Alternative quantization of the Hamiltonian in isotropic loop quantum cosmology
aa r X i v : . [ g r- q c ] A p r Alternative quantization of the Hamiltonian in isotropic loop quantum cosmology
Jinsong Yang , You Ding , , and Yongge Ma ∗ Department of Physics, Beijing Normal University, Beijing 100875, China Centre de Physique Th´eorique de Luminy, Universit´e de la M´editerran´ee, F-13288 Marseille, EU
Since there are quantization ambiguities in constructing the Hamiltonian constraint operator inisotropic loop quantum cosmology, it is crucial to check whether the key features of loop quantumcosmology, such as the quantum bounce and effective scenario, are robust against the ambiguities.In this paper, we consider a typical quantization ambiguity arising from the quantization of thefield strength of the gravitational connection. An alternative Hamiltonian constraint operator isconstructed, which is shown to have the correct classical limit by the semiclassical analysis. The ef-fective Hamiltonian incorporating higher order quantum corrections is also obtained. In the spatiallyflat FRW model with a massless scalar field, the classical big bang is again replaced by a quantumbounce. Moreover, there are still great possibilities for the expanding universe to recollapse due tothe quantum gravity effect. Thus, these key features are robust against this quantization ambiguity.
PACS numbers: 04.60.Kz,04.60.Pp,98.80.Qc
I. INTRODUCTION
An important motivation of the theoretical search for aquantum theory of gravity is the expectation that the sin-gularities predicted by classical general relativity wouldbe resolved by the quantum gravity theory. This expec-tation has been confirmed by the recent study of cer-tain isotropic models in loop quantum cosmology (LQC)[1, 2, 3], which is a simplified symmetry-reduced model ofa full background-independent quantum theory of gravity[4], known as loop quantum gravity (LQG) [5, 6, 7, 8]. Inloop quantum cosmological scenario for a universe filledwith a massless scalar field, the classical singularity getsreplaced by a quantum bounce [3, 9, 10]. Moreover, itis revealed in the effective scenarios that there are greatpossibilities for a spatially flat FRW expanding universeto recollapse due to the quantum gravity effect [11]. How-ever, as in the ordinary quantization procedure, thereare quantization ambiguities in constructing the Hamilto-nian constraint operator. Thus a crucial question arises.Whether the above significant results come from certainparticular treatment of quantization? Before confirmingthe robustness of the key results against the quantizationambiguities, one could not believe that they are not someartifact under particular assumptions.In this paper, we consider a typical quantization ambi-guity arising from the quantization of the field strengthof the gravitational connection. An alternative Hamil-tonian constraint operator is constructed. Semiclassicalstates are then employed to show that the new Hamil-tonian operator has the correct classical limit. The ef-fective Hamiltonian incorporating higher order quantumcorrections is also obtained. In the spatially flat FRWmodel with a massless scalar field, the classical big bangis again replaced by a quantum bounce. Moreover, thereare still great possibilities for the expanding universe to ∗ Electronic address: [email protected] recollapse due to the quantum gravity effect.In the spatially flat isotropic model of LQC, one hasto first introduce an elementary cell V and restrict allintegrations to this cell. Fix a fiducial flat metric o q ab and denote by V o the volume of the elementary cell V inthis geometry. The gravitational phase space variables —the connections A ia and the density-weighted triads E ai —can be expressed as A ia = c V − / o o ω ia and E ai = p V − / o √ o q o e ai , (1)where ( o ω ia , o e ai ) are a set of orthonormal co-triads andtriads compatible with o q ab and adapted to the edges ofthe elementary cell V . The basic (nonvanishing) Poissonbracket is given by { c, p } = 8 πGγ , (2)where G is the Newton’s constant and γ is the Barbero-Immirzi parameter.To pass to the quantum theory, one constructs a kine-matical Hilbert space H gravkin = L ( R Bohr , d µ Bohr ), where R Bohr is the Bohr compactification of the real line andd µ Bohr is the Haar measure on it [12]. The abstract ∗ -algebra represented on the Hilbert space is based on theholonomies of connection A ia . In the Hamiltonian con-straint of LQG, the gravitational connection A ia appearsthrough its curvature F iab . Since there exists no operatorcorresponding to c , only holonomy operators are well de-fined. Hence one is led to express the curvature in termsof holonomies. Similarly, in the improved dynamics set-ting of LQC [3], to express the curvature one employedthe holonomies h (¯ µ ) i := cos ¯ µc I + 2 sin ¯ µc τ i (3)along an edge parallel to the triad o e ai of length ¯ µ p | p | with respect to the physical metric q ab , where I is theidentity 2 × τ i = − iσ i / σ i are the Paulimatrices). Thus, the elementary variables could be takenas the functions exp( i ¯ µc/
2) and the physical volume V = | p | / of the cell, which have unambiguous opera-tor analogs. II. THE ALTERNATIVE HAMILTONIANCONSTRAINT OPERATOR
In general relativity, the dynamics of a gravitationalsystem is determined by the Hamiltonian constraint.Many problems, such as the big-bang singulary, arise inclassical dynamics. One expects that some quantum dy-namics can resolve these problems. Hence, it is importantto have a well-defined Hamiltonian constraint operator inLQC. An improved Hamiltonian constraint operator hasbeen constructed in [3]. However, there are quantizationambiguities in the construction. In this section, we willconstruct an alternative Hamiltonian constraint operatorby a different quantization procedure.Because of spatial flatness and homogeneity, the gravi-tational part of the Hamiltonian constraint of full generalrelativity is simplified to the form C grav = − γ − Z V d xN ǫ ijk F iab e − E aj E bk , (4)where e := p | det E | , the lapse N is constant and we willset it to one.The procedure used in LQC (and specifically in [3, 12])so far can be summarized as follows. The term involvingthe triads can be written as ǫ ijk e − E aj E bk = X k sgn( p )2 πγG ¯ µV / o ǫ abc o ω kc × Tr (cid:16) h (¯ µ ) k n h (¯ µ ) k − , V o τ i (cid:17) . (5)To express the curvature components F iab in terms ofholonomies, one considers a square ✷ ij in the i - j planespanned by a face of V , each of whose sides has length¯ µ p | p | with respect to q ab . Then the ab component of thecurvature is given by F iab τ i = lim Ar ✷ → h (¯ µ ) ✷ ij − µ V / o o ω i [ ao ω jb ] , (6)where Ar ✷ is the area of the square under consideration,and the holonomy h (¯ µ ) ✷ ij around the square ✷ ij is just theproduct of holonomies along the four edges of ✷ ij , h (¯ µ ) ✷ ij = h (¯ µ ) i h (¯ µ ) j h (¯ µ ) i − h (¯ µ ) j − . (7)However, quantization ambiguities arise here, since theapproach to express the curvature components F iab interms of holonomies is not unique. Hence the corre-sponding operators in different approaches will be dif-ferent from each other. In the following, we will con- sider an expression of the curvature components differ-ent from Eq. (6). Taking account of the definition (3) ofholonomies, we have the identitylim ¯ µ → h (¯ µ ) i − h (¯ µ ) i − ¯ µ = lim ¯ µ → µc/ τ i ¯ µ = 2 cτ i . (8)Hence the curvature of connection can be written interms of the holomomies as F kab τ k = c V − / o ǫ ijk o ω ia o ω jb τ k = lim ¯ µ → (cid:16) h (¯ µ ) i − h (¯ µ ) i − (cid:17) (cid:16) h (¯ µ ) j − h (¯ µ ) j − (cid:17) µ V / o o ω i [ a o ω jb ] . (9)Combining Eqs. (5) and (9), the Hamiltonian constraintcan be written as C grav = − lim ¯ µ → sgn( p )4 πγ G ¯ µ ǫ ijk Tr h (cid:16) h (¯ µ ) i − h (¯ µ ) i − (cid:17) × (cid:16) h (¯ µ ) j − h (¯ µ ) j − (cid:17) h (¯ µ ) k n h (¯ µ ) k − , V o i ≡ lim ¯ µ → C R(¯ µ )grav . (10)Since the constraint is now expressed in terms of ele-mentary variables and their Poisson bracket, it can bepromoted to a quantum operator directly. The resultingalternative regulated constraint operator with symmetricfactor-ordering readsˆ C R(¯ µ )grav = sin ¯ µc h i sgn(ˆ p ) πγ ¯ µ ℓ p (cid:16) sin ¯ µc V cos ¯ µc − cos ¯ µc V sin ¯ µc (cid:17)i sin ¯ µc , (11)where, for clarity, we have suppressed hats over the oper-ators h (¯ µ ) i , sin(¯ µc/
2) and cos(¯ µc/ ℓ p = √ G ~ . Todeal with the regulator ¯ µ , we adopt the improved scheme[3]. We shrink the length of holonomy edges, as mea-sured by the physical metric q ab , to the value √ ∆, where∆ = 4 √ πγℓ p is a minimum nonzero eigenvalue of thearea operator [10]. Thus we are led to choose for ¯ µ aspecific function ¯ µ ( p ), given by¯ µ = p ∆ / | p | . (12)It is convenient to work with the v -representation. Inthis representation, states | v i constituting an orthonor-mal basis in H gravkin is more directly adapted to the volumeoperator ˆ V , ˆ V | v i = πγℓ p ! / | v | K | v i , (13)where K = 43 r πγℓ p . (14)The action of \ exp( i ¯ µc/
2) is given by \ exp( i ¯ µc/ | v i = | v + 1 i . (15)Hence the alternative Hamiltonian constraint operator isgiven byˆ C Rgrav = sin ¯ µc h i sgn( v ) πγ ¯ µ ℓ p (cid:16) sin ¯ µc V cos ¯ µc − cos ¯ µc V sin ¯ µc (cid:17)i sin ¯ µc µc A sin ¯ µc . (16)It is easy to show that ˆ A is well defined and | v i is aneigenvector of ˆ A . Furthermore, the eigenvalues of ˆ A arereal and negative. So ˆ A is a negative definite self-adjointoperator on H gravkin . Hence, ˆ C Rgrav is a negative-definiteself-adjoint operator on H gravkin . The action of ˆ C Rgrav onthe basis | v i of H gravkin is given byˆ C Rgrav | v i = f ′ + ( v ) | v + 2 i + f ′ o ( v ) | v i + f ′− ( v ) | v − i , (17)where f ′ + ( v ) = 274 r π Kℓ p γ / ( v + 1) (cid:0) | v + 2 | − | v | (cid:1) ,f ′− ( v ) = f ′ + ( v − , f ′ o ( v ) = − f ′ + ( v ) − f ′− ( v ) . (18)Thus, ˆ C Rgrav is again a difference operator. Recall that bycontrast to Eq. (16), the Hamiltonian constraint opera-tor defined in [3] readsˆ C grav = sin(¯ µc ) h i sgn( v ) πγ ¯ µ ℓ p (cid:16) sin ¯ µc V cos ¯ µc − cos ¯ µc V sin ¯ µc (cid:17)i sin(¯ µc )=: sin(¯ µc ) ˆ A sin(¯ µc ) . (19)This shows a quantization ambiguity arising from thequantization of the field strength of the gravitational con-nection.To identify a dynamical matter field as an internalclock, we take a massless scalar field φ with Hamiltonian C φ = | p | − / p φ /
2, where p φ denotes the momentum of φ . In the standard Schr¨odinger representation, the mat-ter part of the quantum Hamiltonian constraint readsˆ C φ = \ | p | − / c p φ /
2. Thus we get the total constraint asˆ C R = πG ˆ C Rgrav + ˆ C φ . III. THE CLASSICAL LIMIT AND THEMODIFIED FRIEDMANN EQUATION
It has been shown in [11, 13] that the improved Hamil-tonian constraint operator constructed in [3] has the cor-rect classical limit. In this section, we will show that thealternative Hamiltonian constraint operator constructedin last section also has the correct classical limit. More-over, the effective Hamiltonian incorporating higher or-der quantum corrections can also be obtained. In order todo the semiclassical analysis, it is convenient to introducenew conjugate variables by a canonical transformation of( c, p ) as b := √ ∆2 c p | p | and v := sgn( p ) | p | / πγℓ p √ ∆ , (20)with the Poisson bracket { b, v } = 1 / ~ . In terms of thesenew variables, the classical Hamiltonian constraint canbe written as C = C grav πG + C φ = − πG r π Kℓ p γ / b | v | + 12 (cid:18) πγℓ p (cid:19) / K | v | p φ . (21)Let us first consider the gravitational part. Since thereare uncountable basis vectors, the natural Gaussian semi-classical states live in the algebraic dual space of somedense set in H gravkin . A semiclassical state (Ψ ( b o ,v o ) | peakedat a point ( b o , v o ) of the gravitational classical phasespace reads:(Ψ ( b o , v o ) | = X v ∈ R e − [( v − v o ) / d ] e i b o ( v − v o ) ( v | , (22)where d is the characteristic “width” of the coherentstate. For practical calculations, we use the shadow ofthe semiclassical state (Ψ ( b o ,v o ) | on the regular latticewith spacing 1 [14], which is given by | Ψ i = X n ∈ Z h e − ( ǫ / n − N ) e − i ( n − N ) b o i | n + λ i , (23)where λ ∈ [0 , ǫ = 1 /d and we choose v o = N + λ ,here N ∈ Z . Since we consider large volumes and latetimes, the relative quantum fluctuations in the volume ofthe universe must be very small. Therefore we have therestrictions: 1 /N ≪ ǫ ≪ b o ≪
1. One can checkthat the state (22) is sharply peaked at ( b o , v o ) and thefluctuations are within specified tolerance [11, 13]. Thesemiclassical state of matter part is given by the standardcoherent state(Ψ ( φ o ,p φ ) | = Z d φ e − [( φ − φ o ) / σ ] e ip φ ( φ − φ o ) / ~ ( φ | , (24)where σ is the width of the Gaussian. Thus the wholesemiclassical state reads (Ψ ( b o , v o ) | N (Ψ ( φ o ,p φ ) | .The task is to use this semiclassical state to calcu-late the expectation value of the Hamiltonian operatorto a certain accuracy. In the calculation of h ˆ C Rgrav i , onegets the expression with the absolute values, which isnot analytical. To overcome the difficulty we separatethe expression into a sum of two terms: one is analyti-cal and hence can be calculated straightforwardly, whilethe other becomes exponentially decayed out. We thusobtain (see the Appendix for details) h ˆ C Rgrav i = − r π Kℓ p γ / | v o | h e − ǫ sin b o + 12 (cid:16) − e − ǫ (cid:17) i + O ( e − N ǫ ) . (25)In the calculation of h ˆ C φ i , one has to calculate the expec-tation value of the operator \ | p | − / . A straightforwardcalculation gives: h \ | p | − / i = (cid:18) πγℓ p (cid:19) / K | v o | h | v o | ǫ + 59 | v o | + O (1 / | v o | ǫ ) i + O (cid:0) e − N ǫ (cid:1) + O (cid:0) e − π /ǫ (cid:1) . (26)Collecting these results we can express the expectationvalue of the total Hamiltonian constraint, up to correc-tions of order 1 / | v o | ǫ and e − π /ǫ , as follows: h ˆ C R i = − πG r π Kℓ p γ / | v o | h e − ǫ sin b o + 12 (cid:16) − e − ǫ (cid:17) i + 12 (cid:18) πγℓ p (cid:19) / K | v o |× (cid:18) p φ + ~ σ (cid:19) (cid:18) | v o | ǫ + 59 | v o | (cid:19) . (27)Hence the classical constraint (21) is reproduced up tosmall quantum corrections. Therefore, the new Hamil-tonian operator is also a viable quantization of the clas-sical expression. For clarity, we will suppress the label o in the following. Using the expectation value of theHamiltonian operator in Eq. (27), we can further ob-tain an effective Hamiltonian with the relevant quantumcorrections of order ǫ , /v ǫ , ~ /σ as H Reff = − πG r π Kℓ p γ / | v | (cid:18) sin b + 12 ǫ (cid:19) + πγℓ p ! / | v | K ρ ~ σ p φ + 12 v ǫ ! , (28)where ρ = (cid:16) πγℓ p (cid:17) (cid:0) Kv (cid:1) p φ is the density of the mat- ter field. Then we obtain the Hamiltonian evolutionequation for v by taking its Poisson bracket with H Reff as ˙ v = { v, H Reff } = 27 K p πG ~ γ | v | sin b cos b. (29)Further, the vanishing of H Reff impliessin b = ρρ ′ c ~ σ p φ + 12 v ǫ ! − ǫ , (30)where ρ ′ c = 3 / (2 πGγ ∆). The modified Friedmann equa-tion can then be derived from Eqs. (29) and (30) as H = (cid:18) ˙ v v (cid:19) = 8 πG ρ ′ c sin b cos b = 8 πG ρ h − ρρ ′ c ~ σ p φ + 1 v ǫ ! + ~ σ p φ + 12 v ǫ − ǫ ρ ′ c ρ i . (31)Recall that, up to the quantum fluctuation of matterfield, the modified Friedmann equation given in [11] reads H = 8 πG ρ (cid:20) − ρρ c (cid:18) v ǫ (cid:19) + 12 v ǫ − ǫ ρ c ρ (cid:21) , (32)where ρ c = 3 / (8 πGγ ∆). Comparing Eq. (31) with(32), we find that the leading order critical energy densityreads ρ ′ crit = 4 ρ crit . We can also express the new modifiedFriedmann equation by using ρ c as H = 8 πG ρ h − ρ ρ c ~ σ p φ + 1 v ǫ ! + ~ σ p φ + 12 v ǫ − ǫ ρ c ρ i . (33) IV. DISCUSSION
Because the properties of the alternative Hamiltonianconstraint operator in Eq. (16) are similar to the onein Eq. (19), the physical Hilbert space, Dirac observ-ables and so on investigated in [3] can also be straight-forwardly obtained. Although there are quantitative dif-ferences between the two versions of quantum dynam-ics, qualitatively they have the same dynamical features.In the leading order approximation, the universe wouldbounce again from the contracting branch to the expand-ing branch when the energy density of scalar field reachesto the critical ρ ′ crit = 4 ρ crit . The quantum bounce impliedby Eq. (31) is shown in Fig. 1.On the other hand, the key results discussed in [11] effective classical v1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000f K1.2K1.1K1.0K0.9K0.8K0.7K0.6K0.5K0.4 FIG. 1: The effective dynamics represented by the observable v | φ are compared to classical trajectories. In this simulation,the parameters were: G = ~ = 1 , p φ = 10 000 , ǫ = 0 . , σ =0 .
01 with initial data v o = 100 000 . can be carried out similarly. It is easy to see from Eq.(31) that, while the term containing the quantum fluc-tuations of matter field is qualitatively negligible, theasymptotic behavior of the quantum geometric fluctu-ations plays a key role for the fate of the universe. Bythe ansatz ǫ = α ( r ) v − r ( φ ) with 0 ≤ r ( φ ) ≤
1, there aregreat possibilities for the expanding universe to undergoa recollape in the future. The recollape can happen pro-vided 0 ≤ r < /ǫ of v cannot increase as v unboundedly as v approachesinfinity. Thus the recollape is in all probability as viewedfrom the parameter space of r ( φ ). For example, in thescenario when r = 0 asymtotically, besides the quan-tum bounce when the matter density ρ increases to thePlanck scale, the universe would also undergo a recol-lapse when ρ decreases to ρ coll ≈ ǫ ρ ′ c /
2. Therefore, thequantum fluctuations again lead to a cyclic universe inthis case. The cyclic universe in this effective scenariois illustrated in Fig. 2. This is an amazing possibilitythat quantum gravity manifests herself in the large scalecosmology. Nevertheless, the condition that the semi-classicality is maintained in the large scale limit has notbeen confirmed. Hence further numerical and analyticinvestigations to the properties of dynamical semiclassi-cal states in the model are still desirable. It should benoted that in some simplified completely solvable modelsof LQC (see [9] and [15]), the dynamical coherent statescould be obtained, where r ( φ ) approaches 1 in the largescale limit. While those treatments lead to the quantumdynamics different from ours, they raise caveats to theinferred re-collapse. r=0 classical v 10 f K2K10123 FIG. 2: The cyclic model is compared with expanding andcontracting classical trajectories. In this simulation, the pa-rameters were: G = ~ = 1 , p φ = 10 000 , ǫ = 0 . , σ = 0 . v o = 100 000. In conclusion, the key features of LQC in this model,that the big bang singularity is replaced by a quantumbounce and there are great possibilities for an expandinguniverse to recollapse, are robust against the quantiza-tion ambiguity which we have considered.
ACKNOWLEDGMENTS
This work is a part of project 10675019 supported byNSFC.
APPENDIX
Let us calculate the expectation value of ˆ C Rgrav , h ˆ C Rgrav i = (Ψ | ˆ C Rgrav | Ψ ih Ψ | Ψ i . (34)Applying the Poisson resummation formula ∞ X n = −∞ g ( n + x ) = ∞ X n = −∞ e πinx Z ∞−∞ g ( y ) e − πiny d y (35)to the norm of the shadow state (23), one obtains h Ψ | Ψ i = X n ∈ Z e − ǫ n = √ πǫ (cid:16) O ( e − π ǫ ) (cid:17) . (36)By Eq. (17), we obtain the action of the gravitationalHamiltonian operator on the shadow state asˆ C Rgrav | Ψ i = X n ∈ Z e − ( ǫ / n − N ) e − i ( n − N ) b o × h f ′ + ( n + λ ) | n + λ + 2 i + f ′ o ( n + λ ) | n + λ i + f ′− ( n + λ ) | n + λ − i i . Then a straightforward calculation shows that(Ψ | ˆ C Rgrav | Ψ i = X n ∈ Z e − ǫ ( n − N ) h e − ǫ cos(2 b o ) f ′ + ( n + λ − − ( f ′ + ( n + λ ) + f ′− ( n + λ )) i = 274 r π Kℓ p γ / h e − ǫ cos(2 b o ) ¯ S − ( ¯ S + ¯ S − ) i , (37)where f ′ + ( n ) = 274 r π Kℓ p γ / ( n + 1) (cid:0) | n + 2 | − | n | (cid:1) ,f ′− ( n ) = 274 r π Kℓ p γ / ( n − (cid:0) | n | − | n − | (cid:1) , and we have set¯ S m := X n ∈ Z e − ǫ ( n − N ) ( n + λ + m ) × (cid:0) | n + λ + m + 1 | − | n + λ + m − | (cid:1) ,S m :=2 X n ∈ Z e − ǫ ( n − N ) ( n + λ + m )=2( N + λ + m ) X n ∈ Z e − ǫ n . We may bound | ¯ S m,k − S m | by an exponentially sup-pressed term: | ¯ S m − S m | = (cid:12)(cid:12)(cid:12)(cid:12) X n ∈ Z e − ǫ ( n − N ) ( n + λ + m ) × (cid:0) | n + λ + m + 1 |− | n + λ + m − | − (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) X − 6, 5227 (2001).[2] A. Ashtekar, T. Powlowski and P. Singh, Phys. Rev. Lett. , 141301 (2006). [3] A. Ashtekar, T. Pawlowski and P. Singh, Phys. Rev. D , 084003 (2006).[4] M. Bojowald, Living Rev. Rel. , 11 (2005). [5] A. Ashtekar and J. Lewandowski, Class. Quantum Grav. , R53 (2004).[6] C. Rovelli, Quantum Gravity , (Cambridge UniversityPress, Cambridge, England, 2004).[7] T. Thiemann, Modern Canonical Quantum General Rel-ativity , (Cambridge University Press, Cambridge, Eng-land, 2007).[8] M. Han, Y. Ma and W. Huang, Int. J. Mod. Phys. D ,1397 (2007).[9] M. Bojowald, Phys. Rev. D , 081301(R) (2007).[10] A. Ashtekar, Loop quantum cosmology: an overview,arXiv:0812.0177. [11] Y. Ding, Y. Ma and J. Yang, Phys. Rev. Lett. ,051301 (2009).[12] A. Ashtekar, M. Bojowald and J. Lewandowski, Adv.Theor. Math. Phys. , 233 (2003).[13] V. Taveras, Phys. Rev. D , 064072 (2008).[14] A. Ashtekar, S. Fairhurst and J. Willis, Class. QuantumGrav. , 1031 (2003); J. Willis, PhD Thesis, (Pennsyl-vania State University, 2004).[15] A. Ashtekar, A. Corichi and P. Singh, Phys. Rev. D77