Semiclassical and quantum behavior of the mixmaster model in the polymer approach

Orchidea Maria Lecian,1,* Giovanni Montani,1,2,† and Riccardo Moriconi1,‡

1Dipartimento di Fisica (VEF), P. le A. Moro 5 (00185) Roma, Italy2ENEA-UTFUS-MAG, C.R. Frascati, 00044 Rome, Italy(Received 3 June 2013; published 11 November 2013)

We analyze the quantum dynamics of the Bianchi Type IX model, as described in the so-called polymer

representation of quantum mechanics, to characterize the modifications that a discrete nature in the

anisotropy variables of the Universe induces on the morphology of the cosmological singularity. We first

perform a semiclassical analysis, to be regarded as the zeroth-order approximation of a WKB approxi-

mation of the quantum dynamics, and demonstrate how the features of polymer quantum mechanics are

able to remove the chaotic properties of the Bianchi IX dynamics. The resulting evolution towards the

cosmological singularity overlaps the one induced, in a standard Einsteinian dynamics, by the presence of

a free scalar field. Then, we address the study of the full quantum dynamics of this model in the polymer

representation and analyze the two cases in which the Bianchi IX spatial curvature does not affect the

wave-packet behavior, as well as the instance for which it plays the role of an infinite potential confining

the dynamics of the anisotropic variables. The main development of this analysis consists of investigating

how, differently from the standard canonical quantum evolution, the high quantum number states are not

preserved arbitrarily close to the cosmological singularity. This property emerges as a consequence, on

one hand, of the no-longer-chaotic features of the classical dynamics (on which the Misner analysis is

grounded), and, on the other hand, of the impossibility of removing the quantum effect due to the spatial

curvature. In the polymer picture, the quantum evolution of the Bianchi IX model remains always

significantly far from the semiclassical behavior, as far as both the wave-packet spread and the occupation

quantum numbers are concerned. As a result, from a quantum point of view, the mixmaster dynamics loses

any predictivity characterization for the discrete nature of the Universe’s anisotropy.

DOI: 10.1103/PhysRevD.88.103511 PACS numbers: 98.80.Qc, 04.60.Kz, 04.60.Pp

I. INTRODUCTION

Even though the thermal history of the Universe isproperly described by the homogeneous and isotropicRobertson-Walker cosmological model [1], up to the veryearly stage of its evolution, reliably since the inflationaryphase has been completed, the nature of the cosmologicalsingularity is a general property of the Einsteinian dynam-ics, suggesting the necessity of relaxing the high symmetryof the geometry that characterizes the big bang. A veryvaluable insight on dynamical behaviors more general thanthe simple isotropic case, especially in view of the classicalfeatures of the initial singularity and of its quantum ones, isoffered by the Bianchi type IX model [2–10], essentiallyfor the following three reasons: (i) this model is homoge-neous, but its dynamics possesses the same degree ofgenerality as any generic inhomogeneous model; (ii) thecanonical quantization of this model can be performed viaa minisuperspace approach, which reduces the asymptoticevolution near the singularity to the well-defined paradigmof the ‘‘particle in a box’’; (iii) the late (classical) evolutionof the model is naturally reconciled, with or without theinflation [11,12], to that of a closed isotropic Universe [13].

Two years after the derivation of the Wheeler-DeWittequation, C.W. Misner applied this canonical quantizationapproach to the Bianchi IX model, which, in theHamiltonian version he had just provided (restating theoscillatory regime derived by Belinski, Khalatnikov, andLifshitz), constitutes a proper application.The main result of the Misner quantum analysis of the

Bianchi IX dynamics is to provide a brilliant demonstra-tion of the phenomenon for which very high occupationnumbers are preserved in the evolution towards the cos-mological singularity. This result is achieved by using thechaotic properties of the classical Bianchi IX dynamics(the mixmaster model) and by approximating the potentialwell in which the Universe-particle moves with a simplesquare box, instead of the equilateral triangle it indeed is.Many subsequent studies have been pursued on such a

quantum dynamics, both in the Misner variables [14], aswell as in other frameworks (such as the Misner-Chitrescheme [15,16]), but the original semiclassical nature ofthe cosmological singularity, when considered in terms ofhigh occupation numbers, still remains the most strikingprediction of the canonical quantum dynamics provided bythe mixmaster model.In the Misner variables, the dynamical problem is re-

duced to a two-dimensional scheme, in which the role ofthe time variable is played by the Universe’s volume, whilethe two physical degrees of freedom are represented via the

*lecian@icra.it†giovanni.montani@frascati.enea.it‡moriconi.phys@gmail.com

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Universe’s anisotropies. Such a picture is elucidated by anArnowitt-Deser-Misner (ADM, [17]) reduction of the var-iational principle (based on the solution of the super-Hamiltonian constraint), but its physical significanceemerges in a very transparent way already in the directHamiltonian approach to the dynamics.

Here, we address the quantum analysis of the Bianchi IXmodel by using the super-Hamiltonian constraint, which,via the Dirac prescription, leads to the Wheleer-DeWittequation; this procedure is expectedly fully consistent withthe Schrodinger-like dynamics following from the ADMreduction of the Dirac constraint. The new feature weintroduce in the present study is the discrete nature of theanisotropic degrees of freedom, by a two-dimensionalquantization approach, based on the so-called polymerrepresentation of quantum mechanics. The use of such amodified quantization scheme is justified by the requestthat a cutoff in the spatial scale, as expected at thePlanckian level, would induce a corresponding discretemorphology in the configuration space. The reason forretaining the isotropic Misner variables, connected to theUniverse’s volume, as continuous ones relies on the roletime plays in the dynamical scheme.

This way, we discuss first the semiclassical behavior ofthe model, to be regarded to as the zeroth-order approxi-mation of a WKB expansion of the full quantum theory. Inother words, we analyze the classical behavior of a modi-fied Hamiltonian dynamics of the mixmaster model, basedon the prescriptions fixed by the classical limit of the newparadigm. In this respect, we remark that the typical scaleof the polymer discretization is not directly related to thevalue of ℏ; the classical limit for this quantity approachingzero still constitutes a modification of the Einsteinianclassical dynamics. Such a semiclassical study is relevantto the interpretation of the full quantum behavior of thesystem, especially for the analysis of localized wave pack-ets. The main result for this semiclassical analysis is thedemonstration that the chaotic structure of the asymptoticevolution of the Bianchi IX model to the cosmologicalsingularity is naturally removed by a dynamical mecha-nism very similar to the one induced on the same dynamicsby the introduction of a free massless scalar field. In thelimit of small values of the polymer lattice parameter, wecalculate the modified reflection law for the Universe-particle against the potential walls, which are due to thespatial curvature. For the general case, we provide a precisedescription of how the bounce against these walls isavoided and which conditions the free-motion parametersof the model must satisfy for it to take place.

The absence of chaos in the semiclassical behavior ofthe polymer mixmaster model prevents us from directlyimplementing the Misner procedure, which is basic to itsdescription of the states approaching the singularity withvery high occupation numbers. We are therefore led toanalyze separately two cases: one in which the potential

walls can be neglected in the quantum evolution, such thatwe deal with free-particle wave packets, and one in whichwhen the potential walls play the role of a ‘‘box,’’ in whichthe Universe-particle is confined. In both cases, the behav-ior of localized wave packets is analyzed to better under-stand the limit up to which the Misner semiclassical featuresurvives in this modified approach. The main outcome wedevelop here is to identify how the presence of the walls is,sooner or later, relevant for the wave packets’ evolution,such that also the states with high occupation numbers areaccordingly obliged to spread close enough to the singu-larity, simply because the potential box destroys theirsemiclassical nature. The direct comparison with theMisner result is not possible because of the semiclassicalbehavior of the model, but the conclusion of our analysis isotherwise very clear, because there is no chance to build upa localized state that can reach the singularity withoutbouncing against the potential walls. This is because theconditions to fix a direction in the configuration space,which would allow for a free motion, are indeed timedependent and are, sooner or later, violated during theevolution of the wave packets. In this scheme, there is nopossibility of retaining the semiclassical features in thequantum description, and we can therefore claim thatthe singularity of the mixmaster Universe, as viewed inthe present polymer representation, cannot be described bysemiclassical motion, such as for the Einsteinian oscilla-tory regime of the expanding and contracting independentdirections, and that even its quantum relic, i.e., the occur-rence of high occupation numbers close to the singularpoint, is removed in a discrete quantum picture for theanisotropy degrees of freedom.The relevance of this result is enforced by observing

that, as investigated in [11], an oscillatory regime cannotexist before a real classical limit of the Universe is reached.As a consequence, since a simple model for the cutoffphysics is able to cancel also the memory of semiclassicalfeatures in the Planck era (as Misner argued in [18]), we areled to believe that the classical mixmaster dynamics is notfully compatible with the quantum origin of the Universe,and it is indeed a classical dynamical regime reached bythe system when the quantum effects are very small and thephysical and configuration spaces appear as bounded bycontinuous domains.A certain specific interest for the implementation of a

polymer approach to the quantum dynamics of theUniverse was raised by the analogy of this quantum pre-scription to the main issues of loop quantum cosmology,which, in the minisuperspace, essentially reduces to apolymer treatment of the Ashtekar-Barbero-Immirzi vari-ables, as adapted to the cosmological setting [19–21].The first loop quantum cosmological analysis of the

Bianchi IX model was provided in [22], where thenonchaotic nature of the semiclassical dynamics was ar-gued. The main reason for such nonchaotic behavior of the

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Bianchi IX model must be determined in the discretenature of the Universe volume and, in particular, in itsminimal (cutoff) value. In fact, asymptotically to thesingularity, the potential walls can no longer arbitrarilygrow and the Universe-particle confinement is removed.This analysis was based on the so-called inverse volumecorrections and properly accounts for the inducedsemiclassical implications of the loop quantum gravitytheory. Nonetheless, the considered approach is basedon a regularization scheme (the so-called �0 one) thatis under revision, in order to provide a consistent refor-mulation of the Bianchi IX dynamics as done in [23]for the isotropic Robertson-Walker geometry (the �� regu-larization scheme).

A step in this direction has been pursued in [24], wherethe loop quantum dynamics is rigorously restated by adopt-ing the �� regularization scheme, demonstrating that, undercertain circumstances, the chaotic features of the BianchiIX model are removed again. However, this result holdsonly when the quantum picture includes a massless scalarfield, able to remove the chaoticity even on a classicalEinsteinian level. The reason for this striking differencein the two results obtained in these two approaches must beindividualized in the kind of semiclassical corrections dis-cussed in [24]. Indeed, in this analysis only the so-calledholonomy correction contributions are considered and theyare unable to induce on the dynamics the basic feature of avolume cutoff scale. It is just these different types ofquantum corrections, adopted to construct the semiclassi-cal limit, that are the reliable source of the nongenericnature of the chaoticity removal. Although it is expectedthat inverse volume corrections can remove the chaoticbehavior of the Bianchi IX model as in [22], neverthelessthis has not been explicitly demonstrated and it stands as amere conjecture.

It is worth noting that a critical revision of the loopquantum cosmology picture of the primordial Universespace was presented in [25], where the necessity of a gaugefixing in implementing the homogeneity constraint is re-quired. A consistent quantum reformulation of the dynam-ics of a homogeneous model was then constructed in [26],which allows a semiclassical limit of the theory, in closeanalogy to the full loop quantum gravity theory.

The polymer formulation of the canonical approach tothe minisuperspace mimics very well some features of theloop quantum cosmology methodology (de facto a polymertreatment of the restricted Ashtekar-Barbero-Immirzi var-iables for the homogeneity constraint). However, there is acrucial difference between our result and such recent loop-like approaches. In fact, we apply the polymer procedure tothe anisotropic variables only (the real degrees of freedomof the cosmological gravitational field), leaving theUniverse’s volume unaffected by the cutoff physics, inview of its timelike behavior. The removal of the chaos,discussed here, is therefore not related to the volume

discretization, and it is also difficult to characterize itsrelation with the holonomy correction approach (studyingproperties of the edge morphology more than of the nodes,but nondirectly reducible to the anisotropy concept). Inthis respect, the present result must be regarded as essen-tially an independent one with respect to the ones actuallyavailable in loop quantum cosmology.This paper is organized as follows. In Sec. II, we in-

troduce the polymer representation of quantum mechanics,by a kinematical and a dynamical point of view. Then weanalyze the continuum limit and conclude the section byillustrating two fundamental examples of one-dimensionalsystems: the polymer free particle and the particle in a box.In Sec. III, we review the principal (classical and quantum)features of the mixmaster model, as studied by Misner in[18]. Section IV is dedicated to the study of the polymermixmaster model, from a semiclassical point of view. Inparticular, we analyze the modified relational motion be-tween the Universe-particle and the walls, and we derive amodified reflection law for one single bounce against thewall. In Sec. V, we build up the wave packets for the casewhen the wave function of the Universe is related to a freepolymer particle and to a polymer particle in a square box,respectively. Finally, Sec. VI is devoted to the numericalintegrations of the polymer wave packets and to the analy-sis on the quantum numbers related to the anisotropy.Concluding remarks complete the paper.

II. THE POLYMER REPRESENTATION OFQUANTUM MECHANICS

To apply the modified polymer approach to the mix-master quantum dynamics, we briefly summarize the fun-damental features of this modified quantization scheme. Inparticular, after giving a general picture of the model, weconsider the two specific cases of the free particle and ofthe particle in a box, which are relevant for the subsequentcosmological study.

A. Kinematical properties

The polymer representation of quantum mechanics is anonequivalent representation of the usual Schrodingerquantum mechanics, based on a different kind of canonicalcommutation rules. It is a really useful tool to investigatethe consequences of the hypothesis for which the phasespace variables are discretized.For the definition of the kinematics of a simple one-

dimensional system [27], one introduces a discrete set ofkets j�ii, with �i 2 R and i ¼ 1; . . . ; N. These vectorsj�ii are taken from the Hilbert space H poly ¼L2ðRb; d�HÞ, i.e., the set of square-integrable functionsdefined on the Bohr compactification of the real line Rb

with a Haar measure d�H. One chooses for them an innerproduct with a discrete normalization h�j�i ¼ ��;�.

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The state of the system is described by a generic linearcombination of them:

jc i ¼ XNi¼1

aij�ii: (1)

One can identify two fundamental operators in this Hilbertspace: a label operator " and a shift operator sð�Þ. Theyact on the kets as follows:

"j�i ¼ �j�i; sð�Þj�i ¼ j�þ �i: (2)

To characterize our system, described by the phase spacevariables p and q, one assigns a discrete characterization tothe variable q, and chooses to describe the wave function ofthe system in the so-called p polarization. Consequently,the projection of the states on the pertinent basis vectors is

��ðpÞ ¼ hpj�i ¼ e�i�p: (3)

Through the introduction of two unitary operators Uð�Þ ¼ei�q, Vð�Þ ¼ ei�p, ð�;�Þ 2 R which obey the Weyl com-mutation rules (WCRs) Uð�ÞVð�Þ ¼ ei��Vð�ÞUð�Þ, onesees that the label operator is exactly the position operator,while it is not possible to define a (differential) momentumoperator, as a consequence of the discontinuity for sð�Þpointed out in Eq. (2).

B. The dynamical features

For the dynamical characterization of the model,the properties of the Hamiltonian system have to beinvestigated. The simplest Hamiltonian describing aone-dimensional particle of mass m in a potential VðqÞ isgiven by

H ¼ p2

2mþ VðqÞ: (4)

In the p polarization, as a consequence of the discretenessof q, it is not possible to define p as a differential operator.The standard procedure is to define a subspace H �a

of

H poly containing all vectors that live on the lattice of

points identified by the lattice spacing a:

�a ¼ fq 2 Rjq ¼ na;8 n 2 Zg; (5)

where a has the dimensions of a length.Consequently, the basis vectors are of the form j�ni

(where �n ¼ an), and the states are all of the form

jc i ¼ Xn

bnj�ni: (6)

The basic realization of the polymer quantization is toapproximate the term corresponding to the nonexistentoperator (in this case, p), and to find for this approximationan appropriate and well-defined quantum operator. The

operator V is exactly the shift operator s, in both polar-izations. Through this identification, it is possible to ex-ploit the properties of s to write an approximate version ofp. For p � 1

a , one gets

p ’ sin ðapÞa

¼ 1

2aðeiap � e�iapÞ; (7)

and then the new version of p is

paj�ni ¼ i

2aðj�n�1i � j�nþ1iÞ: (8)

One can define an approximate version of p2. For p � 1a ,

one gets

p2 ’ 2

a2½1� cos ðapÞ� ¼ 2

a2½1� eiap � e�iap�; (9)

and then the new version of p2 is

p2aj�ni ¼ 1

a2½2j�ni � j�nþ1i � j�n�1i�: (10)

Remembering that q is a well-defined operator as in thecanonical way, the approximate version of the startingHamiltonian (4) is

Ha ¼ 1

2mp2a þ VðqÞ: (11)

The Hamiltonian operator Ha is a well-defined andsymmetric operator belonging to H �a

.

C. The continuum limit

The polymer representation of quantum mechanics thatis related to Schrodinger representation can now beanalyzed.Starting from a Hilbert space H poly, one needs to

verify a limit operation to demonstrate that the space isisomorphic to the Hilbert space H S ¼ L2ðR; dqÞ.1The natural way to proceed is to start from a lattice �0 ¼fqk 2 Rjqk ¼ ka0;8 k 2 Zg and subdivide each intervala0 in 2n intervals of length an ¼ a0

2n . Unfortunately, this is

not possible because, when densifying the lattice, theelements of H poly have a norm that tends to infinity.

This is because H S and its states cannot be included inH poly. However, it is possible to realize a different proce-

dure. From a continuous wave function, one has to find thebest wave function defined on the lattice that approximatesit, in the limit when the lattice becomes denser. The strat-egy to properly implement this approach is the introductionof a scale Cn, which, in our case, is the subdivision ofthe real line into disjoint intervals of the form �i ¼½ian; ðiþ 1ÞanÞ, where the extrema of the range are thelattice points. On this level, one approximates continuousfunctions with constant intermediate states belonging toH S. So far, one has a whole series of effective theories,depending on the scale Cn, that approximate better andbetter the continuous functions and that have a well-defined Hamiltonian. As in [28], by introducing a cutoff

1L2 is the set of square-integrable functions defined on the realline R with a Lebesgue measure dq.

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for each Hamiltonian defined on the intervals, making theoperation of coarse graining and entering a normalizationfactor in the internal product, one verifies that the existenceof a continuous limit is equivalent to the description of theenergy spectrum (relative to the Hamiltonian defined afterthe cutoff) as tending to the continuous spectrum, such thata complete set of normalized eigenfunctions exists. Thespace obtained this way is isomorphic to the space H S.

D. The free polymer particle

In this subsection, we analyze the simplest one-dimensional system in the presence of a discrete structureof the space variable q, i.e., the free polymer particle [29].When the free polymer particle problem is taken intoaccount, the potential term in Eq. (11) is negligible.Therefore, in the p polarization, the quantum state of thesystem is described by the wave function c ðpÞ via theeigenvalue problem�

1

ma2ð1� cos ðapÞÞ � Ea

�c ðpÞ ¼ 0: (12)

Here, Ea is an eigenvalue depending on the scale a, andone has

Ea ¼ 1

ma2½1� cos ðapÞ� � 2

ma2¼ Emax

a : (13)

From Eq. (13), one sees that, for each scale a, there is abounded and continuous eigenvalue. In the limit a ! 0,i.e., switching the polymer effect off, one obtains the un-

bounded eigenvalue E ¼ p2

2m , typical for a free particle. It is

easy to verify that the solution c ðpÞ for the eigenvalueproblem (12) has the form

c ðpÞ ¼ A�ðp� PaÞ þ B�ðpþ PaÞ; (14)

where A, B are integration constants and

Pa ¼ 1

aarccos ð1�ma2EaÞ (15)

induces the modified dispersion relation in the presence ofa polymer structure.

For an (inverse) Fourier transform for the eigenfunction(14), one obtains the eigenfunction in the q polarizationc ðqÞ as

c ðqÞ ¼Z

c ðpÞeipq ¼ AeiqPa þ Be�iqPa : (16)

The eigenfunction in the q polarization becomes amodified wave plane, due to the dispersion relation (15),which is valid at each scale.

E. The polymer particle in a box

In this subsection, we will analyze the dynamicalfeatures of a one-dimensional particle in a box, withinthe framework of the polymer representation of quantum

mechanics. For a one-dimensional box (i.e., a segment) oflength L ¼ na, n 2 N, the potential VðqÞ ¼ VðnaÞ reads

VðqÞ ¼�1; x > L; x < 0

0; 0< x < L; (17)

i.e., in the case of a potential limited by infinite walls.In this case, the particle behaves as a free particle withinthe segment, and proper boundary conditions for eigen-function (16) have to be imposed. In particular

c ð0Þ ¼ c ðLÞ ¼ 0 !�A ¼ �B

LPa ¼ n; (18)

for which the eigenfunctions c ðqÞ in the q polarization areobtained

c ðqÞ ¼ 2A sin

�nq

L

�: (19)

The corresponding energy spectrum Ea;n is a function of

both the lattice constant a and the quantum number n, suchthat

Ea;n ¼ 1

ma2

�1� cos

�an

L

��: (20)

In the limit a ! 0 one gets the energy spectrum of thestandard case.

III. THE MIXMASTER MODEL: CLASSICALAND QUANTUM FEATURES

In this section, we provide a complete description ofthe most relevant achievements obtained for the dynamicsof the Bianchi IX cosmological model, both in the classicaland the quantum regime towards the cosmological singu-larity, as they are depicted in the two pioneering works[14,18].

A. The classical dynamics

Homogeneous spaces are an important class of cosmo-logical models. These spaces are characterized by thepreservation of the space line element under a specificgroup of symmetry, and are collected in the so-calledBianchi classification [30]. The most general homogene-ous model is the Bianchi IX model. As demonstrated byBelinski, Khalatnikov, and Lifshitz (BKL) [31], when ageneric inhomogeneous space approaches the singularity,it behaves as an ensemble of Bianchi IX independentmodels in each point of space.2 Following the Misnerparametrization [14], the line element for the Bianchi IXmodel is

ds2 ¼ NðtÞ2dt2 � ab!a!b; (21)

2Also the Bianchi VIII has the same degree of generality but itdoes not admit an isotropic limit.

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where !a ¼ !a�dx

� is a set of three invariant differentialforms, NðtÞ is the lapse function and ab is definedas ab ¼ e2�ðe2�Þab. In the Misner picture, � expressesthe isotropic volume of the Universe (for � ! �1, the

initial singularity is reached.), while the matrix �ab ¼diagð�þ þ ffiffiffi

3p

��; �þ � ffiffiffi3

p��;�2�þÞ accounts for the

anisotropy of this model. The introduction of the Misnervariables allows one to rewrite the super-Hamiltonian con-straint (written following the ADM formalism [17]) in thissimple way:

H IX ¼ �p2� þ p2þ þ p2� þ 3ð4Þ4

k2e4�Vð��Þ ¼ 0; (22)

where the ðp�; p�Þ are the conjugated momenta toð�;��Þ, respectively, k ¼ 8G, and Vð��Þ is the potentialterm depending only on ��, i.e., the anisotropies.Respectively to the anistropies the potential term for theBianchi IX model has the form:

Vð��Þ ¼ e�8�þ � 4e�2�þ cosh ð2 ffiffiffi3

p��Þ

þ 2e4�þ½cosh ð4 ffiffiffi3

p��Þ � 1�: (23)

Let us execute now the ADM reduction of the dynamics[32] by solving the super-Hamiltonian constraint withrespect to a specific conjugated momenta and then byidentifying a time variable for the phase space. For thepurposes of this investigation, we choose to solve (22) withrespect to p� and identify � as a time variable. This choiceis justified because, if we choose a time gauge _� ¼ 1 in thesynchronous reference system [NðtÞ ¼ 1], the isotropicvolume � depends on the synchronous time t by therelation � ¼ 1

3 ln t. Then one obtains

�p� ¼ H ADM �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2þ þ p2� þ 3ð4Þ4

k2e4�Vð��Þ

s; (24)

i.e., the so-called reduced Hamiltonian of our problem.From relation (24), one recognizes that, as studied byMisner [18], the dynamics of the Universe towards thesingularity is mapped to the description of the motion ofa particle that lives on a plane inside a closed domain. Thisway, we can study how the anisotropies �� change withrespect to the time variable � through equations of motionrelated to the reduced Hamiltonian

�0� ¼ d��d�

¼ p�H ADM

;

p0� ¼ dp�d�

¼ 3ð4Þ42kH ADM

e4�@Vð��Þ@��

:

(25)

Studying the two opposite approximations of Vð��Þ, i.e.,far from the walls (V ’ 0) and close to the walls(V ’ 1

3 e�8�þ), we can obtain the relative motion between

the particle and the potential wall. It is possible to obtain,for V ’ 0, the behavior of �� as a function of time � via asimple integration of the first equation of motion. This way,one gets

�� / p�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2þ þ p2�

q �: (26)

Moreover, the anisotropy velocity of the particle far fromthe walls is defined as

�0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�d�þd�

�2 þ

�d��d�

�2

s¼ 1 (27)

for each value of p�. On the other hand, the investigationon the motion of one of the equivalent sides allows oneto understand that the walls move towards the ‘‘outer’’direction with velocity j�0

wj ¼ 12 . The particle always

collides against the wall and bounces from one to another.This chaotic dynamics is the analogue of the oscillatoryregime described by BKL in [33].It is worth noting that the regime under which V ’ 0

corresponds to the Bianchi I case of the Bianchi classifi-cation, the so-called Kasner regime, in which the particlemoves as being free and the two constraints

p1 þ p2 þ p3 ¼ 1; p21 þ p2

2 þ p23 ¼ 1; (28)

are satisfied. Here p1, p2, p3 are the Kasner indices, i.e.,three real numbers that express the anisotropy of themodel. Writing the spatial metric of the Bianchi I modelin the synchronous reference system, i.e.,

dl2 ¼ t2p1dx1 þ t2p2dx2 þ t2p3dx3; (29)

the presence of the Kasner indices inside the spatial metricis understood to imply a different behavior along the differ-ent directions, which define the anisotropic directions.On the other hand, when the particle is close to the

wall (V ’ 13 e

�8�þ), the Bianchi II model, i.e., the model

describing one single bounce against the infinite wallpotential [34], is considered. The system (25) can bestudied close to a potential wall, and it is possible toidentify two constants of motion

p� ¼ const; K ¼ 1

2pþ þH ADM ¼ const: (30)

These relations have been obtained for the ‘‘vertical’’potential wall in Fig. 1; it is however necessary to stressthat the bounces against the potential walls are all equiva-lent as far as the dynamics of the system is concerned,as the potential walls can be obtained one from the other,by taking into account the symmetries of the model, asanalyzed in [35].A description of this regime is illustrated in Fig. 1. The

anisotropies can be parametrized as functions of both theincidence angle and of the reflection one, �i and �f,

respectively. This way,

ð�0�Þi ¼ sin �i; ð�0þÞi ¼ � cos�i;

ð�0�Þf ¼ sin �f; ð�0þÞf ¼ cos �f:(31)

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The relations (30) are used to obtain a reflection law for ageneric single bounce

sin �f � sin �i ¼ 1

2sin ð�i þ �fÞ: (32)

However, there is a maximum angle �max after whichno bounce occurs. For the occurrence of a bounce, thelongitudinal component of the velocity �0þ must be greaterthan the wall velocity �0

w. This condition is expressed as

j�ij< j�max j ¼ arccos

��0

w

�0þ

�¼

3: (33)

As a result, the particle, sooner or later, will assume all thepossible directions, regardless of the initial condition.Following the convenience choice used by Misner in[18], and taking advantage of the geometric properties ofthis scheme, in the limit close to the singularity (� ! �1)one finds a conservation law of the form

hH ADM�i ¼ const: (34)

For two successive bounces [the ith and the (iþ 1)-th ofthe sequence], �i expresses the time at which the ithbounce occurs and H i

ADM is the value of the reducedHamiltonian (24) just before the ith bounce: relation (34)states that

H iADM�

i ¼ H iþ1ADM�

iþ1: (35)

In other words, the quantity H ADM� acquires the sameconstant value as just before each bounce towards thesingularity.

B. The quantum behavior

The canonical quantization of the system consists of thecommutation relations

½qa; pb� ¼ i�ab; (36)

which are satisfied for pa ¼ �i @@qa

¼ �i@a, where a,

b ¼ �, �þ, ��. By replacing the canonical variableswith the corresponding operators, the quantum behaviorof the Universe is given by the quantum version of thesuper-Hamiltonian constraint (22), i.e., the Wheeler-DeWitt equation (WDW) for the Bianchi IX model

H IX�ð�;��Þ

¼�@2��@2þ�@2þþ3ð4Þ4

k2e4�Vð��Þ

��ð�;��Þ; (37)

where �ð�;��Þ is the wave function of the Universewhich provides information about the physical state ofthe Universe. A solution of Eq. (37) can be looked for inthe form

� ¼ Xn

�nð�Þ�nð�;�Þ: (38)

The adiabatic approximation consists in requiring thatthe � evolution be principally contained in the �nð�Þcoefficients, while the functions �nð�;�Þ depend on� parametrically only. The adiabatic approximation istherefore expressed by the condition

j@��nð�Þj � j@��nð�;�Þj: (39)

By applying condition (39), the WDW Eq. (37) reduces toan eigenvalue problem related to the reduced HamiltonianH ADM via

H2ADM�n ¼ E2

nð�Þ�n

¼��@2þ � @2� þ 3ð4Þ4

k2e4�Vðc��Þ

��n: (40)

However, even without finding the exact expression of theeigenfunctions, one may gain important information aboutthe system from a quantum point of view near the initialsingularity. From Fig. 1, one can see how the potential (23)can be modeled as an infinitely steep potential well with atriangular base. In [18], the strong hypothesis to replace thetriangular box with a squared box having the same area L2

is proposed. This way, the problem describing a two-dimensional particle in a squared box with infinite wallsis recovered. In this case, the eigenvalue problem becomes

H 2ADM�n;m ¼ 2ðm2 þ n2Þ

L2ð�Þ �n;m; (41)

where m, n 2 N are the quantum numbers associatedto ð�þ; ��Þ. By a direct calculation, we can derive

L2ð�Þ ¼ 3ffiffi3

p4 �2, such that the eigenvalue is

FIG. 1. Equipotential lines of the Bianchi IX model in theð�þ; ��Þ plane [18].

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En;m ¼ 2

33=4�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ n2

p: (42)

As demonstrated in [36], substituting the eigenvalueexpression (42) in Eq. (37), the self-consistence ofadiabatic approximation is ensured. Let us use (42) with(34) to estimate the quantum numbers’ behavior towardsthe singularity. One can see in Eq. (42) that the eigenvaluespectrum is unlimited from above, such that, forsufficiently high occupation numbers, replacing H ADM ’En;m is a good approximation. This way, for � ! �1,

Eq. (34) becomes

hH ADM�i ���!�!�1h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ n2

pi ¼ const: (43)

Since the current state of the Universe anisotropy can be

characterized by a classical nature, i.e.,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ n2

p� 1,

we can say, by Eq. (43), that this quantity is constant whileapproaching the singularity. This way, the quantum state ofthe Universe related to the anisotropies remains classicalfor all the backwards history until the singularity.

IV. SEMICLASSICAL POLYMER APPROACH TOTHE MIXMASTER MODEL

The aim of the present section is to discuss how toapply the polymer approach of Sec. II to the Bianchi IX

model at a semiclassical level and to verify if and how thenature of the cosmological singularity is modified. Here,‘‘semiclassical’’ means that we are working with a modi-fied super-Hamiltonian constraint obtained as the lowestorder term of a WKB expansion for ℏ ! 0. At this level,the modified theory is subject to a deterministic dynamics.Following the procedure in Sec. II B, one can choose, witha precise physical interpretation, to define the anisotropiesof the Universe ð�þ; ��Þ as discrete variables leaving thecharacterization of the isotropic variable � unchanged,which here plays the role of time. This procedure formallyconsists in the replacement

p2� ! 2

a2½1� cos ðap�Þ�: (44)

The super-Hamiltonian constraint (22) becomes

� p2� þ 2

a2½2� cos ðapþÞ � cos ðap�Þ�

þ 3ð4Þ4e4�k2

Vð��Þ ¼ 0: (45)

We define �p� � Hpoly as the reduced Hamiltonian, such

that one gets

� p� � Hpoly ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

a2½2� cos ðapþÞ � cos ðap�Þ� þ 3ð4Þ4e4�

k2Vð��Þ

s: (46)

Starting from the new Hamiltonian formulation (46), wecan get the following set of the Hamiltonian equations as

�0� ¼ d��d�

¼ sin ðap�ÞaHpoly

;

p0� ¼ dp�d�

¼ 3ð4Þ42kHpoly

e4�@Vð��Þ@��

:

(47)

This modification leaves the potential Vð��Þ and theisotropic variable � unchanged. Therefore, even in themodified theory, the walls move in the outer directionwith velocity j�0

wj ¼ 12 and the initial singularity is not

expected to be removed.Let us start by analyzing the system far from the wall,

i.e., with V ’ 0. As one can see in (47) when V ’ 0, theanisotropy velocity is modified if compared to the standardcase. In particular, the behavior of�� is proportional to thetime �, as in the standard theory, but with a differentcoefficient, i.e.,

�� / sin ðap�Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4� 2½cos ðapþÞ þ cos ðap�Þ�

p �: (48)

In particular, by the definition of the anisotropy velocity,Eq. (27), one obtains

�0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sin ðapþÞ2 þ sin ðap�Þ24� 2½cos ðapþÞ þ cos ðap�Þ�

s¼ rða; p�Þ: (49)

It is worth noting that rða; p�Þ is a bounded function(r 2 ½0; 1�) of parameters that remains constant betweenone bounce and the following one. From Eq. (48), we havea Bianchi I model modified by the polymer substitution. Asa consequence of this feature, also in the modified theory,the anisotropies behave with respect to � in a proportionalway. The first important semiclassical result is the relativemotion between wall and particle. From (49), one canobserve the existence of allowed values of ðapþ; ap�Þ,such that the particle velocity is smaller than the wallvelocity �0

w. Therefore, the condition for a bounce is

�0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sin ðapþÞ2 þ sin ðap�Þ24� 2½cos ðapþÞ þ cos ðap�Þ�

s>

1

2¼ �0

w: (50)

It means that the infinite sequence of bounces against thewalls, typical of the mixmaster model, takes place until

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condition (50) is valid. When r < 12 , the particle becomes

slower than the potential wall and reaches the singularitywithout any other bounces.

This feature is confirmed by the analysis on the Kasnerrelations (28). The first Kasner relation is still valid in thedeformed approach, because the sum of the Kasner indicesis linked by the Misner variables just through the isotropicvariable �. Instead, the second Kasner relation is directlyrelated to the anisotropy velocity [34] and its results aremodified into

p21 þ p2

2 þ p23 ¼

1

3þ 2

3½ð�0þÞ2 þ ð�0�Þ2�

¼ 1� 2

3ð1� r2Þ ¼ 1� q2; (51)

where q2 ¼ 23 ð1� r2Þ. We introduce q2 in (51) because

23 ð1� r2Þ � 0 for any values of ðapþ; ap�Þ. The introduc-tion of the polymer structure for the anisotropies acts thesame way as a massless scalar field in the Bianchi IXmodel [37,38]. For this reason, we can choose to describethe Kasner indices with the same parametrization due toBelinski and Khalatnikov in [39]. It is realized through theintroduction of two parameters ðu; qÞ3 and it allows us torepresent all possible values of the Kasner indices. Onegets

p1 ¼ �u

1þ uþ u2;

p2 ¼ 1þ u

1þ uþ u2

�u� u� 1

2

�1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

q ��;

p3 ¼ 1þ u

1þ uþ u2

�1þ u� 1

2

�1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

q ��;

�2 ¼ 2ð1þ uþ u2Þq2ðu2 � 1Þ2 :

(52)

Here, �1< u<þ1 and �ffiffi23

q< q<

ffiffi23

q. The presence of

�2 inside Eq. (52) means that not all values of u, q areallowed. The u, q allowed values are those which respectthe condition �2 < 1. Inside this region of permitted val-ues, there are two fundamental areas where all Kasnerindices are simultaneously positive, i.e., for q > 1ffiffi

2p and

q <� 1ffiffi2

p . When it happens, remembering that the spatial

Kasner metric is dl2 ¼ t2p1dx1 þ t2p2dx2 þ t2p3dx3, thedistances contract along all spatial directions approachingthe singularity (t ! 0). It means that the system behaves asa stable Kasner regime and the oscillatory regime issuppressed.

Furthermore, the relation (35) remains valid until r < 12

or rather when the particle become slower than the

potential wall. When it happens, approaching the singular-ity, � ! �1 while Hpoly remains constant without

changes. In this sense, when the outgoing momenta con-figuration of the jth bounce is such that r < 1

2 , the quantity

Hjpoly�

j is no longer a constant of motion.

As in the standard case, we can introduce a parametri-zation for the particle velocity components, before andafter a single bounce:

ð�0�Þi ¼ ri sin�i; ð�0þÞi ¼ �ri cos �i;

ð�0�Þf ¼ rf sin �f; ð�0þÞf ¼ rf cos�f;(53)

where ð�i; �fÞ are the incidence and the reflection angles

and ðri; rfÞ are the anisotropy velocities before and

after the bounce. Equation (33) states the existence of amaximum angle �max ¼

3 for a bounce to occur.

In the modified model, the condition for a bounce to takeplace is

�i<�polymax ¼ arccos

�1

2ri

�� arccos

�1

2

�¼�max ¼

3: (54)

The new maximum angle �polymax coincides with �max just for

r ¼ 1, i.e., when the standard case is restored (Fig. 2). Thelast semiclassical result is the modified reflection law for asingle bounce: as in the standard case, we can identify twoconstants of motion by studying the system near the po-tential wall. In particular, one has

p� ¼ const; K ¼ 1

2pþ þHpoly ¼ const: (55)

The expression of pþ as function of �0 can be obtainedfrom (47):

pþ ¼ 1

aarcsin ða�0þHpolyÞ: (56)

FIG. 2 (color online). The maximum angle for a bounce �polymax

as a function of r. In the r ! 1 limit, the standard case isrestored. This treatment makes sense only for a configurationin which the particle’s velocity is higher than the wall’s velocity,i.e., for r > 1

2 .

3In the standard case (absence of the scalar field or polymermodification, i.e., q2 ¼ 0), p1, p2, p3, and u are related this way

[35]: p1ðuÞ ¼ � u1þuþu2

, p2ðuÞ ¼ 1þu1þuþu2

, p3ðuÞ ¼ uð1þuÞ1þuþu2

. In

this case 0< u< 1.

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This way, by a substitution of Eq. (56) in Eq. (55), remembering arcsin ð�xÞ ¼ � arcsin ðxÞ and using the parametrization(53), one obtains

1

2aarcsin ð�ariH

ipoly cos �iÞ þHi

poly ¼1

2aarcsin ðarfHf

poly cos�fÞ þHfpoly: (57)

Now we express r and Hpoly as functions of a, pþ, p�:

1

2

�arcsin

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin ðapiþÞ2 þ sin ðapi�Þ2

qcos�i

�þ arcsin

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin ðapiþÞ2 þ sin ðapi�Þ2

q cos �f sin�isin �f

��

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4� 2ðcos ðapiþÞ þ cos ðapi�Þ

q� sin �i

sin �f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin ðapiþÞ2 þ sin ðapi�Þ2sin ðapf

þÞ2 þ sin ðapf�Þ2½4� 2ðcos ðapf

þ þ cos ðapf�Þ�s

: (58)

To perform a direct comparison with the standard case, aTaylor expansion up to second order for ap� � 1 forEq. (58) is needed. This way, after standard manipulation,the reflection law is rewritten as

1

2sin ð�i þ �fÞ ¼ sin �f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2

4

ðpiþÞ4 þ ðpi�Þ4ðpiþÞ2 þ ðpi�Þ2

s

� sin �i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2

4

ðpfþÞ4 þ ðpf�Þ4

ðpfþÞ2 þ ðpf�Þ2

s: (59)

Defining R ¼ a2

4

p4þþp4�

p2þþp2�

, one has

1

2sinð�iþ�fÞ¼ sin�f

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þRi

p �sin�iffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þRf

q: (60)

We obtain for ap� � 1 a modified reflection law that,differently from the standard case, depends on two parame-ters ðR; �Þ. Obviously, in the limit ap� ! 0, i.e., switchingoff the polymer modification, the standard reflection law(32) is recovered.

V. POLYMER APPROACH TO THE QUANTUMMIXMASTER MODEL

We now analyze the quantum properties of the polymermixmaster model. As in Sec. III B, one searches for asolution for the wave function of the form

�ðp�; �Þ ¼ �ð�Þc ð�; p�Þ: (61)

In this case, one can choose to describe the �ð�Þ compo-nent of the wave function in the q polarization and thec ð�; p�Þ component of the wave function in the p polar-ization. As in the semiclassical model, we choose to dis-cretize the anisotropies ð�þ; ��Þ, leaving unchanged thecharacterization of the isotropic variable �. Therefore,as in Sec. II, one applies the formal substitution p2� !2a2½1� cos ðap�Þ�. Of course, the conjugated momenta p�

have a well-defined operator of the form p� ¼ �i@�. Thisway, we can obtain the WDW equation for the polymermixmaster model by writing the quantum version of thesuper-Hamiltonian in (45), that is

��@2� þ 2

a2ð1� cos ðapþÞÞ þ 2

a2ð1� cos ðap�ÞÞ

þ 3ð4Þ4k2

e4�Vð��Þ��ðp�; �Þ ¼ 0: (62)

The conservation of quantum numbers associated to theanisotropies, as obtained byMisner in the standard quantumtheory [see Eq. (43)], is essentially based on a fundamentalpropriety of the mixmaster model: the presence of chaos.Nevertheless, as in Sec. IV, the chaos is removed for dis-cretized anisotropies of the Universe. This way, one cannotobtain for the modified theory a conservation law towardsthe singularity as in the standard case. For a quantumdescription, the polymer wave packets for the theory areneeded. By a semiclassical analysis of the relational motionbetween the wall and the particle, as in Sec. IV, the polymermodification implies for the particle different conditions forthe reach of the potential wall. This way, it behaves as a freeparticle (no potential case V ¼ 0) or as a particle in a box(infinitely steep potential well case). In this section, wemake use of the adiabatic approximation (39) as in thestandard case. Following the same procedure of Sec. III B,the polymerWDWequation reduces to an eigenvalue prob-lem associated to the ADM Hamiltonian.

A. The free motion

In the free particle case, the potential term Vð��Þis negligible in the WDW equation. As in Sec. III B,condition (39) is applied to Eq. (62), and the followingfree-particle eigenvalue problem is obtained:

H2polyc ðp�Þ ¼ k2c ðp�Þ

¼�2

a2ð2� cos ðapþÞ � cos ðap�ÞÞ

�c ðp�Þ:

(63)

From the structure of the eigenvalue problem (63), one can

write cH2poly ¼ cH2þ þ cH2�. As a consequence, it is

possible to describe the anisotropic wave function asc ðp�Þ ¼ cþðpþÞc�ðp�Þ. This way, one obtains thetwo independent eigenvalue problems

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ðdH2þ�k2þÞcþðpÞ¼�2

a2½1�cosðapþÞ��k2þ

�cþðpÞ¼0;

ðdH2��k2�Þc�ðpÞ¼�2

a2½1�cosðap�Þ��k2�

�c�ðpÞ¼0;

(64)

where k2 ¼ k2þ þ k2�. These eigenvalue problems can betreated as in Sec. II D and, by a similar procedure, one caneasily verify that the momentum wave functions cþðpÞand c�ðpÞ have the form

cþðpþÞ ¼ A�ðpþ � pþa Þ þ B�ðpþ þ pþ

a Þ;c�ðp�Þ ¼ C�ðp� � p�

a Þ þD�ðp� þ p�a Þ;

(65)

where A, B, C, D are integration constants and pþa , p

�a are

defined as

pþa ¼ 1

aarccos

�1� k2þa2

2

�;

p�a ¼ 1

aarccos

�1� k2�a2

2

�:

(66)

From Eqs. (64), the eigenvalue k2 is given by

k2 ¼ k2þ þ k2�

¼ 2

a2½2� cos ðapþÞ � cos ðap�Þ� � k2max ¼ 8

a2; (67)

i.e., a bounded and continuous eigenvalue is found.Now one can obtain c ð��Þ by performing a Fourier

transform for c ðp�Þ ¼ cþðpþÞc�ðp�Þ, such that

c kð��Þ¼ZZ

dpþdp�c ðp�Þeipþ�þeip���

¼C1eipþ

a �þeip�a �� þC2e

ipþa �þe�ip�

a ��

þC3e�ipþ

a �þeip�a �� þC4e

�ipþa �þe�ip�

a �� ; (68)

where C1 ¼ AC, C2 ¼ AD, C3 ¼ BC, C4 ¼ BD. We arenow able to build up the polymer wave packet for the wavefunction of the Universe. We choose to integrate the packeton the energies kþ, k�. As a consequence of the modifieddispersion relations (66), the energies eigenvalues kþ, k�can only take values within the interval ½� 2

a ;þ 2a�.

Therefore, we have

�ð��; �Þ ¼ZZ 2

a

�2a

dk�Aðk�Þc k�ð��Þ�ð�Þ; (69)

where Aðkþ; k�Þ ¼ e�ðkþ�k0þÞ2

2 2þ e�ðk��k0�Þ2

2 2� is a Gaussianweighting function, 2� are the variances along the twodirections ð�þ; ��Þ, and k0� are the energies’ eigenvaluesaround which we build up the wave packet. Let us notefrom Eq. (69) that the polymer structure modifies thestandard wave packet related to the plane wave in termsof the anisotropy component as a consequence of Eqs. (66),i.e., the modified dispersion relations.

The shape for the isotropic component of the wave

function in the free particle case is �ð�Þ ¼ e�iR

�

0kdt ¼

e�iffiffiffiffiffiffiffiffiffiffiffiffik2þþk2�

p�. This shape is a solution of the WDWequation

@2�ð�Þ þ k2�ð�Þ ¼ 0 obtained by the application of theadiabatic approximation (39). Furthermore, the self-consistence of this approximation is ensured.

B. Particle in a box

We analyze the problem of a particle in a box accordingto the Misner hypothesis about the substitution of thetriangular box by a square domain having the same areaL2, as in Sec. III B. Furthermore, following the semiclas-sical results in Sec. IV, one takes into account the outsidewall velocity defining the side of square box L as

Lð�Þ ¼ L0 þ j�j; (70)

where L0 is the side of the square box when � ¼ 0.Proceeding in the same way as in Sec. II E, the potentialhas the well-known form

Vð��Þ ¼�1; �� > Lð�Þ

2 ; �� <� Lð�Þ2

0; � Lð�Þ2 <�� < Lð�Þ

2

: (71)

We can obtain a solution for c ð��Þ in the same way ofSec. VA, recalling that the potential form (71) impliesthese kinds of boundary conditions for c ð��Þ along thetwo directions

c���L0

2� �

2

�¼ c�

�þL0

2þ �

2

�¼ 0: (72)

When one applies the conditions (72) separately along thetwo directions ð�þ; ��Þ, one obtains

cþð�þÞ ¼ A½ein�þL0þ� � e

�in�þL0þ� e�in�;

c�ð��Þ ¼ B½eim��L0þ� � e

�im��L0þ� e�im�:

(73)

This way, c ð��Þ is the product of the two separate wavefunctions cþð�þÞ and c�ð��Þ. Thus, one gets4

c n;mð��; �Þ ¼ cþð�þÞc�ð��Þ

¼ 1

2ðL0 þ �Þ ½ein�þL0þ� � e

�in�þL0þ� e�in�

½eim��L0þ� � e

�im��L0þ� e�im�; (74)

where A, B are integration constants and ðn;mÞ 2 Z arequantum numbers associated with anisotropy degrees of

4It is possible to evaluate the constant AB by requesting thatjc n;mð��Þj2 ¼ 1 over the entire square box. This way, AB ¼

12ðL0þ�Þ is obtained.

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freedom. Due to the presence of the integers’ quantumnumbers ðn;mÞ, a bounded and discrete eigenvaluespectrum

k2 ¼ k2þ þ k2�

¼ 2

a2

�2� cos

�an

L0 þ �

�� cos

�am

L0 þ �

��(75)

is obtained.As in the free particle case, one builds the polymer wave

packet. However, in this case, one cannot integrate on alimited domain of energies k�, and a sum over all quantumnumbers n, m between �1 and 1 is necessary. This way,

�ð��; �Þ ¼Xþ1

n;m¼�1Bðn;mÞc n;mð��; �Þ

e�iR�0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a2½2� cos ð anL0þtÞ � cos ðam

L0þtÞ�q

dt;

(76)

where Bðn;mÞ ¼ e�ðn�nÞ2

2 2þ e�ðm�mÞ2

2 2� is a Gaussian weightingfunction and n, m are the quantum numbers aroundwhich we build up the wave packet.

Let us note that, differently from the free particle case,the presence of the polymer structure modifies the standardwave packet related to a particle in a box in terms of theisotropic components. It happens because, in the wavepacket (76), the energies k� are expressed throughðn;mÞ, namely, the quantum numbers associated to theanisotropies.

As from Eq. (76), one chooses a shape for the isotropiccomponent

�ð�Þ ¼ e�iR

�

0kðtÞdt ¼ e

�iR

�

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

a2½2�cos ð anL0þtÞ�cos ð am

L0þtÞ�p

dt: (77)

In this case, Eq. (77) is a solution of the WDW equation@2�ð�Þ þ kð�Þ2�ð�Þ ¼ 0, obtained by means of the adia-batic approximation (39) in the asymptotic limit � ! �1.In this limit, the self-consistence of the adiabatic approxi-mation is ensured. The form of the isotropic component ofthe wave function (77) is also an exact solution for theSchrodinger equation associated to the ADM reduction.

VI. NUMERICAL ANALYSIS OFPOLYMER WAVE PACKETS

We dedicate this section to the discussion of the polymerwave packet for the mixmaster towards the cosmologicalsingularity. Both in the case of a free particle (69) and inthe one of a particle in a box (76), it is not possible toperform an analytic integration for the wave packets. Thisway, in order to obtain the quantum behavior of the wavepackets near the cosmological singularity, we evaluatethem via numerical integrations.

A. Behavior of the free particle

In the case of a free particle, we perform the numericalintegration by choosing the parameters which selectsemiclassical initial conditions concerning a particle withvelocity smaller than the wall one (r < 1

2 ).

One can see, in the first row of Fig. 3, the behaviortowards the singularity (formally for j�j ! 1) of theabsolute value of the wave packet j�ð�;��Þj in Eq. (69)while, in the second row, the behavior towards the singu-larity of the full width at half maximum width. It isinteresting to study the evolution of �m�, i.e., the wavepacket’s maximum position. This way, we can see whichtrajectory the wave packet follows towards the singularity.As we can see in the first graph in Fig. 4, the behavior of themaximum position is completely overlapping the semiclas-sical trajectory selected by our choice of the initial con-ditions. In this sense, the polymer wave packet follows thesemiclassical trajectory until the singularity. This feature isnot undermined by the spread d of the wave packet, i.e., thedelocalization of the wave packet, as expressed by thedistance between the maximum position of the wavepacket and the edge of the region identified by the fullwidth at half maximum. Obviously, one expects that thespread velocity is really smaller than the wall velocity.Otherwise, it would be possible for the wave packet toreach the potential wall. In that case, the description of thequantum system with the wave packets for the free particlewould not be correct. The second graph in Fig. 4 representsthe spread evolution, and we can see it follows a linearbehavior (solid line) with a slope much smaller thanj�0

wj ¼ 12 , i.e., the one related to the behavior of the wall

position (dashed line). This assures that the quantum rep-resentation of the system near the singularity for the freeparticle case is well described by the wave packetrepresentation.

B. Behavior of the particle in a box

The numerical integration related to the polymer wavepacket (76) has to face a significant technical difficulty.As a consequence of Eq. (75), the conjugated momentap� turn into discretized variables. Therefore, we select forthe particle in a box the initial semiclassical conditionconsidering the substitution

apþ ! an

L0 þ �; ap� ! am

L0 þ �: (78)

It is worth noting that the initial condition of the particledepends on �, such that one deals with a time-dependentcondition. In this subsection, the influence of quantumnumbers n, m on the dynamics is investigated. For thisreason, one introduces six data sets with different values ofquantum numbers ðn; mÞ and box side L0

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8>>>>>>>>>>><>>>>>>>>>>>:

a¼ 0:014

n0 ¼ 1000

m0 ¼ 1000

L0 ¼ 17

þ ¼ 50

� ¼ 50

8>>>>>>>>>>><>>>>>>>>>>>:

a¼ 0:014

n1 ¼ 2000

m1 ¼ 2000

L1 ¼ 34

þ ¼ 50

� ¼ 50

8>>>>>>>>>>><>>>>>>>>>>>:

a¼ 0:014

n3 ¼ 3000

m3 ¼ 3000

L3 ¼ 52

þ ¼ 50

� ¼ 508>>>>>>>>>>><>>>>>>>>>>>:

a¼ 0:014

n4 ¼ 6000

m4 ¼ 6000

L4 ¼ 103

þ ¼ 50

� ¼ 50

8>>>>>>>>>>><>>>>>>>>>>>:

a¼ 0:014

n4 ¼ 8000

m4 ¼ 8000

L4 ¼ 137

þ ¼ 50

� ¼ 50

8>>>>>>>>>>><>>>>>>>>>>>:

a¼ 0:014

n5 ¼ 10000

m5 ¼ 10000

L5 ¼ 172

þ ¼ 50

� ¼ 50

:

(79)

They select the same initial condition of a particle slowerthan the potential wall (r < 1

2 ) and we show in Fig. 5 theevolution of j�ð�;��Þj and its full width at half maximumfor the first data set. As in the free particle case, the wavepacket spreads with �; i.e., it delocalizes until it disappears

in a finite � time. The real difference between the freeparticle case and the particle in a box case is the trajectoryfollowed by the wave packet. If we study the evolution ofthe wave packet maximum position�m� for all data sets, weobserve that the wave packet trajectories move away fromthe polymer semiclassical trajectory identified by theinitial condition, as we can see in the first graph ofFig. 6. The separation from the polymer semiclassicaltrajectory depends on the quantum numbers n, m. Inparticular, the larger n, m are, the longer the semiclassi-cal trajectory is followed. Anyway, no matter how largethey are, in a finite time �, the wave packet stops followingthe semiclassical trajectory, is directed to the potentialwall, and reaches it. As in Fig. 7, this behavior is repeatedfor every unexpected bounce against the wall. This way,it is not possible to chose an initial semiclassical state(i.e., large n, m) conserved until the singularity. Thisresult is opposite with respect to the one in Eq. (34), wherein the standard theory the state remains classical until thesingularity. It happens because we have a time-dependentinitial condition [as in Eq. (78), it depends on �] thatchanges the particle velocity. This behavior is explainedif one considers the two different data sets

0 5 10 15 20

0

5

10

15

20

5 10 15 20 255

10

15

20

25

40 45 50 55 6040

45

50

55

60

FIG. 3 (color online). The evolution of the polymer wave packet j�ð�;��Þj (upper row) and its full width at half maximum(lower row) for the free particle case, respectively, for the values of j�j ¼ 0, 50, 150. The numerical integration is done for this choiceof parameters: a ¼ 0:07, kþ ¼ k� ¼ 25, þ ¼ � ¼ 0:7. They select an initial semiclassical condition of a particle with a velocitysmaller than the wall velocity. It is worth noting that the particular choice of the parameter couple ða; �Þ is made because this way thecondition a � 1

�is valid. It is related to the condition that the typical polymer scale a be much smaller than the characteristic width of

the wave packet 1 �

.

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FIG. 4 (color online). The solid line in the first graph represents the polymer semiclassical trajectory identified by the choice of theinitial conditions. The dashed line represents the classical trajectory followed by a wave packet buildup in the same way of Sec. VA,but starting from the classical super-Hamiltonian constraint (22). The points in the second graph represent the evolution of the spread das a function of j�j. The solid line represents the best fit for the points while the dashed line represents the evolution of the wallposition j�wj ¼ 1

2 j�j.

FIG. 5 (color online). The evolution of the polymer wave packet j�ð�;��Þj (the first row) and its full width at half maximum (thesecond row) for the particle in a box case, respectively, for j�j ¼ 0, 20, 200. The numerical integration is done for this choice ofparameters: a ¼ 0:014, n ¼ m ¼ 3000, þ ¼ � ¼ 50, L0 ¼ 52. They select an initial condition of a particle inside a square boxwith velocity smaller than the wall velocity. This time, the particular choice of the parameters ða; �; L0Þ is made because this way the

condition a � Lð�Þ �

is valid. It is related to the condition that the typical polymer scale a is much smaller than Lð�Þ �

, i.e., the correct

dimensional quantity related to the width of the wave packet.

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8>>>>>><>>>>>>:

a1 ¼ 0:014

n1 ¼ m1 ¼ 3000

L1 ¼ 26

þ ¼ � ¼ 50

8>>>>>><>>>>>>:

a2 ¼ 0:014

n2 ¼ m2 ¼ 400

L2 ¼ 26

þ ¼ � ¼ 50

(80)

They respectively select a particle with initial velocityr < 1

2 and with r > 12 . The first one is related to a particle

in a box which semiclassically cannot reach the potentialwall, while the second one is related to a particle in a boxwhich semiclassically reaches the potential wall. For ourpurposes, we take two data sets with the same values of a, �, L0 but with different n and m.

In Fig. 8, the evolution of the distance d between thewave packet maximum position and the potential wall inthe two cases towards the singularity is described.

When the first one is still traveling, the second one hasalready bounced on the wall and it is traveling again. Thered (light grey) points indicate the (expected) velocitychange due to the dynamical initial condition (78).Finally, it is interesting to study the spread for the two

wave packets near the potential wall. In Fig. 9, the twowave packets and the full width at half maximum aresketched. Since the first wave packet should not reachthe wall, one would expect a high rate of delocalizationnear the wall. Instead, as from the second line in Fig. 9, thetwo wave packets near the potential wall have a compa-rable delocalization. Thus, we can conclude that, when the

FIG. 6 (color online). The points in the first graph represent theevolution of the wave packet maximum position �m� as afunction of j�j for all data sets. The solid line represents thepolymer semiclassical trajectory identified by the choice ofthe initial conditions. The points in the second graph representthe evolution of the spread d as a function of j�j for all data sets.The solid line represents the evolution of the wall positionj�wj ¼ 1

2 j�j. As in the free particle case, the spread evolution

follows a linear trend for all data sets and the slopes are reallysmaller than the one related to the trend of the wall position.

FIG. 7 (color online). The points represent the evolution ofthe wave packet maximum position �m� as a function of j�j fora ¼ 00:14, n ¼ m ¼ 3000, þ ¼ � ¼ 50, L0 ¼ 32. Thetwo solid lines represent the � evolution of the position of twoopposite walls of the square box. Finally, the dashed linesrepresent the polymer semiclassical trajectory identified by thechoice of the initial conditions that the wave packet follows aftereach bounce for a finite � time.

FIG. 8 (color online). The red (grey) points represent theevolution of the distance d between the wave packet maximumposition and the potential wall for r < 1

2 . The black points

represent the evolution of the distance d between the wavepacket maximum position and the potential wall for r > 1

2 .

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potential is taken into account as an infinite well, anynotion of a free semiclassical wave packet is lost.

VII. CONCLUDING REMARKS

The mixmaster model, interpreted as the most generaldynamics allowed by the homogeneity constraint, consti-tutes a valuable prototype of the behavior of a genericinhomogeneous model near the cosmological singularity,when referred to sufficiently small space regions, havingroughly the causal size.

Therefore, the characterization of its classical and quan-tum dynamics has a very relevant value in understandingthe general features of the Universe’s birth.

The present work is aimed at generalizing [18], in whichthe classical mixmaster Hamiltonian dynamics is reducedto the motion of a two-dimensional point particle in aclosed trianglelike potential and the corresponding quan-tum behavior is reconducted to the one of a point particle ina box. The main result of the classical picture is the never-ending bouncing of the particle against the potential walls(resulting in a chaotic evolution), while, in the quantumregime, the surprising feature emerges of states havingvery high occupation numbers which can approach theinitial singularity.

This generalization is the reformulation of the quantummixmaster dynamics in the polymer quantum approach.We have applied this procedure to the physical degreesonly, i.e., the Universe’s anisotropies, while the Universe’svolume has been kept in its standard interpretation as atime variable for the system evolution.The semiclassical behavior of the mixmaster model, i.e.,

the classical modified dynamics by means of the polymerfeatures, gives us a result that is chaos free, in formalanalogy with the case in which a massless scalar field isintroduced in the Einsteinian dynamics. As a consequence,the quantum regime loses its property to admit very highoccupation numbers asymptotically to the singularity.Actually, we demonstrated that the absence of chaoticbehavior prevents us from constructing the classical con-stant of the motion that Misner used to infer the quantumproperties for high occupation numbers. Thus, the mostimpressive property of the quantum mixmaster, i.e., its‘‘classicality’’ across the Planckian era, is no longer wellgrounded.In the polymer framework, such impossibility to recover

a quasiclassical behavior near the singularity is enforcedby noting that it is impossible to construct wave packetspeaked around the classical trajectories that do not impactagainst the potential. Such packets can follow the classical

FIG. 9 (color online). The wave packets and the full width half maximum near the potential wall for the two cases with initialcondition (80). The first case is evaluated for j�j ¼ 85, while the second case is evaluated for j�j ¼ 45.

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trajectory for a finite time interval, after which the bounceof the wave packet against the potential walls takes place.We showed that this fact is a direct consequence of thetime dependence of the potential well, resulting in a con-dition on the free motion of the wave packets which iscorrespondingly time dependent and, soon or later, isviolated.

We can conclude that the polymer features of the mix-master model, i.e., the implications of this particular cutoffphysics on the anisotropic degrees of freedom, enforce therelevance of the quantum nature of the model near thecosmological singularity, since they introduce a nonlocaleffect of the potential walls on the behavior of wave

packets, localized around classical trajectories. This resultsuggests that, to better focus on the behavior of a mix-master model near the cosmological singularity, it is nec-essary to implement a more rigorous semiclassicalinterpretation of the wave function toward the cosmologi-cal singularity in the presence of cutoff induced effects,and a full quantum picture in the discretized picture.

ACKNOWLEDGMENTS

This work has been partially developed within theframework of the CGW Collaboration (http://www.cgwcollaboration.it).

[1] E.W. Kolb and M. S. Turner, The Early Universe(Westview Press, New York, 1994).

[2] G. Montani, M.V. Battisti, G. Imponente, and R. Benini,Primordial Cosmology (World Scientific, Singapore,2011).

[3] E.M. Lifshitz and I.M. Khalatnikov, Adv. Phys. 12, 185(1963).

[4] V. A. Belinskii and I.M. Khalatnikov, Sov. Phys. JETP 29,911 (1969).

[5] I.M. Khalatnikov and E.M. Lifshitz, Phys. Rev. Lett. 24,76 (1970).

[6] V. A. Belinski and I.M. Khalatnikov, Sov. Phys. JETP 30,1174 (1970).

[7] V. A. Belinski, E.M. Lifshitz, and I.M. Khalatnikov, Sov.Phys. Usp. 13, 745 (1971).

[8] L. D. Landau and E.M. Lifshitz, Classical Theoryof Fields (Addison-Wesley, New York, 1975), 4th ed.

[9] E.M. Lifshitz, I.M. Lifshitz, and I.M. Khalatnikov, Sov.Phys. JETP 32, 173 (1971).

[10] V. A. Belinski, I.M. Khalatnikov, and E.M. Lifshitz, Sov.Phys. JETP 35, 838 (1972).

[11] G. Montani and A.A. Kirillov, Zh. Eksp. Teor. Fiz. 66, 449(1997).

[12] G. Montani and A.A. Kirillov, Phys. Rev. D 66, 064010(2002).

[13] L. P. Grishchuk, A.G. Doroshkevich, and V.M. Yudin, Zh.Eksp. Teor. Fiz. 69, 1857 (1975).

[14] C.W. Misner, Phys. Rev. Lett. 22, 1071 (1969).[15] D.M. Chitre, Ph.D. thesis, University of Maryland,

Technical Report No. 72-125, 1972.[16] G. P. Imponente and G. Montani, Phys. Rev. D 63, 103501

(2001).[17] R. Arnowitt, S. Deser, and C.W. Misner, Phys. Rev. 117,

1595 (1960).[18] C.W. Misner, Phys. Rev. 186, 1319 (1969).[19] A. Ashtekar, Nuovo Cimento Soc. Ital. Fis. B 122, 135

(2007).[20] A. Ashtekar, Gen. Relativ. Gravit. 41, 707 (2009).[21] A. Ashtekar, M. Bojowald, and J. Lewandowski, Adv.

Theor. Math. Phys. 7, 233 (2003).

[22] M. Bojowald, H. Hernandez, and A. Skirzewski, Phys.Rev. D 76, 063511 (2007); M. Bojowald, G. Date, andG.M. Hossain, Classical Quantum Gravity 21, 3541(2004).

[23] A. Ashtekar, T. Pawlowski, and P. Singh, Phys. Rev. D 74,084003 (2006).

[24] E. Wilson-Ewing, Phys. Rev. D 82, 043508 (2010).[25] F. Cianfrani, A. Marchini, and G. Montani, Europhys.

Lett. 99, 10003 (2012); F. Cianfrani and G. Montani,Phys. Rev. D 85, 024027 (2012).

[26] E. Alesci and F. Cianfrani, Phys. Rev. D 87, 083521(2013); E. Alesci and F. Cianfrani, Europhys. Lett. 104,10001 (2013).

[27] A. Corichi, T. Vukasinac, and J. A. Zapata, Phys. Rev. D76, 044016 (2007).

[28] A. Corichi, T. Vukasinac, and J. A. Zapata, ClassicalQuantum Gravity 24, 1495 (2007).

[29] M.V. Battisti, O.M. Lecian, and G. Montani, Phys. Rev. D78, 103514 (2008).

[30] L. D. Landau, The Classical Theory of Fields(Butterworth-Heinemann, Washington, DC, 1975), Vol. 2.

[31] V. A. Belinski, I.M. Khalatnikov, and E.M. Lifshitz, Adv.Phys. 31, 639 (1982).

[32] R. Arnowitt, S. Deser, and C.W. Misner, The Dynamics ofGeneral Relativity in Gravitation: An Introduction toCurrent Research, edited by L. Witten (Wiley, London,1962), p. 22726.

[33] V. A. Belinski, I.M. Khalatnikov, and E.M. Lifshitz, Adv.Phys. 19, 525 (1970).

[34] G. Montani, M.V. Battisti, R. Benini, and G. Imponente,Int. J. Mod. Phys. A 23, 2353 (2008).

[35] T. Damour and O.M. Lecian, Phys. Rev. D 83, 044038(2011).

[36] G. P. Imponente and G. Montani, AIP Conf. Proc. 861, 383(2006).

[37] B. K. Berger, Phys. Rev. D 61, 023508 (1999).[38] J. K. Erickson, D. H. Wesley, P. J. Steinhardt, and N.

Turok, Phys. Rev. D 69, 063514 (2004).[39] V. A. Belinski and I.M. Khalatnikov, Sov. Phys. JETP 36,

591 (1973).

SEMICLASSICAL AND QUANTUM BEHAVIOR OF THE . . . PHYSICAL REVIEW D 88, 103511 (2013)

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