Regularization of the cosmological sector of loop quantum gravity with bosonic matter and the related problems with general covariance of quantum corrections
aa r X i v : . [ g r- q c ] J a n Regularization of the cosmological sector of loop quantum gravity with bosonic matterand the related problems with the general covariance of quantum corrections
Jakub Bilski ∗ Institute for Theoretical Physics and Cosmology,Zhejiang University of Technology, 310023 Hangzhou, China
This article concerns the problems regarding different lattice regularization techniques for the matterfields Hamiltonian constraints defined in the framework of loop quantum gravity. The analysis isformulated in the phase space-reduced cosmological model of the hypothetical theory of canonicalquantum general relativity. This article explains why a different than links-related lattice smearingof fields leads to a local violation of general covariance. This happens by assuming, for instance, thenodes-related smearing. Therefore, this problem occurs in the case of any polymer-like scalar fieldquantization method by breaking the background independence of the semiclassical predictions. Inconsequence, the diffeomorphism symmetry that depends on a links distribution is broken locallyat the level of generally relativistic corrections. Moreover, by using the phase space-reduced gaugefixing technique to analyze this issue, the results are general and they concern any coupling with thelinks-regularized gravitational degrees of freedom in loop quantum gravity. Therefore, they lead tothe following no-go conclusion. Any lattice smearing of matter, not defined by using the geometricaldistribution specified by the links-fluxes duality, violates the general principle of relativity.
I. INTRODUCTION
Nonperturbative quantum gravity is a theoretical branch of physics, which assumes that the unification of the principlesof general relativity (GR) with a possibly quantum nature of the gravitational field is probable. The quantum theorythat aims to capture the restrictions of the strong equivalence principle is loop quantum gravity (LQG) [1, 2]. In thismodel, the gravitational field is described by the Ashtekar variables [3]. Its kinematical structure is similar to one ofthe SU (2)
Yang-Mills field [4]. By following the example of the regularization in quantum chromodynamics [5], LQGis formulated in terms of su (2) -invariant variables on a lattice [3]. This theory, however, does not lead to the generallycovariant description of experiments and observations. The general postulate of relativity [6] in LQG concerns onlythe gravitational field, which is described equivalently in all coordinate systems. However, a procedure providingmeasurable predictions, which should not depend on any character of an observer’s frame, is not uniquely determinedeven only for this field. This equivalent description in LQG is known as the strong formulation of systems equivalence(SE) for gravitation (the equivalence principle without its standard Einstein’s version for matter fields) [7, 8]. Theobserver-independent predictions are known as the background independence (BI) of the related observations (thelaw of general covariance) [6, 9].The fundamental consistency of the theory requires a unique procedure of the quantum corrections derivation inLQG, which will guarantee the BI of this framework. Almost any research toward possibly detectable predictionsof quantum gravity requires also an extension of SE and BI to matter. Only pure gravitational experiments orobservations would not need this extension. Therefore, physical studies on gravitational waves, which involve theirpossible quantum nature could be described by LQG. However, any cosmological research of the interactions betweenquantum spacetime and matter requires a more general framework.The first unified construction of GR and quantum field theories of matter interactions (QFT), formulated as anextension of LQG, is known as canonical quantum general relativity (CQGR) [2]. In this article, CQGR is goingto have a more general meaning. We use this name for any hypothetical theory, which generalizes the interactionsbetween the Newtonian gravitational field and QFT into a consistent quantum model that satisfies SE and BI, andwhich is quantized in the canonical procedure [10].Although, any complete formulation of CQGR does not exist, one can investigate a coupling between LQG anda simple model of bosonic matter field that is SE and BI. By assuming that both the gravitational and matterfields satisfy the general postulates of relativity and by constructing the quantization of these fields which preservesthese postulates, one can formulate the following hypothesis. The postulated model can correctly describe physicalinteractions between gravity and bosonic matter, by satisfying the physical constructional requirements. Therefore, itcould be worth to study the phenomenological predictions also of the toy model that would be a symmetry-preserving ∗ [email protected] simplification of a more general theory. However, by finding agreements between the predictions and the relatedmeasurements, one could not claim the truthfulness or universality of this model yet. This would only increase theprobability that the toy model is the simplified version of a correct fundamental theory. In consequence, it would beworth to look for a theory, the particular limit of which is this toy model that was found to be in an agreement withdata.Conversely, one can also construct an inconsistent toy model in purpose, for instance, because in this way it wouldbe simpler. The result, however, could be used only for theoretical analysis; for instance to verify the behavior ofa particular mathematical procedure. Using this inconsistent model to study phenomenology would not have anyphysical value. Moreover, if one would do it anyhow and find any agreement between the predictions and data, thiswould tell nothing about the nature of the only apparently ‘predicted’ phenomenon.By following the preceding methodology of the qualitative evaluation of toy models, several proposed cosmologicaltheories based on the framework of LQG are going to be tested as the potential candidates to study phenomenology.This directly involves the consistency verification of the restrictions and methods regarding whether the formulationof quantum gravitational fields and quantum matter fields is SE and BI. Both types of these fields must satisfy bothconditions and be independently quantizable.To demonstrate that a model is inconsistent, is enough to indicate a single violation of the methodological as-sumptions. Hence, the investigation of the less strict and recurrent element of quantum cosmological models is goingto be studied. To find problems regarding this element is more probable. Therefore, the matter sector of differentapproaches to quantum cosmology and the BI of its semiclassical limit is analyzed in this article. The general inves-tigation of this issue would require detailed studies for each model separately. To avoid this complication, a maximalsimplification of these theories is going to be assumed. It what follows it will be enough to consider the formalism ofanisotropic loop quantum cosmology (LQC) [11, 12]. The SE gravitational degrees of freedom description is going tobe coupled with the SE framework of the scalar field [13, 14]. This is the simplest system, which could be consideredas a cosmological phase space reduction [15] of CQGR. As the reader will see, the violation of the general covariancein this system will be found. Therefore, it is worth to introduce one more reference matter field, the quantization ofwhich is more similar to the one of gravity. This is the vector field [13, 16], and its simplest, Abelian version will beconsidered. Finally, the formalism linking the isotropic and anisotropic cosmology with the vector matter field willbe derived to demonstrate how BI, which is violated in the scalar field case, can be preserved in the theory. Theessential part of this analysis is going to be the investigation of the so-called inverse volume corrections in LQG.In this article the standard framework of canonical LQG is considered. The decomposition of a manifold M that represents spacetime is introduced by the foliation into Cauchy hypersurfaces Σ t [17, 18]. The tetrad formalismwith an internal SU (2) symmetry is applied and the time gauge is assumed. In this article, the gravitational couplingconstant is defined as κ = 16 πG , where the speed of light is normalized to c = 1 . The fundamental constant for thecanonical DeWitt quantization [10] is defined as ¯ k := γ ~ κ = 8 πγl P , where γ and l P are the real Immirzi parameterand the Planck length, respectively. The repeated indices written in ( ) brackets are not summed; all the other indicesfollow the Einstein summation convention.The article is organized as follows. In Sec. II the lattice regularization of bosonic fields is introduced. In Sec. IIIthe phase space-reduced cosmological framework of CQGR is defined. Then the verification of general covariance isdone in Sec. IV. The conclusions of the article are that the node-related regularization of the matter sector leads tobackground-dependent predictions. The general postulate of relativity can be preserved by considering only the linkssmearing of all the propagating degrees of freedom in CQGR. II. REGULARIZATIONII.1. Lattice Yang-Mills theory
Two examples of the Yang-Mills field [4] are important concerning the cosmological analysis in this article. Thesimplest representative of the matter vector field is the Abelian Yang-Mills field. Its non-Abelian variant that satisfythe su (2) algebra describes gravity [3].Let l : [0 , → Σ t be a smooth path parameterized by s ∈ [0 , and located inside the constant time surface Σ t constructed by the ADM method [17, 18]. One can define an embedding of l ( s ) in Σ t and introduce a parameter ε such that l ε ( s ) := l ( εs ) . In this article ε has dimension of a length, its maximal value is restricted by the subsequentdefinition and the minimal one by the choice of the so-called shadow states [19] (the coherent states in LQC); thisleads to the inequality > ε > | γ | l P that is implicitly expressed in some length scale units i . The parallel transportequation for a vector u ( s ) along l ε ( s ) reads ∂ ˙ l ε u ( s ) = dds u ( s ) + A (cid:0) ˙ l ε ( s ) (cid:1) u ( s ) = 0 . (1)It has the following solution: u ( s ) = (cid:0) h l ε ( s ) (cid:1) − u (0) , known as a holonomy, where h l ε := P exp (cid:18)Z ds A (cid:0) ˙ l ε ( s ) (cid:1)(cid:19) . (2)The propagating degrees of freedom of the Abelian matter vector field A µ are introduced by the action S ( A ) := − g A Z M d x √− g g µν g ξπ F µξ F νπ , (3)where g µν , g µν , and g are the metric tensor, its inverse, and determinant, respectively. The coupling constant isdenoted by g A and F µξ is the curvature of A µ .The spacetime splitting allows to derive the momomentum E a = √ q g A e µ q ab F µb canonically conjugated to A a .Here, q denotes the determinant of the q ab := e ia e ib metric on Σ t and e µ = (1 /N, − N a /N ) is the upper row of thevierbein matrix e µI (‘ I ’ represents directions in the Minkowski space).The Legendre transform of (3) leads to the completely constrained system with the total Hamiltonian H ( A ) T = V ( A ) + H ( A ) that is composed of two first class constraints. The vector constraint (called also the diffeomorphismconstraint) V ( A ) := Z Σ t d x N a V ( A ) a = Z Σ t d x N a F ab E b (4)imposes the invariance under the spatial diffeomorphism transformations. The Hamiltonian constraint (called alsothe scalar constraint) H ( A ) := Z Σ t d x N H ( A ) = g A Z Σ t d x N √ q q ab (cid:0) E a E b + B a B b (cid:1) (5)generates the time reparametrization symmetry. The last quantity in the preceding equation is the magnetic field B a := g A ǫ abc F bc , where ǫ abc := √ q ˜ ǫ abc and ˜ ǫ abc is the Levi-Civita tensor. It is worth noting that E a and B a are avector density and a pseudovector density, respectively. By being densities, these objects scale properly according tothe scaling of the integration measure d x , where d x √ q is the measure invariant in R and √ q is a weight- scalardensity.In the lattice framework the vector constraint in (4) is added to its gravitational analog and they are solved atthe classical level. The Hamiltonian constraint in (5) is quantized after the regularization of the canonical fields onthe diffeomorphisms-invariant lattice. This leads to the construction of the Hamiltonian constraint operator (HCO),which is the only element of H ( A ) T that is going to be solved at the quantum level.The regularization procedure assumes the introduction of the Wilson loops [5]. In the Abelian case they triviallyreduce to loop holonomies h l (cid:9) l ′ = ε l ε l ′ F ab ˙ l a ˙ l ′ b + O ( ε ) , (6)where the loop begins at the initial point of the l link, goes along a quadrilateral path (in the cosmological frameworkin this article) and returns to the same point along l ′ . The second lattice-regularized variable takes the form of the E a = ǫ abc ∗ E bc field flux, constructed by smearing the two-form ∗ E bc (Hodge dual to E a ) over a two-dimensionalsurface S , f ( S ) := Z S ∗ E = Z S n a E a , (7)where n a := ǫ abc dx b ∧ dx c is the normal to S . i This assumption on the one hand decreases the universality of this analysis, but on the other hand allows to quickly compare theobtained results with the most popular LQC’s framework-related models. It is worth noting that an improved approach to the latticeregularization [20] and the related cosmologically reduced model [21] would lead to the same consclusions concerning the structure ofthe semiclassical corrections. In this case the lower cut-off on ε would not be needed. II.2. GR in terms of Ashtekar variables
The gravitational degrees of freedom are represented by the non-Abelian real vector field A ia := ǫ ijk Γ jka + γK ia knownas the Ashtekar-Barbero connection [3, 22]. Here, Γ jka is the spin connection, K ia := γ Γ i a is the dreibein-contractedextrinsic curvature, and γ denotes the Immirzi parameter. By neglecting the possible coupling of spinors to gravity,the kinematics of the gravitational field can be defined by the Einstein-Hilbert action S ( gr ) := 1 κ Z M d x √− gR , (8)where R is the Ricci scalar and the gravitational coupling constant reads κ = 16 πG . The momentum of A ia is givenby the densitized dreibein E ai := √ qe ai . These fields are in the canonical relation (cid:8) A ia ( t, x ) , E bj ( t, y ) (cid:9) = γκ δ ba δ ij δ ( x − y ) (9)with respect to the ADM variables.The total Hamiltonian H ( A ) T = G ( A ) + V ( A ) + H ( A ) corresponding to (8) is composed of three constraints G ( gr ) := 1 γκ Z Σ t d x A it D a E ai ,V ( gr ) := 1 γκ Z Σ t d x N a F iab E bi ,H ( gr ) := 1 κ Z Σ t d x N (cid:18) √ q (cid:0) F iab − ( γ + 1) ǫ ilm K la K mb (cid:1) ǫ ijk E aj E bk (cid:19) . (10)The operator D a is the covariant derivative of the Ashtekar-Barbero connection and the curvature of this su (2) fieldis specified by F iab := ∂ a A ib − ∂ b A ia + ǫ ijk A ja A kb .As in the case of the matter vector field, the constrains that do not contain propagating degrees of freedom aresolved classically. These are the Gauss constraint G ( A ) and the diffeomorphism one V ( A ) . The scalar constraintis regularized and quantized on the su (2) -invariant and diffeomorphisms-invariant lattice. These procedures lead toHCO for gravity. The fields F iab and E ai are regularized in the way presented in (6) and (7), respectively. Due to theinternal symmetry of the Ashtekar variables, the precise formulas readtr ( τ i h l (cid:9) l ′ ) = − ε l ε l ′ F iab ˙ l a ˙ l ′ b + O ( ε ) (11)and f i ( S ) := Z S n a E ai , (12)respectively.The object absent in (5) but present in H ( gr ) is the extrinsic curvature K ia . However, in the cosmological frameworkdiscussed in this article, the spin connection Γ jka contribution to the constant field A ia vanishes and this latterfield becomes proportional to K ia . Therefore any separated regularization of the extrinsic curvature does have to beintroduced. Finally, the lattice smearing of the Ashtekar-Barbero connection is defined in analogy to (11),tr (cid:0) τ i h l (cid:1) = − ε l A ia ˙ l a + O ( ε ) . (13)It is worth noting that expressions (11) and (13) are not expended up to the same order. We neglect this problem inthis article, although it is an essential issue concerning the general investigation of the lattice regularization procedurein LQG, cf. [20, 23]. II.3. Methods of scalar fields coupling to LQG
The simplest classical representative of the bosonic matter content in cosmology is the real massless scalar field ϕ without internal degrees of freedom. To formulate a diffeomorphism-invariant representation of ϕ , one needs torely on a different strategy than for vector fields. This issue is related to a different geometrical properties of theaforementioned objects. The scalar field and its momentum π are not a one-form density and a vector density,respectively, but a scalar and a pseudoscalar density, respectively. Therefore, their smearing along a link and througha surface would not be correct. The point-solid duality appears to be the right pair of objects that allows to describethe degrees of freedom of ϕ and π on a lattice.The massless Klein-Gordon scalar field is defined by the action S ( ϕ ) := 12 g ϕ Z M d x √− g g µν ∂ µ ϕ ∂ ν ϕ , (14)where g ϕ is the coupling constant. The Legendre transform results in same structure of the total Hamiltonian H ( ϕ ) T = V ( ϕ ) + H ( ϕ ) as in the case of the vector matter field. The diffeomorphism and Hamiltonian analogs of theconstraints in (4) and (5) are V ( ϕ ) := Z Σ t d x N a ∂ a ϕ π (15)and H ( ϕ ) := 12 Z Σ t d x N (cid:18) g ϕ √ q π + √ q g ϕ q ab ∂ a ϕ ∂ b ϕ (cid:19) , (16)respectively. The explicit form of the momentum canonically conjugated to ϕ is π = √ q g ϕ e µ ∂ µ ϕ . It is worth notingthat the aforementioned quantities can be easily extended to the self-interacting field formalism. In this case, thepotential can be given by a polynomial of ϕ . This potential trivially couples to gravity only by multiplication with √ q , hence it does not bring any significant contribution to the analysis in this article.The simplest point-solid symmetry reflecting the lattice representation of the Klein-Gordon field is the following.The holonomy-like representation [13, 24] located at a node v (an intersection of links) is Φ v := exp (cid:0) i ε v ϕ ( x ) (cid:1) . (17)The solid-related momentum representation is Π( R v ) := Z R v d x δ v,x π ( x ) , (18)where the integration was done all over the region R v centered at v . The last quantity in (18) is assumed to be apriori smeared, reading π ( x ) := P y ∈ R δ ( x − y ) Π( y ) .The preceding pair of definitions is related to nodes. Their trivial distribution all over the lattice leads to a simpleconstruction of the related Fock space. In this case one usually consider the polymer representation [25–27] withnodes-located states having a similar form to the definition in (17). II.4. Models contradictive with CQGR
At the end of this section three popular quantum cosmological models that are indirectly related to LQG are goingto be discussed. Each of these models is associated with a different approach to regularize and quantize matter. Byfollowing the review of these theories [28], three quantization procedures can be recognized: the effective constraints,dress metric, and separate universe quantization approach. By concerning the methodology introduced in Sec. I, onecan verify if these models meet the quantum general postulate of relativity criterion. In this way, one can check if anyof these approaches could be considered as a simplification of CQGR, hence as a phenomenology-valued cosmologicalmodel.The effective constraints method [29–31] does not define any QFT for matter. Instead, it introduces unspecifiedperturbations around the classical cosmological matter density derived form LQC [12, 32, 33]. The structure of theseperturbations is then restricted by a closeness of the constraint algebra. Therefore, this effective model is not a priori contradictive with CQGR, unless the formulation of LQC is not a cosmological limit of CQGR. In the latter case,one can always repeat the procedures of the effective constraints method around a different cosmological frameworkobtained from LQG. However, the results of this model, by definition, do not provide any insight into the structureof the matter sector. This approach formulates only an effective description of the cosmological data, but it doesnot describe the mechanism that explains the origin of this data. Hence the effective constraints method may havephysical applications, but not as a phenomenology predicting technique.The dressed metric approach is based on the idea proposed in [34]. It was applied both to the scalar [35] and vector[36] fields, which were described by the method of QFT in curved spacetime. By defining the Fock space for matterfields, by choosing the expectation value of HCO in LQC as a background, one directly violates SE in the constructionof the theory. As a consequence, the approximation of this model omits the corrections that otherwise would bepresent as a result of the quantization of the gravitational degrees of freedom in the HCO of the matter sector. Thesecorrections would be of the same order of significance as the cosmological corrections from the free sector of gravityand the QFT perturbations of matter — see also Sec. IV.1. This argument demonstrates the BI violation indicatesthe related incompleteness of the results. A particular form of the corrections based on the SE formulation of LQCand BI method of the derivation of its semiclassical results is used to define the background on which the Fock spacefor the matter sector is constructed. Then, by definition, these corrections will be absent in the semiclassical limit ofthe matter sector. Therefore, the dressed metric approach is an inconsistently formulated toy model and it cannotbe applied as a physical tool. This model can be used only to study particular theoretical mechanisms. It is worthnoting that a specific variant of this approach, called the hybrid quantization [37, 38], additionally assumes left andright multiplication of the total HCO by the quantized equivalent of the q − / quantity. This affects the gravitationalsector, which, by construction of the mentioned multiplication, cannot be thought as a limit of SE LQG. In the caseof a possibility to compare any results of this model with data, the same predicable inapplicability arguments as inthe case of the dressed metric approach hold.Finally, the separate universe quantization [39] is the the long-wavelength gravitational modes quantization on theLQC background. This method uses the long-wavelength approximation to construct a loop quantization both for thebackground and perturbations. It could be an improvement of the dressed metric approach for particular applications.This model does not assume a separate quantization for the background variables (in the LQG-like method) and theperturbative degrees of freedom (in the Fock space method) like the previously discussed approach. However, theseparate universe quantization generates a different problem by neglecting the specific structure of quantum matterfields with their corresponding corrections. It is difficult to imagine a generalization of this effective approach toall the matter fields in the Standard Model of particle physics. Moreover, from the perspective of a simple effectivecosmological theory, the separate universe quantization neglects the gravitational corrections to the matter sector.Hence, by construction, this model cannot be expected to be the cosmological limit of a fundamental theory of CQGR,which would be constructed as LQG with an analogous quantization of matter fields.Concluding, all the aforementioned models are not fundamental and cannot provide any certain insight into realcosmological processes. They are not compatible with the SE and BI canonical procedures of QFT (including thetheory of gravity) on a lattice. However, they can be used to study particular theoretical or mathematical procedures.Moreover, the first one, the effective constraints method can be used to describe all the cosmological data statistically.The possible physical application of the second and the third model is more limited. However, they can still be usedas the effective tools to describe cosmological data from the epochs in which their incomplete predictions are expectedto be negligible. In general, none of these models is expected to give a deeper insight into the understanding ofcosmology than the standard methods of QFT on curved spacetime. Let us emphasized that this statement is basedon the assumption that the hypothetical fundamental quantum theory of gravity and matter is SE and BI. A moredetailed critical reviews concerning also other problems of the mentioned models can be found in [40–42].In the next section a toy model satisfying the SE condition will be constructed. Then its BI will be tested to set adirection toward future attempts of a fundamental model construction. III. KINEMATICSIII.1. Cosmological models
To discuss general covariance regarding the semiclassical limit of what could be a SE cosmological limit of LQGmethods-based fundamental theory, one needs to consider consistent regularization and quantization procedures.The cosmological phase space reduction of the lattice-regularized gravity formulated in the Ashtekar variables isdescribed in [15]. Here, the cosmological phase space reduction is defined as the SU (2) breakdown of the internalspace invariance into the U (1) case and the breakdown of the spatial diffeomorphisms into the ones that satisfy theBianchi I symmetry. This result of the reduction is identical if it is done before the regularization and if after thereduction, the corresponding lattice structure is adjusted to the symmetry of the reduced Ashtekar variables [15]. Inboth cases of the phase space reduction, applied either to the holonomy-flux description or the Ashtekar variablesformulation, the resulting (classical) lattice-regularized Hamiltonian constraint is equivalent with the one assumed inthe Bianchi I extension of LQC in [11]. The reduction can be also applied to the expectation values of the operatorson the states providing the classical limit of the system. Naturally, the structure of HCO is again the same [15]. Inthis latter case, however, the states are already given by the formalism of LQG (these are the symmetry-reduced spinnetwork states [1, 43, 44]) and are different than the ones assumed in the extended LQC [11, 12].By assuming the expectation value of HCO for the cosmological reduction of LQG and by keeping all the quantumcorrections up to the quadratic order in the regularization parameter ε , one obtains (cid:10) ˆ H ( gr ) (cid:11) = − γ κ X v N v q ¯ E a δ a ¯ E b δ b ¯ E c δ c | ¯ E di δ id | Y k = i sin (cid:0) ε ( k ) ¯ A ke δ ek (cid:1) ε ( k ) " O (cid:0) ¯ j ( i ) (cid:1) ! . (19)Here, ¯ A ( i ) a δ ai is the Ashtekar connection’s diagonal sector that is obtained by a simultaneous fixing of the internaland diffeomorphism symmetries. Analogously, ¯ E ai denotes the diagonal densitized dreibein, reintroduced by thecorrespondence principle, i.e. by replacing the eigenvalue of the ˆ E ai operator with its classical equivalent. It is worthmentioning that the same result was postulated by the incorrectly derived [15] ‘partial quantum reduction procedure’known as Quantum reduced loop gravity (QRLG) [45, 46]. Moreover, the expectation value of HCO in cosmologicalcoherent quantum gravity (CCQG) [47] derived from LQG by assuming a particular selection of states is the same upto the terms of order ε . Moreover, a similar expression with the same structure up to the order ε appears when theLorentzian term (see the subsequent analysis) is taken into account [48]. Finally, the second term in the quadraticbracket in (19) is not written explicitly, because it differs in the mentioned models by a numerical factor. This term isknown as the inverse volume corrections. It was first noticed in the isotropic model of LQC in [49]. As demonstratedin [50], the related corrections have a significant contribution to the dynamics of the primordial universe. Theirstructure is going to be the essential issue studied in this article.By neglecting the differences in the order- ε corrections coming from the expansion of the trigonometric functional inthe gravitational sector of the scalar constraint i , the structure of the expectation value of HCO in the aforementionedmodels, including the inverse volume corrections, remains the same up to a constant factor ii . These latter correctionsappear as a result of the action of the operator ˆ h − a (cid:2) ˆ V , ˆ h a (cid:3) . (20)Analogous corrections to the gravitational degrees of freedom are present also in the matter sector. These correctionsare sourced from a more general expression ˆ h − a (cid:2) ˆ V n , ˆ h a (cid:3) , (21)where n is a positive rational number. It is worth be emphasized that here the volume operator acts on a state thatis initially modified by the gravitational holonomy operator ˆ h a . The latter operator acts by multiplication. Then thedifference in the states on which the volume operator acts, results in the inverse volume corrections.To identify each occurrence of the terms given in (21), as well as each value of the n parameter, in the HCO of theentire system, it is enough to investigate all the classical contributions to the lattice-smeared scalar constraint. Theexplicit expression of this object for the torsionless gravity is given by H ( gr ) = Z Σ t d x N ( x ) (cid:16) H ( gr ) Eucl ( x ) + H ( gr ) Lor ( x ) (cid:17) , (22)where H ( gr ) Eucl ( x ) := 2 γκ lim ε → ǫ abc tr (cid:18) ε (cid:16) h ab ( x ) − h − ab ( x ) (cid:17) ε h − c ( x ) n V ( x ) , h c ( x ) o(cid:19) (23)and H ( gr ) Lor ( x ) := − ( γ + 1) γ κ lim ε → ǫ abc tr (cid:18) ε h − a ( x ) n K ( x ) , h a ( x ) o ε h − b ( x ) n K ( x ) , h b ( x ) o ε h − c ( x ) n V ( x ) , h c ( x ) o(cid:19) . (24)The terms in the form of the expression in (20) are easily recognizable. i The form of the trigonometric functional slightly varies from one model to another. However, this form remains always expandable intoa power series of connections. ii This statement is true in general, as long as the volume operator is an eigenoperator of the states defined on a cuboidal lattice, cf. [51].
The structure of the lattice corrections in the matter sector is also directly readable from the regularized form ofthe Hamiltonian constraint. In the case of the vector field, it is given by H ( A ) = Z Σ t d x N ( x ) (cid:16) H ( A ) elec ( x ) + H ( A ) magn ( x ) (cid:17) , (25)where H ( A ) elec ( x ) = 2 g A ( γκ ) lim ε → E a ( x ) tr (cid:18) τ i ε h − a ( x ) n V ( x ) , h a ( x ) o(cid:19) × Z d y δ ( x − y ) E b ( y ) tr (cid:18) τ i ε h − b ( y ) n V ( y ) , h b ( y ) o(cid:19) (26)and H ( A ) magn ( x ) = 2 g A ( γκ ) lim ε → B a ( x ) tr (cid:18) τ i ε h − a ( x ) n V ( x ) , h a ( x ) o(cid:19) × Z d y δ ( x − y ) B b ( y ) tr (cid:18) τ i ε h − b ( y ) n V ( y ) , h b ( y ) o(cid:19) . (27)Analogously, the regularized scalar field Hamiltonian is expressed by H ( ϕ ) = Z Σ t d x N ( x ) (cid:16) H ( ϕ ) mom ( x ) + H ( ϕ ) der ( x ) + H ( ϕ ) pot ( x ) (cid:17) , (28)where H ( ϕ ) mom ( x ) = 2 g ϕ ( γκ ) lim ε → π ( x ) ǫ ijk ǫ abc Z d z δ ( x − z ) tr (cid:18) τ i ε h − a ( z ) n V ( z ) , h a ( z ) o(cid:19) × tr (cid:18) τ j ε h − b ( z ) n V ( z ) , h b ( z ) o(cid:19) tr (cid:18) τ k ε h − c ( z ) n V ( z ) , h c ( z ) o(cid:19) × Z d y δ ( x − y ) π ( y ) ǫ lmn ǫ def Z d z ′ δ ( y − z ′ ) tr (cid:18) τ l ε h − d ( z ′ ) n V ( z ′ ) , h d ( z ′ ) o(cid:19) × tr (cid:18) τ m ε h − e ( z ′ ) n V ( z ′ ) , h e ( z ′ ) o(cid:19) tr (cid:18) τ n ε h − f ( z ′ ) n V ( z ′ ) , h f ( z ′ ) o(cid:19) , (29) H ( ϕ ) der ( x ) = 2 ( γκ ) g ϕ lim ε → ∂ a ϕ ( x ) ǫ ijk ǫ abc tr (cid:18) τ j ε h − b ( x ) n V ( x ) , h b ( x ) o(cid:19) tr (cid:18) τ k ε h − c ( x ) n V ( x ) , h c ( x ) o(cid:19) × Z d y δ ( x − y ) ∂ d ϕ ( y ) ǫ ilm ǫ def tr (cid:18) τ l ε h − e ( y ) n V ( y ) , h e ( y ) o(cid:19) tr (cid:18) τ m ε h − f ( y ) n V ( y ) , h f ( y ) o(cid:19) (30)and H ( ϕ ) pot ( x ) = 12 g ϕ p q ( x ) V [ ϕ ( x )] ≈ g ϕ lim ε → V [ ϕ ( x )] 1 ε V ( x, ε ) . (31)Here, for clearness, the potential term was introduced to demonstrate the ambiguity in the choice of its form V [ ϕ ( x )] about the presence of the related gravitational corrections. These do not appear, because, after the quantization, thevolume operator does not act on holonomy-modified states. III.2. States space
To discuss the form of the semiclassical quantum-geometrical corrections in the cosmological simplification of CQGRone needs to specify the states space. It is worth to repeat the fact already recalled in the previous subsection.The only terms in the scalar constraint contributing to the next-to-the-leading-order inverse volume corrections havethe same structure independently of the selected cosmological model that does not break SE. These terms dependentirely on the postulated classical action of all the contributing fields and on the power of volume in the followingapproximate identity [1, 2, 13], applied to the regularization procedure, (cid:8) A ia , V n (cid:9) = nγκ E ai (cid:0)p | E | (cid:1) n + O ( ε ) . (32)It is worth noting that neglecting the last lattice correction term is as precise as neglecting the analogous correction in(13). Moreover, in the n = 1 case this term vanishes identically and the additional constraint εn ≪ is required. Fur-thermore, in the limit ε → , this correction vanishes and the whole lattice-regularized system takes an ε -independentfinite form.The fact that the structure of next-to-the-leading-order inverse volume corrections does not depend on the selectedcosmological formulation, is a result of the proper phase space reduction of the hypothetical fundamental theory. Thereduction of variables into the Bianchi I symmetry have to entail the reduction of the lattice structure into the cuboidalform [15]. The volume operator or its power V n , expressed as a functional of the diagonal densitized dreibein fluxes,is an eigenoperator of the states defined on a cuboidal lattice [51]. The modifications of the states by the holonomycontribution to formula (21) generate the inverse volume corrections along directions of these holonomies. Therefore,to investigate the semiclassical structure of the generated inverse volume corrections, it is enough to select the simpleststates that reveal these corrections and derive the semiclassical limit. This last step can be easily done by definingthe coherent states as the states that restore the volume from the eigenvalue of the related operator ˆ V by thecorrespondence procedure.One can consider the system of minimally coupled bosonic matter and gravity with the Hilbert space H kin := H ( gr ) kin ⊗ H ( A ) kin ⊗ H ( ϕ ) kin . (33)The vector matter field sector is labeled by H ( A ) kin and it is defined analogously to the one for the SU (2) -invariantgravitational field in LQG, labeled by H ( gr ) kin , cf. [1, 2]. In both cases one assumes the space of cylindrical functions ofthe gauge connections holonomies. Also, in both cases, the basis states are the invariant spin network states: | Γ; j l i for the Abelian vector field and | Γ; j l , i v i for the SU (2) -invariant gravitational field, respectively. They are labeled byquantum numbers (spins) j l and j l , respectively. These numbers determine the notion of the gauge groups irreduciblerepresentations at each link l . To preserve the gauge invariance in the non-Abelian gravitational case of LQG, thecorresponding intertwiners i v are attached at each node v . The reduced phase space approach allows to fix the internalspace to the Abelian U (1) case [15]. Consequently, one can drop the trivial intertwiners from states; any Hilbert spaceis, by definition, specified up to a number. Concluding, the simplest states for the matter and gravitational vectorfields are defined along the cuboidal lattice links, and are denoted by | Γ; j l i ∈ H ( A ) kin and | Γ; j l i ∈ H ( gr ) kin .The nodes-related states are qualitatively different. The Hilbert space describing the scalar field point holonomyrepresentation is defined as H ( ϕ ) kin := (cid:8) a U π + ... + a n U π n : a i ∈ C , n ∈ N , π i ∈ R (cid:9) , (34)where the wave function reads U π ( ϕ ) := h ϕ | U { v ,..,v n } , { π v ,π vn } i := e i P v ∈ Σ π v ϕ v . (35)This definition explicitly preserves the rotational symmetry of the scalar field at each point. The whole collection ofnodes forms a trivial polymer-like structure. Moreover, by construction, this structure is diffeomophically-independentof any spacetime geometry. The modes of the scalar field do not oscillate in space, but are statically located at pointsand the distance between these points is only trivially coupled with gravity — by multiplication. This isotropicstructure distributed over the lattice does not reflect any possible internal quantum relation between the gravitationaland matter degrees of freedom.By considering the single-point state h ϕ | v ; U π i := e i π v ϕ v located at v , the action of the canonical operators istrivially defined in the exponential form. The point holonomy shifts the state as follows, e i π ′ v ′ ˆ ϕ v ′ | v ; U π i := e i π ′ v ′ ϕ v ′ | v ; U π i = | v ∪ v ′ ; U π + π ′ i . (36)Analogous action for the momentum operator corresponding to the ε -smeared momentum π in the region around the v node is given by the eigenequation ˆΠ( v ′ ) | v ; U π i := − i ~ ∂∂ϕ ( v ′ ) | v ; U π i = ~ π v ′ δ v,v ′ | v ; U π i . (37)0The scalar product definition is simply adjusted to the trivial form of the canonical operators, reading h v ; U π | v ′ ; U π ′ i = δ v,v ′ δ π v ,π ′ v ′ . (38)More details concerning this polymer states for the point holonomy representation are given in [14, 25–27]. h (2) x,y - ,z j (2) x,y - ,z h (2) x,y,z j (2) x,y,z h (3) x,y,z j (3) x,y,z h (3) x,y,z - j (3) x,y,z - h (2) x,y - ,z j (2) x,y - ,z h (2) x,y,z j (2) x,y,z h (3) x,y,z j (3) x,y,z h (3) x,y,z - j (3) x,y,z - e i π x,y,z + φ x,y,z + e i π x,y - ,z φ x,y - ,z e i π x,y,z φ x,y,z e i π x,y + ,z φ x,y + ,z e i π x,y,z - φ x,y,z - x direction ⊙− x direction ⊗ FIG. 1. The normalized basic state of bosonic fields for cubic lattice
Concluding, the basis states are defined by H kin ∋ | Γ; j l , j l , U π i := | Γ; j l i ⊗ | Γ; j l i ⊗ | Γ; U π i . (39)By considering a single hexavalent node state c v ∈ Γ , one can express the related Hilbert space structure in thegraphical form, see FIG. 1. The dashed frame specifies the normalization that allows to tessellate the reduced spacewith the embedded graph structure. This tessellation results in the set of the cuboidal cylindrical functions. Here, j ( i ) p,q,r and j ( i ) p,q,r are the spin numbers associated to the links l ( i ) p,q,r . The scalar field state is represented by the pointholonomy e i π p,q,r ϕ p,q,r at the node v p,q,r ∈ Γ , where π p,q,r is the real coefficient.It is worth mentioning that analogous structure in LQC is represented by a normalized hexavalent node state inFIG. 2. This latter structure also preserves the symmetry of the anisotropic Bianchi I model. However, it does notreflect the symmetry of the volume operator in LQG that acts at nodes. Moreover, the nodes-located distributionof the scalar field polymer structure has to coincide with the symmetry of any separable cellular form of the Fockspace. This observation is based on the fact that the general form of the Fock space for LQG, known as the spinnetwork, is not separable and has no a priori defined centers of symmetry [2]. Conversely, the trivial states space forthe scalar field restricts the modes oscillations to a point. To keep this symmetry at the quantum level, one cannotspecify the elementary cell in the way proposed in FIG. 2. In this latter case, the scalar field contribution would needto be determined as the sum of the degrees of freedom at eight nodes, which would break the classical local rationalsymmetry of this field. Therefore, the specification of the elementary cell for the separable Fock space defined in (39)is uniquely restricted to the form in FIG. 1.Finally, only by staring at the structure of the state represented in FIG. 1, one should recognize a methodologicalinconsistency in the construction of this multi-matter coupling model — the scalar field is lattice-regularized ina qualitatively different manner than the other fields. On the one hand this inconsistency will be the source of thegeneral covariance violation of the quantum corrections. On the other hand the consistency in the lattice regularizationof the gravitational and the vector field will be preserved in the final results. This issue shows how to formulate thegravitational-matter coupling correctly in the future.1 h (2) x,y - ,z j (2) x,y - ,z h (2) x,y,z j (2) x,y,z h (3) x,y,z j (3) x,y,z h (3) x,y,z - j (3) x,y,z - h (2) x,y - ,z j (2) x,y - ,z h (2) x,y,z j (2) x,y,z h (3) x,y,z j (3) x,y,z h (3) x,y,z - j (3) x,y,z - e i π x,y,z + φ x,y,z + e i π x,y - ,z φ x,y - ,z e i π x,y,z φ x,y,z e i π x,y + ,z φ x,y + ,z e i π x,y,z - φ x,y,z - x direction ⊙− x direction ⊗ FIG. 2. The basic state of bosonic fields for cubic lattice in LQC
III.3. Gravitational coherent states
To discuss the semiclassical corrections precisely, one needs to define coherent states. For clearness of the analysis,the notation typical to LQC [11, 12, 32] is going to be used. The reduced canonical variables [15] are specified to ˜ A ia ( t ) := 1 l ( i )0 ˜ c ( i ) ( t ) e ia , (40) ˜ E ai ( t ) := l ( i )0 V ˜ p ( i ) ( t ) p q e ai . (41)The matrices e ai and e ia represent constant orthonormal Cartesian frame and co-frame fields, respectively. Thedeterminant of the fiducial metric q ab in (41) compensates the density weight of ˜ E ai ( t ) . The fiducial length l ( i )0 andthe corresponding volume V := l l l of the fiducial cell are introduced to simplify the symplectic structure of thesystem. This leads to the following Poisson brackets, (cid:8) ˜ c ( i ) ( t ) , ˜ p ( j ) ( t ) (cid:9) = κγ δ ( i )( j ) . (42)The semiclassical dynamics of the cosmologically reduced CQGR is specified by the Ehrenfest theorem i and dependson the coherent states | i ∈ H kin . The form of these states in this article is defined as the tensor product of thecoherent states for different fields. The related Heisenberg equation reads, ddt (cid:10) ˆ O (cid:11) − (cid:28) ∂ ˆ O∂t (cid:29) = 1i ~ (cid:10) [ ˆ O, ˆ H ] (cid:11) , (43)where the states factorize as follows, | i = e | i ( gr ) ⊗ | i ( matt ) = e | i ( gr ) ⊗ O φ | i ( φ ) . (44) i It is worth noting that the operator equation in (43), known as the Ehrenfest theorem, was derived by Heisenberg [52]. φ represents any matter field and the term (cid:10) ∂ ˆ O∂t (cid:11) was neglected by assuming only implicit time dependenceof variables.The normalized Bianchi I coherent states for the gravitational sector are defined as e | i ( gr ) := X v O i (cid:16)(cid:10) c ( i ) v ( ˜ A ) (cid:12)(cid:12) c ( i ) v ( ˜ A ) (cid:11)(cid:17) − (cid:12)(cid:12) c ( i ) v ( ˜ A ) (cid:11) . (45)The last factor is known as the shadow state [19] with a d -width Gaussian distribution around the densitized dreibeinoperator eigenvalue. The form of this state reads (cid:12)(cid:12) c ( i ) v ( ˜ A ) (cid:11) := X µ ( i ) v exp " − d (cid:18) µ ( i ) v − ˜ p ( i ) ¯ k (cid:19) exp " − i (cid:18) µ ( i ) v − ˜ p ( i ) ¯ k (cid:19) ˜ c ( i ) µ ( i ) v (cid:11) . (46)This formula is constructed on the link excitation states [11, 32] (the last factor above) that are given by the expression (cid:12)(cid:12) µ ( i ) v (cid:11) := exp (cid:20) i µ ( i ) v c ( i ) (cid:21) , µ ( i ) ∈ Z . (47)This definition is formulated in a direct analogy to the reduced form of the holonomy, ˜ h ( i ) ν ( v ) := exp (cid:18)Z ν ( i ) v l ( i )0 ds ˜ A ia τ i ˙ l aν ( s ) (cid:19) = e ν ( i ) v ˜ c i τ i , (48) cf. [15]. Then, the actions of the lattice-regularized Bianchi I variables in (40) and (41) read ˆ˜ c ( i ) (cid:12)(cid:12) µ ( i ) v (cid:11) := − ν ( i ) v tr (cid:0) τ ( i ) ˜ h ( i ) ν (cid:1)(cid:12)(cid:12) µ ( i ) v (cid:11) = i ν ( i ) v (cid:16)(cid:12)(cid:12) µ ( i ) v − ν ( i ) v (cid:11) − (cid:12)(cid:12) µ ( i ) v + ν ( i ) v (cid:11)(cid:17) (49)and ˆ˜ p ( i ) (cid:12)(cid:12) µ ( i ) v (cid:11) := − i¯ k ∂∂ ˜ c ( i ) (cid:12)(cid:12) µ ( i ) v (cid:11) = µ ( i ) v k (cid:12)(cid:12) µ ( i ) v (cid:11) , (50)respectively.In this article only the structure of the corrections, not their exact value, is going to be verified. This allows tosimplify the notation even more by replacing the reduced variables in (40) and (41) by ¯ A ia ( t ) := A ( i )( a ) ( t ) e ia = 1 εc ( i ) ( t ) (51)and ¯ E ai ( t ) := E ( a )( i ) ( t ) e ai = 1 ε p ( i ) ( t ) , (52)respectively, where ε is the small regularization parameter. Here, the anisotropy and inhomogeneity of the regulatorsthat depend on links lengths was neglected. This second simplification does not affect the structure of corrections.Moreover, in the properly reduced system, the final form of the Hamiltonian should be regulator-independent (thishas been verified in the recent improved cosmological model in [21]), hence this operation is not going to modifyconclusions. The reduced holonomy becomes h ( i ) ( v ) = exp (cid:18)Z ε ds ¯ A ia τ i ˙ l aν ( s ) (cid:19) = e c i τ i . (53)and the related link excitation states, analogous to (47), takes the form (cid:12)(cid:12) m ( i ) v (cid:11) := exp (cid:20) i m ( i ) v c ( i ) (cid:21) , m ( i ) ∈ Z . (54)3Then, the actions of the lattice-regularized variables on these states are ˆ c ( i ) (cid:12)(cid:12) m ( i ) v (cid:11) := − ε tr (cid:0) τ ( i ) h ( i ) (cid:1)(cid:12)(cid:12) m ( i ) v (cid:11) = i ε (cid:12)(cid:12)(cid:12)(cid:12) m ( i ) v − (cid:29) − (cid:12)(cid:12)(cid:12)(cid:12) m ( i ) v + 12 (cid:29)! (55)and ˆ p ( i ) (cid:12)(cid:12) m ( i ) v (cid:11) := − i¯ k ∂∂c ( i ) (cid:12)(cid:12) m ( i ) v (cid:11) = m ( i ) v ¯ k (cid:12)(cid:12) m ( i ) v (cid:11) . (56)The parameter m ( i ) v is linked to the spin number j ( i ) v by the relation j ( i ) v = (cid:12)(cid:12) m ( i ) v (cid:12)(cid:12) .The last step toward formulation of the node-symmetric toy-model states on which the structure of the cosmologicalsector of CQGR will be tested is needed. To indicate the basic cell states centered at nodes (see FIG. 1), one has tosplit the link states initially formulated to describe the states for LQC (see FIG. 2). This fitting of the well-knownLQC shadow states to the analysis in this article is specified in the following relation, (cid:12)(cid:12) m ( i ) v (cid:11) = exp h i ~m ( i ) v c ( i ) i exp h i ~m ( i ) v + ε ( i ) c ( i ) i = (cid:12)(cid:12)(cid:12) ~m ( i ) v E ⊗ (cid:12)(cid:12)(cid:12) ~m ( i ) v + ε ( i ) E , (57)where the oriented link l ( i ) ( v ) that starts at the point v was split in half, l ( i ) ( v ) = ~l ( i ) ( v ) h ~l ( i ) (cid:0) v + ε ( i ) (cid:1)i − . (58)The quantity v ∓ ε ( i ) labels the nearest node along the negatively/positively-oriented i -th direction. In this way, twopaths, ~l ( i ) ( v ) and (cid:2) ~l ( i ) (cid:0) v + ε ( i ) (cid:1)(cid:3) − , which have the following properties: l ( i ) ( v )(0) = ~l ( i ) ( v )(0) = h ~l ( i ) ( v ) i − (0) ,l ( i ) ( v )(1 /
2) = ~l ( i ) ( v )(1) = h ~l ( i ) (cid:0) v + ε ( i ) (cid:1)i − (0) ,l ( i ) ( v )(1) = h ~l ( i ) (cid:0) v + ε ( i ) (cid:1)i − (1) = ~l ( i ) (cid:0) v + ε ( i ) (cid:1) (0) , (59)were created. Then, the quantum numbers became fitted to this structure by postulating the simple averaging ~m ( i ) v = ~m ( i ) v + ε ( i ) = m ( i ) v i . This completes the definition of the node-centered states that share the symmetry of boththe volume operator and the scalar field distribution. These simple toy-model states are (cid:12)(cid:12)(cid:12) ¯ m ( i ) v E := (cid:12)(cid:12)(cid:12) m ( i ) v − ε ( i ) , m ( i ) v E = (cid:12)(cid:12)(cid:12) ~m ( i ) v E ⊗ (cid:12)(cid:12)(cid:12) ~m ( i ) v E = exp (cid:20) i2 (cid:16) m ( i ) v − ε ( i ) + m ( i ) v (cid:17) c ( i ) (cid:21) = exp h i ¯ m ( i ) v c ( i ) i . (60)The lattice-regularized canonical variables have the following actions on these basis states, ˆ c ( i ) (cid:12)(cid:12) ¯ m ( i ) v (cid:11) = − ε tr (cid:18) τ ( i ) h ( i ) (cid:19)(cid:16)(cid:12)(cid:12)(cid:12) ~m ( i ) v E ⊗ (cid:12)(cid:12)(cid:12) ~m ( i ) v E(cid:17) = i ε (cid:18)(cid:12)(cid:12)(cid:12) ¯ m ( i ) v − ε ( i ) − ε E − (cid:12)(cid:12)(cid:12) ¯ m ( i ) v + ε E(cid:19) (61)and ˆ p ( i ) (cid:12)(cid:12) ¯ m ( i ) v (cid:11) = ¯ m ( i ) v ¯ k (cid:12)(cid:12) ¯ m ( i ) v (cid:11) . (62)Notice that in the former equation in (61), the half-link-adjusted holonomy operator is h ( i ) = e εA ( i )( a )0 e ( i )( a ) τ ( i ) → h ( i ) := e ε A ( i )( a )0 e ( i )( a ) τ ( i ) = e c i τ i . (63)Then the coherent states analogous to (45) are given by the formula (cid:12)(cid:12) (cid:11) ( gr ) := X v O i (cid:20)(cid:16)(cid:10) ~ c ( i ) v ( A ) (cid:12)(cid:12) ~ c ( i ) v ( A ) (cid:11)(cid:17) − (cid:12)(cid:12) ~ c ( i ) v ( A ) (cid:11) ⊗ (cid:16)(cid:10) ~ c ( i ) v ( A ) (cid:12)(cid:12) ~ c ( i ) v ( A ) (cid:11)(cid:17) − (cid:12)(cid:12) ~ c ( i ) v ( A ) (cid:11)(cid:21) . (64) i It is worth noting that this arithmetical mean corresponds to the averaging of the division of the analogous gravitational momentum,which would be constructed by the correspondence principle related to the original shadow states. (cid:12)(cid:12) ~~ c ( i ) v ( A ) (cid:11) := X ~~ m ( i ) v exp " − d (cid:18) ~~m ( i ) v − p ( i ) ¯ k (cid:19) exp " − i (cid:18) ~~m ( i ) v − p ( i ) ¯ k (cid:19) c ( i ) ~~m ( i ) v (cid:11) . (65)The node-centered coherent states, adjusted to (60) are defined analogously, ¯ (cid:12)(cid:12) (cid:11) ( gr ) := X v O i (cid:16)(cid:10) ¯ c ( i ) v ( A ) (cid:12)(cid:12) ¯ c ( i ) v ( A ) (cid:11)(cid:17) − (cid:12)(cid:12) ¯ c ( i ) v ( A ) (cid:11) , (66)where their node-centered shadow state coefficients are (cid:12)(cid:12) ¯ c ( i ) v ( A ) (cid:11) := X ¯ m ( i ) v exp " − d (cid:18) ¯ m ( i ) v − p ( i ) ¯ k (cid:19) exp " − i (cid:18) ¯ m ( i ) v − p ( i ) ¯ k (cid:19) c ( i ) ¯ m ( i ) v (cid:11) . (67)It is worth noting that these states (and the ones in (46), before the symmetrization) satisfy the coherent statesrequirements discussed concerning different aspects of LQG [19, 53–55]. The detailed analysis of the constructions ofanalogous states as the gauge-invariant projection of a product over links of heat-kernels for the complexification ofgroup elements can be found in [46].Finally, the reader more familiar with LQC might be interested whether the simplified, node-symmetrized modelleads to the same expectation values of the canonical operators. By deriving the expectation value of the ˆ c ( i ) operator, (cid:10) ¯ c ( i ) v ( A ) (cid:12)(cid:12) ˆ c ( i ) (cid:12)(cid:12) ¯ c ( i ) v ( A ) (cid:11) = R (cid:10) ~ c ( i ) v ( A ) (cid:12)(cid:12) ⊗ R (cid:10) ~ c ( i ) v ( A ) (cid:12)(cid:12) ˆ c ( i ) (cid:12)(cid:12) ~ c ( i ) v ( A ) (cid:11) R ⊗ (cid:12)(cid:12) ~ c ( i ) v ( A ) (cid:11) R = 2 ε exp " − (cid:18) ε d (cid:19) m ( i ) v exp " − d (cid:18) p ( i ) ¯ k − ¯ m ( i ) v (cid:19) sin (cid:20) ε c ( i ) + i ε (cid:18) p ( i ) ¯ k − ¯ m ( i ) v (cid:19)(cid:21) , (68)one obtains the result analogous to the one know for LQC. The identification would be exact after the replacement ¯ m ( i ) v → µ ( i ) v . (69)By substituting the appropriate correspondence principle ¯ m ( i ) v → p ( i ) ¯ k , (70)the result can be recast in the simple form (cid:10) ˆ c ( i ) (cid:11) = c ( i ) (cid:16) O (cid:0) ε (cid:1)(cid:17) . (71)Analogously, the expectation value of the reduced flux operator becomes (cid:10) ˆ p ( i ) (cid:11) = p ( i ) . (72)The last pair of equations will be enough to discuss the SE quantum matter coupling to LQG concerning the BI ofthe related semiclassical results. IV. QUANTUM CORRECTIONSIV.1. Ehrenfest theorem and Heisenberg equation
In this section the matrix elements on the coherent states of what could be the cosmological reduction of CQGR areanalyzed. All the conclusions are going to be studied in the formalism general enough to be directly related withLQC, QRLG, CCGR, and analogous models.5The semiclassical dynamics of the whole cosmological system is given by the Heisenberg equations d (cid:10) ˆ c (cid:11) ( gr ) dt = 1i ~ (cid:18)D(cid:2) ˆ c, ˆ H ( gr ) (cid:3)E ( gr ) + D(cid:2) ˆ c, ˆ H ( matt ) (cid:3)E ( gr ) (cid:19) = 1i ~ D(cid:2) ˆ c, ˆ H ( gr ) (cid:3)E ( gr ) + ∆ H ( matt ) c , (73) dpdt = 1i ~ D(cid:2) ˆ p, ˆ H ( gr ) (cid:3)E ( gr ) , (74) d (cid:10) ˆ φ (cid:11) ( matt ) dt = 1i ~ (cid:28)D(cid:2) ˆ φ , ˆ H ( matt ) (cid:3)E ( matt ) (cid:29) ( gr ) = 1i ~ D(cid:2) ˆ φ , ˆ H ( matt ) (cid:3)E ( matt ) + ∆ | i ( gr ) φ . (75)Analogously to (44), the symbol φ represents any matter field. Precisely, it denotes either the canonical field variableor the corresponding conjugate momentum. The indices labeling the directions and position of operators were omittedfor simplicity. It was also assumed that the gravitational and matter fields are not explicitly time-dependent and theirevolution is encoded only in the equations of motion.The quantum GR corrections both in (73) and (75) are of the same order in the inverse spin number, precisely ∆ H ( matt ) c ∝ ∆ | i ( gr ) φ ∝ m . (76)Here, the large quantum number approximation | ¯ m | ≫ is needed i . The detailed derivation and the exact numericalvalue of these SE-sourced corrections is model-dependent — see for instance [16, 46]. In what follows, only the structureof the gravitational degrees of freedom contributing to the corrections from equation (75) is going to be used in thegeneral covariance verification. Consequently, the decomposition into the gravity- and matter-related expressionswill be later introduced (in (87)). Finally, the explicit derivation of the structure of the matter sector-related GRcorrections, needed for the verification procedure, will be given in Sec. IV.3.The second type of the classical dynamics perturbations that come from the quantum gravitational corrections willbe denoted by δ ˙ c , δ c , and δ p . These quantities are sourced by the terms d (cid:10) ˆ c (cid:11) ( gr ) dt = dcdt (cid:0) δ ˙ c (cid:1) , (77) D(cid:2) ˆ c, ˆ H ( gr ) (cid:3)E ( gr ) = i ~ δH ( gr ) δp (cid:0) δ c (cid:1) , (78)and D(cid:2) ˆ p, ˆ H ( gr ) (cid:3)E ( gr ) = − i ~ δH ( gr ) δc (cid:0) δ p (cid:1) , (79)respectively. They have a qualitatively different structure than the quantum GR corrections in (76), by satisfying δ ˙ c ∝ δ c ∝ δ p ∝ ε . (80)Another difference between these corrections is in the fact that the gravitational corrections are functionals of theconnection, δ ˙ c = δ ˙ c [ c ] , δ c = δ c [ c ] , δ p = δ p [ c ] , and are related only to the regularization of the gravitational sector.The GR corrections are related to the SE restriction imposition and depend only on quantum numbers. Thus, by thecorrespondence principle in (70), they indirectly depend on the reduced flux, which is directly related to the spatialmetric tensor. However, they are independent of the gravitational correction. This feature can be written as ∂∂c ∆ H ( matt ) c = ∂∂c ∆ | i ( gr ) φ = 0 . (81)Notice also that by neglecting the evolution of the gravitational degrees of freedom in (75), this Heisenberg equationtakes the form d (cid:10) ˆ φ (cid:11) ( matt ) dt = 1i ~ D(cid:2) ˆ φ , ˆ H ( matt ) (cid:3)E ( matt ) , ∆ | i ( gr ) φ = 0 . (82) i This approximation relates the single fiducial cell formula with the Hamiltonian on Σ t in the continuum limit [2, 56]. Generalization toany value of ¯ m is possible, but it would require a redefinition of the coherent states. Heuristically, this could be done replacing ¯ m with ¯ m ε := ¯ m/ε in the definition (60). Precise approach would require redefinition of LQG and the appropriate phase space reduction. Thefirst articles concerning the former issue was recently announced, cf. [20]. The regulator-independent formulation of the lattice reducedtheory is based on [23] and is given in [21]. q ± / factorsand/or by the contraction with the q ab metric tensor — see the expressions in (5) and (16). After the latticeregularization, these recalled expressions take the forms given in formulas (25) and (28), respectively. The GRcorrections in these formulas will be sourced by the quantized version of the termstr (cid:16) τ i h − a ) ( v ) (cid:8) V n ( v ) , h ( a ) ( v ) (cid:9)(cid:17) , n ∈ Q + (83)that were constructed by using the relation in (32). At the quantum level, the aforementioned quantity becomes thetrace of the product of the su (2) generator and the operators in (21), and its structure varies for different matter fields.Moreover, even in the Hamiltonian constraint for a given field, the elements with different power of volume in (83)are present — compare (29), (30), and (31). To study these differences, the expression in (75) has to be decomposedmore specifically.One should first observe the following relation, (cid:28)D(cid:2) ˆ φ , ˆ H ( matt ) (cid:3)E ( matt ) (cid:29) ( gr ) = (cid:28)h ˆ φ , (cid:10) ˆ H ( matt ) (cid:11) ( gr ) i(cid:29) ( matt ) . (84)This leads to the conclusion that the structure of ∆ | i ( gr ) φ depends only on the matrix element (cid:10) ˆ H ( matt ) (cid:11) ( gr ) . Then,by splitting the matter sector Hamiltonian into the contributions from different fields φ α i , one finds the followingdecomposition, H ( matt ) = X α H ( φ α ) = X α (cid:16) H ( φ α ) one + H ( φ α ) two + ... (cid:17) =: X α (cid:18) X elements H ( φ α ) element (cid:19) . (85)The second splitting in the formula above is given by the introduction of the terms H ( φ α ) one , H ( φ α ) two , ... that label differentelements in the φ α field Hamiltonian. For instance, the Hamiltonian of φ A decomposes as follows: H ( A ) =: H ( φ A ) = H ( φ A ) elec + H ( φ A ) magn .The matrix element derivation is a linear operation, thus without loss of generality it is enough to focus on a singleelement D ˆ H ( φ α ) element E ( gr ) = H ( φ α ) element (cid:16) δ ( φ α ) element + δ ′ ( φ α ) element + ... (cid:17) , (86)where δ ( φ α ) element ∝ / ¯ m , δ ′ ( φ α ) element ∝ / ¯ m , etc . For simplicity, the terms of order / ¯ m and smaller are going to beneglected. Consequently, the quantum GR corrections to the matter sector are expressible by ∆ | i ( gr ) φ = 1i ~ X α X elements (cid:28)h ˆ φ α , ˆ H ( φ α ) element i(cid:29) ( matt ) δ ( φ α ) element , (87)where the linearity of a commutator was used. Finally, it should be pointed out that when the correspondenceprinciple in (70) is applied, the corrections become explicitly dreibein-dependent, thus also metric tensor-dependent.In the case of the vector field in the cosmological framework, the structure of this dependence is readable from theexpression (cid:28)D(cid:2) ˆ φ , ˆ H ( A ) (cid:3)E ( A ) (cid:29) ( gr ) = X a (cid:28)D(cid:2) ˆ φ , ˆ H ( A )( a ) (cid:3)E ( A ) (cid:29) ( gr ) = X a (cid:28)h ˆ φ , ˆ H ( A )( a ) i(cid:29) ( A ) (cid:16) δ ( A )( a ) (cid:17) , (88) i In this general approach , φ α represents any matter field. In the case of the simplified cosmological model with bosonic matter and withthe Hilbert space given in (33), only two different matter fields are considered: φ A := A and φ ϕ := ϕ . δ ( A )( a ) ∝ ¯ k (cid:0) p ( a ) (cid:1) . (89)The form of the preceding outcome reflects the symmetry between the regularized elements in the Hamiltoniancontributions in (26) and (27). The analogous matrix element of the scalar field leads to the result (cid:28)D(cid:2) ˆ φ , ˆ H ( ϕ ) (cid:3)E ( ϕ ) (cid:29) ( gr ) = (cid:28)h ˆ φ , ˆ H ( ϕ ) mom i(cid:29) ( ϕ ) (cid:16) δ ( ϕ ) mom (cid:17) + X a (cid:28)h ˆ φ , ˆ H ( ϕ )( a ) der i(cid:29) ( ϕ ) (cid:16) δ ( ϕ )( a ) der (cid:17) + (cid:28)h ˆ φ , ˆ H ( ϕ ) pot i(cid:29) ( ϕ ) , (90)where δ ( ϕ ) mom ∝ X a ¯ k (cid:0) p ( a ) (cid:1) , (91) δ ( ϕ )( a ) der ∝ X b = a ¯ k (cid:0) p ( b ) (cid:1) . (92)This outcome is not symmetric with respect to the metric tensor structure, hence the BI of the indicated quantumGR corrections is explicitly broken. IV.2. General quantum relativity
Before describing the general covariance breaking in more details by the use of inconsistently selected regularizationmethods, it is worth to state more precisely what is the general postulate of relativity [6] in the context of quantumphysics.This postulate was originally formulated as follows.
The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates,that is, are covariant with respect to any substitutions whatever (generally co-variant). . . . For the sum of all substitutions in any case includes those which correspond to all relative motionsof three-dimensional systems of co-ordinates. . . . Moreover, the results of our measuring are nothing butverifications of such meetings of the material points of our measuring instruments with other materialpoints, coincidences between the hands of a clock and points of the clock dial, and observed point-eventshappening at the same place at the same time. [6]The postulate and its explanation (see also their earlier formulations in [7–9]) consists of two restrictions on a physicaltheory. The theory has to be SE, i.e. the equations must be equivalently expressed in all systems of coordinates. Italso has to be BI, i.e. the predictions of the theory must be invariant under any substitution of a reference frame.It should be emphasized that SE does not require any independence of coordinates or a metric involving formalism.The SE model can be explicitly formulated in a particular system of coordinates, however, the choice of any othersystem has to lead to the equivalent formulation. Furthermore, BI does not assume any restriction on a formalism.The BI theory has to provide universal predictions independently of any particular characteristics of an observer.In this article it is assumed that a hypothetical CQGR model satisfies SE and BI. This does not provide any a priori reference to gravity. Whatever content is included in the theory, it has to be SE and BI. This clarifies what GR meansin the context of CQGR. Now, the meaning of quantum should be specified. This term is going to be understood inthe sense of the quantum mechanical formalism adjusted to each component of CQGR. In particular, this assumesthe probabilistic distributional framework, noncommutative relations and the predictions formulated as expectationvalues. The description of these features is well understood in terms of interacting quantum fields. Therefore, CQGRis going to be understood as QFT that satisfies SE and BI. It is not difficult to realize how to impose these latterrestrictions on QFT.The SE formulation of QFT means that the formulation of interacting quantum fields is equivalent in any systemof coordinates. Therefore the action of any operator on any state cannot in reality depend on any reference frame.However, an apparent dependence is not excluded. This means that operator equations cannot be localized at anyfixed spacetime points and the scalar product cannot depend on the position in a Fock space. All the interactionsmust consist of the relations between certain characteristics of quantum operators and states and cannot depend on8any fixed reference frames that would classify these characteristics. Simply speaking, the framework of QFT has tobe generally relative.It is easy to see that the phenomenological models discussed in Sec. II.4 violate the SE requirement. This excludesthese models from being a candidate for a cosmological limit of CQGR funded by using the LQG’s framework. Itis then surprising that there are several studies toward formulation of phenomenological predictions by using theseeffective models. These models directly break SE in their constructions, which are based on the LQG’s formalismthat was created to describe the QFT of gravity in the SE way. Thus, these mentioned phenomenological applicationsas the predictions that could explain reality are methodologically inconsistent.The framework satisfying the SE requirement was introduced in Sec. III.1. In Sec. IV.3 details of the BI verificationare explained. In general, the BI formulation of QFT means that the predictions of the theory are independent of anyreference frame. In the context of quantum physics this restriction is directly related to the notion of observables.These are the indirectly measurable quantities in QFT. However, the predictions are not formulated in terms ofobservables. Only the eigenvalues of these operators are directly measurable. Therefore the BI of a quantum theorymeans BI spectra of all observables. These are the only quantities in which an observer verifies the laws of nature.They are described relatively if their predictions are formulated independently of how, when, and where can be tested. /c ~ G Galilei-Newtoniantheory non-relativisticquantum mechanicsclassical mechanics(Newtonian gravitation) non-relativisticquantum theoryspecial relativity quantum fieldtheorygeneral relativity quantumgeneral relativityFIG. 3. Bronstein cube in c − ~ G orientation Eventually, the model of quantum GR (see FIG. 3) that is based on the framework of LQG assumes canonicalquantization procedure. This last property defines how the classical and quantum descriptions are related. Aspointed in [20], this relation is not clear in LQG, which is a separate problem and is not going to be discussed in thisarticle. In general, the canonical quantization is defined as a replacement of canonical variables with their operatorrepresentations and a replacement of Poisson brackets with commutators. Then, the change of variables into operatorsshould preserve the relative orientations and positions of these objects accordingly to all the frames indicated in atheory (this is not properly implemented in the original canonical formulation of LQG [1, 2], cf. [20]). In this way,the gauge invariance is properly preserved both locally and globally.Concluding, the properly formulated candidate for CQGR has to satisfy the following restrictions.a) Quantization is performed in a canonical procedure that preserves gauge invariance both locally and globally.b) Equivalence principle is satisfied strongly for all the fields and all the coordinates systems.c) All the predictions are background independent, thus are the same for any observer.The first condition allows to analyze phase space-reduced versions of CQGR and obtain results related to the generalmodel. The second restriction is imposed already in the construction of theories, hence one can focus only on themodels that satisfy this condition. To verify the last restriction, kinematics of the theory has to be derived and itsdynamics has to be formulated.9Finally, it is worth to emphasize that besides the theoretical notion of the analysis in this article, one can also indicateits practical value. In order to describe the physical meaning of the correct formulation of CQGR cosmologicalreduction, an independent consideration of the matrix element of HCO only with respect to the matter or to thegravitational degrees of freedom is going to be discussed. This will be presented on the Bronstein cube [57] illustrativediagram in FIG. 3.The semiclassical low energy approximation of quantum matter excitations corresponds to the classical matter fieldstheory on quantum geometry. The semiclassical slightly curved approximation of quantum geometry coincides withQFT on classical curved spacetime. The first approximation corresponds to taking the ~ → limit and the secondone to taking the G → limit. These approximations can be relevant in different physical processes — see FIG. 4. c = 1 ~ energy-dominated G geometry-dominatedgeneralrelativity classical fieldtheoryon quantumgeometry quantumgeneralrelativityquantum field theoryon curved spacetimespecialrelativity quantum fieldtheoryFIG. 4. Planck hyperbola, l P = constant The classical and flat limit of CQGR can be (not rigorously) understood as l P → . To study only the semiclassicalcorrections of the theory, one can expand the results around a small (but not zero) value of the Planck length. Thisexpansion can be represented by the hyperbola in FIG. 4. This figure expresses the c = 1 face of the Bronstein cubein FIG. 3 and the sketched curve is going to be called the Planck hyperbola.From the cosmological perspective, the well-known QFT on curved spacetime approach [58] can be applied forinstance to explain the details of the inflation process [59]. The classical field theory on quantum geometry can beused as an approximation of the early phase in the Universe evolution. This model predicts for instance a big bouncescenario at the origin of the Universe [60, 61]. Therefore, to understand the whole Universe evolution even onlyapproximately, by studying the semiclassical corrections of CQGR, one needs to be able to smoothly move along thePlanck hyperbola. Then, to be sure that this move is smooth, the cosmological reduction of CQGR has to be properlyconstructed. Although, if the general theory is not completely formulated, one cannot be absolutely sure that thecosmological limit of CQGR can be precisely applied as a model of the Universe. This argumentation provides thephysical motivation for the verification of the covariant structure of the corrections indicated in Sec. IV.1. IV.3. Violation of general covariance
The explicit form of the quantum GR corrections indicated in expressions (88) and (90) depends on the state onwhich the operator constructed from the term in (83) acts. The symmetrized shadow states in (67), which satisfybasic properties of correctly formulated coherent states, are based on the links excitations states in (60). Thus onlythe results of the reduced operators actions on these links excitations states have to be verified.The holonomy operator in (53) of the constant Abelian connection resulting from the phase space reduction of0CQGR leads to the following simplification of the corrections generating operator,tr (cid:18) τ i ˆ h − a ) h ˆ V n , ˆ h ( a ) i(cid:19) = sin ˆ c i ( a ) ! ˆ V n cos ~ ˆ c ( a ) ! − cos ~ ˆ c ( a ) ! ˆ V n sin ˆ c i ( a ) ! . (93)For clearness, the projection of the spatial directions into the internal ones and the related correction of variablesweights [15, 32] is not explicitly written in this expression. The vector symbol over the connection in ~ ˆ c ( a ) indicatesthe direction-independent series representation of the cosine operator functional cos ~ ˆ c ( a ) ! = ∞ X k =0 ( − k (2 k )! (cid:18) ˆ c i ( a ) ˆ c i ( a ) (cid:19) k . (94)By assuming only cuboidal cells, the action of the volume operator in (93) expressed in terms of the reducedmomenta in (52) simplifies into the operator constructed from the following quantity, ¯ V := 1 ε Z ε d x p p p p = p p p p = ε √ ¯ q . (95)The square root of operators after quantization is going to be derived by the expansion of the radicand around thecoherent state, resulting in the expression ˆ¯ V n = ε n (cid:0) h ˆ¯ q i (cid:1) n ∞ X k =0 (cid:18) n/ k (cid:19)(cid:18) ˆ¯ q − h ˆ¯ q ih ˆ¯ q i (cid:19) k , (96)where the expectation value of the ˆ¯ q operator is h ˆ¯ q i = ¯ k ¯ m ¯ m ¯ m . (97)The reduced holonomy in (93) leads to the states modifications indicated in (55). The action of the volume operatorin (96) on these modified states can be expressed in the form of a power series, ˆ¯ V n (cid:12)(cid:12)(cid:12)(cid:12) ¯ m ( i ) v ± (cid:29) = (cid:0) ¯ k ¯ m v ¯ m v ¯ m v (cid:1) n " ± n m ( i ) v + n ( n − (cid:0) ¯ m ( i ) v (cid:1) ± n ( n − n − · (cid:0) ¯ m ( i ) v (cid:1) + O (cid:0) ¯ m ( i ) v (cid:1) ! ¯ m ( i ) v ± (cid:29) . (98)Here, the large quantum numbers assumption was needed. Then, the action of the quantum GR corrections generatingoperator is easy calculable and readstr (cid:18) τ ( i ) ˆ h − j ) h ˆ V n , ˆ h ( j ) i(cid:19) O k (cid:12)(cid:12)(cid:12) ¯ m ( k ) v E = i n m ( i ) v (cid:0) ¯ k ¯ m v ¯ m v ¯ m v (cid:1) n δ ( i )( j ) " n − n + 82 · (cid:0) ¯ m ( i ) v (cid:1) + O (cid:0) ¯ m ( i ) v (cid:1) ! O k (cid:12)(cid:12)(cid:12) ¯ m ( k ) v E . (99)Consequently, the values of the dimensionless corrections in (89), (91), and (92) are δ ( A )( i ) = 72 (cid:0) ¯ m ( i ) v (cid:1) → π γ l P ε q ( i )( i ) | ¯ q | , (100) δ ( ϕ ) mom = 72 X i (cid:0) ¯ m ( i ) v (cid:1) = X i δ ( A )( i ) , (101)and δ ( ϕ )( i ) der = 652 · X j = i (cid:0) ¯ m ( j ) v (cid:1) = 6584 X j = i δ ( A )( j ) , (102)respectively.1For the analysis in this article, only the precise structure of corrections is needed. However, the reader interested inthe explicit values can easily derive each correction by using the correspondence principle in (70) — as demonstratedon the right-hand side of formula (100). The l P /ε ratio must be regularized by a cut-off on the value of the regulator,which has to be consistent with the large spin approximation in (98). The condition ε > | γ | l P is acceptable, butit should be replaced with ε ≫ γ l P . Otherwise, keeping the trigonometric form of the reduced holonomy in (93)has no sense and the approximation sin (cid:0) ˆ c i ( a ) / (cid:1) ≃ ˆ c i ( a ) / , cos (cid:0) ˆ c i ( a ) / (cid:1) ≃ is indistinguishable from that form. Thisoccurs for instance by using the so-called area gap, ε ≈ γ l P , cf. [12]. This cut-off imposed on the phase space-reduced CQGR leads to the domination of the inverse volume corrections over any other quantum corrections andHCO becomes almost exactly an eigenvector of the basis states.Finally, one can test BI of the cosmologically reduced CQGR. By applying the phase space reduction, all the gaugesymmetries of the theory became restricted to their reduced versions, but not violated or modified. Therefore by theinspection of the reduced diffeomorphism transformations of the CQGR semiclassical limit, general covariance can beverified.The main vector field observables are the ˆ E a and ˆ B a operators. Their expectation values are the electric vector fielddensity and the magnetic pseudovector field density. They are the weight physical quantities that are the measurablemodes of an electromagnetic wave. Another observable is HCO. Its expectation value is the weight scalar densitythat represents the energy density. The difference between the energy densities related to different spacetime pointsis also an explicitly measurable quantity. The semiclassical limit of the related HCO reads (cid:10) ˆ H ( A ) (cid:11) = X i (cid:10) ˆ H ( A )( i ) (cid:11) = X i (cid:10) ˆ H ( A )( i ) (cid:11) ( A ) (cid:16) δ ( A )( i ) (cid:17) . (103)It is clear that the reduced diffeomorphism transformations describable by the metric tensor contraction are thesame in the electric and magnetic elements of the scalar constraint. Moreover, the corrections in (103) are preservedconcerning the temporal diffeomorphism transformation, i.e. these corrections are independent of dynamics — see(88). Therefore, although the quantum GR corrections apparently change the contraction with the metric tensor, therelative diffeomorphism symmetry along all the reduced directions is not modified.Considering then the quantum system of the vector field and gravity, one could wish to be able to restore the explicitclassical form of the general covariance specifying spatial metric tensor. This would allow to apply the methods of QFTon curved spacetime to the expectation values of the matter degrees of freedom in an effective model. This effectiveprocedure would be possible by assuming that the Hamiltonian density of the whole system equals zero. Then, theeffective ‘covarianization’ method can be defined as the multiplication of all the scalar constraint elements by theinverse of (cid:16) δ ( A )( i ) (cid:17) . In this way, the GR corrections will be moved to the Hamiltonian gravitational contribution.This contribution will effectively represent the gravitational sector of the scalar constraint coupled with the standardQFT representation of the electromagnetic field on classical curved spacetime.In the case of the scalar field, the situation is completely different. It is worth emphasizing that the variable π is a weight pseudoscalar density, but ∂ a ϕ and ϕ are a one-form and a scalar, respectively. This explains why thesemiclassical limit of HCO has different GR corrections for each element, (cid:10) ˆ H ( ϕ ) (cid:11) = X i (cid:20)D ˆ H ( ϕ )( i ) mom E + D ˆ H ( ϕ )( i ) der E + D ˆ H ( ϕ )( i ) pot E(cid:21) = X i (cid:20)D ˆ H ( ϕ )( i ) mom E ( ϕ ) (cid:16) δ ( A )( i ) (cid:17) + D ˆ H ( ϕ )( i ) der E ( ϕ ) (cid:18) − δ ( A )( i ) + 6584 X j = i δ ( A )( j ) (cid:19) + D ˆ H ( ϕ )( i ) pot E ( ϕ ) (cid:16) − δ ( A )( i ) (cid:17)(cid:21)(cid:16) δ ( A )( i ) (cid:17) + O (cid:18)(cid:16) δ ( A )( i ) (cid:17) (cid:19) , (104)where D ˆ H ( ϕ )1 mom E = D ˆ H ( ϕ )2 mom E = D ˆ H ( ϕ )3 mom E = D ˆ H ( ϕ ) mom E and D ˆ H ( ϕ )1 pot E = D ˆ H ( ϕ )2 pot E = D ˆ H ( ϕ )3 pot E = D ˆ H ( ϕ ) pot E . Thereforethe relative diffeomorphism symmetry is not preserved at the level of corrections. These GR corrections are thesemiclassical predictions, hence they are potentially measurable quantities. This asymmetry indicates the backgrounddependence of the result and thus breaks general covariance. Even by neglecting the selfinteraction terms, the relativelocal diffeomorphism symmetry of the momentum and derivative sectors is not equivalent. Consequently, the effectivecovarianization procedure is also not applicable to the expression (104).The last result demonstrates that the node representation leads to the background-dependent structure of GRcorrections. The simplest resolution of this problem is to apply the isotropic vector field representation to describethe quantity that classically is expressed by the scalar field.By assuming this representation, the effective Hamiltonian of the bosonic system on a cuboidal lattice can be2written as the following sum, ¯ H = X i (cid:16) ¯ H ( gr )( i ) + ¯ H ( A )( i ) + ¯ H ( ϕ )( i ) (cid:17) =: X i ¯ H ( i ) . (105)One can assume that the selfinteracting scalar field is represented by the isotropic Proca Hamiltonian. Then, byfixing the total energy to zero by setting H = 0 , the effective covarianization method will be the removal of the GRcorrections in the procedure defined by ¯ H ( i ) covar. −→ ¯ H ( i ) (cid:16) δ ( A )( i ) (cid:17) − . (106)As a result, the whole free matter sector becomes corrections-independent. However, the mass term and any otherpotential contribution becomes shifted down by the factor (cid:0) − P i δ ( A )( i ) (cid:1) . This is an interesting prediction, however,it is not a fundamental theory result. Therefore, as in the case of the effective models recalled in Sec. II.4, one shouldnot consider this outcome as a prediction, for instance, of the inflaton field’s real mass loss. It could be only used toeffectively describe this phenomenon if it would be observed.An even more interesting observation concerns the inclusion of the fermionic sector. The scalar constraint ofthe system that describes all fundamental interactions in the cosmologically reduced framework can be effectivelyexpressed by H = Z Σ t d x N (cid:20) κ √ q (cid:16)(cid:0) F icd − ( γ + 1) ǫ ilm K lc K md (cid:1) ǫ ijk E aj E bk (cid:17) + g A √ q q cd (cid:0) E aI E bI + B aI B bI (cid:1) + 2 √ q κ Λ q cd q ab + H ( ϕ ) cd q ab + H ( ψ ) abcd (cid:21) δ ca δ db =: H abcd δ ca δ db . (107)Here, all the matter sector is assumed to be smeared along the links of the cuboidal lattice. The torsional contributionfrom the fermionic sector is assumed to by given by the procedure in [62, 63] and the regularization of the Diraccontribution follows either the method in [2, 13], or the one in [62, 63] adjusted to the links smearing of the fermionicfield. Then the analog of (25) and (28) is the following Hamiltonian constraint H ( ψ ) = Z Σ t d x N √ q (cid:20) ǫ ijk ǫ abe tr (cid:18) τ i ε h − c ( x ) n V ( x ) , h c ( x ) o(cid:19) tr (cid:18) τ j ε h − d ( x ) n V ( x ) , h d ( x ) o(cid:19) ( fermionic ) ke (cid:21) δ ca δ db . (108)Here, ‘ ( fermionic ) ’ denotes the Dirac field’s degrees of freedom. Consequently, the related quantum GR correctionstake analogous form to (103), but they are antisymmetric. Conversely, the structure of the bosonic fields Hamiltonianhas the form H abcd δ ca δ db , hence it is symmetric in the pairs of spatial-internal indices (the external and internal direc-tions are indistinguishable after the reduction). This breaks the BI of the system, however, this violation occurredin the expected manner. One can anticipate that in the antisymmetric sector of CQGR, the corresponding diffeo-morphism invariance will be correctly preserved. To complete this conjecture it is worth to mention that the relatedcovarianization will be given by the expression H abcd ( v ) covar. −→ H abcd ( v ) (cid:18) δ ( A )( a ) ( v ) (cid:19) − (cid:18) δ ( A )( b ) ( v ) (cid:19) − . (109) V. CONCLUSIONS
This article revealed the problems with the accurate implementation of general covariance in the matter sector ofCQGR, where the theory is assumed to be constructed by using the LQG’s framework. By general covariance, the BIcondition originally postulated by Einstein was considered. This quantity together with SE, called also the equivalenceprinciple, forms the general principle of relativity.The BI violation was a consequence of using inconsistent regularization methods. This inconsistency was regardingthe local spatial diffeomophism symmetry breakdown in the continuous to discrete transition of the multi-field system.Then, the lack of general covariance was revealed in the structure of the semiclassical corrections of the cosmologicallyreduced CQGR.In the LQG’s framework, the symmetry of the canonical fields lattice smearing is the symmetry of the links of thislattice. The links structure specifies the discrete diffeomorphism transformations directions distribution. Therefore,3it is not surprising that by using the locally diffeomophisms breaking representation of a field located at nodes, thegeneral covariance of the system is violated. It should be emphasized that the diffeomophism symmetry becomeslocally broken in the following series of steps. First the propagating gravitational degrees of freedom are smearedby using the holonomy-flux representation, where the relation in (32) is assumed. Next, the phase space reduction,which preserves all the reduced symmetries, is implemented. Then, the theory is quantized and the semiclassicallimit is derived on the Gaussian states that are picked at the momenta (or volume) eigenvalues. Finally, by thecorrespondence principle, the original metric structure is restored and its asymmetry in the scalar field Hamiltonianelements is revealed. What needs to be added to this list, is the fact that the scalar field degrees of freedom werelattice-regularized at nodes, conversely to all of the other variables, which were smeared accordingly to the linksstructure. Moreover, all but the first step were exact, however, this step considered only the essential techniques ofLQG. Furthermore, the approximations in this step (before quantization) were reproducing the original, continuousformulation of the theory exactly, by taking the limit ε → . Anything that could be questioned in the analysis inthis article, concerns the methods of LQG. The phase space reduction was implemented in the standard manner [64]in which the Dirac brackets take the form of the Poisson ones, cf. [15].Concluding, the following no-go theorem concerning the lattice regularization in the framework of LQG can beformulated. Let a model of quantum general relativity be considered, where the loop quantum gravitational techniquesare used to regularize and quantize gravitational degrees of freedom. By assuming the systems-equivalent descriptionand background-independent predictions of this model, the lattice regularization of matter minimally coupled togravity is restricted. The matter variables selected for the lattice smearing should be represented by vector densitiesto ensure that all the coupled gravitational degrees of freedom are written in an appropriate form. Moreover, thisrepresentation allows to express the matter degrees of freedom on the lattice in terms of the holonomy-flux formalism,which is also the representation of the gravitational variables. By choosing a nodes smearing, the general covarianceof the theory predictions will be violated.
Furthermore, it is worth to add that in the case of the properly lattice-regularized electromagnetic field, the smeared variables are the ones that have the explicit and real physical meaning.They are the electric and magnetic fields.One more comment is worth to be added at the end. In this article it was not certainly demonstrated that thefermionic matter must be lattice-regularized accordingly to the aforementioned theorem. However, so suggesting indi-cations were found. Therefore, it is probable that the fermionic variables proposed in the context of LQG (representedby the Grassmann-valued scalar half-densities) [2, 13, 24, 62] should be replaced by appropriate vector half-densities.The weight / would reflect the fermionic otherness from the weight of the vector representations of bosons. ACKNOWLEDGEMENTS
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