used for a reduction to minisuperspace models [9].Note that this is not a standard symmetry reduction ofthe classical theory because symmetric states are inter-preted as generalized states in the full kinematical quan-tum theory. Only the Hamiltonian constraint has to bequantized and solved after the reduction. An immediateand striking consequence is that, in contrast to standardquantum cosmological models, spatial Riemannian ge-ometry is discrete leading to a discrete volume spectrum[10]. Furthermore, in contrast to standard quantum cos-mology, the same techniques as in the full theory [11] can

be used for the quantization of the reduced Hamiltonianconstraint of the cosmological models [12]. This impliesanother difference, namely that the evolution equation isnot a differential equation in time, but a difference equa-tion manifesting the discreteness of time [13].

Structure of isotropic models. According to [8,9]states for isotropic models in the connection representa-tion are distributional states of the full kinematical quan-tum theory supported on isotropic connections of theform Aia = c

iI

Ia where I is an internal SU(2)-dreibein

and I are the left-invariant one-forms on the transla-tional part of the symmetry group acting on the space

manifold . The momenta are densitized triads of theform Eai = p

Ii X

aI with left-invariant densitized vector

fields XI fulfilling I(XJ) =

IJ. Besides gauge freedom,

there are only the two canonically conjugate variables{c, p} = /3 ( = 8G is the gravitational constantand > 0 the BarberoImmirzi parameter) which havethe physical meaning of extrinsic curvature and squareof radius (a =

|p| is the scale factor). The kinemati-cal Hilbert space Hkin = L2(SU(2), dH) is the space offunctions of isotropic connections which are square inte-

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with action

H(E)|n = 3l2

P

(V|n|/2 V|n|/21)(|n + 4 2|n + |n 4) . (6)

In order to unfreeze dynamics and interpret solu-tions as evolving states, as usual [15,16] we have to in-

troduce an internal time which we choose as the dreibeincoefficient p. Accordingly, we transform states |s intoan adapted representation by expanding |s =n sn|nin eigenstates |n of p. This will allow us to find an in-terpretation of physical states as evolving histories. Fur-thermore, discrete geometry imples that eigenvalues of pare discrerte, whence time evolution is now discrete (see[13] for details). Moreover, since we chose a geometricalquantity as time which can be negative and is zero forvanishing volume, we will be able to test the possibilityof a quantum evolution through the classical singularity.

To realize dynamics, we need to extend the model withmatter degrees of freedom which can evolve with this in-

ternal time. Matter can be incorporated by using co-efficients sn() depending on the matter field in anappropriate fashion, the details of which is not impor-tant for what follows. The Hamiltonian constraint canthen be written down using a matter Hamiltonian H(as in [14]) which is diagonal in the gravitational degreesof freedom (and can also contain a cosmological term).The resulting quantum constraint equation can then beregarded as an evolution in discrete time:

(V|n+4|/2 V|n+4|/21)sn+4()2(V|n|/2 V|n|/21)sn()+(V|n4|/2 V|n4|/21)sn4() = 13l2P H sn() (7)

(Vj are the eigenvalues (2) of the volume operator withV1 = 0) which is a difference equation for the coeffi-cients sn() depending on the discrete label n (our dis-crete time).

Fate of the singularity. Given initial data sn() forsome negative n, we can use (7) in order to determinelater values for higher n. This, however, is possibleonly as long as the highest order coefficient V|n+4|/2 V|n+4|/21 is nonzero, which is the case if and only ifn = 4. So all coefficients for n < 4 are determined bythe initial data. However, (7) does not determine s0 andinstead leads to a consistency condition for the initial

data. So the quantum evolution appears to break-downjust at the classical singularity, i.e. at the zero eigenvalueof p. But this is not the case; in fact all sn for n > 0are determined by (7) from the initial data. This oc-curs because for n = 0 we have: i) V|n|/2 V|n|/21 = 0,and ii) Hsn() = 0; thus s0 completely drops out ofthe iterative evolution. E.g., s4 is determined solely bys4 because the coefficient of sn vanishes for n = 0. Sowe can evolve through the singularity and determine allsn for n = 0. (The vanishing of Hs0() follows from

the quantization of matter Hamiltonians [14] similarly asdescribed for the inverse scale factor.)

Of course, in order to determine the complete state wealso have to know s0, but a closer analysis reveals thats0 is fixed from the outset: The Hamiltonian constraintalways has the eigenstate sn = s0n0 with zero eigenvaluewhich is completely degenerate and not of physical inter-est. All evolving solutions are orthogonal to this stateand have s0 = 0 which already fixes the coefficient s0 leftundetermined by using the evolution equation. We seethat the complete state is determined by initial data fornegative n, and so there is no singularity in isotropic loopquantum cosmology. The intuitive picture is as follows:Since for n < 0 the volume eigenvalues V(|n|1)/2 decreasewith increasing n, there is a contracting branch for neg-ative n leading to a state of zero volume (in general,s1 = 0 and the volume vanishes for n = 1 which cor-responds to j = 0) in which the universe bounces off lead-ing to the expanding branch for positive n which only canbe seen in the classical theory and in standard quantum

cosmology. This conclusion holds true for any kind ofmatter and cosmological constant, and is a purely quan-tum gravitational effect. In particular, we do not need tointroduce matter violating energy conditions and therebyevade the singularity theorems. However, our result cru-cially depends on the factor ordering of the constraintwhich was chosen as one of the standard possibilities or-dering all triad components to the right.

The semiclassical regime. We have seen that the clas-sical singularity is removed in loop quantum cosmol-ogy. But we need more for a viable cosmological model,namely we also need the correct behavior in the semi-classical regime. Classical behavior can only be present

for large volume and small extrinsic curvature, i.e. if|n| is large, c is small and the wave function does notvary strongly between successive times n (otherwise thestate would have access to the Planck scale). In thisregime we can interpolate between the discrete labelsn and define a wave function (a) := sn(a), n(a) :=6a2/l2P with a ranging over a continuous range (using

a =|p| lP|n|/6 for large |n| as interpola-

tion points). The difference operator then becomes(s)n := sn+1 sn1 = 16l2Pa1d/da + O(l5P/a5)leading to an approximate constraint operator H(E) 96(i/2)2 a/4 62l4P( i3d/d(a2))2a for large a.

This is exactly what one obtains from the classical con-straint H(E) = 6c2|p| in standard quantum cosmol-

ogy [17] by quantizing 3c = il2Pd/dp. In our frame-work, however, this is only an approximate equation validfor large scale factors. For this equation one can useWKB-techniques in order to derive the correct classicalbehavior.

Going to smaller a one has to include more and morecorrections in the expansion of the difference operatorsand also of the volume eigenvalues. By doing so one can

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derive perturbative corrections for an effective Hamilto-nian including higher derivative terms. The closer wecome to the classical singularity, the more correctionswe have to include; and at the singularity we need toknow all corrections which, as we know from our non-perturbative solution, have to add up to yield the dis-crete time behavior. So in these models higher orderterms arise from the non-locality in discrete time of the

fundamental theory. But even knowing all perturbativecorrections, it would be very hard to see the correct be-havior without knowing the non-perturbative quantiza-tion.

Quantum Euclidean space. In the simplest case, theEuclidean constraint for a spatially flat model withoutmatter, it is possible to find an explicit solution to theconstraint. The constraint equation is of order eight withone consistency condition as described above, so one ex-pects seven independent solutions. But we are interestedonly in solutions which have a classical regime in the pre-vious sense, i.e. no strong dependence on j for large j.Under this condition one can see that there is a unique(up to a constant factor) solution

(c) =j

2j + 1

Vj+ 12

Vj 12

j(c) (8)

in the connection representation. In standard quantumcosmology the constraint equation is c2

|p|(c) = 0 witha solution

|p|(c) = (c) which is not unique. In or-der to compare the solutions we quantize a by aj =2i(l2P)

1(Vj+ 12

Vj 12

)j leading to a

j(2j + 1)jwhich in fact is the delta function on the configurationspace SU(2). Therefore, we have a unique solution which

incorporates the characterization of Euclidean space tohave vanishing extrinsic curvature of its flat spatial slices.Conclusions. We have shown in this paper that

canonical quantum gravity is well-suited to analyze thebehavior close to the classical singularity. For this, it isimportant to use only techniques which are applicable inthe full theory. This leads to a discrete structure of spaceand time which cannot be seen in standard quantum cos-mology. In our framework, the standard quantum osmo-logical description arises only as a limit for large volumewhere the discreteness is unimportant. For small vol-ume, quantum geometry leads to new effects which areresponsible for the removal of the classical singularity. In

contrast to earlier attempts this is not achieved by intro-ducing matter which violates energy conditions; it is apure quantum gravity effect. It also does not avoid thezero volume state present in the classical singularity be-cause in general the wave function is not orthogonal tostates with zero volume eigenvalue. Nevertheless there isno sign of a singularity because in quantum geometry itis possible to have vanishing volume but non-diverginginverse scale factor, which in isotropic models dictates

all curvature blow-ups. Besides removing the singular-ity, the fact that an evolution through a state of zerovolume is possible without problems could lead to topol-ogy change in quantum gravity. Technically, the removalof the singularity is achieved by using Thiemanns strat-egy [11] of absorbing inverse powers of V into a Poissonbracket which also lead to densely defined matter Hamil-tonians [14]. So it is the same mechanism which regular-

izes ultraviolet divergences in matter field theories andwhich removes the classical cosmological singularity. Wehave also seen that non-perturbative effects are solelyresponsible for this behavior and a purely perturbativeanalysis could not lead to these conclusions.

Acknowledgements. The author is grateful to A.Ashtekar for suggesting a study of the implications ofdiscrete volume and time close to the classical singu-larity, for discussions and for a critical reading of themanuscript, and to H. Kastrup for comments. This workwas supported in part by NSF grant INT9722514 and theEberly research funds of Penn State.

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absence of singularities in loop quantum cosmology

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