Singularity resolution from polymer quantum matter
aa r X i v : . [ g r- q c ] A ug Singularity resolution from polymer quantum matter
Andreas Kreienbuehl ∗ and Tomasz Paw lowski † Theoretical High Energy Physics, Radboud University,Mailbox 79, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands. Departamento de Ciencias F´ısicas, Facultad de Ciencias Exactas,Universidad Andres Bello, Av. Rep´ublica 220, Santiago de Chile. Katedra Metod Matematycznych Fizyki, Universytet Warszawski, ul. Ho˙za 74, 00-681 Warszawa, Poland.
We study the polymeric nature of quantum matter fields using the example of a Friedmann-Lemaˆıtre-Robertson-Walker universe sourced by a minimally coupled massless scalar field. Themodel is treated in the symmetry reduced regime via deparametrization techniques, with the scalefactor playing the role of time. Subsequently, the remaining dynamic degrees of freedom correspond-ing to the matter are polymer quantized. The analysis of the resulting genuine quantum dynamicshows that the big bang singularity is resolved, although with the form of the resolution differing sig-nificantly from that in the models with matter clocks: dynamically, the singularity is made passablerather than avoided. Furthermore, this analysis exposes crucial limitations to the so-called effectivedynamic in loop quantum cosmology when applied outside of the most basic isotropic settings.
PACS numbers: 98.80Qc, 04.60Kz, 04.60Pp
I. INTRODUCTION
Einstein’s theory of general relativity (GR) successfullydescribes gravitational phenomena, predicting with highprecision all large scale observations made to date. It ishowever expected to fail in the ultraviolet regime due tothe quantum nature of the reality at the Planck energyscale. To obtain accurate predictions for such situationsone has to resort to quantum gravity (QG).Despite many attempts [1–6], no general, complete,and working (quantitatively) formulation of QG exists.In particular, in the context of the canonical quantizationprograms, so far it was possible to complete the quantiza-tion program only in certain situations, where gravity iscoupled to specific matter fields (for irrotational dust see[7, 8] and also the earlier, well defined implicit construc-tions in [9, 10]). In order to generalize these frameworks(or complete the alternative approaches) it is crucial tofirst study in detail the simplified mini- and midisuper-space settings.In this paper we consider the minisuperspacemodel, which represents a Friedmann-Lemaˆıtre-Robert-son-Walker (FLRW) universe – an isotropic and flatspacetime admitting a massless scalar field as source.This model is widely used as a testing ground for QGmethods and is at the same time of particular interest incosmology.In the context of QG the model has been studied in de-tail using tools of loop quantum cosmology (LQC – seethe references in the second paragraph and [11–17]). InLQC, an application of the polymer quantization [18–22]to the geometric degrees of freedom results in a dynam-ical singularity resolution [17], whereby the big bang is ∗ [email protected] † [email protected] replaced by a big bounce. This result was later confirmed(at the genuine quantum level) for different matter fields,in particular the Maxwell field [23] and dust [24].The above-mentioned big bounce result was howeverobtained via a somewhat “hybrid” approach: the ge-ometry is quantized via loop techniques, while the mat-ter (the scalar field) is treated by methods of standardquantum mechanics (Schr¨odinger representation). Fur-thermore, the system was analyzed by methods dedi-cated to theories with a time reparametrization freedom.Namely, the evolution was implicitly defined by meansof the formalism of partial observables [26]. The aboveapproach was also applied outside of the isotropic set-tings, both for homogeneous models (like various Bianchimodels [27]) and inhomogeneous spacetimes (in particu-lar Gowdy models [28]) as well as in the context of per-turbation theory about the cosmological sectors [29–31]. The results for the inhomogeneous settings are howeverbased on heuristic methods and the dynamic is not sys-tematically investigated.An alternative approach is presented in [32] (see also[33]), where the investigated model is the one consid-ered here but including a nonvanishing cosmological con-stant. The analysis is carried out in the context of quan-tum geometrodynamic, which is based on the standardSchr¨odinger quantization of the metric Hamiltonian for-mulation of Arnowitt, Deser, and Misner (ADM). Morespecifically, the system is treated via the so-called de-parametrization technique (for an example starting from Recently, the results of [17] were confirmed [25] through the anal-ysis of the same model, where both the geometric and the matterdegrees of freedom are quantized via polymer techniques. Thiswork used one of several possible in this context loop quantiza-tion schemes (see [19] and the discussion in Sec. III). The list of references given here contains only selected examplesrepresenting the current state of development for each model.For a more complete list we refer the reader to [13, 15]. the full theory see [9]): one of the dynamical variables– in this case the scale factor – is selected as clock atthe classical level, after which the system can be quan-tized and regarded as freely evolving with respect to thisclock. The obtained results are (by the majority) con-sistent with those of the studies of the same systems inthe framework of geometrodynamic in [34–36], where thepartial variable formalism was applied [26].A consistent treatment requires the quantization of thegeometry and the matter in the same way. In the contextof LQC the intermediate step towards this goal is theanalysis of a loop quantized scalar field coupled to gravityquantized via standard techniques. This step is necessaryto identify the physical effects arising specifically due tothe polymer nature of matter.In full loop quantum gravity (LQG) a consistent quan-tization of the scalar field was proposed in [37], where oneof two possible (and inequivalent) implementations of thepolymer representation [19] was used. The same choicewas later made in [25] to derive the symmetry reduceddescription and to determine the LQC dynamic of (thisform of) the polymer scalar field.The alternative (in a certain sense dual to the above)consistent quantization prescription was applied in [38],where again the FLRW isotropic universe is investigatedby scalar quantum mechanics on a classical cosmologi-cal background. Elements of a semiclassical analysis ledto the construction of an effective approximation of thedynamic, of which the study showed that the big bangsingularity is replaced by a past-eternal de Sitter phase(“eternal inflation”) with graceful exit.The mathematical formalism characteristic to this pre-scription was later successfully extended to the inhomo-geneous setting in the context of quantum field theoryon Minkowski space and on a cosmological background[39]. Both of these extensions were built via Bojowald’slattice refinement techniques [40].
Since the nature of the studies mentioned just now issemi-heuristic, a comparison with the genuine quantumdynamic is indispensable. This is exactly the goal of thework presented here.
To provide the precise quantumtheory we first perform a deparametrization analogous tothe one in [32], choosing the scale factor cubed as clock.Then, we quantize the scalar field via loop techniques,applying the prescription originally provided in [38].In our work we focus on the precise construction of thequantum model, that is in particular the correct defini-tion of the Hilbert space, and on the analysis of the phys-ical consequences: the dynamic, the existence of a semi-classical sector, and the correct GR limit. Surprisingly,the requirement of the latter will have a critical impacton the form of the Hilbert space and, consequently, onthe domain of applicability of the heuristic constructionof the so-called effective dynamic from LQC.The numerical analysis of the dynamic shows thatthere is no quantum big bang. However, instead ofbouncing back, the quantum state (the wave packet)transits deterministically through the point marking the singularity in GR. The quantum evolution picture ap-pearing here resembles thus the one advertised in the“early LQC epoch” [11]. Consequently, this work (seealso the results in [25]) suggests that the big bounce isan effect arising solely due to the polymeric (discrete)quantum nature of the geometry .The paper is structured as follows: in Sec. II we in-troduce the details of the classical FLRW model that weanalyze. Then we proceed in Sec. III with the construc-tion of the precise quantum theory. Finally, in Sec. IVwe analyze the physical results and conclude in Sec. Vwith a general discussion.In our studies we select the natural units c ≡ ̷ h ≡ L ≡ ( πG N ) / for a lengthscale. Later in the paper we further restrict our attentionto the case L = II. CLASSICAL THEORY
In this paper we focus on the case of an isotropic andflat FLRW universe with a minimally coupled masslessscalar field φ ≡ φ ( T ) ∈ R as source. The metric tensor ofsuch a universe can be expressed as g ≡ −( N d T ) + a δ ij d X i d X j , (2.1)where N ≡ N ( T ) ∈ R + is the lapse function and a ≡ a ( T ) ∈ R is the scale factor (we are working in an “extended”minisuperspace [41]). Since the “symmetric criticalityprinciple” [3, 42] is valid in the present situation, thecanonical action A = ∫ T F T I ( P a ˙ a + P φ ˙ φ − H [ N ]) d T (2.2)can be directly and conveniently derived from the re-duced Einstein-Hilbert action. The canonical momentaappearing in (2.2) are P a = V C L ∣ a ∣ ˙ aN , P φ = V C ∣ a ∣ ˙ φN , (2.3a) { a, P a } = , { φ, P φ } = , (2.3b)whereas the scalar Hamiltonian constraint takes the form H [ N ] = N V C ∣ a ∣ L ⎡⎢⎢⎢⎢⎣ − ( L P a V C a ) + ( LP φ V C a ) ⎤⎥⎥⎥⎥⎦ . (2.4)Note that to arrive at the form (2.2) of the action we hadto first introduce the 1+3 splitting M ≃ R × N , where forthe flat universe (considered here) N = R . Due to thenoncompactness of the spatial slices we had to then intro-duce in the process of deriving (2.2) an infrared regulator We note that the early results were derived for a different system,where the geometry instead of the matter was polymeric. – a cube or “cell” V ⊂ R of finite size (see [34, 43] andthe discussion in [44]). Its physical volume is V = V C ∣ a ∣ ,where V C ≡ ∫ V d X is the comoving coordinate volume of V .This infrared regulating step introduces an additionalcomplication into the treatment, as one has to make surethat the resulting model has a well defined (unambigu-ous) regulator removal limit. The classical FLRW theoryis invariant under the rescaling ⃗ X ↦ ζ ⃗ X , ζ ∈ R + , whichincreases V by a factor of ζ (an active diffeomorphism).This invariance is a natural requirement for the descrip-tion of the model to remain well defined when the reg-ulator is removed [45]. We stress that this requirementis however by no means sufficient in the quantum theory(see in particular [44]). Furthermore, in the quantum the-ory the ζ -invariance is not given trivially [13, 34, 43] andhence imposing it as a condition for consistency affectsthe choice of the canonical variables for the quantization[see (2.11), (2.12), and the paragraph prior to them].The first -class [46–48] Hamiltonian constraint H [ N ] ≈ infinitesimal transformations of T . As canbe seen from (2.1), there also is the possibility of areparametrization of the scale factor a and the Euclideanmetric δ by an η ∈ R ± such that a ↦ a / η and δ ↦ η δ , re-spectively. This residual η -symmetry corresponds to thefreedom of fixing the coordinate scale (a passive diffeo-morphism by ∣ η ∣ ) and orientation ( a “large gauge trans-formation” [13, 17, 34, 43, 49] by sig ( η ) ) . Just like itis required for the ζ -transformation, the physics has tobe invariant under an η -transformation. Among the η -invariant quantities are the action A , the volume V , andratios of the scale factor such as the Hubble parameter h ≡ ˙ a /( N a ) .In the next step we fix the time reparametrization free-dom and time orientation by implementing the second -class [46–48] gauge G ≡ T − V C a L . (2.5)Given the equation of motion˙ a = { a, H [ N ]} = − N L P a V C ∣ a ∣ , (2.6)the form of G implies in particular that P a is negative.This and the form of the constraint H [ N ] then lead tothe reduced canonical action A = ∫ T F T I ( P φ ˙ φ − H R ) d T, (2.7)where again { φ, P φ } =
1. The reduced Hamiltonian takesthe form H R ≡ − P T = − L P a V C a = ∣ L P a V C a ∣ = ∣ P φ LT ∣ . (2.8)Finally, the consistency condition ∂ T G + { G, H [ N ]} = N = L ∣ P φ ∣ . (2.10)We emphasize that the time-gauge G becomes T − sig ( η ) V C a / L under an η -transformation. This meansthat the orientation of a relative to T changes if η ∈ R − .Replacing T by − T has no impact on the space of so-lutions to the Wheeler-DeWitt equation resulting from(2.4) and amounts to a “time-reversal” operation [50].However, the reduced classical formalism derived here isthe result of singling out one of ± T [see (2.5)] so that itis not time-reversal invariant. If (2.5) defines a future-directed clock, the past orientation would be given bythe gauge constraint G = T + V C a / L . These considera-tions are relevant in the construction of the initial statefor the quantum evolution [see (4.14) and the paragraphcontaining this equation].The reduced canonical formalism we constructed justnow has the deficiency of explicitly depending on the in-frared regulator V since, according to (2.5), the clockvariable T scales like ζ . This would make the removalof the infrared regulator V from the resulting quantumtheory a rather tedious task. As the initial step in ad-dressing this problem we replace T by the dimensionlessvariable t ≡ T /∣ T ♢ ∣ , where T ♢ ∈ R ± is some fixed but oth-erwise arbitrary reference value. Furthermore, since P φ also scales like ζ , we analogously define p φ ≡ P φ /∣ P ♢ φ ∣ with P ♢ φ ∈ R ± being a fixed reference value for P φ . Tech-nically, the replacement of T by t can be brought aboutby a change of the integration variable in (2.7), whereasthe replacement of P φ by p φ is realized by an “extendedcanonical” or “scale transformation” [51]. Altogether,this procedure yields the canonical action A = ∣ P ♢ φ ∣ L ∫ ∣ T ♢ ∣ t F ∣ T ♢ ∣ t I ( Lp φ φ ′ − H S ) d t, (2.11)where φ ′ ≡ dφ / dt , { φ, p φ } = / L , and H S ≡ ∣ p φ t ∣ , ∣ p φ t ∣ d t = L ∣ P ♢ φ ∣ H R d T, (2.12)is the Hamiltonian related to H R by a scaling . We stressthat t and p φ are dimensionless variables.At this point it is necessary to mention that the abovemodification does not yet completely remove the depen-dence of the theory on the infrared regulator. Indeed,while the constants T ♢ and P ♢ φ are fixed, no particularvalue of V can be distinguished on physical grounds. Inconsequence, the particular “physical” universe is repre-sented by classes of solutions rather than by single ones.This nonuniqueness can be easily shown at the level ofspecifying the initial data. There, the single universe reg-ulated by different cells V will correspond to the entireset (equivalence class) of the initial data at a chosen ini-tial time t ⋆ . This dependence will propagate through tothe quantum theory.For initial t ⋆ , p ⋆ φ ∈ R ± the solutions to Hamilton’s equa-tions of motion derived from (2.11) and (2.12) are p φ ( t ) = p ⋆ φ , φ ( t ) = φ ⋆ + sig ( tp ⋆ φ ) L ln (∣ tt ⋆ ∣) . (2.13)Therefore, the lapse function N = ∣ T ♢ ∣/∣ P ♢ φ p φ ∣ is a con-stant [positive because of the chosen gauge constraint(2.5)]. The canonical representation of the spacetimeRicci scalar takes then the form R = − h , which in turnimplies − R ( t ) R ♢ = [ p φ ( t ) t ] = ( p ⋆ φ t ) , R ♢ ≡ ( P ♢ φ LT ♢ ) . (2.14)The form of the reference value R ♢ suggests the nat-ural and simplifying choice ∣ P ♢ φ ∣ = ∣ T ♢ ∣ , which corre-sponds to fixing the initial curvature value to be R ( t ⋆ ) =− ( / )[ p ⋆ φ /( Lt ⋆ )] . The value of ∣ P ♢ φ ∣ relative to ∣ T ♢ ∣ isnow fixed but the implicit ξ -dependence discussed earlieris still present.In order to simplify the expressions, from now on wewill use the Planck units normalized by L =
1. With thischoice the energy density ̺ and the pressure p are ̺ ( t ) = p ( t ) = − R ( t ) = [ p φ ( t ) t ] , (2.15)and the spacetime singularity occurs for either of t → ± . The positivity of N implies that the future-pointingevolution of the scalar field is “into” a big crunch for t ∈ R − and “away from” a big bang for t ∈ R + . From(2.13) it is evident that for an element of the branch ofthe solution space admitting a big crunch the value of φ ( t ) − φ ⋆ for p ⋆ φ ∈ R − is related to the analogous value for p ⋆ φ ∈ R + by an overall sign-change. The same holds foran element of the big bang branch of the solution space.Furthermore, we have the correspondence ( sig ( p ⋆ φ )[ φ ( t ) − φ ⋆ ]) − = − ( sig ( p ⋆ φ )[ φ ( t ) − φ ⋆ ]) + , (2.16)which relates big crunches (the left-hand side for t ∈ R − )with big bangs (the right-hand side for t ∈ R + ). In thecanonical formalism at hand these identities are manifes-tations of the invariance of the covariant action under areplacement of φ with − φ .Finally, we note again that the variables t and p φ are dimensionless but still not invariant under a ζ -transformation. The observable scalar field in (2.13) andthe spacetime Ricci scalar in (2.14) – along with its re-lated scalars in (2.15) – are inheriting this implicit non-invariance. This fact will play a crucial role in singlingout the correct regularization scheme in the quantum the-ory. III. QUANTUM THEORY
Our goal here is to build the precise quantum mechan-ical representation of the model introduced above. This means in particular the construction of a suitable Hilbertspace H and the representation of the Hamiltonian H S [see (2.12)] as a self-adjoint operator acting on a suit-able domain in H . The quantum evolution will then bedetermined by a Schr¨odinger equation i ∂∂t ψ = ̂ ∣ p φ t ∣ ψ = ˆ H S ψ (3.1)for ψ ≡ ψ ( t, φ ) . A. The scalar field momentum operator
To begin, let us recall that the canonical formalism in-troduced in the previous section describes a freely evolv-ing isotropic and flat FLRW model. The evolution is gov-erned by the Hamiltonian H S , which by (2.5) depends onthe scale factor clock t = V C a /∣ T ♢ ∣ . The scalar field φ isthus the only object subject to a quantization. Here, wehave several possibilities to proceed.The most obvious way to construct the quantum de-scription is to apply the Schr¨odinger representation as in[32]. As shown there, this representation leads neither toa singularity avoidance nor to a singularity resolution asthe semiclassical wave packets simply follow the classicaltrajectories.An alternative approach that is pursued here is theimplementation of the polymer representation [19, 20].As we will see, the requirement of the existence of aninfrared regulator removal limit (see the previous section)forces this representation to be time- dependent . To initiate the detailed specification of the polymerquantization procedure, let us briefly recall the stan-dard Schr¨odinger representation. It is characterized bythe Stone-von Neumann uniqueness theorem [19, 20, 53],which implies that among all the irreducible regular re-alizations of the Weyl formˆ I λ ˆ J µ = e i λµ ˆ J µ ˆ I λ , λ, µ ∈ R + , (3.2)of the canonical commutation relation [ ˆ φ, ˆ p φ ] = i ˆ1 (3.3)on the space L ( R , d φ ) of Lebesgue square-integrablefunctions, the Schr¨odinger representationˆ I λ ≡ e i λ ˆ φ , ˆ J µ ≡ e − i µ ˆ p φ , (3.4)is unique up to unitary transformations. The regularityproperty says that the mappings of λ to ˆ I λ and µ to ˆ J µ are This situation is analogous to the one in the loop quantizationof the geometric degrees of freedom, where consistency require-ments label the improved dynamic construction as the correctone [52]. (strongly) continuous, which holds if for ψ, ω ∈ L ( R , d φ ) the mappings λ ↦ ⟨ ψ ∣ ˆ I λ ∣ ω ⟩ , µ ↦ ⟨ ψ ∣ ˆ J µ ∣ ω ⟩ , (3.5)are (weakly) continuous.To generalize the above formalism, let us now con-sider the space of exponentiated operator labels, furtherparametrized in the following way λ ≡ λ t ≡ νν t , µ ≡ µ t ≡ ρρ t , ν, ν t , ρ, ρ t ∈ R + . (3.6)The subscript t can be seen as parametrizing the param-eters λ and µ of the groups of unitary operators ˆ I λ andˆ J µ , respectively. That is, the Weyl algebra given in (3.2) depends now on time and so do the unitary operatorsdefined in (3.4) . However, because of the (strong) conti-nuity of the operators ˆ I λ and ˆ J µ in λ and µ , respectively,the operators ˆ φ and ˆ p φ areˆ φ ≡ i lim λ t → ˆ1 − ˆ I λ t λ t = i lim ν → ˆ1 − ˆ I λ t λ t = φ ˆ1 , (3.7a)ˆ p φ ≡ − i lim µ t → ˆ1 − ˆ J µ t µ t = − i lim ρ → ˆ1 − ˆ J µ t µ t = − i ∂∂φ , (3.7b)thus they are independent of t . To conclude, in theSchr¨odinger quantization the operators ˆ φ and ˆ p φ arenot changing if the group parameters are themselvesparametrized by t . In particular, they remain time-independent.The situation changes drastically if the regularity con-dition in the Stone-von Neumann uniqueness theorem isdropped. In this case a possible faithful realization of(3.2) is given by the so-called “polymer representation”.For the sake of generality we will further allow it to betime- dependent (in a yet unspecified way as it was thecase above). The non -separable Hilbert space H for thisrepresentation consists of functions ψ ∈ l ( R , φ ) satis-fying the square- summation requirement ∥ ψ ∥ ≡ ∑ φ ∈ D µt ( ψ ) ∣ ψ ( φ )∣ < ∞ . (3.8)The inner product on H providing this norm is ⟨ ψ ∣ ω ⟩ ≡ ∑ φ ∈ D µt ( ψ,ω ) ψ ( φ ) ω ( φ ) , (3.9)where ω is another element of l ( R , φ ) . We denote by φ the measure that maps a subset of R to its cardinality(the so-called “counting measure”) and by D µ t ( ψ ) ≡ ⋃ φ ∈ S ( ψ )/≋ L µ t ( φ ) , D µ t ( ψ, ω ) ≡ ⋃ φ ∈ [ S ( ψ ) ∩ S ( ω )]/ ≋ L µ t ( φ ) , (3.10)domains defined in terms of (necessarily countable) sup-ports S of ψ and ω . For φ, χ ∈ R we define by φ ≋ χ an equivalence relation such that φ and χ are equiva-lent if and only if there exists an integer k ∈ Z such that φ = χ + kµ t [recall (3.6)]. The domains D µ t are thendisjoint unions of uniform lattices L µ t ( φ ) ≡ { φ } + Z µ t , φ ∈ [ , µ t ) . (3.11)Orthonormal basis states of H are “half-deltas” δ φ ∶ χ ↦ δ φ ( χ ) ≡ δ φχ ≡ { , φ = χ, , otherwise , (3.12)extending the definition of the Kronecker delta symbolto the real line.The polymer representation is now given byˆ I λ t δ φ ≡ e i λ t φ δ φ , ˆ J µ t δ φ ≡ δ φ + µ t , (3.13)which characterizes again a multiplication and a trans-lation operator, respectively. The “ λ -mapping” given in(3.5) is once more continuous in λ t so that by Stone’stheorem [19, 53] the scalar field multiplication opera-tor remains to be given by (3.7). The difference tothe Schr¨odinger representation in (3.4) is that the “ µ -mapping” in (3.5) is no longer continuous in µ t . Thereis therefore no self-adjoint momentum operator gener-ating infinitesimal translations. On l ( R , φ ) there isonly an operator generating finite translations. We aretherefore forced to regularize it, for which we employ thetechnique introduced by Thiemann in the context of fullLQG [5, 54]. In essence this technique is approximatingthe undefined ˆ p φ by well-defined translation operators.Following [20, 39, 55], we chooseˆ p φµ t ≡ − i µ t ( ˆ J † µ t − ˆ J µ t ) , (3.14a) ̂ p φµ t ≡ µ t ⎛⎝ ˆ1 − ˆ J † µ t + ˆ J µ t ⎞⎠ . (3.14b)The action of the former on a state ψ ∈ H isˆ p φµ t ψ ( φ ) = − i µ t [ ψ ( φ + µ t ) − ψ ( φ − µ t )] (3.15)so that, if we could send µ t to 0 [or according to(3.6) send ρ to 0, thereby taking the limit at the kine-matic level], we would get back the differential opera-tor − i ∂ /( ∂φ ) . We observe that the representation of themomentum operator ˆ p φµ t is highly non-unique, in thesame way the representation of finite difference opera-tors in numerical analysis is. We stress that, unlike inthe Schr¨odinger representation, the momentum operatoris now time- dependent [see (3.6)].At this point it is necessary to emphasize that the pre-sented polymer quantization is not the only possible one.Essentially, by replacing the roles of φ and p φ we arriveat another polymer representation, inequivalent (and in The role of these sets will become evident in the next subsection. a sense “dual”) to ours (see the discussion in [19]). Sucha dual representation was used in the quantization ofthe scalar field in full LQG [37]. Its symmetry reducedversion was applied to the LQC model of an FLRW uni-verse [25] filled with a massless scalar field. The subse-quent analysis of the spectral decomposition of the evolu-tion operator (playing the role of the Hamiltonian) showsthat the dynamic of such a system is exactly the same as the one of the system with the scalar field quantized viastandard methods of quantum mechanics [34]. Both ap-proaches, ours and the one of [25], are equally viable froma mathematical point of view. Therefore, choosing oneof them requires a physical input.
B. The Hamiltonian
The next step is the construction of the quantumHamiltonian ˆ H S and the determination of its action,which generates the unitary evolution. To do so, weswitch to the scalar field momentum space, which is againthe Pontryagin dual of the real line but this time the lat-ter is equipped with the discrete topology. In short, itis the Bohr-compactified real line R B . The Hilbert spacedefined in the previous subsection is then equivalent tothe space H ♮ , which consists of Bohr square-measurablefunctions ψ ♮ ( p φ ) ≡ ∑ φ ∈ D µt ( ψ ) ψ ( φ ) e − i φp φ ∈ L ( R B , ( d p φ ) B ) (3.16)satisfying ∥ ψ ♮ ∥ ≡ ∫ ∣ ψ ♮ ( p φ )∣ ( d p φ ) B ≡ lim C →∞ C ∫ C − C ∣ ψ ♮ ( p φ )∣ d p φ < ∞ . (3.17)The inner product (between ψ ♮ and another ω ♮ ∈ H ♮ )generating this norm is ⟨ ψ ♮ ∣ ω ♮ ⟩ ≡ ∫ ψ ♮ ( p φ ) ω ♮ ( p φ ) ( d p φ ) B . (3.18)The basis orthonormal with respect to it is formed by theplane waves e φ ∶ p φ ↦ e φ ( p φ ) ≡ e − i φp φ = δ ♮ φ ( p φ ) . (3.19)We observe that for a uniform lattice D µ t ( ψ ) = L µ t ( φ ) [see (3.11)] the Bohr measure ( d p φ ) B becomes theLebesgue measure d p φ with an integration over the fixedinterval ( − π / µ t , π / µ t ] . The momentum space polymertheory defined here would then be that of Fourier withdiscreteness in position rather than momentum space.In the general polymer theory at hand, the action ofthe multiplication and translation operator on the basisstates e φ is unchanged in comparison to (3.13) so that thescalar field operator is given by i ∂ /( ∂p φ ) . On the other hand, the (undefined) operator ˆ p φ has become the reg-ularized ˆ p φµ t (see the previous subsection), which is ap-proximated by translation operators according to (3.14).Since e φ + µ t = e µ t e φ we obtainˆ J µ t = e − i µ t p φ ˆ1 , ˆ p φµ t = sin ( µ t p φ ) µ t ˆ1 , (3.20)because of which the action of the Hamiltonian operatorcan be explicitly given byˆ H S ψ ♮ = ∣ sin ( µ t p φ ) µ t t ∣ ψ ♮ , (3.21)where ψ ♮ ≡ ψ ♮ ( t, p φ ) . The fact that ∥ ψ ∥ = ∥ ψ ♮ ∥ allowsus now to prove the conservation of the norm under anaction of ˆ H S . To show this we write explicitly the timederivative of the norm i ∂∂t ∥ ψ ∥ = i ∂∂t ∥ ψ ♮ ∥ = ∫ i ∂∂t ∣ ψ ♮ ∣ ( d p φ ) B = ∑ φ,χ ∈ D µt ( ψ ) ψ ( t, φ ) ψ ( t, χ ) ∫ ˆ H S e − i p φ ( φ − χ ) ( d p φ ) B . (3.22)To evaluate the right-hand side we first observe that theintegral can be expressed as the integrallim C →∞ C C ∫ − C = lim n →∞ µ t nπ n ∑ k = ⎛⎝ − ( k − ) π / µ t ∫ − kπ / µ t + − ( k − ) π / µ t ∫ − ( k − ) π / µ t + ( k − ) π / µ t ∫ ( k − ) π / µ t + kπ / µ t ∫ ( k − ) π / µ t ⎞⎠ . (3.23)The specific form of this integral allows us to drop theabsolute value in (3.21), replacing it in (3.22) insteadwith a sign appropriate for each integration domain in(3.23). Next, we apply some trigonometric identities, the µ t -translation invariance of D µ t ( ψ ) , and (see [56]) n ∑ k = cos ( ( k + ) πµ t ( φ − χ )) = sin ( nπµ t ( φ − χ )) ( πµ t ( φ − χ )) . (3.24)Finally, if we divide this by n and take the limit n → ∞ [see (3.23)], we get i ∂ ∥ ψ ∥/ ∂t = µ t inthe approximated ˆ p φ operator has been arbitrary. At themathematical level the situation is analogous to the onein the loop quantization of the geometry (see [57]), wherethe fiducial holonomy length could be an arbitrary func-tion on the phase space. There, however, the physicalconsistency requirements restricted the possible choicesto just one class of functions [52]. We expect that thesame situation occurs in our model. To show that thisexpectation is indeed realized let us recall the followingfacts.The particular moment of the universe’s evolution canbe represented by various points on the phase space cor-responding to different choices of the regulator cell. Fur-thermore, once we ask about the locally measurable prop-erties of the universe (observables) at this moment, therehas to exist their nontrivial limit as we remove the regu-lator.One such local observable is the energy density (2.15)determined by (2.14). According to (2.12) and (2.15),the quantum operator corresponding to it is related tothe Hamiltonian ˆ H S in the following wayˆ ̺ = ̂ H . (3.25)From (3.21) it follows that at a fixed point in time t thespectrum of this operator equalsSp ( ˆ ̺ ) = [ , µ t t ] ⊂ R . (3.26)The most natural way to satisfy the consistency require-ments discussed in the previous paragraph is to requirethat Sp ( ˆ ̺ ) be time- independent . This implies µ t ∝ / t so that we can fix the function ρ t in (3.6) by ρ t ≡ ∣ t ∣ . (3.27)This in turn gives µ t = ρ /∣ t ∣ [see again (3.6)], which for the“volume clock” v ≡ t / ρ results in the momentum spaceSchr¨odinger equation i ∂∂v ψ ♮ = ∣ sin ( p φ v )∣ ψ ♮ (3.28)for ψ ♮ ≡ ψ ♮ ( v, p φ ) .Note that this method of fixing µ t is almost a full ana-log of the conditions used for the geometry degrees offreedom in [52]. There, however, the reasoning exploitedthe existence of “nicely” behaving semiclassical sectorsthrough the use of the so called effective dynamic . Here,as we have not yet investigated the dynamical sector,implementing that reasoning directly would be risky. In-stead, we managed to fix µ t through considerations ofthe genuine quantum formalism.In the next section we solve the Schr¨odinger equation(3.28) in order to analyze the dynamic and to discuss thephysical properties of the system. IV. THE DYNAMIC
The Hilbert space and the explicit action of the Hamil-tonian operator constructed just now allow us to deter-mine the system’s dynamic. At this level the requirementof the theory to be physically meaningful becomes cru-cial. The principal requirement is that the theory musthave the proper low energy limit. Here, this means thatin the distant past and future the quantum evolution ought to agree with the predictions of GR. In our casean inability of the model-description to realize this prop-erty would imply that the formulation should be furtherand adequately corrected. In fact, as we will see below,this is precisely what is required here.To begin, let us investigate the dynamic of the theoryexactly as specified in the previous section.
A. Single lattice Hilbert space
Once we select p φ as the configuration variable, theSchr¨odinger equation given in (3.28) becomes an ordinarydifferential equation, which we can solve for v ∈ R ± . Thesolution reads ψ ♮ ( v, p φ ) ≡ ˆ E vv ⋆ ψ ♮ ( v ⋆ , p φ )≡ e − i [ F ( v,p φ ) − F ( v ⋆ ,p φ )] ψ ♮ ( v ⋆ , p φ ) , (4.1a) F ( v, p φ ) ≡ vS ( v, p φ ) [ sin ( p φ v ) − Ci (∣ p φ v ∣) p φ v ] , (4.1b) S ( v, p φ ) ≡ sig ( sin ( p φ v )) , (4.1c)where we set v ⋆ = t ⋆ / ρ . For ∣ arg ( z )∣ < π the cosine inte-gral function isCi ( z ) ≡ γ + ln ( z ) + ∫ z cos ( y ) − y d y (4.2)with γ being the Euler-Mascheroni number [58]. Thisdefinition implies Ci ( z ) ∼ γ + ln ( z ) for z →
0, suggest-ing semiclassical behavior of sufficiently sharply peakedinitial states in the limit ∣ v ∣ → ∞ .The operator ˆ H S is not defined at v = v → ± F ( v, p φ ) = v ≠ v = F ( , p φ ) ≡ there exists a unique unitary operator ˆ E v ⋆ ≡ e i F ( v ⋆ ,p φ ) ˆ1 (4.4) that evolves states to the instant v = . In consequence,there exists a preferred extension of the evolution through v =
0, defined by the requirement of continuity of ψ ♮ at v =
0. The global solution is thus given by (4.1) with(4.4).It appears that the existence of such a preferred exten-sion is sufficient for singularity resolution. However, aswe will see below, this is not the case. To explain whatis missing, we consider any unit-normalized initial state ψ ♮ ( v ⋆ , p φ ) such that the expectation value of the scalarfield operator is finite ⟨ ψ ♮ , v ⋆ ∣ ˆ φ ∣ ψ ♮ , v ⋆ ⟩ = φ ⋆ . (4.5)The expectation value of ˆ φ at any value of v is then givenby the formula ⟨ ˆ φ ⟩ ψ ♮ ( v ) ≡ ⟨ ψ ♮ , v ∣ ˆ φ ∣ ψ ♮ , v ⟩ = ⟨ ψ ♮ , v ⋆ ∣ ˆ φ − [ S ( v, p φ ) Ci (∣ p φ v ∣)− S ( v ⋆ , p φ ) Ci (∣ p φ v ⋆ ∣ )] ˆ1 ∣ ψ ♮ , v ⋆ ⟩ . (4.6)Since the cosine integral function defined in (4.2) belongsto L ( R , d p φ ) , this expectation value is in fact equal to φ ⋆ . That is to say the evolution is frozen . This result isthen in direct disagreement with the predictions of GR.In consequence, our states exhibit an unphysical behaviorin the low energy (large ∣ v ∣ ) limit.Our model then still lacks an appropriate physical Hilbert space. To explore the possibilities of construct-ing it, let us first go back to analyzing the solutions to(3.28) but this time by considering the wave functionson the configuration space as opposed to the momen-tum one used in (4.1). On the configuration space, theevolution of a state ψ v ≡ ψ ( v, ⋅ ) can be viewed as an as-signment v ↦ ψ v ∈ H v , where v ∈ R . The Hilbert space H v is spanned by eigenstates of ˆ H S for a fixed value of v .However, per analogy with the loop quantization of thegeometry [34] we can distinguish sectors that are invari-ant with respect to the action of ˆ H S at v . These sectorsconsist of functions that are supported on the lattices L ( ϕ ) µ v , where µ v = /∣ v ∣ [see (3.6), (3.27), and the def-inition of v in the sentence prior to (3.28)] and L ( ϕ ) ≡ L µ v ( φ )/ µ v ≡ { ϕ } + Z , ϕ ∈ [ , ) . (4.7)We can then regard at the initial v = v ⋆ the subspaces H vϕ ≡ H v ∣ µ v L ( ϕ ) as the superselection sectors and evolvethem independently. Such a decomposition can be per-formed at each v independently. Let us now probewhether there exists any relation between the spaces H vϕ for different values of v . The answer is given by the formof (4.1): since the cosine integral function is non-periodic,the unitary evolution to any v ⋆ + v ε , where v ε ∈ R ± , in-stantaneously couples an infinite number of these lattices.In consequence, the sectors H vϕ of H v are not true su-perselection sectors in the sense of [13, 17, 34, 43, 49].Therefore, we are forced to work with the original non-separable Hilbert space H v without access to previouslyavailable tools that allow for a distinction of separablesubspaces. The form of (4.6) suggests then that in or-der to provide a nontrivial evolution, the physical Hilbertspace needs to be equipped with a continuous rather thana discrete inner product. B. Integral Hilbert space
A similar situation appeared in LQC already in a differ-ent context during the studies of the FLRW universe with a massless scalar field and a positive cosmological con-stant [36]. There, following the choice of a lapse adoptedto using the scalar field as time variable, the evolution op-erator admitted a family of self-adjoint extensions, eachwith a discrete spectrum. However, a different choice ofthe lapse – corresponding to parametrizing the evolutionby the cosmic time variable – led to a unique self-adjointgenerator of the evolution with a continuous spectrum[60]. The physical Hilbert space corresponding to thelatter case (the “cosmic time case”) appeared, further-more, to be an integral of all the Hilbert spaces corre-sponding to the particular self-adjoint extensions of theformer case (the “matter clock case”), with the Lebesguemeasure determined by the group averaging procedure.Motivated by this observation, we introduce the ana-log of the integral Hilbert space in our case. First, wenote that on the domain [ , ) of ϕ one can introducea natural (quite general and time dependent) Lebesguemeasure M ( v, ϕ ) d ϕ . Next, we introduce a decompo-sition of the non-separable Hilbert space H into spaces H vϕ at the initial time v ⋆ . This leads to the follow-ing definition of the decomposition of the initial data at v = v ⋆ H v ⋆ ∋ ψ ( v ⋆ , φ ) ↦ ψ ϕ ( v ⋆ , φ ) ≡ ψ ( v ⋆ , φ )∣ L µv ⋆ ( φ ) ∈ H v ⋆ ϕ . (4.8)This initial data is then extended to the solutions to(3.28) via (4.1). We thus have a decomposition of thephysical Hilbert space into explicitly separable (at leastat v = v ⋆ ) subspaces.Now, we can define the new physical Hilbert space H P v at v = v ⋆ via H P v ⋆ ≡ ∫ H v ⋆ ϕ M ( v ⋆ , ϕ ) d ϕ (4.9)and equip it with the inner product ⟨ ψ v ∣ ω v ⟩ ≡ ∫ ⟨ ψ vϕ ˆ E vv ⋆ ∣ ˆ E v ⋆ v ω vϕ ⟩ M ( v ⋆ , ϕ ) d ϕ . (4.10)We used the abbreviation ψ vϕ ≡ ψ ϕ ( v, φ ) ∈ H vϕ . Thisis our candidate for the physical inner product: betweeneach pair of solutions it is evaluated on the initial dataslice at v = v ⋆ . On that initial slice it can be written as ⟨ ψ v ⋆ ∣ ω v ⋆ ⟩ ≡ ∫ R ψ ( v ⋆ , φ ) ω ( v ⋆ , φ ) M ( v ⋆ , ϕ ( φ ))/ µ v ⋆ d φ. (4.11)It is by definition time-independent but a priori it maynot have a local form analogous to (4.11) at v ≠ v ⋆ , whichcan potentially complicate the evaluations of the expec-tation values of the observables.We note that the construction performed for v = v ⋆ can be repeated at each value of v , giving rise to poten-tially inequivalent constructions of the candidate physicalHilbert space. One can then consider a function P ( ψ v ∣ ω v ) ≡ ∫ R ψ ( v, φ ) ω ( v, φ ) M ( v, ϕ ( φ ))/ µ v d φ. (4.12)On each slice of constant v this function equals the innerproduct of the candidate Hilbert space constructed forthis slice. One can then ask under which condition theseHilbert spaces will be equivalent and their inner productsequal. A condition necessary and sufficient for it is that ∂P ( ψ v ∣ ω v )/ ∂v =
0. The form of the unitary evolutionoperator [see in particular (4.1a)] implies however thatthis condition will be satisfied if and only if we require M ( v, ϕ ) ≡ µ v m ( ϕ ) . (4.13)Following this choice, our candidate Hilbert space be-comes (up to a rescaling ψ v ↦ ψ v / m / on H v ) the space L ( R , d φ ) with the standard L -inner product. Also, the momentum space is now L ( R , d p φ ) with the correspond-ing Lebesgue measure.Using (4.4), we can now consider an initial state ψ ♮ ( v ⋆ , p φ ) ≡ ˆ E v ⋆ ψ ♮ ( , p φ ) (4.14)with a real unit- L -normalized Gaussian ψ ♮ ( , p φ ) ≡ √ w √ π e − w ( p φ − p ⋆ φ ) / . (4.15)This class of states is “special” in the sense that thequantum evolution they undergo is semiclassical bothfor v ∼ v ⋆ and v ∼ − v ⋆ (see below). Furthermore,the states ˆ E v ψ ♮ ( , p φ ) with unit- L -normalized ψ ♮ ( , p φ ) span the solution space of the Wheeler-DeWitt equationdefined by the Hamiltonian constraint in (2.4) for theclock V C a = T . The set of the complex conjugate ofthese states represents the analogous states for the clock − V C a = T (see [50] and recall that we set L = v ⋆ =
250 and for (4.15) the width w = p ⋆ φ = ∣ v ∣ ≫
1. In fact, theclassical solution that is well approximating the quantumtrajectory is characterized by φ ⋆ = ∓ ⟨ S ( v ⋆ , p φ ) [ Ci (∣ p φ v ⋆ ∣) + ln (∣ v ⋆ ∣)] ˆ1 ⟩ ψ ♮ , (4.16)where the overall sign “ ∓ ” corresponds to v ∈ R ± and,where v ⋆ = S in(4.1), we observe that ∣ v ⋆ ∣ ≫ ∣ p ⋆ φ ∣/ π is a necessary require-ment for semiclassicality. As we can see, the physicalstate is indeed passing in a continuous manner through −
200 0 200 − − v = t / ρ ⟨ ˆ φ ⟩ ψ ♮ ± ( △ ˆ φ ) ψ ♮ p ⋆ φ = p ⋆ φ = − (a) Quantum evolution for two values of p ⋆ φ . − − − v = t / ρφ ⟨ ˆ φ ⟩ ψ ♮ ± ( △ ˆ φ ) ψ ♮ (b) Classical and quantum evolution for p ⋆ φ = FIG. 1. An illustration of the quantum evolution of the scalarfield is presented in Fig. 1(a). In Fig. 1(b) we display the p ⋆ φ = v -interval used in Fig. 1(a).The circles, triangles, and squares correspond to actual mea-surements made using the software Matlab and GNU Oc-tave. The time interval between two consecutive measure-ments becomes smaller as ∣ v ∣ approaches 0. In this plot, thesmallest value of ∣ v ∣ is 0 .
25 but values as small as 10 − havebeen considered with the same outcome. Namely, there issemiclassicality for ∣ v ∣ ≫ v = the point v =
0, corresponding in the classical theory tothe big bang singularity [which is particularly clear fromFig. 1(a) and also from (4.3)]. This happens regardlessof the sign of the initial momentum so that the quan-tum evolution is effectively respecting (2.16), which inthe classical theory specifies the relation between the so-lutions for negative and positive t .To examine more closely the issue of the singularity res-olution we also analyzed the expectation values of the op-erator corresponding to the spacetime Ricci scalar. Thequantum trajectory is presented in Fig. 2. The measure-0 −
200 0 200 − − v = t / ρ ⟨ ˆ R ⟩ ψ ♮ p ⋆ φ = FIG. 2. Expectation values of the spacetime Ricci operatorfor various values of v are plotted in this figure. As in Fig. 1,measurements are taken the more often the closer ∣ v ∣ is to 0with the smallest values given by ∣ v ∣ = .
25. It follows that thebig bang curvature singularity is resolved and the branches v ∈ R ± are connected. −
200 0 2000 . . . v = t / ρ ( △ ˆ φ ) ψ ♮ p ⋆ φ = FIG. 3. In this figure an illustration of the fluctuations of thescalar field operator is presented. For ∣ v ∣ → ∞ the fluctuationsincrease but remain nonetheless finite. ments are independent of the overall sign of p ⋆ φ , thus onlythe case p ⋆ φ > v . This confirms theanalytical result of (3.26) and, thus, implies the globalboundedness of the spectrum of the Ricci scalar operatoronce µ v is fixed via (3.27). One can thus conclude thatthe big bang singularity is resolved.Finally, in Figs. 3 and 4 the fluctuations of the scalarfield and polymer momentum operator are depicted, re-spectively. They both quickly approach ( / ) / as ∣ v ∣ in-creases, which is the value expected for a Gaussian with w =
1. This confirms the semiclassical nature of the statefor ∣ v ∣ ∼ v ⋆ . What is interesting in the near-singularityregion is the fact that the fluctuations of the scalar field −
200 0 2000123 v = t / ρ ( △ ˆ p φ µ v ) ψ ♮ p ⋆ φ = FIG. 4. This figure provides an illustration of the fluctuationsof the polymeric momentum operator. Just like for the scalarfield fluctuations in Fig. 3, the behavior for ∣ v ∣ ≫ operator are in fact decreasing for ∣ v ∣ →
0. This may hap-pen because the state gets “squeezed” towards the originin order to “fit through” the point v =
0. This howeverrequires a more detailed analysis of the nature of thestate there, which may be the subject of a subsequentinvestigation.
V. DISCUSSION
In this paper we investigated the quantum dynamicof the isotropic and flat FLRW universe of infinite extentand sourced by a minimally coupled massless scalar field.Our focus was on the modifications to the dynamic fol-lowing from the polymeric nature of the matter, and inparticular on the issue of the singularity resolution. Toidentify these effects we implemented one of two possi-ble loop quantization schemes of the scalar field. Thisscheme is the analog of the one used so far in LQC toquantize the geometry degrees of freedom.
Unlike inthe most of the existing works in LQC, instead of im-plementing the Dirac program to solve the Hamiltonianconstraint, we performed a complete deparametrizationof the system by choosing a time variable that dependson the scale factor. As a result, the physical evolutionis described by a free Hamiltonian. The quantizationof such a deparametrized system is implicitly equivalentto selecting the Schr¨odinger quantization for the geom-etry when applying the Dirac program. Therefore, theeffects of the geometry discreteness are not featured inour model. Rather, the matter degrees of freedom arediscreet.In the process of constructing the correct descriptionof the quantum system we encountered several obstacles:First, the noncompactness of the universe’s spatialslices forced us to introduce an infrared regulator. The1necessary consistency condition, that the theory has toadmit a well defined and nontrivial regulator removallimit, restricted then the Hamiltonian to a particularform, which happened to be explicitly time-dependent.Second, the Hilbert space to which the physical statesbelong occurred to be non-separable. This is a standard(and treatable) problem in LQC. Here, however, the ex-plicit time dependence of the Hamiltonian prevented usfrom implementing the known technique of subdividingthe (too big) Hilbert space onto separable superselectionsectors. An idea to naively proceed by determining thedynamic on that space led to a model significantly dis-agreeing with GR predictions at the low curvature limit.Indeed, the quantum evolution of the scalar field was frozen.To cure this defect we performed a specific constructionof the separable Hilbert space out of the nonseparableone, taking as the guideline the relation between Hilbertspaces corresponding to the models with different choicesof the lapse function in LQC in the presence of a posi-tive cosmological constant. As a result, we were able toconstruct a certain integral Hilbert space equipped witha continuous rather than a discrete (as usual in LQC)inner product.Such a construction of the Hilbert space was then usedto investigate the dynamic. To do so we selected a classof Gaussian initial states and evolved them numerically.The resulting quantum trajectories showed a good con-vergence to the classical trajectories predicted by GR atlow energies. At high curvatures (small ∣ v ∣ ) however weobserved a significant departure from GR. Indeed, themost critical feature of the model is the existence of aunique unitary evolution operator evolving to/from thetime slice v = deter-ministic evolution through the classical singularity. Fur-thermore, the quantum counterparts of the Ricci scalar,energy density, or pressure are explicitly bounded oper-ators. In consequence, the listed quantities remain finitethroughout the entire evolution, including in particular v = unlikein previous contributions to the literature on this model,here the quantum features responsible for singularity res-olution originate from the matter rather than the geomet-ric sector. Therefore, the form of the singularity resolu-tion differs from that in the literature: instead of be- ing avoided, the surface v = both the matter and the geometry, inwhich situation the results may change qualitatively. Theanalysis of this scenario is a task for the future.Finally, let us comment on an important lesson learnedfrom this model: the predicted dynamic depends criti-cally on the construction of the physical Hilbert spaceof the model, even though the regularized form of theHamiltonian remains the same. This implies in particu-lar that the regularized form of the classical Hamiltonianor Hamiltonian constraint is not sufficient to robustlydetermine or even well approximate the quantum evolu-tion. The problem in Hilbert space construction appearsnot only when the matter degrees of freedom are quan-tized “ `a la loop” but already in “standard” LQC in themodels as simple as a Bianchi I universe [61]. This is-sue is particularly critical in all the studies of models inLQC performed via the so called effective dynamic tech-niques without prior specification of the elements of thegenuine quantum theory that the effective formulation issupposed to mimic.In a further project we intend to take a closer lookat the behavior of the state near the singularity. Whydo the quantum fluctuations of the scalar field decreasetowards the origin of the time-axis? Of interest is alsothe inclusion of a non-zero cosmological constant. Fi-nally, and this is most intriguing, we would like to addressthe question of how the quantization procedure presentedhere can be combined with that of the geometric sectordiscussed in the LQC works [13, 17, 34, 43, 49]. ACKNOWLEDGMENTS
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