Time evolution in deparametrized models of loop quantum gravity
aa r X i v : . [ g r- q c ] F e b Time evolution in deparametrized models of loop quantum gravity
Mehdi Assanioussi, ∗ Jerzy Lewandowski, † and Ilkka M¨akinen, ‡ Faculty of Physics, University of Warsaw,Pasteura 5, 02-093 Warsaw, Poland
An important aspect in understanding the dynamics in the context of deparametrizedmodels of LQG is to obtain a sufficient control on the quantum evolution generated by agiven Hamiltonian operator. More specifically, we need to be able to compute the evolutionof relevant physical states and observables with a relatively good precision. In this article,we introduce an approximation method to deal with the physical Hamiltonian operators indeparametrized LQG models, and apply it to models in which a free Klein-Gordon scalar fieldor a non-rotational dust field is taken as the physical time variable. This method is basedon using standard time-independent perturbation theory of quantum mechanics to definea perturbative expansion of the Hamiltonian operator, the small perturbation parameterbeing determined by the Barbero-Immirzi parameter β . This method allows us to definean approximate spectral decomposition of the Hamiltonian operators and hence to computethe evolution over a certain time interval. As a specific example, we analyze the evolutionof expectation values of the volume and curvature operators starting with certain physicalinitial states, using both the perturbative method and a straightforward expansion of theexpectation value in powers of the time variable. This work represents a first step towardsachieving the goal of understanding and controlling the new dynamics developed in [25, 26]. The Hamiltonian formulation of general relativity encodes the dynamics in constraints. Thistranslates in a frozen picture of the dynamics where there is no time flow nor evolution of physicalquantities. This specific aspect raises several serious issues in the interpretation of a quantumtheory of gravity, as one fails to make the link to the experimental setup with definite instantsof time. One of the approaches to circumvent this problem of time is the deparametrizationof gravity [1–11]. This approach however carries the drawback of choosing a specific globalreference frame to parametrize either time or both space and time, hence the description andinterpretation of the physics derived within the framework would be tied to this choice of theframe. Nevertheless, this approach turns out to be technically very efficient in constructingcomplete quantum models where gravity is fully quantized [10, 21, 25], bypassing the difficultiesencountered in the case of the standard vacuum theory. Those models then become a veryrich playground to test the many technical steps of the quantization procedures along with thedevelopment and analysis of new methods and ideas to answer even more complex questionsconcerning the semi-classical and coherent states, the quantum observables and the continuumlimit of the quantum theory.In this article we consider two LQG [14–17] models where deparametrization of the scalarconstraint is performed with respect to the free Klein-Gordon scalar field [1, 10, 21, 25] andthe non-rotational dust field [9–11]. The main difference between the two models is the finalform of the physical Hamiltonian which dictates the dynamics of the gravitational degrees offreedom with respect to the relational time provided by the considered scalar field. Following thequantization presented in [25, 26], the Hamiltonian operators in the quantum theories are denselydefined on a non-separable physical Hilbert space. Since a complete spectral decomposition ofthose Hamiltonian operators is so far unavailable, it is imperative to develop and use approximatemethods in the analysis of the dynamics induced by these operators. A straightforward approachis to consider the expansion of the evolution operator in powers of the time parameter and ∗ [email protected] † [email protected] ‡ [email protected] introduce a truncation of the expansion at a certain fixed order of time. In this case the evolutionoperator reduces to a finite sum of terms where each one is calculated through a finite number ofsuccessive actions of the Hamiltonian operator. Such truncation forms a valid approximation ofthe time evolution when the time interval under consideration is sufficiently small. Nevertheless,this method is not enough to concretely calculate the evolution in the deparametrized model witha massless scalar field. An interesting alternative, which we propose here, is to use standard time-independent perturbation theory of quantum mechanics to introduce a perturbative expansion ofthe Hamiltonian operator, the small perturbation parameter being determined by the Barbero-Immirzi parameter β , which requires β ≫
1. This method allows us to define an approximatespectral decomposition of the Hamiltonian operators and hence to compute the evolution in acertain time interval.
I. DEPARAMETRIZED MODELS AND PHYSICAL HAMILTONIANS
In the following we will present a short overview of the two deparametrized models we areinterested in. Then we will briefly present the LQG quantization of the two models and thecomplete quantum theories. Note that in both models the spatial diffeomorphism constraintsare not solved classically and they will be carried to the quantum theory.
A. Gravity coupled to a massless scalar field
As any covariant theory, general relativity minimally coupled to a scalar field φ in its Hamil-tonian formulation is set as a fully constrained system. Using Ashtekar-Barbero canonical vari-ables ( A ia , E bj ) for the gravitational field ( a, b are spatial indices while i, j internal SU (2) indices)[12, 13], the Hamiltonian analysis reveals the following constraints [20, 21]: G i ( x ) = ∂ a E ai + ǫ ijk A ja E ak , (I.1) C a ( x ) = C gr a ( x ) + π ( x ) φ ,a ( x ) , (I.2) C ( x ) = C gr ( x ) + 12 π ( x ) p q ( x ) + 12 p q ( x ) E ai ( x ) E bi ( x ) φ ,a ( x ) φ ,b ( x ) , (I.3)the Gauss constraints, the spatial diffeomorphism constraints and the scalar constraints respec-tively. The quantity π ( x ) is the conjugate momentum to φ ( x ), q ( x ) := (cid:12)(cid:12)(cid:12) ǫ abc ǫ ijk E ai ( x ) E bj ( x ) E ck ( x ) (cid:12)(cid:12)(cid:12) is the determinant of the densitized triad E ai ( x ), and the functionals C gr a ( x ) and C gr ( x ) in theabove constraints are the gravitational parts. These have the following expressions C gr a ( x ) = 1 kβ F iab ( x ) E bi ( x ) , (I.4) C gr ( x ) = − πGβ (cid:18) ǫ ijk E ai ( x ) E bj ( x ) F kab ( x ) p | det E ( x ) | + (1 + β ) p | det E ( x ) | R ( x ) (cid:19) , (I.5) G being the Newton constant, β the Barbero-Immirzi parameter, F iab ( x ) the curvature of theconnection A ia ( x ), and R ( x ) the Ricci scalar obtained from the metric tensor q ab on the 3-dimensional Cauchy hypersurface Σ, the relation between E ai and the inverse metric q ab beinggiven by q ab ( x ) = q ( x ) E ai ( x ) E bi ( x ). In this article we call Euclidean part the part of the scalarconstraint containing F iab ( x ), and Lorentzian part the part containing R ( x ).The deparametrization procedure sums up to rewriting the scalar constraint C ( x ) so that itis linear in the momenta π ( x ) and the explicit dependence on the field φ ( x ) is removed. Thescalar field φ ( x ) can be then chosen to be the physical time. Assuming that the constraints (I.2)are satisfied, the scalar constraints take the form C ( x ) = C gr ( x ) + 12 π ( x ) p q ( x ) + 12 π ( x ) p q ( x ) E ai ( x ) E bi ( x ) C gr a ( x ) C gr b ( x ) . (I.6)Solving this equation for the momenta π ( x ) leads to a new form of the scalar constraint C ′ ( x )equivalent to (I.3) in a specific region of phase space, namely C ′ ( x ) = π ( x ) − r −√ qC gr + √ q q ( C gr ) − q ab C gra C grb =: π ( x ) − h SF ( x ) . (I.9)The constraints C ′ ( x ) strongly commute [5] and a Dirac observable O ( x ) on the phase space,i.e. a function which commutes with the new set of constraints, would satisfy d O dφ ( x ) = { O , h SF ( x ) } , (I.10)This equation shows precisely how the quantity h SF ( x ) arises as a physical Hamiltonian densityin the reference frame of the scalar field φ , that it is the foliation with slices of constant valueof the scalar field. Note that h SF ( x ) is a functional of the gravitational variables only, henceall the redundant degrees of freedom in the scalar constraints (I.9) are absorbed in the scalarfield φ . The dynamics of the system is then promoted from imposing constraints, to describingevolution of the gravitational degrees of freedom with respect to the physical time set by thescalar field. B. Gravity coupled to non-rotational dust
The model of gravity coupled to non-rotational dust [6, 9–11, 22] is to some extent very similarto the one of gravity coupled to a massless scalar field described above. The difference arisesfrom adding a potential term in the action of the system which is analogous to a cosmologicalconstant term. While the Gauss and the spatial diffeomorphism constraints are identical to themassless scalar field case, the mentioned difference appears explicitly in the scalar constraintsof the theory, namely C ( x ) = C gr ( x ) + 12 ρ π ( x ) p q ( x ) + ρ p q ( x ) E ai ( x ) E bi ( x ) φ ,a ( x ) φ ,b ( x ) + ρ p q ( x ) , (I.11)where ( φ, π ) are the dust field variables and ρ is a Lagrange multiplier appearing in the action ofthe system and which must satisfy certain second class constraints. Replacing ρ in (I.11) by itsexplicit form obtained from solving the second class constraints, and using the diffeomorphism Sign choices arise in the expression of π ( x ) when solving (I.6). Those choices corresponds to treating differentregions of the phase space: π ≥ / ≤ q ab ( x ) φ ,a ( x ) φ ,b ( x ) q ( x ) . (I.7)We choose the phase space region corresponding to π ≥ (cid:12)(cid:12)(cid:12) q ab ( x ) φ ,a ( x ) φ ,b ( x ) q ( x ) (cid:12)(cid:12)(cid:12) , and − √ qC gr + √ q q ( C gr ) − q ab C gra C grb ≥ , (I.8)which interestingly contains the sector of spatially homogeneous spacetimes [21]. The second condition on thegravitational constraints must also be implemented in the quantum theory. constraints as in the previous case, we obtain the new simplified scalar constraints C ′ ( x ) = π ( x ) + C gr =: π ( x ) − h D ( x ) . (I.12)This equation, similarly to (I.9), presents the quantity h D ( x ) as the physical Hamiltonian densityfor the dynamics of the gravitational degrees of freedom in the reference frame of the dust. C. Quantum theory
The quantization is performed along the canonical program of LQG (see [21, 25] for the scalarfield case). In both models, the kinematical Hilbert space H kin is defined as the completion(with respect to the norm defined by a natural scalar product [18]) of the space of cylindricalfunctions of the connection variable A , i.e. functions depending on the differential 1-form A = A ia τ i ⊗ dx a (with τ i the generators of the su (2) algebra) through finitely many holonomies ofthe connection, which are SU (2) group elements. The space H kin admits a basis called the spinnetwork basis, where each element is labeled by a closed embedded graph, spins on the edgesof the graph corresponding to SU (2) representations of the holonomies, and SU (2) invarianttensors at the vertices of the graph intertwining the representations meeting at those vertices.The fundamental operators are holonomies ˆ h ( l ) γ , acting as multiplicative operators, defined witharbitrary embedded curves γ and in arbitrary SU (2) irreducible representations l , and derivativeoperators ˆ J x,γ,i associated to curves γ starting at a point x in Σ and acting in the su (2) algebra.The Gauss and spatial diffeomorphism constraints are then implemented through a groupaveraging procedure [19]. The resulting space is a Hilbert space of SU (2) gauge invariant andspatial diffeomorphism invariant states, we denote it H G Diff , with a scalar product induced fromthe scalar product on H kin . The space H G Diff is then the physical Hilbert space of the quantumtheory in both models.The last ingredient to complete the quantization program is to define a quantum Hamiltonianoperator which would generate the quantum dynamics through a Schr¨odinger-like equation i ~ ddT | ψ i = ˆ H | ψ i (I.13)for any state | ψ i ∈ H G Diff , where T is the physical time equal to the value of the deparametriza-tion field, either the scalar field or the dust field. This task can be achieved in a satisfactorymanner through a careful regularization of the classical expressions of R Σ d x h SF ( x ) =: H SF and R Σ d x h D ( x ) =: H D . Following [25] for the massless scalar field case, and [26] for the dust fieldcase , in which different regularizations than the one due to Thiemann [27, 28] were proposed,symmetric Hamiltonian operators acting on H G Diff with a dense domain can be defined in bothmodels. The ambiguity of the choice between various valid operators arises from the differentavailable symmetric extensions of the non-symmetric operator derived from the regularizationprocedure. An ultimate criterion which could remove this ambiguity would be obtained throughconfronting the semi-classical physics induced by a certain choice of Hamiltonian operator withthe predictions of the classical model.In this article we will proceed with a specific choice of symmetric extension. Explicitly, thechosen Hamiltonian operators are as follows: In the present work we modify the Euclidean and Lorentzian parts of the Hamiltonian compared to the oneintroduced in [26]: on one hand we change the ordering of the operators in the Lorentzian part, on the otherother hand, instead of using Thiemann’s trick in defining the Euclidean operator, we use the “inverse volume”operator in the final expression, in a similar way that is used in the curvature operator [24]. Some for which there are proof of self-adjointness. • For the massless scalar field ˆ H SF : D ( ˆ H SF ) ⊂ H G Diff −→ H G Diff ˆ H SF := s β πGβ X x ∈ Σ q ( ˆ C x,SF + ˆ C † x,SF ) | R + (I.14)= s β πGβ X x ∈ Σ r (cid:16) ˆ C x,SF + ˆ C † x,SF + (cid:12)(cid:12)(cid:12) ˆ C x,SF + ˆ C † x,SF (cid:12)(cid:12)(cid:12)(cid:17) , with ˆ C x,SF := 11 + β ˆ C Ex + ˆ C Lx , (I.15)such that ˆ C Ex : = κ X I,J ǫ ijk ǫ ( ˙ e I , ˙ e J ) ˆ h k ( l ) α IJ ˆ J x,e I ,i ˆ J x,e J ,j , (I.16)ˆ C Lx : = κ X I,J ǫ ( ˙ e I , ˙ e J )2 q δ ii ′ ( ǫ ijk ˆ J x,e I ,j ˆ J x,e J ,k )( ǫ i ′ j ′ k ′ ˆ J x,e I ,j ′ ˆ J v,x J ,k ′ ) (I.17) × πα − π + arccos δ kl ˆ J x,e I ,k ˆ J x,e J ,l q δ kk ′ ˆ J x,e I ,k ˆ J x,e I ,k ′ q δ ll ′ ˆ J x,e J ,l ˆ J x,e J ,l ′ , where D ( ˆ H SF ) is the domain of the operator ˆ H SF which contains the span of the spinnetwork basis. The sum in (I.14) is over the all points x of Σ, but it reduces to a fi-nite sum over the vertices of a graph when the operator acts on a spin network state.ˆ C † x,SF is the adjoint operator of ˆ C x,SF . The operators ˆ C Ex and ˆ C Lx represent the Eu-clidean and Lorentzian parts of the Hamiltonian operator respectively. The operator ˆ C Lx is graph-preserving while the operator ˆ C Ex is graph-changing. Here κ and κ are aver-aging coefficient and α an unfixed constant, resulting from the regularization procedures[24–26]. The two sums in (I.16) and (I.17) are over pairs of curves { e I , e J } meeting ata point x with tangent vectors { ˙ e I , ˙ e J } . The coefficients ǫ ( ˙ e I , ˙ e J ) is 0 if ˙ e ′ I and ˙ e ′ J arelinearly dependent or 1 otherwise. It is important to note that the square root present inthe definition of the operator ˆ H SF is to be understood as taking the square root of theoperator ˆ C x,SF + ˆ C † x,SF restricted to the positive part of its spectrum. • For the dust field ˆ H D : D ( ˆ H D ) ⊂ H G Diff −→ H G Diff ˆ H D := 1 + β πGβ X x ∈ Σ ˆ C x,D + ˆ C † x,D (I.18) Notice that the term q ab C gra C grb in (I.9) is dropped from the physical Hamiltonian expression as it is assumedto vanish on spatial diffeomorphism invariant states, which is the case for states in the physical Hilbert space H G Diff . This reduces the second condition in (I.8) on the gravitational constraints to C gr ≤
0, which can beimplemented on the operator level by introducing an absolute value term as shown in the second line of (I.14). with ˆ C x,D := 11 + β q d V − ˆ C Ex q d V − + q d V − ˆ C Lx q d V − , (I.19)where d V − is the “inverse volume” operator [29] defined in terms of the LQG volumeoperator [23] as d V − := lim s → (cid:0) ˆ V + s l p (cid:1) − ˆ V . (I.21) D ( ˆ H D ) is the domain of the operator ˆ H D which contains the span of the spin networkbasis and ˆ C † x,D is the adjoint operator of ˆ C x,D .Schematically, given a spin network state with a closed graph Γ, the two operators ˆ C Ex and ˆ C Lx defined in (I.16) and (I.17) act on the vertex x of the graph Γ as follows:ˆ C Ex = + + . . . , (I.22)ˆ C Lx = . (I.23) II. APPROXIMATION METHODS FOR LQG DYNAMICS
Having fully completed the quantization of both systems, we naturally turn to the questionof testing the quantum theories. An important aspect of this question is to obtain a sufficientcontrol on the quantum dynamics. More specifically, we need to be able to compute transi-tion amplitudes and the evolution of observables with a relatively good precision. Consideringthe Hamiltonian operators on the non-separable physical Hilbert space H G Diff defined above, aderivation of their complete spectral decomposition has not been achieved so far. It becomesthen imperative to develop and use approximate methods in the analysis of the dynamics.
A. Expansion of expectation values in powers of time
As mentioned in the introduction, one can consider the expansion in powers of the timevariable of the evolution operator and introduce a truncation of the expansion at a certain fixedorder of time. Such truncation forms a valid approximation of the time evolution when the timeinterval under consideration is sufficiently small. We investigate this approach through a coupleof examples within the dust field model in section III B. In this context, the most convenientway to compute the time evolution of the expectation value of an operator A is by evaluating In other words, given the spectral decomposition of the volume operator ˆ V = P i v i | v i ih v i | , we have d V − | v i i = (cid:26) v − i | v i i if v i = 0 , the coefficients in the power series expansion of the expectation value, h A ( T ) i = X n a n T n . (II.1)The coefficients are given by expectation values of repeated commutators of A with the Hamil-tonian in the initial state | ψ i , a n = ( − i ) n n ! (cid:10) [ H, . . . , [ H, [ H, A ]] . . . ] | {z } n commutators (cid:11) ψ . (II.2)The advantage of directly considering the expansion of an expectation value h A ( T ) i , as opposedto computing the evolved state vector | Ψ( T ) i truncated at some order T n , is that the expectationvalue can be determined up to order T n without having to compute all the components of thetruncated state vector | Ψ( T ) i in the spin-network basis. For example, if the initial state is basedon a single graph containing no “special loops” (of the kind created by the Euclidean part of theHamiltonian), and the operator A is graph-preserving , and one wants to find the expectationvalue h A ( T ) i to order T n , then states containing more than ⌊ n/ ⌋ special loops do not enter thecalculation of the numbers a k ( k = 1 , . . . , n ), even though the state vector | Ψ( T ) i truncated atorder n has components containing up to n special loops.However, this method is not appropriate to deal with the Hamiltonian operator ˆ H SF presentin the model with a massless scalar field (I.14). The reason being the presence of the squareroot in the expression of ˆ H SF , which requires an access to the spectral decomposition of theoperator under the square root. B. Perturbation theory with the Barbero-Immirzi parameter
An alternative solution to the problem is provided by standard time-independent perturba-tion theory of quantum mechanics. In the following we introduce a perturbative expansion ofthe Hamiltonian, the small perturbation parameter being determined by the Barbero-Immirziparameter β . This approach allows us to define an approximate spectral decomposition of thephysical Hamiltonian ˆ H SF , and hence the time-evolution operator U SF ( T ), on appropriate timeintervals.Recall that the expression of the operator ˆ C x,SF isˆ C x,SF := 11 + β ˆ C Ex + ˆ C Lx , (II.3)and hence ˆ C x,SF + ˆ C † x,SF := 11 + β ( ˆ C Ex + ˆ C E † x ) + 2 ˆ C Lx . (II.5)Since the operator ˆ C Lx is graph preserving and acts locally on the vertices of the graph withoutchanging the SU (2) representations [24, 25], its spectral decomposition breaks down to stable Which is the case in the examples we present in Section III B. The operator ˆ C Lx and the curvature operator p d V − ˆ C Lx p d V − are self-adjoint operators [24], thereforeˆ C L † x = ˆ C Lx , ( q d V − ˆ C Lx q d V − ) † = q d V − ˆ C Lx q d V − (II.4) finite dimensional blocks. Each block corresponds to the Hilbert space of a fixed graph withfixed coloring (spins) and takes the form of a tensor product over the vertices of stable sub-blocks, each representing a separate intertwiner space assigned to each vertex of the coloredgraph. Given a colored graph, the dimension of each intertwiner space is then fixed, hence onecan proceed with the diagonalization of the (self-adjoint) Lorentzian part of the Hamiltonianthat is the operator ˆ C Lx .Having the spectral decomposition of this operator, the idea is to treat the Euclidean partof the operator, β ( ˆ C Ex + ˆ C E † x ), as a perturbation to the Lorentzian part with 1 / (1 + β )being the perturbation parameter. This means that we will assume that the Barbero-Immirziparameter is significantly large, β ≫
1, large enough so that the perturbative expansion in1 / (1 + β ) gives a good approximation for the eigenvalues and eigenstates of the Hamiltonian.The condition for this is that the corrections to the eigenvalues and eigenvectors should besmall in norm, compared to the corresponding eigenvalues and eigenvectors of the unperturbedHamiltonian (in our case, the Lorentzian part of the Hamiltonian).The procedure is then as follows: given an intertwiner space I v of dimension d v associatedto a vertex v of a given colored graph, the Lorentzian part operator is put in a diagonal form2 ˆ C Lv = d v X i =1 λ i | λ i ih λ i | . (II.6)For β sufficiently large, we can write2 ˆ C Lv + 11 + β ( ˆ C Ev + ˆ C E † v ) = d v X i =1 λ ′ i | λ ′ i ih λ ′ i | , (II.7)and replace the eigenvalues λ ′ i and the eigenstates | λ ′ i i with their approximate expressions givenby perturbation theory to second order in 1 / (1 + β ). We have λ ′ i = λ i + (cid:18)
11 + β (cid:19) d ′ v X k =1 λ k = λ i (cid:12)(cid:12) h λ i | ˆ C Ev + ˆ C E † v | λ k i (cid:12)(cid:12) λ i − λ k + O (cid:0) (1 + β ) − (cid:1) , (II.8)and | λ ′ i i = | λ i i + 11 + β d ′ v X k =1 λ k = λ i h λ k | ˆ C Ev + ˆ C E † v | λ i i λ i − λ k | λ k i + (cid:18)
11 + β (cid:19) d ′ v X k =1 λ k = λ i (cid:18) d ′ v X n =1 λ n = λ i h λ k | ˆ C Ev + ˆ C E † v | λ n ih λ n | ˆ C Ev + ˆ C E † v | λ i i ( λ i − λ k )( λ i − λ n ) (cid:19) | λ k i + (cid:18)
11 + β (cid:19) (cid:18) − d ′ v X k =1 λ k = λ i (cid:12)(cid:12) h λ k | ˆ C Ev + ˆ C E † v | λ i i (cid:12)(cid:12) ( λ k − λ i ) (cid:19) | λ i i + O (cid:0) (1 + β ) − (cid:1) . (II.9)Because the Euclidean part does not preserve each of the stable subspaces of the Lorentzianpart separately, as it modifies the graph structure at the vertex v , the first-order correction tothe eigenvalue λ i vanishes. Also, the sums in (II.8) and (II.9) are over the eigenstates of the Since we expect to be dealing with unbounded operators, it is not clear to us yet if, given a fixed value of β ,the perturbative expansion would be valid for all eigenstates of ˆ C LSF, sym ,x or ˆ C LD, sym ,x on H G Diff . Lorentzian part in the new intertwiner spaces at v , which together contain the image of thespace I v by the Euclidean part. The upper limit of the summation d ′ v is then the finite sum ofdimensions of the new intertwiner spaces at the vertex v .The derivation of the corrections to the eigenstate requires some care, due to a degeneracyof the unperturbed operator that is not removed by the perturbation (at least to second orderin the perturbation parameter), and is therefore discussed in the Appendix.It is then straightforward to obtain the explicit expression of the square root operator andthe evolution operator: q ( ˆ C v,SF + ˆ C † v,SF ) | R + = r (cid:16) ˆ C v,SF + ˆ C † v,SF + (cid:12)(cid:12)(cid:12) ˆ C v,SF + ˆ C † v,SF (cid:12)(cid:12)(cid:12)(cid:17) = d v X i =1 λ ′ i ≥ q λ ′ i | λ ′ i ih λ ′ i | , (II.10) U SF ( T ) := exp (cid:18) − i ~ T ˆ H SF (cid:19) = Y x ∈ Σ exp (cid:18) − i ~ T s β πGβ q ˆ C x,SF + ˆ C † x,SF (cid:19) = Y x ∈ Σ d x X i =1 λ ′ i ≥ exp (cid:18) − i ~ T s (1 + β )16 πGβ λ ′ i (cid:19) | λ ′ i ih λ ′ i | . (II.11)It follows that given an operator A and an initial state | Ψ i , the state at time T is given by | Ψ( T ) i = U SF ( T ) | Ψ i and the expectation value h A ( T ) i is computed as h A ( T ) i = h Ψ( T ) | A | Ψ( T ) i (II.12)= Y x ∈ Σ d x X i,j =1 λ ′ i ≥ λ ′ j ≥ exp (cid:18) − i ~ T s (1 + β )16 πGβ (cid:18)q λ ′ i − q λ ′ j (cid:19)(cid:19) h Ψ | λ ′ j ih λ ′ j | A | λ ′ i ih λ ′ i | Ψ i . In order to compute the expectation value of the volume or the curvature operator to secondorder in 1 / (1 + β ), some parts of the expression (II.9) for the corrected state vector can bediscarded, because they do not contribute to the expectation value at the specified order inthe perturbation. If the initial state | Ψ i is based on a single graph Γ , which will be the casein the examples in the following section, then the following simplifications can be made. Forunperturbed eigenstates based on Γ , one has to take the first-order correction, and the partof the second-order correction which is based on Γ . For unperturbed eigenstates whose graphis Γ decorated with one special loop, it suffices to take the part of the first-order correctionbased on Γ , and the second-order correction can be discarded entirely. Unperturbed eigenstateswhose graph contains more than one special loop do not enter the calculation at second order in1 / (1 + β ). A more detailed expression of the expectation value (II.12), up to the second orderin perturbation theory, is given in Appendix B.All that was mentioned above for the operator ˆ H SF can be similarly applied to the operatorˆ H D in the dust model. Later in the examples within the dust model, we separately test theBarbero-Immirzi parameter pertubative expansion (the β -expansion), and the approximationobtained by the short time truncation in the time expansion of the evolution operator.0 III. EXAMPLES AND NUMERICAL ANALYSIS
In the following graphics we present the evolution of the expectation values of the volumeoperator and the curvature operator [24]. In the scalar field deparametrized model we use the β -expansion with certain values of β , while in the dust model we consider both the β -expansionand the time expansion approximation. We consider initial states corresponding to certaineigenvectors of the volume operator with a graph consisting of a single non-degenerate 4-valentvertex v .In all the calculations, we fix all the constants in the operators as follows16 πG = ~ = κ = 1 , α = 3 , (III.1)where κ is the averaging constant present in the definition of the volume operator [23]. Addi-tionally, the SU (2) representation of the holonomies associated to the special loops created bythe Euclidean part operator is fixed to 1 / A. Perturbation theory in the scalar field and dust field models
The β -expansion is taken to second order, because with our choice of initial states, theEuclidean part of the Hamiltonian does not contribute to the time evolution of the expectationvalues of volume and curvature at first order of the expansion. The expectation values h V ( T ) i and h R ( T ) i are computed from equation (II.12), with the eigenvalues λ ′ i and eigenstates | λ ′ i i being given by equations (II.8) and (II.9).In all the graphics below, the parameter T stands for the standard time given either by thescalar field or the dust field depending on the considered case. The parameters T ′ and T ′′ inthe embedded graphics stand for the rescaled times given by T ′ := p β T , T ′′ := 1 + β | β | / T . (III.2) • Perturbation theory in the scalar field model: – Eigenvectors with spins j = 2: T X V \(cid:144) Β (cid:144)
10 20 T’ Β= Β= Β= Figure 1: Evolution of the expectation value h V i of the volume operator with an initial eigenvector witheigenvalue v = 0 . T X R \(cid:144) Β (cid:144)
10 20 T’ Β= Β= Β= Figure 2: Evolution of the expectation value h R i of the curvature operator with an initial eigenvectorwith eigenvalue v = 0 . – Eigenvectors with spins j = 10: T X V \(cid:144) Β (cid:144)
10 20 T’ Β= Β= Β= Figure 3: Evolution of the expectation value h V i of the volume operator with an initial eigenvector witheigenvalue v = 8 . T X V \(cid:144) Β (cid:144)
10 20 T’ Β= Β= Β= Figure 4: Evolution of the expectation value h R i of the curvature operator with an initial eigenvectorwith eigenvalue v = 5 . • Perturbation theory in the dust field model: – Eigenvectors with spins j = 2: T X V \(cid:144) Β (cid:144) T’’ Β= Β= Β= Figure 5: Evolution of the expectation value h V i of the volume operator with an initial eigenvector witheigenvalue v = 0 . T X R \(cid:144) Β (cid:144) T’’ Β= Β= Β= Figure 6: Evolution of the expectation value h R i of the curvature operator with an initial eigenvectorwith eigenvalue v = 0 . – Eigenvectors with spins j = 10: T X V \(cid:144) Β (cid:144) T’’ Β= Β= Β= Figure 7: Evolution of the expectation value h V i of the volume operator with an initial eigenvector witheigenvalue v = 5 . T X R \(cid:144) Β (cid:144) T’’ Β= Β= Β= Figure 8: Evolution of the expectation value h R i of the curvature operator with an initial eigenvectorwith eigenvalue v = 8 . Discussion: • Since the initial states we are considering in this numerical analysis correspond to specificspin-network states, and the evolution operators in this approximation contain a finiteorder of the graph changing Euclidean operators, the expectation values of the volumeand curvature operators are both bounded throughout the time evolution of those states.It is possible that unbounded expectation values can be obtained in this approximation byconsidering initial states which take the form of infinite linear combination of spin-networkstates. • The degeneracy of the volume and curvature eigenvalues is preserved under time evolution,in the sense that two degenerate initial states give rise to the same function h V ( T ) i or h R ( t ) i . Furthermore, the degeneracy present in the eigenvalues of the Lorentzian partof the Hamiltonian is not removed by the perturbation provided by the Euclidean part,at least to second order in perturbation theory, suggesting that the degeneracy might bepreserved exactly. These observations strongly indicate the existence of some symmetryshared by the volume operator and the Lorentzian and Euclidean operators. • The large fluctuations of the expectation values curves for β = 5 with respect to β = 50(which can be seen as the limit where the perturbations is totally negligible) in the figuresabove demonstrate that the value β = 5 of the Barbero-Immirzi parameter is not goodenough to make sense of the perturbation method, suggesting that, at least according tothese examples, the range of β consistent with the pertubative treatment is | β | & • In the case of the scalar field model, when the perturbation from the Euclidean part is small(e.g. spin 2 case, or 4 for spin 10), one can notice a periodic evolution of the expectationvalues of the volume and curvature operators. This periodicity seems to manifest forall eigenvectors of the volume operator, independently of the intertwiner space. This isanother piece of evidence pointing towards the presence of a certain symmetry betweenthe volume operator, the curvature operator and the Lorentzian operator ˆ C SF . We leavethe investigation of the symmetry properties of our operators as a question for futurestudy. • Figures 6 and 8, for the expectation value of the curvature operator in the dust model,show practically constant curves for β = 10 and β = 50. This is expected because whenthe perturbation is small, the dust model Hamiltonian reduces to almost the curvatureoperator itself, hence the constant expectation value.4 • Finally, the embedded graphics on the right of each figure display the evolution withrespect to the rescaled time. Comparing those graphics to the graphics for the evolutionwith respect to the standard time exhibits how the overall factors depending on β in theevolution operators affect the phases in the evolution curves. Those overall factors areobtained by factorizing out all the dependence on β in the Lorentzian and Euclidean partsof the Hamiltonian operator, i.e one write the Hamiltonian in the formˆ H = f ( β ) (cid:18) ˆ C L + 11 + β ˆ C E (cid:19) , (III.3)such that ˆ C L and ˆ C E are independent of β . f ( β ) is then the rescaling factor which equals p β for ˆ H SF and (1 + β ) / | β | / for ˆ H D . B. Time-expansion approximation in the dust field model
The time-expansion in the following examples is taken up to fourth order. The expectationvalues of the volume and curvature operators are computed according to eqs. (II.1) and (II.2).At order T , the set of graphs that enters the computation consists of the graph of the initialstate (a single four-valent node), and of the graphs generated by no more than two actions ofthe Euclidean part of the Hamiltonian on the initial state.The computation of the coefficients of the power series expansion of the expectation values h V ( T ) i and h R ( T ) i reveals the following properties: • Only even powers of T are present in the expansion of the functions h V ( T ) i and h R ( T ) i .The coefficients of the odd powers of T ( T and T ) vanish up to numerical roundingerror. This seems to suggest the invariance of the Hamiltonian, the volume and curvatureoperators under time reversal. • Degeneracy of eigenvalues is again preserved under time evolution, in the sense that for agiven degenerate eigenvalue of the volume or the curvature, the function h V ( T ) i or h R ( T ) i does not depend on which eigenstate belonging to the degenerate eigenvalue is selected asthe initial state.To determine the range of validity of the time expansion, one should estimate the value of T at which the magnitude of the first neglected term in the expansion of an expectation value (weexpect this to be the term of order T ) starts being comparable to the terms included in theapproximation. This criterion can be tested in a toy example in which the Hamiltonian consistsonly of the Lorentzian part, and the dynamics can be evaluated exactly. In this case we find thatthe criterion correctly predicts the order of magnitude of the time at which an expectation valuecomputed from the fourth-order time expansion begins to diverge from the exact expectationvalue.5 • Eigenvectors with spins j = 2: T X V \(cid:144) Β (cid:144) T’’ Β= Β= Β= Figure 9: Evolution of the expectation value h V i of the volume operator with an initial eigenvector witheigenvalue v = 0 . T X V \(cid:144) Β (cid:144) Β= Β= Figure 10: Evolution of the expectation value h V i of the volume operator with an initial eigenvector witheigenvalue v = 0 . T X R \(cid:144) Β (cid:144) T’’ - Β= Β= Β= Figure 11: Evolution of the expectation value h R i of the curvature operator with an initial eigenvectorwith eigenvalue v = 0 . T X R \(cid:144) Β (cid:144) Β= Β= Figure 12: Evolution of the expectation value h R i of the curvature operator with an initial eigenvectorwith eigenvalue v = 0 . • Eigenvectors with spins j = 25: T X V \(cid:144) Β (cid:144) T’’ Β= Β= Β= Figure 13: Evolution of the expectation value h V i of the volume operator with an initial eigenvector witheigenvalue v = 32 . T X V \(cid:144) Β (cid:144) Β= Β= Figure 14: Evolution of the expectation value h V i of the volume operator with an initial eigenvector witheigenvalue v = 32 . T X R \(cid:144) Β (cid:144) T’’ Β= Β= Β= Figure 15: Evolution of the expectation value h R i of the curvature operator with an initial eigenvectorwith eigenvalue v = 20 . T X R \(cid:144) Β (cid:144) Β= Β= Figure 16: Evolution of the expectation value h R i of the curvature operator with an initial eigenvectorwith eigenvalue v = 20 . In figures 17 and 18, we compare the results given by the time expansion and the β -expansionfor h V ( T ) i in a particular volume eigenstate, for j = 2 and j = 10 respectively with β = 10. Forthis value of β , the β -expansion presumably provides an accurate description of the dynamicsover a longer time interval than the time expansion does. In both figures we observe that arounda certain time T , different in each case, the expectation value given by the time expansionbegins to differ significantly from the expectation value given by the β -expansion. At this timewe expect the latter to still be a very close approximation to the exact expectation value. T X V \(cid:144) Β (cid:144) Β- expansionTime expansion Figure 17: Comparative plot of the evolution of the volume expectation value h V i between the timeexpansion and the β -expansion, with an initial eigenvector with eigenvalue v = 0 . β = 50. T X V \(cid:144) Β (cid:144) Β- expansionTime expansion Figure 18: Comparative plot of the evolution of the volume expectation value h V i between the timeexpansion and the β -expansion, with an initial eigenvector with eigenvalue v = 5 . β = 100. IV. CONCLUSION
The simulations presented in this article show that the proposed β -approximation can in-deed be applied fully and consistently in a certain sector of the physical Hilbert space in bothdeparametrized models we considered. Therefore, this perturbation method presents itself as apromising tool in the investigation of the dynamics in LQG models.A very interesting outcome is the periodic character of the evolution of the expectation val-ues of the volume and curvature operators, in the deparametrized model with a free scalar field.While it is a rather unexpected result, it is clearly reminiscent to the particular physical Hamilto-nian present in the model, and the choice of the volume and curvature operators as observables.This indeed suggests the presence of a special relation between the spectral decompositions ofthe mentioned operators which is yet to be understood.As a future work, the focus will be on establishing more accurately to which extent one couldapply this approximation with respect to the admissible range of the Barbero-Immirzi parameter β and the choice of initial states. Acknowledgments
This work was supported by the grant of Polish Narodowe Centrum Nauki nr2011/02/A/ST2/00300. I.M. would like to thank the Jenny and Antti Wihuri Foundation forsupport.9
Appendix A: Second-order perturbation theory of a degenerate energy level
In section II, we use time-independent perturbation theory to obtain an approximate spectraldecomposition of the physical Hamiltonian, treating the Euclidean part of the Hamiltonian as aperturbation over the Lorentzian part. In this case some of the eigenvalues of the unperturbedHamiltonian are degenerate, and all matrix elements of the perturbation vanish between thedegenerate (unperturbed) eigenstates. In such a situation the derivation of the corrections tothe eigenstates is not entirely standard, hence we give the full treatment of the perturbativeproblem up to second order in this appendix.To conform to the standard notation, in this appendix we write the Hamiltonian as H = H + ǫV (A.1)where H stands for the Lorentzian part and V for the Euclidean part. The perturbationparameter is ǫ ≡ / (1 + β ).Let us denote the eigenvalues and the corresponding eigenstates by λ (0) n and (cid:12)(cid:12) λ (0) n (cid:11) for theunperturbed Hamiltonian H , and λ n and (cid:12)(cid:12) λ n (cid:11) for the full Hamiltonian H . (For the sake ofclarity, we deviate here from the notation of section II, where the eigenstates and eigenvaluesof the unperturbed operator were denoted by λ n and (cid:12)(cid:12) λ n (cid:11) , and those of the full operator by λ ′ n and (cid:12)(cid:12) λ ′ n (cid:11) .) To determine λ n and (cid:12)(cid:12) λ n (cid:11) approximately up to second order in ǫ , we write (cid:12)(cid:12) λ n (cid:11) = (cid:12)(cid:12) λ (0) n (cid:11) + ǫ (cid:12)(cid:12) λ (1) n (cid:11) + ǫ (cid:12)(cid:12) λ (2) n (cid:11) + O ( ǫ ) , (A.2) λ n = λ (0) n + ǫλ (1) n + ǫ λ (2) n + O ( ǫ ) . (A.3)Inserting these into eq. (A.1), we obtain (cid:0) H + ǫV (cid:1)(cid:0)(cid:12)(cid:12) λ (0) n (cid:11) + ǫ (cid:12)(cid:12) λ (1) n (cid:11) + ǫ (cid:12)(cid:12) λ (2) n (cid:11) + . . . (cid:1) = (cid:0) λ (0) n + ǫλ (1) n + ǫ λ (2) n + . . . (cid:1)(cid:0)(cid:12)(cid:12) λ (0) n (cid:11) + ǫ (cid:12)(cid:12) λ (1) n (cid:11) + ǫ (cid:12)(cid:12) λ (2) n (cid:11) + . . . (cid:1) (A.4)as the equation from which the corrections to the eigenvalues and eigenstates will be determined.The derivation of the corrections to the eigenvalues presents no special problems. As is wellknown, for a degenerate eigenvalue, the first-order corrections are given by the eigenvalues ofthe matrix of the perturbation in the degenerate subspace. Therefore, λ (1) n = 0 . (A.5)For the second-order corrections one finds, considering the ǫ terms of eq. (A.4), λ (2) n = (cid:10) λ (0) n (cid:12)(cid:12) V (cid:12)(cid:12) λ (1) n (cid:11) = X k (cid:12)(cid:12)(cid:10) λ (0) n (cid:12)(cid:12) V (cid:12)(cid:12) λ (0) k (cid:11)(cid:12)(cid:12) λ (0) n − λ (0) k . (A.6)In the second equality we used eq. (A.7) to obtain the explicit expression for λ (2) n . This stepis correct because we are interested in the case where the perturbation V vanishes within thedegenerate subspace, implying that the state V (cid:12)(cid:12) λ (0) n (cid:11) has no non-vanishing components on un-perturbed eigenstates with eigenvalue λ (0) n .Let us then go on to the corrections to the eigenstates. The projections of (cid:12)(cid:12) λ (1) n (cid:11) and (cid:12)(cid:12) λ (2) n (cid:11) onto unperturbed eigenstates outside the degenerate subspace are easily found by considering Precisely speaking, the operator H in eq. (A.1) is the Hamiltonian only in the case of the dust field model,while for the scalar field model the physical Hamiltonian is the square root of an operator of the form (A.1). (cid:10) λ (0) k (cid:12)(cid:12) λ (1) n (cid:11) = (cid:10) λ (0) k (cid:12)(cid:12) V (cid:12)(cid:12) λ (0) n (cid:11) λ (0) n − λ (0) k ( λ (0) n = λ (0) k ) (A.7)and, recalling that the first-order correction to the eigenvalue vanishes, (cid:10) λ (0) k (cid:12)(cid:12) λ (2) n (cid:11) = (cid:10) λ (0) k (cid:12)(cid:12) V (cid:12)(cid:12) λ (1) n (cid:11) λ (0) n − λ (0) k ( λ (0) n = λ (0) k ) . (A.8)Below we will show that the correction (cid:12)(cid:12) λ (1) n (cid:11) has no non-vanishing components on unperturbedeigenstates having eigenvalue λ (0) n – see eq. (A.15). Therefore, using eq. (A.7), we obtain (cid:10) λ (0) k (cid:12)(cid:12) λ (2) n (cid:11) = X l (cid:10) λ (0) k (cid:12)(cid:12) V (cid:12)(cid:12) λ (0) l (cid:11)(cid:10) λ (0) l (cid:12)(cid:12) V (cid:12)(cid:12) λ (0) n (cid:11)(cid:0) λ (0) n − λ (0) k (cid:1)(cid:0) λ (0) n − λ (0) l (cid:1) ( λ (0) n = λ (0) k ) . (A.9)To find the projections (cid:10) λ (0) n ′ (cid:12)(cid:12) λ (1) n (cid:11) and (cid:10) λ (0) n ′ (cid:12)(cid:12) λ (2) n (cid:11) , where (cid:12)(cid:12) λ (0) n ′ (cid:11) is another unperturbed eigenstatehaving eigenvalue λ (0) n under the unperturbed Hamiltonian, requires more care. The first-orderterms of eq. (A.4) do not give any information about (cid:10) λ (0) n ′ (cid:12)(cid:12) λ (1) n (cid:11) ; they merely reproduce (cid:10) λ (0) n ′ (cid:12)(cid:12) V (cid:12)(cid:12) λ (0) n (cid:11) = 0 (A.10)as a consistency condition for the perturbative expansion. In our case this condition is satisfiedirrespectively of the choice of basis in the degenerate subspace. If we turn to the second-orderterms of eq. (A.4), we again find no information on (cid:10) λ (0) n ′ (cid:12)(cid:12) λ (1) n (cid:11) , because the first-order correctionto the eigenvalue vanishes. Instead, we obtain another consistency condition, (cid:10) λ (0) n ′ (cid:12)(cid:12) V (cid:12)(cid:12) λ (1) n (cid:11) = X k (cid:10) λ (0) n ′ (cid:12)(cid:12) V (cid:12)(cid:12) λ (0) k (cid:11)(cid:10) λ (0) k (cid:12)(cid:12) V (cid:12)(cid:12) λ (0) n (cid:11) λ (0) n − λ (0) k = 0 . (A.11)By a numerical evaluation of the sum, we find that this condition also seems to be satisfied forany choice of basis in the degenerate subspace.To determine the projections (cid:10) λ (0) n ′ (cid:12)(cid:12) λ (1) n (cid:11) , we must therefore look at the third-order terms ineq. (A.4). We find (cid:10) λ (0) n ′ (cid:12)(cid:12) V (cid:12)(cid:12) λ (2) n (cid:11) = λ (2) n (cid:10) λ (0) n ′ (cid:12)(cid:12) λ (1) n (cid:11) . (A.12)A computation of the second-order corrrections λ (2) n from eq. (A.6) confirms that all of them arenon-vanishing, at least in the examples considered in section III. Therefore the above equationdetermines the projections (cid:10) λ (0) n ′ (cid:12)(cid:12) λ (1) n (cid:11) as (cid:10) λ (0) n ′ (cid:12)(cid:12) λ (1) n (cid:11) = 1 λ (2) n (cid:10) λ (0) n ′ (cid:12)(cid:12) V (cid:12)(cid:12) λ (2) n (cid:11) . (A.13)When the perturbation V is the Euclidean part of the Hamiltonian, the matrix element on theright-hand side actually vanishes. To see this, let us resolve the matrix element in the basis ofthe unperturbed eigenstates as (cid:10) λ (0) n ′ (cid:12)(cid:12) V (cid:12)(cid:12) λ (2) n (cid:11) = X k (cid:10) λ (0) n ′ (cid:12)(cid:12) V (cid:12)(cid:12) λ (0) k (cid:11)(cid:10) λ (0) k (cid:12)(cid:12) λ (2) n (cid:11) . (A.14)1Here the unperturbed eigenstate (cid:12)(cid:12) λ (0) n ′ (cid:11) is a spin-network state based on a single graph, whichconsists of some number L of special loops attached to a loopless ”initial” graph (this is becausethe Lorentzian part of the Hamiltonian is a graph-preserving operator). The Euclidean partof the Hamiltonian changes the number of loops by one; hence the intermediate states (cid:12)(cid:12) λ (0) k (cid:11) entering the sum in eq. (A.14) have L − L + 1 special loops. On the other hand, by eq.(A.9), the second-order correction to the state (cid:12)(cid:12) λ (0) n (cid:11) is composed of states having L − L and L + 2 special loops . Therefore the scalar product (cid:10) λ (0) k (cid:12)(cid:12) λ (2) n (cid:11) on the right-hand side of eq.(A.14) is always zero, and we conclude that (cid:10) λ (0) n ′ (cid:12)(cid:12) λ (1) n (cid:11) = 0 . (A.15)It still remains to determine the components of the second-order correction (cid:12)(cid:12) λ (2) n (cid:11) within thedegenerate subspace. The projection of (cid:12)(cid:12) λ (2) n (cid:11) on the uncorrected eigenstate (cid:12)(cid:12) λ (0) n (cid:11) can be foundby requiring that the corrected eigenstate (cid:12)(cid:12) λ (0) n (cid:11) + ǫ (cid:12)(cid:12) λ (1) n (cid:11) + ǫ (cid:12)(cid:12) λ (2) n (cid:11) is normalized up to secondorder in ǫ . In this way we find (cid:10) λ (0) n (cid:12)(cid:12) λ (2) n (cid:11) = − (cid:10) λ (1) n (cid:12)(cid:12) λ (1) n (cid:11) = X k (cid:12)(cid:12)(cid:10) λ (0) n (cid:12)(cid:12) V (cid:12)(cid:12) λ (0) k (cid:11)(cid:12)(cid:12) (cid:0) λ (0) n − λ (0) k (cid:1) . (A.16)The projection (cid:10) λ (0) n ′ (cid:12)(cid:12) λ (2) n (cid:11) , where (cid:12)(cid:12) λ (0) n ′ (cid:11) is another unperturbed eigenstate sharing the degenerateeigenvalue λ (0) n , is not uniquely determined by any normalization or orthogonality conditions.In our application to the physical Hamiltonian, this projection is also not determined by theequations obtained from (A.4), at least up to sixth order in ǫ , apparently reflecting the factthat the degeneracy present in the eigenvalues of the Lorentzian part of the Hamiltonian is notremoved by the Euclidean part at second order of perturbation theory. We resolve this situationby choosing (cid:10) λ (0) n ′ (cid:12)(cid:12) λ (2) n (cid:11) = 0 , (A.17)this choice being the simplest, and consistent with the normalization and orthogonality of thecorrected eigenstates up to second order in ǫ . This completes the derivation of the correctedeigenvalues and eigenstates up to second order, the solution being given by eqs. (A.5), (A.6),(A.7), (A.9), (A.15), (A.16) and (A.17). Appendix B: Perturbative expansion of expectation values
In equation (II.12), the general and compact expression for the time-dependent expectationvalue of an operator A in the scalar field model was given. Here we display explicitly the differentcontributions to this expectation value, organized order by order in the perturbation. In general, The use of eq. (A.9) in eq. (A.14) is correct, because we find no degeneracy in the eigenvalues of the Lorentzianpart between states based on graphs having a different number of special loops – with the exception of theeigenvalue zero, which occurs in the dust field model for every graph. However, in this case each eigenstate ofthe Lorentzian part having eigenvalue zero is also annihilated by the Euclidean part, implying that the matrixelement (cid:10) λ (0) n ′ (cid:12)(cid:12) V (cid:12)(cid:12) λ (0) k (cid:11) vanishes when λ (0) n = λ (0) k = 0. Hence the summation index k in eq. (A.14) always runsonly over states whose unperturbed eigenvalue is different from λ (0) n . h A ( T ) i = Y x ∈ Σ d x X i,j =1 λ ′ i ≥ λ ′ j ≥ exp (cid:20) − i ~ T s β πGβ (cid:18)q λ (0) i + ǫλ (1) i + ǫ λ (2) i − q λ (0) j + ǫλ (1) j + ǫ λ (2) j (cid:19)(cid:21)h h Ψ | λ (0) j ih λ (0) j | A | λ (0) i ih λ (0) i | Ψ i ++ ǫ (cid:16) h Ψ | λ (1) j ih λ (0) j | A | λ (0) i ih λ (0) i | Ψ i + h Ψ | λ (0) j ih λ (1) j | A | λ (0) i ih λ (0) i | Ψ i ++ h Ψ | λ (0) j ih λ (0) j | A | λ (1) i ih λ (0) i | Ψ i + h Ψ | λ (0) j ih λ (0) j | A | λ (0) i ih λ (1) i | Ψ i (cid:17) ++ ǫ (cid:16) h Ψ | λ (2) j ih λ (0) j | A | λ (0) i ih λ (0) i | Ψ i + h Ψ | λ (0) j ih λ (2) j | A | λ (0) i ih λ (0) i | Ψ i ++ h Ψ | λ (0) j ih λ (0) j | A | λ (2) i ih λ (0) i | Ψ i + h Ψ | λ (0) j ih λ (0) j | A | λ (0) i ih λ (2) i | Ψ i ++ h Ψ | λ (1) j ih λ (1) j | A | λ (0) i ih λ (0) i | Ψ i + h Ψ | λ (0) j ih λ (1) j | A | λ (1) i ih λ (0) i | Ψ i ++ h Ψ | λ (1) j ih λ (0) j | A | λ (1) i ih λ (0) i | Ψ i + h Ψ | λ (0) j ih λ (1) j | A | λ (0) i ih λ (1) i | Ψ i ++ h Ψ | λ (1) j ih λ (0) j | A | λ (0) i ih λ (1) i | Ψ i + h Ψ | λ (0) j ih λ (0) j | A | λ (1) i ih λ (1) i | Ψ i (cid:17)i . (B.1)However, a large number of terms in this general expression actually vanish in the case thatis of interest to us. When the operator A is graph-preserving, the initial state | Ψ i is basedon a single graph, and the unperturbed Hamiltonian and the perturbation are respectively theLorentzian and the Euclidean operators, the above expression for the expectation value simplifiesto the following: h A ( T ) i = Y x ∈ Σ d x X i,j =1 λ ′ i ≥ λ ′ j ≥ exp (cid:20) − i ~ T s β πGβ (cid:18)q λ (0) i + ǫ λ (2) i − q λ (0) j + ǫ λ (2) j (cid:19)(cid:21) (B.2) h h Ψ | λ (0) j ih λ (0) j | A | λ (0) i ih λ (0) i | Ψ i ++ ǫ (cid:16) h Ψ | λ (2) j ih λ (0) j | A | λ (0) i ih λ (0) i | Ψ i + h Ψ | λ (0) j ih λ (2) j | A | λ (0) i ih λ (0) i | Ψ i ++ h Ψ | λ (0) j ih λ (0) j | A | λ (2) i ih λ (0) i | Ψ i + h Ψ | λ (0) j ih λ (0) j | A | λ (0) i ih λ (2) i | Ψ i ++ h Ψ | λ (0) j ih λ (1) j | A | λ (1) i ih λ (0) i | Ψ i + h Ψ | λ (1) j ih λ (0) j | A | λ (0) i ih λ (1) i | Ψ i (cid:17)i . [1] C. Rovelli, L. Smolin, The Physical Hamiltonian in non-perturbative quantum gravity , Phys. Rev.Lett. , 446 (1994).[2] K. Kuchaˇr, C. G. Torre, Gaussian reference fluid and interpretation of quantum geometrodynamics ,Phys. Rev. D The Harmonic gauge in canonical gravity , Phys. Rev. D Dust as a standard of space and time in canonical quantum gravity , Phys.Rev. D , 5600-5629 (1995).[5] K. Kuchaˇr, J. D. Romano, Gravitational constraints that generate a Lie algebra , Phys. Rev. D Null dust in canonical gravity , Phys. Rev. D Gravitational Constraint Combinations Generate a Lie Algebra , Class. Quant.Grav. , 2577-2584 (1996).[8] I. Kouletsis, Action functionals of single scalar fields and arbitrary weight gravitational constraintsthat generate a genuine Lie algebra , Class. Quant. Grav. , 3085-3098 (1996).[9] V. Husain, T. Pawlowski, Time and a physical Hamiltonian for quantum gravity , Phys. Rev. Lett.
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