Quantum-Reduced Loop Gravity: Cosmology Emanuele Alesci∗ Institute for Quantum Gravity, FAU Erlangen-Nu¨rnberg, Staudtstr. 7, D-91058 Erlangen, Germany, EU and Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, ul. Hoz˙a 69, 00-681 Warszawa, Poland, EU Francesco Cianfrani† Instytut Fizyki Teoretycznej, Uniwersytet Wroc lawski, pl. M. Borna 9, 50-204 Wroc law, Poland, EU. We introduce a new framework for loop quantum gravity: mimicking the spinfoam quantization procedureweproposetostudythesymmetricsectorsofthetheoryimposingthereductionweaklyon thefullkinematicalHilbertspaceofthecanonicaltheory. AsafirstapplicationofQuantum-Reduced Loop Gravity we study the inhomogeneous Bianchi I model. The emerging quantum cosmological 3 model represents a simplified arena on which the complete canonical quantization program can be 1 tested. The achievements of this analysis could elucidate the relationship between Loop Quantum 0 Cosmology and thefull theory. 2 n a J I. INTRODUCTION 4 1 The realization of a quantum theory for the gravitationalfield must provide an explanation to the current puzzles ] of General Relativity (GR), i.e. the presence of mathematical singularities. These singularities have been shown c to be unavoidable in some symmetry reduced models describing relevant physical situations, such as the collapse of q - standard matter and the beginning (eventually also the end) of the Universe evolution [1]. Hence, it is demanded to r a quantum formulation of gravity to answer to the questions posed by the unpredictability of GR in these cases. g [ Loop Quantum Gravity (LQG) [2, 3] constitutes the most advanced model which pursues the quantization of geometric degrees of freedom. It is based on a canonical quantization a la Dirac of the holonomy-flux algebra 2 associated with Ashtekar-Barbero variables [4] in the Hilbert space of distributional connections. One first defines a v kinematicalHilbert space in whichthe Gauss constraintis then solved. The resulting basis elements are the so-called 5 spinnetworks: these are labeled by graphs Γ and belong to L2(SU(2)E/SU(2)V), E and V being the total number of 4 2 edges and vertexes of Γ, respectively. The invariance under diffeomorphisms is then implemented by summing over 2 the orbit of the associated operator, which gives the so-called s-knots [5]: these are distributional states representing . the equivalence class of spinnetworks under diffeomorphisms. In the space of s-knots the superHamiltonian operator 1 0 can be regularized [6, 7] and thanks to diff-invariance the regulator can be safely removed leading to an anomaly 3 free quantization of the Dirac algebra. However, particularly in view of the presence of the volume operator [8, 9], 1 the explicit analytical expression for the matrix elements of the superHamiltonian and the properties of the physical : Hilbert space are still elusive. For these reasons other approaches such as the master constraint program [10] or the v i more recent deparametrized system in terms of matter fields [11] have been introduced in the canonical framework. X Cosmology is a natural arena to test the theory and its dynamics due to the high degree of symmetry of the r configurationspace. The cosmologicalimplementation of LQG has been realizedin the frameworkof Loop Quantum a Cosmology (LQC) [12, 13] (see [14–16] for alternative proposals). This is based on the implementation of a minisu- perspace quantization scheme, in which the phase space is reduced on a classicallevel accordingwith the symmetries of the model. Because the Universe is described by a homogeneous (and eventually isotropic) space-time manifold, the resulting configurationspaceis parametrizedby three spatial-independentvariables. These variablesdescribe the connections and the momenta of the reduced model after a gauge-fixing of both the SU(2) gauge symmetry and dif- feomorphismsinvariancehas been performed. As a consequence, the regularizationofthe superHamiltonianoperator canbe accomplishedbyfixinganexternalparameterµ¯ relatedwiththe existenceofanunderlyingquantumgeometry [17](see[18]foracriticaldiscussiononthe regularizationinLQC).Theresultingtheoryisawellestablishedresearch field with several remarkable features and physical consequences, the main ones being a bounce replacing the initial singularity [17, 19–22], the generationof initial conditions for inflation to start [23, 24] and the prediction of peculiar effects on the cosmic microwave background radiation spectrum [25–30] (see also [31–33]). ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] 2 However LQC has not yet been shown to be the cosmological sector of LQG and in order to solve the tension between the regularization procedures of the two theories, new approaches have been recently envisaged in order to provide an alternative definition of the superHamiltonian operator in the full theory (see [34] that bring it closer to the µ¯ scheme of LQC). In this paper, we give a detailed presentation of the procedure introduced in [35], in which we adopt the opposite view-point assuming LQG as the correct theory obtained by quantizing GR and then we look for its cosmological sector imposing a symmetry-reduction at the quantum level. This way we construct a theory in which we first quantize and then reduce instead of first classically reducing and then quantizing as it’s usually done in LQC. This approach is not expected to work only in cosmology, but it can be extended also to other symmetric sectors of the theory. This way, we define a new framework for the analysis of the implications of LQG in relevant (symmetry-reduced) physical cases (Quantum-reduced Loop Gravity). Our cosmologicalquantum model will then be a proper truncation of the full kinematical Hilbert space of LQG. The virtue of our approach mainly consists in the possibility to realize a fundamental description of a cosmological space-time, which fills the gap with the full theory and on which Thiemann’s regularizationprocedure for the superHamiltonian [6] can be applied. The paper is organized as follows: InsectionsIIwequicklyreviewthemaintoolsoftheLQGquantizationofGR,whileinsectionIIIthehomogeneous Bianchimodels arepresentedandthe LQC frameworkis shortlydiscussed. Thenin sectionIV we performa classical analysis and we outline how, by considering a proper inhomogeneous extension, it is possible to retain a certain dependence from spatial coordinates into the reduced variables describing a Bianchi I model. Within this scheme, we get the following set of additional symmetries: i) three independent U(1) gauge transformation,denoted by U(1) i (i=1,2,3),definedinthe 1-dimensionalspacegeneratedby fiducialvectorsω =∂ , andii)reduceddiffeomorphisms, i i which act as 1-dimensional diffeomorphisms along a given fiducial direct i and rigid translations along the other directions j 6=i. We also outline how a similar formulation will be relevant within the BKL conjecture [36] scheme. In section V we discuss the implications of this formulation in a reduced quantization scheme. The elements of the associatedHilbert space are defined overreduced graphs,whose edges are parallel to fiducial vectors and to each edge e //∂ is associateda U(1) groupelement. Within this scheme, a proper quantum implementation can be given i i i to the algebra of reduced holonomy-flux variables. The additional symmetries can then be implemented as in full LQG and they imply the conservationof U(1) quantum numbers along the integral curves of fiducial vectors ∂ and i i that states have to be defined over reduced s-knots. However, we will note that no meaningful expression for the superHamiltonian operator can be given. The failure ofreducedquantizationto accountfor the properdynamicsis the motivationfor consideringadifferent approach,in which a truncationof full LQG is performed. This is done in sectionVI where the truncationis realized such that 1. the elements of the full Hilbert space are defined over reduced graph: this is implemented via a projection and this implies the restriction of arbitrary diffeomorphisms to reduced ones. 2. TheSU(2)gaugegroupisbrokentothe U(1) subgroupsalongeachedgee : thisisrealizedbyimposingweakly i i a gauge-fixing condition on each group element over an edge e . i Aproperquantum-reducedkinematicalHilbertspaceisfoundbymimickingtheanalogousprocedureadoptedinSpin- Foammodelstosolvethesimplicityconstraints[37]. Inparticular,wedevelopprojectedU(1) -networks[38]bywhich i we can embed functionals over the U(1) group into functionals over the SU(2) group. Hence, we impose strongly a i Masterconstraintconditionobtainedbysquaringandsummingallthegauge-fixingconditions. Thisrequirementfixes the relationbetween SU(2)andU(1) quantumnumbers and the resulting projectedU(1) networkssolvethe gauge- i i fixingconditionsweakly. Attheend,thereducedU(1) elementsareobtainedfromfullSU(2)onesbyprojectingover i thestateswithmaximummagneticnumberalongtheinternaldirectioni. TheprojectiontoU(1) elementscanthenbe i applieddirectly to SU(2)-invariantstates. As a resultsome non-trivialintertwinersareinduced betweenU(1) group i elements for different values of the index i. These intertwiners coincide with the projection of the coherent Livine- Speziale intertwiners [39] on the usual intertwiners base. Hence, the U(1) states are not kinematically independent, i but they realize a true three-dimensional vertex structure. This result allows us to implement the superHamiltonian operator according with Thiemann regularization scheme [6]. In fact, by defining states over reduced s-knots it is possible to removethe regulatorandgeta well-definedexpression. Moreover,thanks to the simplificationsdue to the reduced Hilbert space structure (the volume operator is diagonal!), we evaluate in section VII the explicit expression ofthe superHamiltonianmatrixelementsinthe caseofa3-valencevertex. ConcludingremarksfollowinsectionVIII. II. LOOP QUANTUM GRAVITY The kinematical Hilbert space of LQG Hkin is developed by quantizing the holonomy-flux algebra of the corre- sponding classical model, whose phase space is parametrized by Ashtekar-Barbero connections Ai and densitized a 3 triads Ea. In particular, the space of all holonomies is embedded into the space of generic homomorphisms from the i setofallpiecewiseanalyticalpathsofthespatialmanifoldintothetopologicalSU(2)groupX¯ [40]. Onsuchaspacea regular Borel probability measure is induced from the SU(2) Haar one and the kinematical Hilbert space for a graph Γ is the tensor product of L2(X¯,dµ) for each edge e. A basis in this kinematical Hilbert space can be obtained using the Peter-Weyl theorem. Introducing an SU(2) matrix element in representation j, hg|j,α,βi=Dj (g), the generic αβ basis element of Hkin for a given graph Γ with edges e will be of the form: Γ hh |Γ,j ,α ,β i= Dje (h ), (1) e e e e αeβe e e∈Γ O from which we can reconstruct the whole kinematical Hilbert space as Hkin = Hkin. Γ Γ Fluxes E (S) across a surface S are quantized such that a faithful representation of the holonomy-flux algebra is i L realized and they turn out to act as left (right)-invariant vector fields of the SU(2) group. In particular, given a surface S which intersects Γ in a single point P belonging to an edge e such that e = e e and e ∩e = P, the 1 2 1 2 action of Eˆ (S) reads i S Eˆ (S)D(je)(h )=8πγl2 o(e,S)Dje(h )jeτiDje(h ). (2) i e P e1 e2 γ and l being the Immirzi parameter and the Planck length, respectively, and the factor o(e,S) is equal to 0,1,−1 P according to the relative sign of e and the normal to S, while jeτi denotes the SU(2) generator in je-dimensional representation. The set of GR constraints in Ashtekar variables, i.e. the Gauss constraint G, generating SU(2) gauge symme- try, the vector constraint V , generating 3-diffeomorphisms, and the Hamiltonian constraint H, generating time- a reparametrizations,areimplementedinHkin accordingwiththeDiracprescriptionforthequantizationofconstrained systems[41],namelypromotingtheconstraintstooperatorsactingonHkin andlookingforthePhysicalHilbertspace HPhys where the operator equations Gˆ= 0,Vˆ = 0,Hˆ =0 hold. We quickly review how these constraints are imple- a mented in LQG: • G mapsh inh′ =λ h λ−1 , s(e)andt(e) being the initialandfinalpoints ofe,respectively,while λ denotes e e s(e) e t(e) SU(2) group elements and the condition G = 0 is solved implementing a group averaging procedure. To this aim one introduces a projector P to the SU(2)-invariant Hilbert space GHkin, by integrating over the SU(2) G group elements λ and λ for each edge. Basis elements of GHkin are then the so called spinnetworks: s(e) t(e) <h|Γ,{j },{x }>= x ·Dje(h ), (3) e v v e v∈Γe∈Γ Y Y x being the SU(2) invariant intertwiners at the nodes v and they can be seen as maps between the represen- v tations associated with the edges emanating from v and · means index contraction. • The action of finite diffeomorphisms ϕ maps the original holonomy into the one evaluated on the transformed path, h →h : states invariantunder this action canbe found in the dual of Hkin and they arethe so-called e ϕ(e) s-knots [5], namely equivalence class of spinnetworks under diffeomorphisms. • The hamiltonian constraint Hˆ in the gauge and diffeomorphisms invariant Hilbert space can be regularized by adopting the standard prescription given by Thiemann [6] or an alternative recent proposal [7], but at present only the first one has been shown to reproduce the Dirac algebra without anomalies. We resume Thiemann construction because it will be adapted to the cosmologicalmodel of interest in this article. We restrictourattentiontothe so-calledEuclideanpartofthe Hamiltonianconstraint,whichcanbe writtenas H[N]= d3xN(x)H(x)=−2 N Tr(F ∧{A,V}), (4) ZΣ ZΣ V being the volume operator of the full space, while A and F denote the connection 1-form and the curvature 2-form,respectively. The regularizationis basedondefining a triangulationT adaptedto the graphΓ onwhich theoperatoracts. Inparticular,foreachpairoflinkse ande incidentatanodevofΓwechooseasemi-analytic i j arcs a whose end points s ,s are interior points of e and e , respectively, and a ∩Γ={s ,s }. The arc ij ei ej i j ij ei ej s (s ) is the segment of e (e ) from v to s (s ), while s , s and a generate a triangle α :=s ◦a ◦s−1. i j i j ei ej i j ij ij i ij j Three (non-planar) links define a tetrahedra (see Fig. 1). The full triangulation T contains the tetrahedra 4 a jk a ki sk aij FIG. 1: An elementary tetrahedron ∆ ∈ T constructed by adapting it to a graph Γ which underlies a cylindrical function. s j v si Γ obtained by considering all the incident links at a given node and all the possible nodes of the graph Γ. Now we can decompose (4) into the sum of the following term per each tetrahedra ∆ of the triangulation T H[N]= −2 d3xN ǫabc Tr(F {A ,V}) . (5) ab c ∆∈T Z∆ X The connection A and the curvature F are regularized by writing them in terms of holonomy h(m) := h[s] ∈ s SU(2) in a general representation m along the segments s and the loop α , respectively. This yields i ij N(n) Hm[N]:= ǫijkTr h(m)h(m)−1 h(m),V , (6) ∆ N2 αij sk sk m h (cid:8) (cid:9)i the trace being in an arbitrary irreducible representation m: Tr [D] = Tr[D(m)(U)] where D(m) is a matrix m representation of U ∈SU(2), while N2 =Tr [τiτi]= −(2m+1)m(m+1) and h(m) = D(m)(h). As shown in m m [43],theright-handsideofEq. (6)convergestotheHamiltonianconstraint(5)ifthetriangulationissufficiently fine. The expression (6) can finally be promoted to a quantum operator, since the volume and the holonomies have corresponding well-defined operators in Hkin and replacing the Poisson brackets with the commutator {,}→−i[,] we get: ~ Hˆm[N]:=N(n)C(m) ǫijkTr hˆ(m)ˆh(m)−1 hˆ(m),Vˆ . (7) ∆ αij sk sk h (cid:2) (cid:3)i where C(m) = −i . The lattice spacing ǫ of the triangulation T acts as a regularization parameter and it 8πγl2N2 p m canbe removedinasuitable operatortopologyinthe spaceofs-knots,see[6]fordetails. This isessentiallydue to the fact that via a diffeomorphisms it is possible to change ǫ, thus the result of the computation of Hm over ∆ diffeomorphisms-invariantstates does not depend on such a regulator. Remarkably it’s possible to formally write solutions to the quantum Hamiltonian constraint: these are linear combinations of spinnetworks based on graph with “dressed” nodes (see [3]) characterized by “extraordinary links”, i.e. links with three-valent nodes as boundary attached to two collinear links. Due to the particular nature of the “dressed” spinnetworks the procedure described gives an anomaly free quantization of the Dirac algebra. However thesesolutionsareonlyformalbecausetheexplicitexpressionofthematrixelementsofHˆ isverycomplicated[42]and it’s unknown in a closed form because of the presence of the volume operator (for which only numerical calculations are available for arbitrary valence and spins [44]). In the quantum-reduced model that we are going to introduce, insteadthe volumeoperatoris diagonalandthis willallowus toexplicitly compute the matrixelementofHˆ,opening the way to construct the physical quantum states. III. BIANCHI MODELS The early phases of the Universe are described by skipping the assumptions of the FRW model, i.e. isotropy and homogeneity. The relaxing of the former leads to the Bianchi models for the Universe (see [45] for a recent review), which are described by the following line element ds2 =N2(t)dt2−e2α(t)(e2β(t)) ωi⊗ωj , (8) ij 5 α, N and β depending on time coordinates. α determines the total volume, while the matrix β describes local ab ab anisotropies and it can be taken as diagonal and with a vanishing trace, such that two independent components remain. The fiducial 1-forms ωi determine the fiducial metric on the spatial manifold. Fora Bianchimodel,the homogeneityofthe fiducialmetric allowsto define some structureconstantCi asfollows jk dωi =Ci ωj ∧ωk. (9) jk Each model is determined by Ci and the Bianchi type I, II and IX are characterized by Ci = {0,δiǫ1 ,ǫi }, jk jk 1 jk jk respectively. In the following, we will restrict our attention to the so-called class A models for which Ci =0. ij Densitized 3-bein vectors can be determined from the expression of the spatial metric tensor in Eq. (8). However, it is not possible to fix uniquely Ea because one is alwaysfree to perform a rotation in the internal space which does i notmodify the metric tensor. A useful choiceis to setEa parallelto the vectorsω , definedas ωi(ω )=δi, suchthat i i j j it is possible to separate gauge and dynamical degrees of freedom [46]. It is worth noting how this choice implies a gauge fixing of the symmetry under internal rotations. The associated gauge-fixing condition reads [47, 48] χ =ǫ kEaωj. (10) i ij k a At the end, the following expression for densitized 3-bein vectors is inferred Ea =pi(t)ωωa, pi =e2αe−βii, (11) i i ω being the determinant of ωj, while the index i is not summed. In the following, repeated gauge indexes will not b be summed while the Einstein convention will still be applied to the indexes in the tangent space. The associated Ashtekar-Barbero-ImmirziconnectionscanbeinferredbyevaluatingtheextrinsiccurvatureK andthe3-dimensional ab spin connections ω . The extrinsic curvature involves time-derivatives of the 3-metric and Ki =K eib reads ija a ab 1 1 Ki = ∂ h = (α˙ +β˙ )eαeβiiωi, (12) a 2N t ab 2N ii a while the expression of the spin connection ω is given by ija 1 ω = a−1(a a−1a−1Ci +a a−1a−1Cj −a a−1a−1Ck), (13) ija 2 k i j k jk j k i ki k i j ij where ai =eα+βii. The connection Ai is given by the sum of γKi and 1ǫijlω , and it can be written as a a 2 jla γ Ai =c (t)ωi, c = (α˙ +β˙ )+α eαeβii, (14) a i a i N ii i (cid:16) (cid:17) where α depends on the kind of Bianchi model adopted (1 ǫijkω =α ωi). i 2 j,k jka i a P A. Loop Quantum Cosmology TheLQCformulationofhomogeneousBianchimodelsimplementsthequantizationprocedureinthereducedphase space parametrized by {c ,pj} [50]. i The induced symplectic structure leads to the following Poisson brackets 8πG {pi(t),c (t)} = γδi, (15) j PP V j 0 the other vanishing, where V denotes the volume of the fiducial cell on which the spatial integration occurs. 0 The Hilbert space is defined by addressing a polymer-like quantization and it turns out to be the direct product of three Bohr compactifications of the real line, H = L2(R3 ,d~µ), one for each fiducial direction. A generic basis Bohr element is thus the direct product of three quasi-periodic functions, i.e. ψ (c ,c ,c )=⊗ eiµici, (16) µ~ 1 2 3 i ~µ={µ } being real numbers. The operators associated with momenta pi act as follows i pˆiψ (c ,c ,c )=8πγl2µ ψ (c ,c ,c ). (17) µ~ 1 2 3 P i µ~ 1 2 3 6 The scalar constraint is derived by rewriting the one of LQG (6) in terms of the holonomies associated with the connections (14) and of the reduced volume operator V =V p1p2p3. However,the area of the additional plaquette 0 α cannotbesentto0. Thedifferencewithrespecttothefulltheorycanbetracedbacktothelossofdiffeomorphisms ij p symmetry, which was responsible for the restriction to s-knots. This issue has been solved by evaluating the scalar constraint at some fixed non-vanishing values µ¯ µ¯ for the area of the plaquettes α . These values are related with i j ij the scale at which the discretization of the geometry in LQG occurs [17]. The resulting dynamics has been analyzed for Bianchi I, II and IX models [17, 19–22] and the presence of µ¯’s provides a nontrivialevolution for the early phase of the Universe, whose most impressive consequence is the replacement of the initial singularity with a bounce. Therefore, in LQC the parameters µ¯’s contain all the information on the quantum geometry underlying the con- tinuous spatial picture and, at the same time, they are responsible for the departure from the standard Big Bang paradigm. However this construction only mimics the original LQG quantization and even if it is well defined on physical ground there is still a gap between the full theory and this scheme. The formalism that we are going to introduce is insteadobtainedbyadirectreductionfromthe fulltheoryataquantumlevelanditcouldshedlightonthe µ¯ scheme at the base of LQC. IV. INHOMOGENEOUS VARIABLES Our aim is to consider a weaker classical reduction of the full phase-space with respect to the one used in LQC, in suchawaythatareduceddiffeomorphismsinvarianceisretainedandthereisthenmorefreedomintheregularization of the superhamiltonian operator. In this respect, we will consider an inhomogeneous extension of the Bianchi I model. TheBianchiImodeldescribesaspatialmanifoldisomorphictoa3-dimensionalhyperplane. Thestructureconstants Ci vanishandthe 1-formsωi canbetakenasωi =δidxa. ThemetricofBianchiImodelcanbewritteninCartesian jk a coordinates as follows ds2 =N2dt2−a2(t)dx1⊗dx1−a2(t)dx2⊗dx2−a2(t)dx3⊗dx3, (18) I 1 2 3 a (i=1,2,3) being the three scale factors depending on the time variable only. i Let us now consider the following inhomogeneous extension of the line element (18) ds2 =N2(x,t)dt2−a2(t,x)dx1⊗dx1−a2(t,x)dx2⊗dx2−a2(t,x)dx3⊗dx3, (19) I 1 2 3 in which each scale factor a is a function of time and of the spatial coordinates. As soon as the gauge condition i (10) holds the densitized inverse 3-bein vectors read a a a Ea =pi(t,x)δa, pi = 1 2 3, (20) i i a i i.e. they take the same expression as in the relation (11), the only difference being that now reduced variables pi depend also on spatial coordinates. A similar result is obtained for the projected extrinsic curvature, i.e. 1 Ki = a˙ (t,x)δi, (21) a N i a while the spin connections ω for the inhomogeneous model are given by ija ω =a−2a−1δiδb∂ a −a−2a−1δjδb∂ a . (22) ija i j a j b i j i a i b j At this point let us consider two different cases: 1) the reparametrized Bianchi I model and 2) the generalized Kasner solution within a fixed Kasner epoch. InareparametrizedBianchiImodelweassumethateachscalefactorisafunctionoftimeandofthecorresponding Cartesian coordinate xi only, i.e. a =a (t,xi), (23) i i such that ∂ a ∝ δi and the spin connections ω vanish identically. Obviously, the dependence on xi is fictitious b i b ija and it can always be avoided by a diffeomorphisms, so finding the homogeneous Bianchi I model. However, the reparametrizedmodel is endowed with an additional gauge symmetry, which will have a key-role in the development of the quantum theory. 7 The same result concerning the vanishing ofspin connections canalso be obtainedin the limit in whichthe spatial gradientsofthemetriccomponentscanbeneglectedwithrespecttothetime derivatives. Thisapproximationscheme corresponds to the notion of “local homogeneity”, which is implemented when the BKL mechanism is extended to the generic cosmological solution [36, 45]. This is done by considering the generalized Kasner model [51], which describes the behavior of the generic cosmological solution during each Kasner epoch. This model has been realized byconsideringanextensionoftheKasnersolution,inwhichtheKasnerexponentsarefunctionsofspatialcoordinates. Indeed, in general the fiducial vectors do not coincide with the ones of the homogeneous Bianchi I model and they are subjected to a rotationsignaling the transition to a new epoch. Nevertheless, within each epoch, one can neglect the rotation of Kasner axes and take at the leading order the fiducial vectors ωi =δi. a a Therefore, in both cases 1) and 2) the connections retain the same expression as in the homogeneous case, but reduced variables depend on spatial coordinates as follows γ Ai(t,x)=c (t,x)δi, c (t,x)= a˙. (24) a i a i N i The Poissonbrackets between Ai and Ea induce the following Poissonalgebra a i {pi(x,t),c (y,t)}=8πGγδiδ3(x−y), (25) j j the other vanishing. Since we did not impose homogeneity, the SU(2) Gauss constraint G and the super-momentum constraint H do i a not vanish identically. In particular, G reads i G =δa∂ pi =∂ pi, (26) i i a i while the generator of 3-diffeomorphisms takes the following expression D[ξ~]= ξa[H −AiG ]d3x= [ξapi∂ c +(∂ ξi)pic ]d3x, (27) a a i a i i i Z i Z X ξa being arbitrary parameters, while ξi =ξaδi. a V. REDUCED QUANTIZATION FOR INHOMOGENEOUS BIANCHI MODEL Let us now discuss how the quantization of inhomogeneous Bianchi models can be performed in reduced phase space. Inthiscase,oneshoulddefinetheHilbertspaceforfunctionalsofreducedvariablesc ,whoseconjugatevariablesare i pi,andconsiderthesetofreducedconstraints. Inparticular,theSU(2)Gaussconstraintisreplacedbytheconditions (26), which for a given i can be regarded as a U(1) Gauss constraint along the one-dimensional space generated by the vector dual to ωi =δidxa i.e ∂ =δa∂ . We denote the U(1) groupof transformations generatedby G as U(1) . a i i a i i Since {G ,G }=0, the U(1) transformations are all independent from each other. i j i A convenient choice of variables for the loop quantization is to consider the U(1) holonomies for the connections i c along the edges e parallel to ∂ , i.e. i i i redhei =P(eiReicidxi). (28) Hence, we are not dealing with a U(1)3 gauge theory on a 3-dimensional space, since holonomies associated with different U(1) have support on different edges e . What we have is the direct product of three 1-dimensional U(1) i i gauge theories. The Hilbert space can be labeled by reduced graphs Γ, which are cuboidal lattices made by the union of (at most) 6-valent vertices with the ingoing and outgoing edges of the kind e , and it can be defined as the direct product of i the space of square integrable functionals over the U(1) group elements associated with each e , i.e. i i 3 redH= L2(U(1) ,dµ ), (29) i i Oi=1eOi∈Γ dµ being the U(1) Haar measure. i i 8 e3 n3 e′1 n′1 n′2 n′2 e′2 e2 n1 e1 n′3 e′3 FIG. 2: The representation of a vertex in reduced quantization: the quantum numbers n1,n2,n3 are conserved along the directions i=1,2,3, respectively. A generic element is given by taking the direct product of U(1) networks over e and they read i i 3 ψ = ψ , (30) Γ ei Oi=1eOi∈Γ where ψ is a U (1) function, which can be expanded in U(1) irreducible representations as follows ei i i ψ = einiθiψni, (31) ei ei Xni θi being the parameter over the U(1) group, while n denotes the U(1) quantum number. i i i Momenta pi have to be smeared over the surfaces Si dual to e and the associated operators can be inferred by i quantizing the Poissonalgebra (15), so finding pˆl(Si)ψ =8πγl2δl n einiθiψni. (32) ei p i i ei Xni In order to develop the gauge-invariant Hilbert space redHGi in which the conditions (26) are solved, one must inserttheinvariantintertwinersassociatedwiththethreeU(1) groups. TheseintertwinersmapU(1) groupelements i i each other for a fixed value of i. This means that they do not provide us with a real 3-dimensional vertex structure, since they connect only group elements defined over intersecting edges parallel to the same vector field ∂ . i At a single vertex v one can haveat most two U(1) groupelements for a giveni: the ones associatedwith the two i edges e and e′ emanating form v (see Figure 2). i i As soon as ψei and ψe′i are expanded in irreducible representations ψenii and ψen′i′i (31), respectively, the invariant intertwiner selects those representations for which n =n′. i i Therefore, the projection to redHGi implies that the U(1)i quantum numbers are preserved along each integral curve of the vectors ∂ . i A. Diffeomorphisms Theconditions(20)and(24)implyapartialgaugefixingofthediffeomorphismssymmetry. Infact,underageneric 3-diffeomorphisms connections and momenta transform as follows δ Ai =ξb∂ Ai +∂ ξbAi, δ Ea =ξb∂ Ea−∂ ξaEb. (33) ξ a b a a b ξ i b i b i Starting from the expression (14), one gets δ Ai =ξb∂ c δi +ξbc ∂ δi +∂ ξbδic =ξb∂ c δi +∂ ξic . (34) ξ a b i a i b a a b i b i a a i 9 It is worth noting that for arbitrary ξa the connection cannot be written as in (14). This feature signals that by choosing connections as in (24) we are actually performing a partial gauge-fixing of the diffeomorphisms group. The same result is obtained for Ea. However, there is a residual set of admissible transformations which preserve the i conditions (14) and (11) and they are those ones for which ∂ ξi ∝δi →ξi =ξi(xi). (35) a a As soon as the condition above holds, each ξi is the infinitesimal parameter of an arbitrary translation along the direction i and a rigid translationalong other directions. We denote this transformations as reduced diffeomorphisms ϕ˜ . We are going to show how the constraint (27) implies the invariance under reduced diffeomorphisms. ξ In reduced phase-space, the constraint (27) acts on a reduced holonomy (28) as follows Dˆ[ξ~]redhei =8πγlP2 redhei(0,s′)(ξb∂bci+(∂iξi)ci)redhei(s′,1)dxi(s′), (36) Zei e (0,s′) and e (s′,1) being the edges from s=0 to s=s′ and from s=s′ to s=1, respectively. i i Thetransformation(36)hastobecomparedwiththechanginginducedbyareduceddiffeomorphismsϕ˜ :xa(s)→ ξ x′a(s)=xa(s)+ξa under the condition (35). A diffeomorphisms ϕ maps an edge e into one which is generically not i of the reduced class. In fact the tangent vector at the leading order is given by the following expression dx′a dxa dxb = +∂ ξa ∝δa+∂ ξaδb =δa+∂ ξjδaδb. (37) ds ds b ds i b i i b j i Thesecondtermontherightsidegetscontributionsalsofromthefiducialvectors∂ withj 6=i,suchthatthetangent j vector of ϕ(e ) is not proportional to ∂ . However, if one considers the reduced class of transformations (35), these i i additional contributions vanish and the tangent vector of ϕ˜(e ) is parallel to ∂ . Hence, reduced diffeomorphisms ϕ˜ i i map reduced edges e each others. i The holonomy along ϕ˜ is thus given by ξ hϕ˜ξ(ei) =P(eRci(x′)δaidx′a), (38) and by computing the integrand one gets c (x′)δidx′a =c (x)δidxa+ξb∂ c δidxa+c (x)∂ ξidxa. (39) i a i a b i a i a From the expression above and by considering that dxa =δadxi(s) the following relation follows i hϕ˜ξ(ei)−hei =P(eRci(x′)δaidx′a)−P(eRci(x)δaidxa)= redhei(0,s′)(ξb∂bci+(∂iξi)ci)redhei(s′,1)dxi(s′). (40) Zei whichcoincideswiththeexpression(36). Therefore,thereduceddiffeomorphismsϕ˜(35)mapreducedholonomiesinto reduced holonomies and they are associated with the action of the relic diffeomorphisms constraint (27) in reduced phase-space. This residual symmetry can be used to define reduced knot classes as in the full theory. B. Dynamics The superHamiltonian operator in reduced-phase space takes the following form p1p2 p2p3 p3p1 H[N]= d3xN c c + c c + c c , (41) "s p3 1 2 s p1 2 3 s p2 3 1# Z and the quantization of this expression requires to i)give a meaning to the operator 1/ pi, ii) replace c with i some expression containing holonomies. These are the standard issues one encounters in LQG, which are solved by p quantizingtheexpression(6). Therefore,thequantizationofthesuperHamiltonianoperatorinthereducedmodelcan be realized by implementing in the reduced Hilbert space the procedure adopted in the full theory. This can be done formally by replacing SU(2) group elements with U(1) ones and by defining a cubulation of the spatial manifold, i such that the loop α is a rectangle with edges along fiducial vectors. Unfortunately, the resulting expression for ij the superHamiltonian operator regularized `a la Thiemann is not defined in redHGi. This is due to the fact that the operator h increases (decreases) the U(1) (U(1) ) quantum number associated with the segment s (s ). As a αij i j i j consequence, the U(1) quantum number is not conserved along the edge e and the U(1) symmetry is broken (see i i i figure 3). Therefore, it cannot be given a proper definition of the superHamiltonian operator in reduced quantization. This is due to the lack of a real 3-dimensional vertex structure, which instead can be inferred starting from the full LQG theory. 10 6 6 p p n n n n+1 n - - - - 6 6 - 1 1 p+1 p p FIG.3: Theaction oftheoperator associated withthecurvaturechanges U(1)i quantumnumberssuchthat itmapsthestate out of thegauge-invariant Hilbert space (we did not drawn theedges along thethird direction). VI. COSMOLOGICAL LQG Letus nowdiscusshowto realizeinthe SU(2) kinematicalHilbertspaceofLQGHkin the conditions(20)and(24) via a reduction from SU(2) to U(1) group elements. Atfirst,weimposetherestrictiontoedgese paralleltofiducialvectors∂ andwediscussthefateofdiffeomorphisms i i invariance. Then, we will deal with the restriction from SU(2) to U(1) group elements and with the relic features of the original SU(2) invariance. A. Quantum Diff-Constraint The restrictionto cylindricalfunctionals overedges e implies the kindof restrictiononthe diffeomorphisms trans- i formations which we discussed in section VA. We can implement this feature on a quantum level via the action of a projectorP ontothespaceH madeofholonomiesalongreducedgraphs(edgese adaptedtothe ω ). Thisprojector P i i P acting on Hkin is then nonvanishing only for holonomies along edges e . i Letusconsideragenericdiffeomorphismsϕ ,whoseassociatedoperatorU(ϕ )inthespaceofcylindricalfunctional ξ ξ acts on a generic holonomy h along an edge e as follows e Uˆ(ϕ )h =h . (42) ξ e ϕξ(e) The projection of U(ϕ ) in the graph-reduced Hilbert space H is given by ξ P redUˆ(ϕ )=PUˆ(ϕ )P, (43) ξ ξ where Ph =h if e=e for some i,otherwise it vanishes. The actionofredU(ϕ) ona graph-reducedholonomyh e e i ei reads then redUˆ(ϕ)h =PUˆ(ϕ )Ph =PUˆ(ϕ )h =Ph . (44) ei ξ ei ξ ei ϕξ(ei) As we pointed out in section VA, ϕ (e ) is parallel to ω if ϕ is a reduced diffeomorphisms ϕ˜. Hence, the relation ξ i i (44) is nonvanishing only if ϕ=ϕ˜ and one finds redUˆ(ϕ)=Uˆ(ϕ˜). (45) Therefore, in H the relic diffeomorphisms are reduced ones. The development of knot classes with respect to P reduced diffeomorphisms will allow us to regularizethe expressionof the super-Hamiltonianoperator`a la Thiemann.