aa r X i v : . [ g r- q c ] J a n Lattice classical cosmology
Jakub Bilski ∗ and Anzhong Wang
1, 2, † Institute for Theoretical Physics and Cosmology,Zhejiang University of Technology, 310023 Hangzhou, China GCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USA
This article presents the lattice-smeared gravity phase space reduction defined by the cosmologi-cal gauge-fixing conditions. These conditions are specified to reduce the SU (2) symmetry and thespatial diffeomorphism invariance of the loop quantum gravity’s Fock space, known as the spinnetwork. The internal symmetry is fixed to the Abelian case and the diffeomorphism invariance issimultaneously reduced to spatial translations. By rigorously satisfying the reduced gauge symme-tries, the resulting cosmological model is precisely the limit of the gravitational theory expressed interms of holonomies and fluxes. Moreover, the obtained Hamiltonian constraint is finite (withoutany cut-off introduction) and exact (without the holonomy expansion around short links). Further-more, it has the form of the sum over elementary cuboidal cells. Finally, the simple structure of itshomogeneities and anisotropies should allow to describe the quantum cosmological evolution of theUniverse in terms of transition amplitudes, instead of using perturbative approximations.
I. MOTIVATION
The idea to formulate a cosmological theory on a lat-tice was inspired by the recent results in the develop-ment of an analogous approach to gravity [1–4]. Con-sequently, the model described in terms of holonomiesof the Ashtekar-Barbero connections and fluxes of densi-tized dreibeins [5, 6] was constructed. These are the vari-ables, with which loop quantum gravity (LQG) is formu-lated [7, 8]. However, the theory presented in this articleis not formulated by assuming a simplified form of theAshtekar variables, which describe cosmological degreesof freedom. It is neither constructed by applying thisform to the LQG’s variables.The model, called loop quantum cosmology [9, 10],based on this latter construction, requires an UV cut-off and the holonomies expansion around a length of thelattice’ links. Moreover, this cosmological theory is notdefined as a phase space reduction of LQG.The strategy presented in this article is more rigorous.It concerns the standard lattice gauge theory approach[11] and the phase space reduction of the lattice-smearedcanonical variables [12], which are the classical variablesbefore the canonical quantization in the LQG procedure[7, 8]. As a direct result, the system with a finite Hamil-tonian (without a cut-off or geometrical expansion) isobtained. Moreover, the canonical quantization of thislattice-derived phase space-reduced scalar constraint andthe so-obtained canonical fields directly corresponds tothe quantum lattice gravity [4]. In particular, the vari-ables of the lattice cosmological model presented in thisarticle satisfy the equivalently reduced gauge symmetriesof the LQG’s Fock space [7, 13, 14], known as the spinnetwork. ∗ [email protected] † [email protected]
II. LATTICE GRAVITY
The ADM formulation of gravitation [15] imposes a par-ticular gauge on the time reparametrization anisotropy.The unique direction of the spatial evolution allows toconstruct a quantity identified as the momentum in thestandard canonical relation to the gravitational field rep-resentative. Considering a Yang-Mills-like [16] formula-tion of gravitation, the field is represented by the su (2) -valued connection A a := A ia τ i . The related momentumis given by the dentitized dreiben E a := E ai τ i . The gaugegenerators are normalized as [ τ j , τ k ] = ǫ ijk τ i . (1)This ensures the reality of the canonical fields pair,known as the Ashtekar variables. Their Poisson alge-bra regarding the original ADM variables [15] q ab and p ab reads (cid:8) A ia ( x ) , E bj ( y ) (cid:9) q,p = γκ δ ba δ ij δ ( x − y ) , (2)where the normalization of the Einstein constant is κ :=16 πG/c and γ denotes the Immirzi parameter. Thisallows to define the symplectic structure of the spatialmanifold concerning the A a and E a fields. Consequently,the Poisson brackets become { X, Y } := γκ { X, Y } A,E .The gravitational theory formulation, equivalent to theEinstein-Hilbert action [17] expressed in the Ashtekarvariables, is given by the Holst action [18]. The Leg-endre transform leads to the total Hamiltonian withoutdynamical contribution. This Hamiltonian consists ofthree first-class constraints that generate the su (2) , spa-tial diffeomorphisms, and time diffeomorphism symme-tries. They are called the Gauss, diffeomorphism (orvector), and Hamiltonian (or scalar) constraints. Thelast element is H = R d x N ( x ) H ( x ) , where N is the La-grange multiplier called the lapse function. The solutionof this constraint reveals the dynamics of the system.Therefore, one can first derive the Gauss and vector con-straints. This leads to the physical subspace of the phasespace and it is the constraints-solving order in the lat-tice regularization procedure in the LQG construction.In this case, one defines the su (2) - and diffeomorphisms-invariant Fock space [7, 13, 14] founded on the lattice, onwhich the scalar constraint operator acts. This operatoris formulated by regularizing the Hamiltonian constraintdensity H := 1 κ | E | − (cid:0) F iab − ( γ + 1) ǫ ilm K la K mb (cid:1) ǫ ijk E aj E bk , (3)where E is the densitized dreibein determinant, F ab is thecurvature of A a , and K a denotes the dreibein-contractedextrinsic curvature.The regularization procedure depends on a particularchoice of the graph on which the lattice is founded. Inthe established formulation of LQG [7, 8], one assumesa tetrahedral tessellation of the spatial manifold. Thischoice, however, does not reflect the Cartesian symmetryof the differential form in the following transition fromthe continuous into a discrete framework, Z d x f ( x ) = lim ¯ l → X R c h v i f (cid:0) R c h v i (cid:1) ¯ l h v i . (4)Here, ¯ l h v i := L ¯ ε h v i is the elementary cell volume,where the isotropic regulator is given by the relation ¯ ε h v i := ǫ ( p )( q )( r ) ε ( p ) h v i ε ( q ) h v i ε ( r ) h v i . The quantity L isan auxiliary length scale and ε ( p ) h v i denotes the regu-larization parameter assigned to a particular link. Theindices written in brackets are not summed. The sum-mation in (4) goes over elementary cells R c h v i , the edgesof which, ∂R c h v i ∈ Γ , are identified with the Γ graph’sedges, where c h v i := v + l x / l y / l z / denotes the R ’scentroid.The mentioned transition problem, indicated in [2], ledto a different selection of the regularization framework.This problem (as well as several other constructionalshortcomings of LQG) was resolved by choosing the lat-tice over the Γ graph defined on the edges of the quadri-laterally hexahedral honeycomb [2, 4]. Consequently, thestarting point toward the formulation of the cosmologicalframework is the relativistic gravitational theory on thequadrilaterally hexahedral lattice [4, 19].In this model, likewise in the former theory in [7, 8],the connection’s degrees of freedom are smeared alongthe graph’s edges and are represented by holonomies h − p h v i [ A ] := P exp (cid:18) − Z l p h v i ds ˙ ℓ a ( s ) A a (cid:0) ℓ ( s ) (cid:1)(cid:19) . (5)The densitized dreibeins are expressed in terms of thefluxes through the surfaces orthogonal to their spatialdirections. For the hexahedral cell’s face F p h v i , orientedaccording to the right-hand rule-related multiplication l q ∧ l r , the flux is defined as f (cid:0) F p h v i (cid:1) [ E ] := ǫ pqr Z l q h v i dy Z l r h v i dz ǫ bcd E b ∂ y x c ∂ z x d . (6) The holonomies-and-fluxes-dressed graph is going to benamed the lattice and its edges l p h v i emanated frompoints v (called nodes), are going to be named links.The lattice smearing of the canonical variables adjuststhe spatial directions x a into links l p and results in A a ( x ) → A ( l p ) := A p (cid:2) h ( p ) (cid:3) ,E a ( x ) → E ( F p ) := E p (cid:2) f ( p ) (cid:3) . (7)This map is constructed by the expansion of definitions(5), (6) and inversion of the obtained results. How-ever, to preserve the symmetry of cells R c h v i , the point-located fields A a ( x ) and E a ( x ) must be equally dis-tributed around the c h v i centroid. The simplest methodis to integrate their distribution densities along the spa-tial directions that are not present in (5) and (6), ¯ A p (cid:0) R c h v i (cid:1) := Z F ( p ) h v i A p (cid:2) h ( p ) (cid:3) = ε ( p ) h v i L ¯ ε h v i Z F ( p ) h v i A p (cid:2) h ( p ) (cid:3) , ¯ E p (cid:0) R c h v i (cid:1) := Z l ( p ) h v i E p (cid:2) f ( p ) (cid:3) = 1 L ε ( p ) h v i Z l ( p ) h v i E p (cid:2) f ( p ) (cid:3) . (8)These expressions formally assign the canonical variablesto the lattice in the manner both symmetric with itsstructure and consistent with the transition in (4). III. ASHTEKAR-BARBERO CONNECTION ASA CANDIDATE FOR AN OBSERVABLE
Considering a quantum theory formulated in terms ofthe SU (2) holonomies of the su (2) Ashtekar connec-tions, these fields reflect the group-representation duality.Moreover, the Hamiltonian structure in (3) is similar toits Yang-Mills analog [16]. Therefore, it is natural to ap-ply the same quantization strategy to A p and h p as theone selected for the Standard Model fields.The Maurer-Cartan equation [20, 21] for the internalspace of the Ashtekar-Barbero connection reads ∂ ( p ) A ( p ) = − (cid:2) A ( p ) , A ( p ) (cid:3) , (9) cf. [1]. The Lie brackets for the su (2) generators aregiven in (1). By comparing the normalization of thegenerators with the structure constants (proportional tothe Levi-Civita tensor coefficients) and by applying theWigner theorem [22], the SU (2) group element (a holon-omy) can be expanded into a power series of the param-eters θ i ( p ) = R L ε ( p ) A i / . This expansion around the unitelement of the group [1] leads to h ∓ p ) h v i [ θ ] = ∓ θ i ( p ) h v i τ i + 2 θ i ( p ) h v i θ j ( p ) h v i τ i τ j ± ǫ ijk θ j ( p ) h v i θ k ( p ) h v i τ i + O ( θ ) . (10)This quadratic polynomial contains all the relevant termsrelating the Lie group with its representation for a quan-tum gauge field [1, 22]. Consequently, the quantities oforder O ( θ ) and higher are going to be neglected. IV. PHASE SPACE REDUCTION
The internal space of the Ashtekar variables is Euclidean.The same geometry is often considered in cosmology. Themost general Euclidean geometry model, the homoge-neous and isotropic limit of which has the Friedmann-Lemaître-Robertson-Walker metric, is the Bianchi I cos-mology. It can be constructed by imposing constraintson the form of the lattice gravity variables. Its sim-plest realization is using only the diagonal A ia and E ai coefficients [10]. This assumption can be formally de-rived as a solution of the phase space reduction of thelattice-assigned variables in (7), cf. [12]. In this con-struction, the gauge-fixing conditions are imposed glob-ally. The su (2) invariance is broken into the Abelian caseand the spatial diffeomorphism invariance is analogouslyrestricted into linear translations. This latter reductionis transferred into the piecewise linear structure of thelattice gravity in [3, 4, 19]. The quadrilaterally hexahe-dral form of the lattice becomes rectangularly hexahedral(cuboidal).The advantages of this formal phase space reduction[12] are revealed in quantization. Dirac brackets, re-stricted by the gauge conditions to the reduced phasespace variables, turn into Poisson brackets. This ensuresthat the reduced quantum theory remains the gauge-fixedvariant of the general model and no information is lost.Concerning this analysis, the semiclassical limit of the so-constructed lattice cosmology equals the reduced limit ofthe related quantum gravity. It is worth emphasizingthat to preserve this quality, the global reduction mustrestrict also the states supporting the quantum evolutionand any perturbative analysis.The reduced phase space fields are indeed diagonal [12].It is useful to express these diagonal coefficients of theAshtekar variables in the following normalization [9], A ( l p ) → c i (cid:0) l ( i ) (cid:1) [ h ] τ i := L ε ( i ) h v i A ip (cid:2) h ( p ) (cid:3) e p ( i ) τ i , E ( F p ) → p i (cid:0) F ( i ) (cid:1) [ f ] τ i := ε ( i ) h v i L ¯ ε h v i E pi (cid:2) f ( p ) (cid:3) e ( i ) p √ q τ i . (11)The diagonal constant unit matrices e pi and e ip are theprojector from the internal space basis into lattice di-rections and its reciprocal, respectively. The squareroot of the diagonal constant metric tensor determinanthas the standard frame fields representation, √ q := ǫ ijk ǫ pqr e ip e jq e kr / .The linear maps in (11) between the internal and ex-ternal spaces are restricted by one more condition. Theysatisfy the basis-mixing gauge field reduction theoremin [1]. Consequently, the diagonal form of the variables c i (cid:0) l ( i ) (cid:1) [ h ] and p i (cid:0) F ( i ) (cid:1) [ f ] has to be preserved also locallyto ensure the local vanishing of their curvatures. By theviolation of this law, the variables would be no longer Liealgebra representations of the holonomy group.Finally, by expanding the properly selected combina-tions ( cf. [2, 4]) of the lattice quantities in (5) and (6), re-stricted to the reduced variables c i ( l ( i ) )[ h ] and p i ( F ( i ) )[ f ] , respectively, one obtains h i h v i [ c ] − h − i h v i [ c ] = 2 c i (cid:0) l ( i ) (cid:1) τ ( i ) ,f (cid:0) F i h v i (cid:1) [ p ] = p i (cid:0) F ( i ) (cid:1) τ ( i ) . (12)This is the canonical pair of the reduced variables latticerepresentations. In the upper equation, the higher-orderterms in the exponential map in (10) were omitted. Letus emphasize that in the case of the constant connectionobtained in (11), the parameters in (10) simplify to θ i ( p ) = 12 L ε ( p ) A i ( p ) → L ε ( i ) c i (cid:0) l ( i ) (cid:1) [ h ] . (13)In the lower formula in (12), the probability distributionwas integrated with the trapezoidal rule for the constantdensitized dreibein along a linear path.It is worth to recall that the Poisson brackets of thecontinuous analogs of the reduced variables (correspond-ing to (2)) are { c i , p j } = γκ δ ji [9]. These, however, arenot the variables related to the lattice phase space reduc-tion. The Poisson algebra of the canonical pair in (11)is (cid:8) c i ( l h v i )[ h ] , p j ( F h v ′ i )[ f ] (cid:9) = tr (cid:18) τ ( i ) n f (cid:0) F j h v ′ i (cid:1) [ p ] , h i h v i [ c ] − h − i h v i [ c ] o(cid:19) = γκ (cid:0) h ( i ) h v i [ c ] − h − i ) h v i [ c ] (cid:1) δ l h v i l h v ′ i δ ji = γκ c ( i ) (cid:0) l ( i ) (cid:1) τ ( i ) δ l h v i l h v ′ i δ ji . (14)As expected, this result is the Abelian limit of the equiva-lent Poisson brackets of the gravitational variables [3, 4]. V. LATTICE COSMOLOGY
The cosmological fields c i (cid:0) l ( i ) (cid:1) [ h ] and p i ( F ( i ) )[ f ] are link-and face-defined objects, respectively. They do not re-flect the cuboidal lattice symmetry from the point-relatedperspective of the Ashtekar variables. This problem canbe easily resolved in the same way as in the general the-ory [3] — see formulas (8). Consequently, we define thecell-symmetrized probability distributions of the reducedfields ¯ c i (cid:0) R c h v i (cid:1) [ h ] := L ε ( i ) h v i Z F ( p ) h v i A ip (cid:2) h ( p ) (cid:3) e p ( i ) τ ( i ) , ¯ p i (cid:0) R c h v i (cid:1) [ f ] := ε ( i ) h v i L ¯ ε h v i Z l ( p ) h v i E pi (cid:2) f ( p ) (cid:3) e ( i ) p √ q τ ( i ) . (15)Analogously to the derivation of the lower equation in(12), the densities are integrated with the trapezoidalrule, resulting in ¯ c i (cid:0) R c h v i (cid:1) [ h ] = 14 X v ∈ F ( i ) h v i tr (cid:16) τ ( i ) (cid:0) h − i h v i [ c ] − h i h v i [ c ] (cid:1)(cid:17) , ¯ p i (cid:0) R c h v i (cid:1) [ f ] = 12 X v ∈ l ( i ) h v i f (cid:0) F i h v i (cid:1) [ p ] . (16)These objects are the cell-related distributions of thephase space-reduced Ashtekar variables. They are thefundamental expressions of canonical fields in the latticecosmology.Eventually, the Hamiltonian constraint density corre-sponding to a single cell of Γ , expressed in terms of thefields derived in (16), equals H (cid:0) R c h v i (cid:1) = − γ κ ¯ p i ( R ) ¯ p j ( R )( | ¯ p ( R ) ¯ p ( R ) ¯ p ( R ) | ) × (cid:0) ¯ c i ( R ) ¯ c j ( R ) − ¯ c j ( R ) ¯ c i ( R ) (cid:1) . (17)The Gauss and spatial diffeomorphism constraints, writ-ten in terms of constant variables, vanish identically.This confirms the correctness of the imposition of thegauge-fixing conditions [12].Then, the total lattice scalar constraint is given by theelementary cell summation H (Γ) = X R c h v i N (cid:0) R c h v i (cid:1) H (cid:0) R c h v i (cid:1) . (18) This result is finite, hence it does not require any ad-ditional regularization or a cut-off. It expresses the en-tire kinematics of the lattice cosmology. The isotropicinhomogeneities are represented by different lapse func-tion elements N (cid:0) R c h v i (cid:1) . The anisotropies are much morestrictly constrained by the cuboidal structure of the Γ graph, i.e. the face-adjacent cells are restricted to haveparallel nodes-adjacent links. ACKNOWLEDGMENTS
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