First-Order Quantum Correction in Coherent State Expectation Value of Loop-Quantum-Gravity Hamiltonian: II. Detailed Derivations
FFirst-Order Quantum Correction in Coherent StateExpectation Value of Loop-Quantum-GravityHamiltonian: II. Detailed Derivations
Cong Zhang Shicong Song Muxin Han , Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland Department of Physics, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431-0991, USA Institut f¨ur Quantengravitation, Universit¨at Erlangen-N¨urnberg, Staudtstr. 7/B2, 91058 Erlangen, Ger-many
E-mail: czhang(AT)fuw.edu.pl , ssong2019(AT)fau.edu , hanm(At)fau.edu Abstract:
Given the non-graph-changing Hamiltonian (cid:92) H [ N ] in Loop Quantum Gravity (LQG), (cid:104) (cid:92) H [ N ] (cid:105) , the coherent state expectation value of (cid:92) H [ N ], admits an semiclassical expansion in (cid:96) . Inthis paper, as presenting the detailed derivations of our previous work [1], we explicitly compute theexpansion of (cid:104) (cid:92) H [ N ] (cid:105) to the linear order in (cid:96) on the cubic graph with respect to the coherent statepeaked at the homogeneous and isotropic data of cosmology. In our computation, a powerful algo-rithm is developed, supported by rigorous proofs and several theorems, to overcome the complexityin the computation of (cid:104) (cid:92) H [ N ] (cid:105) . Particularly, some key innovations in our algorithm substantiallyreduce the complexity in computing the Lorentzian part of (cid:104) (cid:92) H [ N ] (cid:105) . Additionally, some quantumcorrection effects resulted from (cid:104) (cid:92) H [ N ] (cid:105) in cosmology are discussed at the end of this paper. a r X i v : . [ g r- q c ] F e b ontents F α ··· α m ιab with ι = 0 123.1.4 Expectation value of ˆ F α ··· α m ιab with ι (cid:54) = 0 133.2 The cases when all flux indices vanish 23 (cid:104) ψ ge | ψ g (cid:48) e (cid:105)(cid:107) ψ ge (cid:107)(cid:107) ψ g (cid:48) e (cid:107) , C ) and SU(2) groups 52B the Clebsch-Gordan coefficients with negative parameters 54C Graphical method 56D Proof of Proposition 2 58E derivative of the matrix element of p i ( e ) and D ab ( h e ) – 1 – Introduction
Loop Quantum Gravity (LQG) is an approach toward the background independent and nonper-turbative quantum gravity theory in four and higher dimensions [2–6]. Several recent progresseshave been made by the active research of the quantum dynamics of LQG [7–17]. Particularly,tremendous progresses have been made in both canonical and covariant LQG on the semiclassicallimit and the consistency with respect to classical gravity e.g. [8, 9, 16, 18–28]. However, regardingthe full theory of LQG dynamics, less progress has been made on its quantum corrections (see e.g.[29–32] for some results, and [33, 34] for some results in the covariant approach). As a candidateof quantum gravity theory, it is important that LQG should shed light on quantum corrections tothe classical theory of gravity.The present paper focuses on the canonical aspects of LQG. Due to the non-polynomial Hamil-tonian constraint operator (cid:92) H [ N ] = (cid:92) H E [ N ] + (1 + β ) (cid:92) H L [ N ], there has been persistent confusionthat the quantum dynamics of LQG might not be computable analytically [35]. A previous work[9] partially resolves this confusion, where it has been schematically shown that the coherent stateexpectation value of the Hamiltonian/master constraint are computable order-by-order by the semi-classical expansion in (cid:126) . It is remarkable that the proposed scheme in [9] can also be applied to awide class of non-polynomial operators used in the study of LQG dynamics. Although this schemewas proposed as early as when [9] firstly published in 2006, the expectation value of (cid:92) H [ N ] has onlybeen computed at its classical limit, i.e. the 0-th order (in (cid:126) ). However, due to the complexity ofthe operator, especially the the Lorentzian part of (cid:92) H [ N ] (denoted as (cid:92) H L [ N ]), the O ( (cid:126) ) quantumcorrection has not been studied in the literature.The goal of the present work is to fill this gap by providing an explicit computation of the O ( (cid:126) ) quantum correction in (cid:104) (cid:92) H [ N ] (cid:105) with respect to a certain coherent state. In this paper, in orderto compute the quantum correction in (cid:104) (cid:92) H [ N ] (cid:105) , a powerful algorithm is developed to overcome thecomplexity of (cid:92) H [ N ] that is the non-graph-changing Hamiltonian on a cubic lattice γ . We explicitlyexpand (cid:104) (cid:92) H [ N ] (cid:105) to linear order in (cid:126) by applying the algorithm, with respect to the coherent statethat is peaked at the homogeneous and isotropic data of cosmology. Namely we explicitly compute H and H in (cid:104) (cid:100) H [1] (cid:105) = H + (cid:96) H + O ( (cid:96) ) , (cid:96) = (cid:126) κ (1.1)where κ = 8 πG Newton and the lapse function N = 1. H , representing the 0-th order, reproducesthe cosmological effective Hamiltonian in the µ -scheme [16, 17, 36, 37]. Whereas, H gives the firstorder quantum correction, which is presented by our result in this work. The explicit expression of H is given in Section 6. It is worth noting that the coherent state used for computing (cid:104) (cid:92) H [ N ] (cid:105) isnot SU(2) gauge invariant (the motivation is stated below).This work is closely related to the reduced phase space formulation of LQG (see e.g. [10, 38, 39].In this formulation, some matter fields that are known as the clock fields is coupled to gravity.These matter fields are regarded as material reference frames used to transform gravity variablesto gauge invariant Dirac observables. This procedure resolves the Diffeomorphism constraint andHamiltonian constraint at the classical level resulting in the reduced phase space P red of Diracobservables. The dynamics of the gravity-clock system is described by the material-time evolutiongenerated by the physical Hamiltonian H on the reduced phase space P red . As an interestingmodel, Gaussian dust is chosen to be our clock fields [39, 40]. Then the resulting reduced phasespace, P red , is identical to the pure-gravity unconstrained phase space. This identification definesthe pure-gravity Hamiltonian constraint with unit lapse (i.e. N = 1) H [1] on the resulting reducedphase space P red , which indictaes that the physical Hamiltonian H equals to H [1] for the casewhen gravity is coupled to Gaussian dust. In this model, the quantization of P red is the same asquantizing the pure-gravity unconstrained phase space, which leads to the physical Hilbert space– 2 – that is identical to the kinematical Hilbert space in the usual LQG. H is unconstrained becauseit is from the quantization of P red . The physical Hamiltonian operator is obtained by (cid:98) H = ( (cid:100) H [1]+ (cid:100) H [1] † ) with the usual LQG quantization of (cid:92) H [ N ] [7, 41–43]. Therefore from the perspectiveof reduced-phase-space LQG, our work computes the (cid:104) (cid:98) H (cid:105) with respect to the coherent state peakedat cosmological data on the graph γ , which is given by the real part of (cid:104) (cid:100) H [1] (cid:105) (1.1).Recent works have been focused on building models of LQG on a single graph γ [16, 21, 22,29, 44–47]. Particularly, the quantum dynamics in the reduced-phase-space LQG framework isformulated on a cubic lattice γ with a path integral [17, 23] A [ g ] , [ g (cid:48) ] = (cid:90) d h [d g ] ν [ g ] e S [ g,h ] /(cid:96) , (1.2)which is the canonical-LQG analog of the spinfoam formulation. A [ g ] , [ g (cid:48) ] is regarded as the transitionamplitude of (cid:98) H between the initial and final SU(2) gauge invariant coherent states, denoted as | [ g ] (cid:105) , | [ g (cid:48) ] (cid:105) respectively. The integration variables contains both trajectories of g ∈ P red,γ andSU(2) gauge transformations h on γ , with ν [ g ] being a measure factor. Due to a feature of the pathintegral (1.2), the SU(2) gauge invariant amplitude A [ g ] , [ g (cid:48) ] is expressed as an integral of SU(2)gauge non-invariant variables g and h . The action S [ g, h ] is linear to (cid:104) (cid:98) H (cid:105) at SU(2) gauge non-invariant coherent states when the trajectories of g are continuous in time. In contrast to the usualpath integrals in quantum field theories, S [ g, h ] contains the O ( (cid:126) ) correction from (cid:104) (cid:98) H (cid:105) . Our workprecisely computes this O ( (cid:126) ) correction in S [ g, h ] of cosmological dynamics.A general study of the equation of motion provided by the semiclassical limit of A [ g ] , [ g (cid:48) ] ispresented in [23]. The application of it in cosmology is presented in [17, 47]. The cosmologicaldynamics in the limit of (cid:126) → µ -scheme effective cosmological dynamics which reducesto the classical FRLW cosmology at low energy density. Next, it is equally important to discover the O ( (cid:126) ) correction of the effective cosmological dynamics. The effective dynamics with O ( (cid:126) ) correctioncan be obtained by the quantum effective action [48], denoted as Γ, from the path integral definedin (1.2). Perturbatively, the O ( (cid:126) ) correction in Γ for cosmology contains 3 contributions: (1) O ( (cid:126) )correction in S [ g, h ] which is computed in this work, (2) O ( (cid:126) ) correction in log ν [ g ] where ν [ g ] hasbeen given explicitly in [17], and (3) O ( (cid:126) ) correction in log det( H ) where the “1-loop determinant”det( H ) is the determinant of the Hessian matrix H of S [ g, h ]. The g - g matrix elements in H has beencomputed in [47]. A continuous study of log det( H ) is postponed for future work. Therefore,interms of the quantum correction in the effective cosmological dynamics, the present work computesan significant part in the O ( (cid:126) ) correction of the quantum effective action Γ.After introducing motivations above, let us summarize several key steps in the computation ofthe present work: First of all, an important complication in (cid:92) H [ N ] is the volume operator ˆ V v = (cid:113) | ˆ Q v | which contains the square-root and absolute-value, indicating that the (cid:92) H [ N ] is non-polynomial.When (cid:104) (cid:92) H [ N ] (cid:105) is studied with respect to the coherent state, this issue is overcome by substitutingˆ V v with the semiclassical expansion [9]ˆ V ( v ) GT = (cid:104) ˆ Q v (cid:105) q (cid:34) k +1 (cid:88) n =1 ( − n +1 q (1 − q ) · · · ( n − − q ) n ! (cid:32) ˆ Q v (cid:104) ˆ Q v (cid:105) − (cid:33) n (cid:35) + O ( (cid:126) k +1 ) (1.3)where ˆ Q v is formulated as a polynomial of flux operators and q = 1 /
4. Truncating ˆ V ( v ) GT with afinite k and substituting it back into (cid:92) H [ N ] allows us to express (cid:104) (cid:92) H [ N ] (cid:105) by a expectation value of apolynomial operator.The resulting polynomial sums over a huge number of terms ( ∼ ), each of which is amonomial of holonomy and fluxe operators. Computing expectation values of all terms would leadto a large computational complexity. The major complexity is encoded in the Lorentzian part of (cid:92) H [ N ], denoted as (cid:92) H L [ N ]. Several key methods are used to reduce the number of computations:– 3 – The expectation value of every monomial term can be factorized into expectation values ofholonomy-flux monomials with respect to different edges. Only certain types of expectationvalues of monomials on a single edge shall be computed. We further reduce the number oftypes by using the commutation relations, and several general formulae are derived for theexpectation values of the resulting types (see Section 3). • We develop a power-counting argument in order to specifically locate each power of (cid:126) , expres-sion in O ( (cid:126) ) represents the leading order behavior of each expectation value of the monomialoperator (see Section 4.3). Since we are only interested in expanding (cid:104) (cid:92) H [ N ] (cid:105) to the its linearorder in (cid:126) , a substantial amount of expectation values of monomials can be neglected due tothe fact that they are only contributing to higher order in (cid:126) . • When the coherent states are peaked at homogeneous and isotropic data. A large amountof symmetries that identify different terms are realized, which can be used to reduce thecomputational complexity.(see Secton 5).Our method exponentially reduces the computational complexity. In particular, it is useful incomputing the expectation value of Lorentzian part in (cid:92) H [ N ].In Section 5.2, In order to present the reduction methodology more concretely, an examplethat contains 3 m − ( m can be large) monomials is demonstrated. By applying our method, only5 monomials’ expectation values need to be computed.The purpose of the present paper is to give detailed derivations for the results presented in [1].Computations in this paper are carried out by using Mathematica on the High Performance Com-putation server with two 48-Core Processors (AMD EPYC 7642). One can find the Mathematicacodes at [49].The explicit resulting expression of O ( (cid:126) ) quantum correction in (cid:104) (cid:92) H [ N ] (cid:105) is summarized in Sec-tion 6. In order to demonstrate the physical significance of our results and effects from the O ( (cid:126) )correction to the classical limit of (cid:60)(cid:104) (cid:100) H [1] (cid:105) , the proposal in [16] is adopted: We view (cid:60)(cid:104) (cid:100) H [1] (cid:105) in (1.1)as the effective Hamiltonian on the 2-dimensional phase space, denoted as P cos , of homogeneousand isotropic cosmology. (cid:60)(cid:104) (cid:100) H [1] (cid:105) generates the Hamiltonian time evolution on the 2-dimensionalphase space P cos . Time evolution of the homogeneous spatial volume is plotted, and is comparedwith the evolution generated by (cid:104) (cid:100) H [1] (cid:105) at the limit of (cid:126) →
0. The comparison demonstrates the ef-fects on (cid:104) (cid:100) H [1] (cid:105) , which is generated from the O ( (cid:126) ) correction contribution (see Section 6 for details).We emphasize that the proposal that we adopt for the cosmological evolution is not as rigorous asthe path integral formula (1.2). Nevertheless, we have argued that the O ( (cid:126) ) correction in (cid:104) (cid:100) H [1] (cid:105) only contributes partially to the quantum correction in Γ which ultimately determines the quantumeffect in the dynamics. The cosmological dynamics studied in Section 6 only aims for displayingthe effect of the O ( (cid:126) ) correction in (cid:104) (cid:100) H [1] (cid:105) , and is not a rigorous prediction from the principle of LQG.The structure of the present paper the followings. Section 2 reviews the theory of LQG ona cubic lattice, including the Hamiltonian and the coherent state. Section 3, we demonstrate thecomputations of the expectation value of operators defined at a single edge. Section 4, we developa power-counting argument in order to reduce the computational complexity. Section 5 discusses (cid:104) (cid:100) H [1] (cid:105) with respect to the coherent states peaked at homogeneous and isotropic data, and thesymmetries which reduce the computational complexity. Section 6 presents the explicit results ofthe quantum correction in (cid:104) (cid:100) H [1] (cid:105) . Section 7, we conclude and discuss a few outlooks of the presentwork. – 4 – Preliminaries
Classically general relativity can be formulated with the Ashtekar-Barbero variables ( A ia , E ai ) con-sisting of SU(2) connection A ia and canonically conjugate densitized triad field E ai defined on thespatial manifold Σ [50]. We denote the coordinate on Σ by ( x, y, z ) . Let γ ⊂ Σ be a finite cubiclattice whose edges are parallel to the axes of the coordinates. The sets of edges and vertices in γ are denoted by E ( γ ) and V ( γ ) respectively. Taking advantage of γ , we define holonomies along theedges of γ , h e ( A ) = P exp (cid:90) e A = 1 + ∞ (cid:88) n =1 (cid:90) d t n (cid:90) t n d t n − · · · (cid:90) t d t A ( t ) · · · A ( t n ) , ∀ e ∈ E ( γ ) , (2.1)and gauge covariant fluxes on the 2-faces S e in the dual lattices γ ∗ , p is ( e ) := − βa tr (cid:20) τ i (cid:90) S e ε abc h ( ρ se ( σ )) E c ( σ ) h ( ρ se ( σ ) − ) (cid:21) , (2.2)where S e ∈ γ ∗ is the 2-face, ρ s ( σ ) : [0 , → Σ is a path connecting the source point s e ∈ e to σ ∈ S e such that ρ se ( σ ) : [0 , / → e and ρ se ( σ ) : [1 / , → S e . a is a length unit (e.g. a = 1mm) to make p s ( e ) dimensionless. Alternatively, one can choose the target point t e ∈ e rather than s e to define p it ( e ) := 2 βa tr (cid:20) τ i (cid:90) S e ε abc h ( ρ te ( σ )) E c ( σ ) h ( ρ te ( σ ) − ) (cid:21) . (2.3)where ρ t ( σ ) : [0 , → Σ is a path connecting the target t e ∈ e to σ ∈ S e such that ρ te ( σ ) : [0 , / → e and ρ te ( σ ) : [1 / , → S e . Given ( A ia , E ai ), Eqs. (2.1) and (2.2) lead to a map from the E ( γ ) toSL(2 , C ), g : e (cid:55)→ g e = e ip ks ( e ) τ k h e . (2.4)Because of the relation between p s and p t p ks ( e − ) τ k = p kt ( e ) τ k = − h − e p ks ( e ) τ k h e (2.5)we obtain that g e − = g − e . (2.6)Thus the map g : E ( γ ) → SL(2 , C ) generate a homomorphism from the groupoid of the graph γ toSL(2 , C ). The LQG phase space based on γ is SL(2 , C ) | E ( γ ) | and consists of all such homomorphisms[51]. Given a SU(2)-valued scalar field G : Σ → SU(2) on Σ, G defines a gauge transformation on g ,taking g to G (cid:46) g with ( G (cid:46) g )( e ) = G ( s e ) g ( e ) G ( t e ) − , ∀ e ∈ E ( γ ) . (2.7)The quantization of this classical lattice theory gives us LQG based on the graph γ . TheHilbert space H γ consists of the square integrable functions of the holonomies. Given two functions ψ i : { h e } e ∈ E ( γ ) → C , the inner produce is (cid:104) ψ | ψ (cid:105) = (cid:90) SU(2) | E ( γ ) | d µ h ψ ( { h e } e ∈ E ( γ ) ) ψ ( { h e } e ∈ E ( γ ) ) (2.8)where | E ( γ ) | denote the number of elements (i.e. cardinality) of E ( γ ) and µ h is the Haar measure. H γ is the kinematical Hilbert space of the canonical LQG with the operator-constraint formalism.Moreover, H γ modulo gauge transformations represents the physical Hilbert space of the reduced-phase-space LQG , where any gauge invariant function of h e ( A ) and p is,t ( e ) are Dirac observables,realized from the deparametrization by coupling to clock fields [38].– 5 –n H γ , the holonomy becomes the multiplication operator and, p is ( e ) and p it ( e ) are quantizedas the right- and left-invariant vector field, namely(ˆ p is ( e ) ψ )( h e (cid:48) , · · · , h e , · · · , h e (cid:48)(cid:48) ) = it dd (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 ψ ( h e (cid:48) , · · · , e (cid:15)τ i h e , · · · , h e (cid:48)(cid:48) )(ˆ p it ( e ) ψ )( h e (cid:48) , · · · , h e , · · · , h e (cid:48)(cid:48) ) = − it dd (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 ψ ( h e (cid:48) , · · · , h e e (cid:15)τ i , · · · , h e (cid:48)(cid:48) ) (2.9)where t = κ (cid:126) /a =: (cid:96) /a (if a = 1mm, t (cid:39) . × − ) and τ j = ( − i/ σ j with σ j being thePauli matrix. The commutators between the basic operators are[ h ( e ) , h ( e (cid:48) )] = 0 = [ˆ p is ( e ) , p jt ( e (cid:48) )][ˆ p is ( e ) , ˆ p js ( e (cid:48) )] = − itδ ee (cid:48) (cid:15) ijk ˆ p ks ( e ) , [ˆ p it ( e ) , ˆ p jt ( e (cid:48) )] = − itδ ee (cid:48) (cid:15) ijk ˆ p kt ( e ) , [ˆ p is ( e ) , h ( e (cid:48) )] = itδ ee (cid:48) τ i h ( e ) , [ˆ p it ( e ) , h ( e (cid:48) )] = − itδ ee (cid:48) h ( e ) τ i . (2.10)It is useful to introduce the flux operators with respect to the spherical basis. We defineˆ p ± v ( e ) := ∓ √ p xv ( e ) ± i ˆ p yv ( e )) , ˆ p v ( e ) = ˆ p zv ( e ) (2.11)with v = s, t . In the following context, α, β, · · · = 0 , ± i, j, k · · · = 1 , ,
3, the indices in the Cartesian basis.Taking advantage of the basic operators, one can define operators representing geometric ob-servables such as areas and volumes [52–54]. The volume operator plays an important role in thepresent work. Let
R ⊂
Σ be a region in Σ. The volume operator of R is defined byˆ V R := (cid:88) v ∈ V ( γ ) ∩R ˆ V v = (cid:88) v ∈ V ( γ ) ∩ R (cid:113) | ˆ Q v | (2.12)where ˆ Q v = ( βa ) ε ijk ˆ p is ( e + x ) − ˆ p it ( e − x )2 ˆ p js ( e + y ) − ˆ p js ( e − y )2 ˆ p ks ( e + z ) − ˆ p ks ( e − z )2 (2.13)where e ± i with i = x, y, z are the edges along the i th axis such that v is the source point of e + i and the target point of e − i . The total volume is denoted by ˆ V = (cid:80) v ∈ V ( γ ) ˆ V v . In terms of the fluxoperators with respect to the spherical basis (2.11), the operator ˆ Q v defined in Eq. (2.13) becomesˆ Q v = − i ( βa ) ε αβγ ˆ p αs ( e + x ) − ˆ p αt ( e − x )2 ˆ p βs ( e + y ) − ˆ p βs ( e − y )2 ˆ p γs ( e + z ) − ˆ p γs ( e − z )2 (2.14)where ε αβγ is defined by ε − , , = 1.In the operator-constraint formalism, the dynamics of LQG is encoded in the Hamiltonianconstraint, which can be written as (cid:92) H [ N ] = (cid:92) H E [ N ] + (1 + β ) (cid:92) H L [ N ] (2.15)where (cid:92) H E [ N ] is called the Euclidean part and (cid:92) H L [ N ] is the Lorentzian part. N is the smearedfunction. (cid:92) H [ N ] is constructed by using the Thiemann’s trick [7, 42]. The operator correspondingto the Euclidean part is (cid:92) H E [ N ] = 1 iβa t (cid:88) v ∈ V ( γ ) N ( v ) (cid:88) e I ,e J ,e K at v (cid:15) IJK tr( h α IJ h e K [ ˆ V v , h − e K ]) (2.16)– 6 –here e I , e J and e K are oriented to be outgoing from v , (cid:15) IJK = sgn[det( e I ∧ e J ∧ e K )], α IJ is theminimal loop around a plaquette consisting of e I and e J , where it goes out via e I and comes backthrough e J , taking v as its end point. With the same notion, the Lorentzian part reads (cid:92) H L [ N ] = − iβ a t (cid:88) v N ( v ) (cid:88) e I ,e J ,e K at v ε IJK tr([ h e I , [ ˆ V , ˆ H E ]] h − e I [ h e J , [ ˆ V , ˆ H E ]] h − e J [ h e K , ˆ V v ] h − e K ) . (2.17)In the reduced-phase-space LQG where the diffeomorphism and Hamiltonian constraints aresolved classically, the quantum dynamics is governed by the physical Hamiltonian (cid:98) H . When wedeparametrize gravity by coupling to the Gaussian dust [39, 40], the classical physical Hamiltonian H is formally the same as the Hamiltonian constraint with unit lapse, except all quantities in H are understood as Dirac observables. The quantization gives the Hamiltonian operator (cid:98) H = 12 (cid:18) (cid:100) H [1] + (cid:100) H [1] † (cid:19) (2.18) (cid:98) H is defined on H γ , which can be understood from a similar perspective of quantizing Dirac observ-ables. Note that here we consider the non-graph-changing version of the Hamiltonian (constraint).If H [ N ] is understood as constraint, the discretization and quantization on γ cause the constraintanomaly. However in the reduced-phase-space LQG, the constraint anomaly is absent, because H [ N ] is not a constraint anymore. . The self-adjoint extension of ˆ H exists [41, 55], so we choosethe extension and define the self-adjoint Hamiltonian which is still denoted by ˆ H . Choosing a canonical orientation for each edge e ∈ E ( γ ), the classical phase space based on thegraph γ is Γ γ ∼ = [SL(2 , C )] | E ( γ ) | . (2.19)The complexifier coherent state Ψ g is [18]Ψ g = (cid:79) e ∈ E ( γ ) ψ tg e , ψ tg e ( h e ) = (cid:88) j d j e − t j ( j +1) χ j ( g e h − e ) (2.20)where ψ tg e is the SU(2) coherent state at the edge e . The character χ j ( g e h − e ) is the trace of the j -representation of g e h − e . The property χ j ( g e h − e ) = χ j ( g − e h e ) leads to the useful relation ψ g e ( h e ) = ψ g e − ( h e − ) . Given g ∈ SL(2 , C ), it can be decomposed as g = e ip k τ k u = n s e i ( η + iξ ) τ ( n t ) − , (2.21)where η = −√ (cid:126)p · (cid:126)p and n s , n t ∈ SU(2), as well as ξ ∈ R , are given by n s τ ( n s ) − = − p k √ (cid:126)p · (cid:126)p τ k ,n s e − ξτ ( n t ) − = u. (2.22)Although n s and n t are not uniquely defined by this equation, each of them relates to a uniquevector through the equation, with v = s, t , n v τ ( n v ) − = (cid:126)n v · (cid:126)τ . (2.23) Sometimes, H [ N ] relates to conserved charges, then (cid:92) H [ N ] on γ may break the classical symmetry. – 7 –t is shown in [20] and is revisited shortly that (cid:104) ψ tg e | (cid:126) ˆ p s ( e ) | ψ tg e (cid:105) = − η e (cid:126)n se + O ( t ) , (cid:104) ψ tg e | (cid:126) ˆ p t ( e ) | ψ tg e (cid:105) = η e (cid:126)n te + O ( t ) (2.24)which indicates that η e (cid:126)n te is the classical limit of the flux operator at e .The following properties of the ψ tg [18–20] are useful in our analysis. Firstly, the inner productof these states read (cid:104) ψ tg | ψ tg (cid:105) = ψ tg † g (1) = 2 √ πe t/ t / ζ e ζ t sinh( ζ ) + O ( t ∞ ) (2.25)where tr( g † g ) = 2 cosh( ζ ) and (cid:61) ( ζ ) ∈ [0 , π ] with (cid:61) ( ζ ) the imaginary part of ζ . Consequently,the norm of the coherent state is (cid:104) (cid:105) g := (cid:104) ψ tg | ψ tg (cid:105) = 2 √ πe t/ t / pe p t sinh( p ) + O ( t ∞ ) , (2.26)where p = √ (cid:126)p · (cid:126)p . Secondly, ψ tg satisfy the completeness condition (cid:90) d ν t ( g ) | ψ tg (cid:105)(cid:104) ψ tg | = I , (2.27)where the measure d ν t ( g ) isd ν t ( g ) = 2 √ e − t/ (2 πt ) / sinh( p ) p e − p t d µ H ( u )d p = 2 (cid:104) (cid:105) g πt d µ H ( u )d p. (2.28)Let us complete this section with some discussions on the volume operator contained in theHamiltonian operator (cid:92) H [ N ]. Because of the square root in the definition of the volume operator,matrix elements of these operators are difficult to compute analytically. However, as far as thecoherent state expectation value is concerned, the volume operators ˆ V v in (cid:92) H [ N ] can be replaced byGiesel-Thiemann’s volume [9] ˆ V ( v ) GT which is a semiclassical expansionˆ V ( v ) GT = (cid:104) ˆ Q v (cid:105) q (cid:34) k +1 (cid:88) n =1 ( − n +1 q (1 − q ) · · · ( n − − q ) n ! (cid:32) ˆ Q v (cid:104) ˆ Q v (cid:105) − (cid:33) n (cid:35) + O ( t k +1 ) (2.29)where q = 1 /
4. By making use of ˆ V ( v ) GT , firstly truncating ˆ V ( v ) GT at finite k and replacing ˆ V v by ˆ V ( v ) GT , (cid:92) H [ N ] can be expressed by a polynomial of holonomies and fluxes. Up to higher order in t , it is nowmanageable to compute the expectation value of (cid:92) H [ N ], through computing the expectation valueof a polynomial of holonomies and fluxes. As becoming clear in a moment, computing the coherent state expectation value of (cid:92) H [ N ] can bereduced to computing expectation values of operator monomials on individual edges. In this section,let us firstly focus on the expectation value of operators on one edge.It is convenient to introduce the graphical method for our calculation. The basic idea of thegraphical method is to use graphs to represent the intertwiners which are some particular tensorson the space · · · ⊗ H j i ⊗ · · · ⊗ H ∗ j k ⊗ · · · with H j be the unitary j -representation space of SU(2).Some basic knowledges are introduced in the Appendix C. One can refer to [56–58] for more details Here we used the following result shown in [20]. For any complex number z = R + iI , there exist real numbers s ∈ R and φ ∈ [0 , π ] such that cosh( s + iφ ) = z . s and φ are uniquely determined except in the case I = 0 and | R | > s is determined up to its sign. – 8 –n this method. It should be remarked that, in the current work, lines with arrows correspond tothe upper indices which can be contracted with dual vectors in H ∗ j , and the lines without arrowscorrespond to the lower indices which can be contracted with the vectors in H j .In the graphical method, the cohere state ψ g e with g e = n se e iz e τ ( n te ) − can be represented as ψ tg e ( h e ) = (cid:88) j d j e − t j ( j +1) χ j ( g e h − e ) = (cid:88) j d j e − t j ( j +1) z e
Proposition 1.
Let ˆ O be defined as ˆ O = ˆ O ˆ O · · · ˆ O m . (3.13) Denote I := { , · · · , m } . Let I k := { i , i , · · · , i k } with i < i < · · · < i k be a sublist of I whichcontains k elements. Then, by the definition of commutator, it has ˆ O = ˆ O · · · ˆ O m ˆ O + m − (cid:88) k =1 (cid:88) I k (cid:32) (cid:89) l ∈I−I k ˆ O l (cid:33) [[ · · · [[ ˆ O , ˆ O i ] , ˆ O i ] · · · ] , ˆ O i k ] . (3.14)The proof is quite straightforward with using the relation ˆ A ˆ B = ˆ B ˆ A + [ ˆ A, ˆ B ] iteratively. In Eq.(3.14), the terms at k carry k -fold commutator. Due to the factor t in the right hand side of thecommutation relation (2.10), the k -fold commutator produces a factor t k in the final results, whichimplies that the contributions of these terms to the expectation value of ˆ O are at least at t k -order.Now let us see how to use Proposition 1 to move the holonomies to the right precisely. Assumethat ˆ O = D ιab ( h e ) in Eq. (3.14) and that all of the other operators are fluxes. Then according toEq. (3.14), we need to calculate[ · · · , [[ · · · [[ D ιab ( h e ) , p α s ( e )] , p α s ( e )] · · · p α m s ( e )] , p α t ( e )] · · · , p α n t ( e )] . – 10 –he result can be derived by[ · · · [[ D ιab ( h e ) , p α s ( e )] , p α s ( e )] · · · p α m s ( e )] = ( − it ) m D (cid:48) ιaa ( τ α ) D ι (cid:48) a a ( τ α ) · · · D (cid:48) ιa m − a m ( τ α m ) D ιa m b ( h e )(3.15)with a k = a − (cid:80) ki =1 α i , and[ · · · [[ D ιab ( h e ) , p α t ( e )] , p α t ( e )] · · · p α m t ( e )] = ( it ) m D ιab m ( h e ) D (cid:48) ιb m b m − ( τ α m ) D (cid:48) ιb m − b m − ( τ α m − ) · · · D (cid:48) ιb b ( τ α )(3.16)with b k = (cid:80) ki =1 b i + b , where we used that D ιab ( τ α ) ∝ δ a,b + α . Taking advantage of Eqs. (3.15) and(3.16), we have, for instance, D ιab ( h e ) (cid:16) m (cid:89) i =1 p α i s ( e ) (cid:17)(cid:16) n (cid:89) j =1 p β j t ( e ) (cid:17) = (cid:16) m (cid:89) i =1 p α i s ( e ) (cid:17)(cid:16) n (cid:89) j =1 p β j t ( e ) (cid:17) D ιab ( h e ) − it m (cid:88) k =1 (cid:16) (cid:89) i (cid:54) = k p α i s ( e ) (cid:17)(cid:16) n (cid:89) j =1 p β j t ( e ) (cid:17) D (cid:48) ιac ( τ α k ) D ιcb ( h e )+ ( − it ) (cid:88) k
0, one has (cid:88) j (cid:48) + d ≥ ι f ( d j (cid:48) ) F ι ( j (cid:48) + d, j (cid:48) , ∂η d j (cid:48) η )sinh( η ) + (cid:88) j (cid:48) − d ≥ ι f ( d j (cid:48) ) F ι ( j (cid:48) − d, j (cid:48) , ∂η d j (cid:48) η )sinh( η )= (cid:88) − j (cid:48) − d ≥ ι f ( d j (cid:48) ) F ι ( − j (cid:48) − d, − j (cid:48) − , ∂η − d j (cid:48) η )sinh( η ) + (cid:88) j (cid:48) − d ≥ ι f ( d j (cid:48) ) F ι ( j (cid:48) − d, j (cid:48) , ∂η d j (cid:48) η )sinh( η )= (cid:88) n ∈ Z n/ ∈ [ d − ι,d + ι ] f ( d j (cid:48) ) F ι ( n − − d, n − , ∂η nη )sinh( η ) . Here the conditions 2 j (cid:48) ± d ≥ ι are from the the triangle condition i.e. j (cid:48) + j ≥ ι , of the 3 j -symbol. The second equality is because of the replacement j (cid:48) → − j (cid:48) − j (cid:48) = 2 j (cid:48) + 1 ≡ n and f is an arbitrary function. Therefore, (cid:104) ˆ F α ··· α m ιab (cid:105) z e = t m e bz e (cid:88) ≤ d ≤ ιd + ι ∈ Z − δ ( d, e − t ( d − ) (cid:88) n ∈ Z n/ ∈ [ d − ι,d + ι ] e − t ( n − dn ) F ι ( n − − d, n − , ∂ η nη )sinh( η )(3.45)To calculate F ι ( j, j (cid:48) , k (cid:48) ), we notice that − w j (cid:48) (cid:15) j (cid:48) n i − k i (cid:18) j (cid:48) j (cid:48) k i α i n i (cid:19) = δ ( k i + α i + n i ,
0) ( α i j (cid:48) − n i ) (cid:115) ( j (cid:48) − n i − )! ( n i − + j (cid:48) )!(1 − α i )! ( α i + 1)! ( j (cid:48) − n i )! ( n i + j (cid:48) )! . (3.46)Moreover, by applying Eq. (A.16) and the expression of the Clebsch-Gordan coefficients (Eq. (5),Section 8.2.1 in [59]) , we have (cid:15) j (cid:48) n m k m +1 (cid:18) j (cid:48) ι jk m +1 a (cid:48) k (cid:19) (cid:18) j (cid:48) ι jk (cid:48) b k (cid:48)(cid:48) (cid:19) (cid:15) jk (cid:48)(cid:48) k = ( − j (cid:48) − ι − j + a (cid:48) + b δ a (cid:48) + k,n m δ k (cid:48) + b,k × ( j + ι − j (cid:48) )!( j − ι + j (cid:48) )!( − j + ι + j (cid:48) )!( j + ι + j (cid:48) + 1)! (cid:18) ( j (cid:48) + n m )!( j (cid:48) − n m )!( j (cid:48) + k (cid:48) )!( j (cid:48) − k (cid:48) )!( ι + a (cid:48) )!( ι − a (cid:48) )!( ι + b )!( ι − b )! (cid:19) / (cid:88) z ( − z ( j (cid:48) + ι + k − z )!( j − k + z )! z !( j (cid:48) − j + ι − z )!( j (cid:48) + k + a (cid:48) − z )!( j − ι − k − a (cid:48) + z )! (cid:88) z ( − z ( j + ι + k (cid:48) − z )!( j (cid:48) − k (cid:48) + z )! z !( j − j (cid:48) + ι − z )!( j + k − z )!( j (cid:48) − ι − k + z )! . (3.47)Eqs. (3.46) and (3.47) lead to F ι ( j (cid:48) − d, j (cid:48) , k (cid:48) ) = δ ( m (cid:88) i =1 α i − a + b, m (cid:89) i =1 | α i | ) / (cid:18) ι + a )!( ι − a )!( ι + b )!( ι − b )! (cid:19) / ( − d − a + b (2 j (cid:48) + 1)(2 j (cid:48) − d + 1)(2 j (cid:48) − d + ι + 1) ι +1 m (cid:89) n =1 ( α i j (cid:48) − k (cid:48) + n (cid:88) i =1 α i ) (cid:88) z ( − z ( d + ι ) z ( j (cid:48) + k (cid:48) + b − z + ι ) ι + a ( j (cid:48) − d − k (cid:48) − b + z ) ι − a z ! (cid:88) z ( − z ( ι − d ) z ( j (cid:48) − d + k (cid:48) − z + ι ) ι − b ( j (cid:48) − k (cid:48) + z ) ι + b z ! (3.48)where ( x ) n := x ( x − · · · ( x − n + 1) is the falling factorial.There is subtle issue on the summation over k and k (cid:48) in Eq. (3.40). In order to obtain thefactor sinh((2 j (cid:48) + 1) η ) / sinh( η ) in (3.41), one has to consider a summation of e k (cid:48) η over all valuesof k (cid:48) in [ − j (cid:48) , j (cid:48) ]. However, except the constraint of | k (cid:48) | ≤ j (cid:48) for k (cid:48) , there exists another conditionof | k (cid:48) + b | = | k | ≤ j implied by k = k (cid:48) + b , which seemingly narrows the range of k (cid:48) . In order topreserve the range of k (cid:48) , one can always “artificially engineer” F ι ( j, j (cid:48) , k (cid:48) ) such that it vanishes for | k (cid:48) + b | > j , which has actually been done by the triangle inequality of 3 j -symbols in the RHS Eq.(3.42). Thus the subtlety is now encoded in the condition that F ι ( j, j (cid:48) , k (cid:48) ) vanishes for | k (cid:48) + b | > j .We need to verify this for the algebraic expression of F ι ( j, j (cid:48) , k (cid:48) ) in the RHS of Eq. (3.48). To checkit, assume k (cid:48) + b = j + δ = j (cid:48) − d + δ with 0 < δ ≤ d + b . Then ( j (cid:48) − k (cid:48) + z ) ι + b , in the secondsummand over z , vanishes because j (cid:48) − k (cid:48) + z ≥ d + b − δ ≥ , j (cid:48) − k (cid:48) + z − ( ι + b ) + 1 ≤ δ < ≤ z ≤ ι − d is applied. Therefore, we conclude that the RHS of Eq.(3.48) vanishes for j − b < k (cid:48) ≤ j (cid:48) . Similarly, it can be checked that ( j (cid:48) − d + k (cid:48) − z + ι ) ι − b vanishes, indicating thevanishing of the RHS of Eq.(3.48) for − j (cid:48) ≤ k (cid:48) + b < − j − b . We thus claim that the RHS of– 15 –q.(3.48) vanishes for | k (cid:48) + b | > j , which indicates that the expression of the RHS of Eq.(3.48) canbe analytically extended to give F ι ( j, j (cid:48) , k (cid:48) ) for the full range of k (cid:48) , i.e. | k (cid:48) | ≤ j .Replacing j (cid:48) and k (cid:48) in Eq. (3.48) by n − and ∂ η respectively, we have F ι ( n − − d, n − , ∂ η δ ( m (cid:88) i =1 α i − a + b, m (cid:89) i =1 | α i | ) / (cid:18) ι + a )!( ι − a )!( ι + b )!( ι − b )! (cid:19) / ( − d − a + b n ( n − d )( n − d + ι ) ι +1 m (cid:89) n =1 ( α i n − − ∂ η n (cid:88) i =1 α i ) ι + d (cid:88) z =0 ( − z ( ι + d ) z ( n − + ∂ η + b − z + ι ) ι + a ( n − − ∂ η − d − b + z ) ι − a z ! ι − d (cid:88) z =0 ( − z ( ι − d ) z ( n − + ∂ η − d − z + ι ) ι − b ( n − − ∂ η + z ) ι + b z ! . (3.50)In order to apply the Poisson summation formula to (3.45) to calculate the summation over n , weneed to extend the summation to entire Z , while Eq. (3.45) sums n over Z \ [ d − ι, d + ι ]. However,because of the term ( n − d + ι ) ι +1 in the denominator of F ι , this extension is not trivial. We needto prove that F ι ( n − − d, n − , ∂ η ) sinh( nη )sinh( η ) is well-defined at the points where F ι itself does not.Once it is proven, we have (cid:104) ˆ F α ··· α m ιab (cid:105) z e = t m e bz e (cid:88) ≤ d ≤ ιd + ι ∈ Z − δ ( d, e − t ( d − ) (cid:90) d x e − t ( x − dx ) +2 ikπx F ι ( x − − d, x − , ∂ η xη )sinh( η ) − (cid:88) n ∈ [ d − ι,d + ι ] ∩ Z e − t ( n − dn ) F ι ( n − − d, n − , ∂ η nη )sinh( η ) . (3.51)Moreover, the integrals in the last equation usually produce a factor e η /t . Thus the second termgiven by the summation over n ∈ [ d − ι, d + ι ] ∩ Z decays exponentially as t → (cid:104) ˆ F α ··· α m ιab (cid:105) z e = t m e bz e (cid:88) ≤ d ≤ ιd + ι ∈ Z − δ ( d, e − t ( d − ) (cid:90) d x e − t ( x − dx ) +2 ikπx F ι ( x − − d, x − , ∂ η xη )sinh( η ) + O ( t ∞ ) . (3.52)In order to prove that F ι ( n − − d, n − , ∂ η ) sinh( nη )sinh( η ) is well-defined at the poles of F ι , the followingtheorem will be helpful. Theorem 3.1.
Let f ( x, k ) and g ( x, k ) be a polynomials of x and k and n ∈ Z be some integer.Then x = n is a removable singularity of the function x − n f ( x, ∂ η ) sinh( xη )sinh( η ) provided that f ( n, ∂ η ) = g ( n, ∂ η ) (cid:32) n − (cid:89) k = − n +1 ( ∂ η + k ) (cid:33) (3.53) Proof.
We will prove that (cid:32) n − (cid:89) k = − n +1 ( ∂ η + k ) (cid:33) sinh( xη )sinh( η ) = (cid:88) l g l ( x, η )( x − n ) l (3.54)– 16 –ith g l some function of x and η , and g taking the form g ( x, η ) = ˜ g ( x, η ) sinh(( x − n ) η ) . (3.55)Let y = x − n . Then we havesinh( xη )sinh( η ) = 1sinh( η ) (cosh( yη ) sinh( nη ) + sinh( yη ) cosh( nη )) . (3.56)Substitute the above expression into the LHS of Eq. (3.54) and expand the result. One then ex-presses the LHS of Eq. (3.54) as a linear combination of terms taking the forms of ∂ lη ( h ( η ) sinh( yη ))and ∂ lη ( h ( η ) cosh( yη )), where h and h are some arbitrary functions. Expanding the action ofthe differential operator by Leibniz’s rule, we obtain linear combinations of q ( η ) cosh( yη ) y m and q ( η ) sinh( yη ) y m , where q and q are some arbitrary functions. Thus, (3.54) is obtained.To get g , one just act ∂ η on neither cosh( yη ) nor sinh( yη ). Thus g = (cid:32) n − (cid:89) k = − n +1 ( ∂ η + k ) sinh( nη )sinh( η ) (cid:33) cosh( yη ) + (cid:32) n − (cid:89) k = − n +1 ( ∂ η + k ) cosh( nη )sinh( η ) (cid:33) sinh( yη ) (3.57)Since sinh( nη )sinh( η ) = (cid:80) jk = − j e − kη with 2 j + 1 = n and ( ∂ η + k ) e − kη = 0, we have n − (cid:89) k = − n +1 ( ∂ η + k ) sinh( nη )sinh( η ) = 0 . (3.58)Therefore g = (cid:32) n − (cid:89) k = − n +1 ( ∂ η + k ) cosh( nη )sinh( η ) (cid:33) sinh( yη ) (3.59)Now let us consider the expectation value of ˆ F α ··· α m ιab case by case. ι = 1 / ι = 1 /
2, the value of d in Eq. (3.45) can only be 1 /
2. Substituting the result of the function F , we finally obtain (cid:104) ˆ F α ··· α m ab (cid:105) z e = δ ( m (cid:88) i =1 α i − a + b, − a − b t m e t/ bz e (cid:32) m (cid:89) i =1 | α i | ) / (cid:33) (cid:88) n ∈ Z e − t n ( n − m (cid:89) k =1 (cid:32) α k n − − ∂ η k (cid:88) i =1 α i (cid:33) ( n − − b∂ η ) sinh( nη )sinh( η ) (3.60)where the summation has been trivially extended to entire Z . Applying the Poisson summationformula, we achieve (cid:104) F α ··· α m ab (cid:105) z e = δ ( m (cid:88) i =1 α i − a + b, − a − b t m e t/ bz e (cid:32) m (cid:89) i =1 | α i | ) / (cid:33) (cid:90) ∞−∞ d xe − t x ( x − m (cid:89) k =1 (cid:32) α k x − − ∂ η k (cid:88) i =1 α i (cid:33) ( x − − b∂ η ) sinh( xη )sinh( η ) + O ( t ∞ ) . (3.61)– 17 –ecause of the fact that ∂ nη e ± ηx sinh( η ) = ( ± x + ∂ y ) n e ± ηx sinh( y ) (cid:12)(cid:12)(cid:12) y = η (3.62)we split sinh( ηx ) into ( e xη − e − xη ) / (cid:104) ˆ F α ··· α m ab (cid:105) z e = δ ( m (cid:88) i =1 α i − a + b, − a − b t m m (cid:89) i =1 | α i | ) / e t/ − β z e (cid:110)(cid:90) ∞−∞ d x m (cid:89) i =1 (cid:32) α i − x − ∂ y i (cid:88) k =1 α k − α i (cid:33) (cid:18) − b x − b∂ y − (cid:19) e − t x ( x − ηx y ) (cid:12)(cid:12)(cid:12) y = η − (cid:90) ∞−∞ d x m (cid:89) i =1 (cid:32) α i + 12 x − ∂ y i (cid:88) k =1 α k − α i (cid:33) (cid:18) b x + − b∂ y − (cid:19) e − t x ( x − − ηx y ) (cid:12)(cid:12)(cid:12) y = η (3.63)which leads us to study the integral of x n e − t x ( x − ± xη . We have (cid:90) ∞−∞ d xx n e − t x ( x − ± xη = te t/ sinh( η ) η (cid:104) (cid:105) e ± η + t (cid:98) n/ (cid:99) (cid:88) k =0 (cid:18) n k (cid:19) (cid:18) ± η + t (cid:19) n − k (cid:18) t (cid:19) k − n (2 k − (cid:104) (cid:105) e ± η − t (cid:18) t (cid:19) − n +1 sinh( η ) η (cid:18) ( ± η ) n + n ( ± η ) n − t n ( n − ± η ) n − t · · · (cid:19) . (3.64)The leading-order term of this integral is O ( t − n +1 ) with n the power of x in the integrand. There-fore, the highest-power term in the RHS of (3.63) eventually leads to the leading-order term of (cid:104) ˆ F α ··· α m ab (cid:105) z e , and the highest-power and the next-highest-power terms in the RHS of (3.63) even-tually leads to the next-to-leading order term of (cid:104) ˆ F α ··· α m ab (cid:105) z e , and so on.Finally, we have (cid:104) ˆ F α ··· α m ab (cid:105) z e = (cid:104) (cid:105) δ ( m (cid:88) i =1 α i − a + b, − a − b m (cid:89) i =1 | α i | ) / e − t/ bz e + η/ (cid:110) (cid:32) m (cid:89) i =1 ( α i − (cid:33) − b (cid:18) η m + ( m + 1) η m − t m ( m + 1)2 η m − t (cid:19) + − (cid:0) − b coth( η ) (cid:1) (cid:32) m (cid:89) i =1 ( α i − (cid:33) + m (cid:88) k =1 (cid:89) i (cid:54) = k ( α i − (1 − b )( coth( η ) + α k k − (cid:88) i =1 α i η m − t (cid:111) + (cid:104) (cid:105) δ ( (cid:88) α − a + b, − a − b m (cid:89) i =1 | α i | ) / e − t/ bz e − η/ (cid:110) (cid:32) m (cid:89) i =1 ( α i + 1) (cid:33) b (cid:18) ( − η ) m + ( m + 1)( − η ) m − t m ( m + 1)2 ( − η ) m − t (cid:19) + − (cid:0) − b coth( η ) (cid:1) (cid:32) m (cid:89) i =1 ( α i + 1) (cid:33) + m (cid:88) k =1 (cid:89) i (cid:54) = k ( α i + 1) (1 + 2 b )( coth( η ) + α k k − (cid:88) i =1 α i ) ( − η ) m − t (cid:111) + O ( t ) (3.65)– 18 – .1.4.2 Case 2: ι = 1By setting ι = 1 in (3.45), we can obtain (cid:104) ˆ F α ··· α m ab (cid:105) z e = − δ (cid:32) m (cid:88) i =1 α i − a + b (cid:33) ( − a t m e bz e (cid:32) m (cid:89) i =1 (cid:112) | α i | (cid:33) (cid:112) (1 + | a | )(1 + | b | ) (cid:88) n ∈ Z n/ ∈ [ − , e − t ( n − m (cid:89) k =1 (cid:32) α k n − − ∂ η k (cid:88) i =1 α i (cid:33) n (cid:16) a n − + b + ∂ η (cid:17) (cid:16) b n − − ∂ η (cid:17) n − sinh( nη )sinh( η )+ δ (cid:32) m (cid:88) i =1 α i − a + b (cid:33) ( − b − a t m e bz e (cid:32) m (cid:89) i =1 (cid:112) | α i | (cid:33) (cid:112) (1 + | a | )(1 + | b | ) (cid:88) n ∈ Z n/ ∈ [ − , e − t ( n +1) m (cid:89) k =1 (cid:32) − α k n + 12 − ∂ η k (cid:88) i =1 α i (cid:33) (cid:16) n +12 − (1 − | b | − b ) ∂ η (cid:17) (cid:16) n +12 − ( | b | − − b ) ∂ η + | b | (cid:17) n +12 sinh( nη )sinh( η ) . (3.66)where in the second term we replaced n by − n for the further convenience. It is easy to check thatTheorem 3.1 can be applied to extend the summation in the last equation to all n ∈ Z . Therefore,we obtain that (cid:104) ˆ F α ··· α m ab (cid:105) z e = − ( − a δ (cid:32) m (cid:88) i =1 α i − a + b (cid:33) t m e bz e (cid:32) m (cid:89) i =1 (cid:112) | α i | (cid:112) | a | (cid:112) | b | (cid:33) (cid:90) ∞−∞ d x e − t ( x − m (cid:89) k =1 (cid:32) α k x − − ∂ η k (cid:88) i =1 α i (cid:33) (cid:18) a x −
12 + ∂ η b (cid:19) x ( b ( x − − ∂ η ) x − xη )sinh( η )+( − b − a δ (cid:32) m (cid:88) i =1 α i − a + b (cid:33) t m e bz e (cid:32) m (cid:89) i =1 (cid:112) | α i | (cid:112) | a | (cid:112) | b | (cid:33) (cid:90) ∞−∞ d xe − t ( x +1) m (cid:89) k =1 (cid:32) − α k n + 12 − ∂ η k (cid:88) i =1 α i (cid:33) (cid:18) n + 12 − ( | b | − − b ) ∂ η | b | (cid:19) ( x + 1 − (1 − | b | − b ) ∂ η ) x + 1 sinh( nη )sinh( η )+ O ( t ∞ ) . (3.67)To show the algorithms to compute Eq. (3.67), m (cid:54) = 0 will be assumed without the loss ofgenerality. For the case of m = 0, the algorithm can be applied very similarly.– 19 –o begin with, we will rewrite Eq. (3.67) as (cid:104) ˆ F α ··· α m ab (cid:105) z e = − ( − a δ (cid:32) m (cid:88) i =1 α i − a + b (cid:33) t m e bz e (cid:32) m (cid:89) i =1 (cid:112) | α i | (cid:112) | a | (cid:112) | b | (cid:33) (cid:90) ∞−∞ d x e − t ( x − m (cid:89) k =2 (cid:32) α k x − − ∂ η k (cid:88) i =1 α i (cid:33) (cid:18) a x −
12 + ∂ η b (cid:19) x (cid:18) b − ∂ η x − (cid:19) (cid:18) α − ∂ η x + 1 (cid:19) sinh( nη )sinh( η )+( − b − a δ (cid:32) m (cid:88) i =1 α i − a + b (cid:33) t m e bz e (cid:32) m (cid:89) i =1 (cid:112) | α i | (cid:112) | a | (cid:112) | b | (cid:33) (cid:90) ∞−∞ d x e − t ( x +1) m (cid:89) k =1 (cid:32) − α k x + 12 − ∂ η k (cid:88) i =1 α k (cid:33) (cid:18) x + 12 − ( | b | − − b ) ∂ η | b | (cid:19) (cid:18) − (1 − | b | − b ) ∂ η x + 1 (cid:19) sinh( xη )sinh( η ) . (3.68)Then for the first integral, it is I = (cid:90) ∞−∞ d x e − t ( x − m (cid:89) k =2 (cid:32) α k x − − ∂ η k (cid:88) i =1 α i (cid:33) (cid:18) a x −
12 + ∂ η b (cid:19) x (cid:18) b − ∂ η x − (cid:19) (cid:18) α − ∂ η x + 1 (cid:19) sinh( xη )sinh( η ) . (3.69)Because (cid:18) b − ∂ η x − (cid:19) (cid:18) α − ∂ η x + 1 (cid:19) sinh( xη )sinh( η )= (cid:32) bα sinh( xη )sinh( η ) − α (cid:18) cosh( ηx )sinh( η ) − sinh( η ( x − ( η )( x − (cid:19) − b (cid:18) cosh( ηx )sinh( η ) − sinh( η ( x + 1))sinh ( η )( x + 1) (cid:19) + (cid:18) sinh( ηx )sinh( η ) + cosh( η )sinh( η ) sinh( η ( x − x − − cosh( η )sinh( η ) sinh(( x + 1) η ) x + 1 (cid:19) (cid:33) , (3.70) I can be deduced further as the following I = (cid:90) ∞−∞ d x e − t ( x − m (cid:89) k =2 (cid:32) α k x − − ∂ η k (cid:88) i =1 α i (cid:33) (cid:18) a x −
12 + ∂ η b (cid:19) x ( bα + 1) sinh( xη ) − ( α + b ) cosh( ηx )sinh( η )+ (cid:90) ∞−∞ d y e − t ( y +2 y ) m (cid:89) k =2 (cid:32) α k y − ∂ η k (cid:88) i =1 α i (cid:33) (cid:18) a y ∂ η b (cid:19) ( y + 1) α sinh( η ) + cosh( η )sinh( η ) sinh( ηy ) y − (cid:90) ∞−∞ d y e − t ( y − y ) m (cid:89) k =2 (cid:32) α k y − ∂ η k − (cid:88) i =1 α i (cid:33) (cid:18) a y ∂ η b − a (cid:19) ( y −
1) cosh( η ) − b sinh( η )sinh( η ) sinh( yη ) y . (3.71)Then by using the tricky ∂ nη e ± xη f ( η ) = ( ± x + ∂ z ) n e ± xη f ( z ) (cid:12)(cid:12) z → η , (3.72)– 20 –e have I = (cid:88) s = ± (cid:90) ∞−∞ d x e − t ( x − m (cid:89) k =2 (cid:32) α k − s x − α k + ∂ z k (cid:88) i =1 α i (cid:33) (cid:18) a + s x + − a + ∂ z b (cid:19) x ( − ( α + b ) + s ( bα + 1)) e sx z ) (cid:12)(cid:12)(cid:12) z → η + (cid:88) s = ± (cid:90) ∞−∞ d y e − t ( y +2 y ) m (cid:89) k =2 (cid:32) α k − s y − ∂ z k (cid:88) i =1 α i (cid:33) (cid:18) a + s y + ∂ z b (cid:19) ( y + 1) se sηy y α sinh( z ) + cosh( z )sinh( z ) (cid:12)(cid:12)(cid:12) z → η − (cid:88) s = ± (cid:90) ∞−∞ d y e − t ( y − y ) m (cid:89) k =2 (cid:32) α k − s y − ∂ z k − (cid:88) i =1 α i (cid:33) (cid:18) a + s y + ∂ z b − a (cid:19) ( y − se sηy y cosh( z ) − b sinh( z )sinh( z ) (cid:12)(cid:12)(cid:12) z → η (3.73)By defining C = m (cid:89) k =2 (cid:32) − ∂ η k (cid:88) i =1 α i (cid:33) (cid:18) ∂ z b (cid:19) α sinh( η ) + cosh( η )sinh( η ) C = − m (cid:89) k =2 (cid:32) − ∂ η k − (cid:88) i =1 α k (cid:33) (cid:18) ∂ η b − a (cid:19) cosh( η ) − b sinh( η )sinh( η ) (3.74) I can be finally simplified as I = (cid:88) s = ± (cid:90) ∞−∞ d x e − t ( x − m (cid:89) k =2 (cid:32) α k − s x − α k + ∂ z k (cid:88) i =1 α i (cid:33) (cid:18) a + s x + − a + ∂ z b (cid:19) x ( − ( α + b ) + s ( bα + 1)) e sx z ) (cid:12)(cid:12)(cid:12) z → η + (cid:88) s = ± (cid:90) ∞−∞ d y e − t ( y +2 y ) (cid:32) m (cid:89) k =2 (cid:32) α k − s y − ∂ z k (cid:88) i =1 α i (cid:33) (cid:18) a + s y + ∂ z b (cid:19) ( y + 1) α sinh( z ) + cosh( z )sinh( z ) (cid:12)(cid:12)(cid:12) z → η − C (cid:33) se sηy y − (cid:88) s = ± (cid:90) ∞−∞ d y e − t ( y − y ) (cid:32) m (cid:89) k =2 (cid:32) α k − s y − ∂ z k − (cid:88) i =1 α i (cid:33) (cid:18) a + s y + ∂ z b − a (cid:19) ( y −
1) cosh( z ) − b sinh( z )sinh( z ) (cid:12)(cid:12)(cid:12) z → η − C (cid:33) se sηy y + C (cid:90) ∞−∞ e − t ( y +2 y ) sinh( ηy ) y d y − C (cid:90) ∞−∞ e − t ( y − y ) sinh( ηy ) y d y. (3.75)In this expression, the integrands of first three terms involve only polynomials of the integralvariables, which can be computed by using the same procedure as that to compute (cid:104) ˆ F α ··· α m ab (cid:105) . Forthe last two terms, by using the formula (cid:90) ∞−∞ d ye − t ( y + by ) sinh( yη ) y = 12 π erfi tb η (cid:113) t η + erfi − tb η (cid:113) t η , (3.76)– 21 –nd erfi (cid:18) x (cid:19) = e x xπ ∞ (cid:88) n =0 Γ( 12 + n ) x n , (3.77)a straightforward calculation can be performed as the following (cid:90) ∞−∞ d y e − t ( y ± y ) sinh( yη ) y = (cid:104) (cid:105) √ π sinh( η ) η ∞ (cid:88) m =0 m (cid:88) n =0 Γ( 12 + n ) 2 n η n + m (cid:18) n + mm − n (cid:19)(cid:40) e − η + e η ( − m − n (cid:41) (cid:18) t (cid:19) m +2 , (3.78)which complete our computation for I .Secondly, I can be simplified as the following I = (cid:90) ∞−∞ d xe − t ( x +1) m (cid:89) k =1 (cid:32) − α k x + 12 − ∂ η k (cid:88) i =1 α i (cid:33) (cid:18) x + 12 − ( | b | − − b ) ∂ η | b | (cid:19) sinh( xη )sinh( η ) − (cid:90) ∞−∞ d xe − t ( x +1) m (cid:89) k =1 (cid:32) − α k x + 12 − ∂ η k (cid:88) i =1 α i (cid:33) (cid:18) x + 12 + ( | b | − − b ) ∂ η | b | (cid:19) (1 − | b | − b ) ∂ η x + 1 sinh( xη )sinh( η ) . (3.79)Because ∂ η x + 1 sinh( xη )sinh( η ) = 1sinh( η ) (cid:18) sinh( η ) cosh( xη ) − sinh(( x + 1) η ) x + 1 (cid:19) (3.80)we have I = (cid:90) ∞−∞ d xe − t ( x +1) m (cid:89) k =1 (cid:34) − α k x + 12 − ∂ η k (cid:88) i =1 α i (cid:35) (cid:20) x + 12 − ( | b | − − b ) ∂ η | b | (cid:21) × sinh( xη ) − (1 − | b | − b ) cosh( xη )sinh( η )+ (cid:90) ∞−∞ d ye − t y m (cid:89) k =1 (cid:34) ] − α k y − ∂ η k (cid:88) i =1 α i (cid:35) (cid:20) y − ( | b | − − b ) ∂ η | b | (cid:21) (1 − | b | − b )sinh( η ) sinh( yη ) y . (3.81)By defining C = (cid:18) α − ∂ η (cid:19) (cid:18) α + α − ∂ η (cid:19) · · · (cid:18) ( α + · · · + α m − ∂ η (cid:19) (cid:18) | b | − ( | b | − − b ) ∂ η (cid:19) − | b | − b sinh( η ) , (3.82)we obtain I = (cid:90) ∞−∞ d xe − t ( x +1) m (cid:89) k =1 (cid:34) − α k x + 12 − ∂ η k (cid:88) i =1 α i (cid:35) × (cid:20) x + 12 − ( | b | − − b ) ∂ η | b | (cid:21) sinh( xη ) − (1 − | b | − b ) cosh( xη )sinh( η )+ (cid:90) ∞−∞ d ye − t y (cid:40) m (cid:89) k =1 (cid:34) − α k y − ∂ η k (cid:88) i =1 α k (cid:35) (cid:20) y − ( | b | − − b ) ∂ η | b | (cid:21) − C (cid:41) (1 − | b | − b )sinh( η ) sinh( yη ) y + C (cid:90) ∞−∞ d ye − t y sinh( yη ) y . (3.83)– 22 –imilar as I , the first two integrals are computable because the integrands therein are just poly-nomials multiplied by the Gaussian functions. For the last term, we have C (cid:90) ∞−∞ d ye − t y sinh( yη ) y = C sgn( η ) π erfi (cid:113) t η = (cid:104) (cid:105) C t √ πe t/ sinh( η ) η ∞ (cid:88) n =0 Γ( 12 + n ) (cid:18) tη (cid:19) n . (3.84)Therefore, I is computable.In summary, we obtain (cid:104) ˆ F α ··· α m ab (cid:105) z e = − ( − a δ (cid:32) m (cid:88) i =1 α i − a + b (cid:33) t m e bz e (cid:32) m (cid:89) i =1 (cid:112) | α i | (cid:112) | a | (cid:112) | b | (cid:33) I +( − b − a δ (cid:32) m (cid:88) i =1 α i − a + b (cid:33) t m e bz e (cid:32) m (cid:89) i =1 (cid:112) | α i | (cid:112) | a | (cid:112) | b | (cid:33) I + O ( t ∞ ) (3.85)where I and I can be computed by using the algorithm introduced above. Taking the operatorˆ F ab = D ab ( h e ) as an example, we can have (cid:104) ˆ F ab (cid:105) z e = e iξ − (cid:18) e iξ tanh ( η ) η + e iξ (cid:19) t + O ( t ) , a = b = 1 , − tanh ( η ) η t + O ( t ) , a = b = 0 ,e − iξ − (cid:18) e − iξ tanh ( η ) η + e − iξ (cid:19) t + O ( t ) , a = b = − . (3.86) ι = 3 / and ι = 2In the current work, it is enough to consider ι ≤
2. For ι = 3 / ι = 2, one can easily showthat Theorem 3.1 can still be applied. Hence, the integrands in Eq. (3.52) are well-defined for all x ∈ R . Moreover, according to Eqs. (3.54) and (3.55), the integrands in Eq. (3.52) are in terms ofeither e ay + by y n or e ay + by sinh( ηy ) /y , where one can take the case with ι = 1 as a specific examplefor more details. Thus, for ι = 3 / ι = 2, the results is computable efficiently with the sametechniques as that for the case of ι = 1. In our computation, we often use the operator D ιa b ( h e )[ p s ( e )] m [ p t ( e )] n · · · D ιa k b k ( h e )[ p s ( e )] m k [ p t ( e )] n k . (3.87)To deal with this kind of operators, let us consider the operator D ιab ( h e )( p s ( e )) m ( p t ( e )) n . Byapplying Proposition 1, it can be simplified to D ιab ( h e )[ p s ( e )] m [ p t ( e )] n =[ p s ( e )] m [ p t ( e )] n D ιab ( h e ) − atm [ p s ( e )] m − [ p t ( e )] n D ιab ( h e ) + btn [ p s ( e )] m [ p t ( e )] n − D ιab ( h e ) + O ( t ) . (3.88)– 23 –hen, for the operator (3.87), it has D ιa b ( h e )[ p s ( e )] m [ p t ( e )] n · · · D ιa k b k ( h e )[ p s ( e )] m k [ p t ( e )] n k =[ p s ( e )] (cid:80) ki =1 m i [ p t ( e )] (cid:80) ki =1 n i k (cid:89) i =1 D ιa i b i ( h e ) − t (cid:32) k (cid:88) i =1 a i (cid:34) k (cid:88) l = i m i (cid:35) [ p s ( e )]( (cid:80) ki =1 m i ) − [ p t ( e )] (cid:80) ki =1 n i k (cid:89) i =1 D ιa i b i ( h e ) (cid:33) + t (cid:32) k (cid:88) i =1 b i (cid:34) k (cid:88) l = i n i (cid:35) [ p s ( e )] (cid:80) ki =1 m i [ p t ( e )]( (cid:80) ki =1 n i ) − k (cid:89) i =1 D ιa i b i ( h e ) (cid:33) + O ( t ) (3.89)By (3.22), we finally have D ιa b ( h e )[ p s ( e )] m [ p t ( e )] n · · · D ιa k b k ( h e )[ p s ( e )] m k [ p t ( e )] n k =( − (cid:80) ki =1 n i [ p s ( e )] (cid:80) ki =1 ( m i + n i ) k (cid:89) i =1 D ιa i b i ( h e ) − t ( − (cid:80) ki =1 n i (cid:32) k (cid:88) i =1 a i (cid:34) k (cid:88) l = i m l (cid:35) + k (cid:88) i =1 b i (cid:34) k (cid:88) l = i n l (cid:35)(cid:33) × [ p s ( e )] (cid:80) ki =1 ( m i + n i ) − k (cid:89) i =1 D ιa i b i ( h e ) . (3.90)Recalling the derivation of (cid:104) ˆ F α ··· α m ιab (cid:105) z e , we get (cid:104) ( p s ( e )) m D ιab ( h e ) (cid:105) z e = e bη (cid:18) − t ∂ η (cid:19) m e − bη (cid:104) D ιab ( h e ) (cid:105) z e (3.91)where the result of (cid:104) ( p s ( e )) m (cid:105) z e is given by setting ι = 0 = a = b . It can be verified that, (cid:104) D ιab ( h e ) (cid:105) z e takes the form that (cid:104) D ιab ( h e ) (cid:105) z e = (cid:104) (cid:105) z e ( g + tg ( η ) + O ( t )) = f ( t ) e η t η sinh( η ) ( g + tg ( η ) + O ( t )) (3.92)with some functions g , g and f . Therefore, with Fa`a di Bruno’s formula, we can have that (cid:104) ( p s ( e )) m D ιab ( h e ) (cid:105) z e = e bη (cid:18) − t ∂ η (cid:19) m e − bη (cid:104) D ιab ( h e ) (cid:105) z e = (cid:104) (cid:105) z e ( − η ) m [ g + tg ( η )] + (cid:104) (cid:105) z e m ( m + 1)4 ( − η ) m − g t + (cid:104) (cid:105) z e m − η ) m − (coth( η ) + b ) g t + O ( t ) (3.93)Based on these formula, we can propose a faster algorithm to deal with these cases. In the current work, we finally need to deal with operators of the form (cid:80) (cid:126)α T α α ··· α m ˆ O α α ··· α m with T being some numerical factors and ˆ O being some polynomial operators of holonomies andfluxes. In principle, we would need to compute the expectation values of ˆ O α ··· α m for all indices α , · · · , α m . The computational complexity comes from the huge amount of terms in the sum over (cid:126)α . Since we are only interested in the expectation value up to O ( t ), the complexity can be reducedby certain power-counting argument: we count the least power of t contains in each (cid:104) ˆ O α ··· α m (cid:105) before explicit computation, then we omit those terms only contribute to higher order than O ( t )in (cid:104) (cid:92) H [ N ] (cid:105) . It turns out that a large degree of complexity can be reduced in this manner. Thisargument will be proven rigorously in this section.– 24 –n this section, we will denote Ψ g defined in (2.20) by | Ψ (cid:126)g (cid:105) with (cid:126)g = { g e } e ∈ E ( γ ) , namely | Ψ (cid:126)g (cid:105) = (cid:79) e ∈ E ( γ ) | ψ g e (cid:105) . (4.1)Similarly, | Ψ (cid:126)g ( i ) (cid:105) denotes the coherent state that at the edge e is | ψ g ( i ) e (cid:105) . Let ˆ O take the form ofˆ O = ˆ O ˆ O · · · ˆ O k (4.2)with ˆ O i being arbitrary polynomial of fluxes and holonomies. Inserting the resolution of identity(2.27), we have (cid:104) Ψ (cid:126)g | ˆ O | Ψ (cid:126)g (cid:105) = (cid:90) k − (cid:89) m =1 d ν ( (cid:126)g ( m ) ) k (cid:89) i =1 (cid:104) Ψ (cid:126)g ( i − | ˆ O i | Ψ (cid:126)g ( i ) (cid:105) (4.3)where | Ψ (cid:126)g (0) (cid:105) = | Ψ (cid:126)g ( k ) (cid:105) := | Ψ (cid:126)g (cid:105) and the measure d ν ( (cid:126)g ( m ) ) isd ν ( (cid:126)g ( m ) ) = (cid:89) e ∈ E ( γ ) d ν ( g ( m ) e ) (4.4)with d ν ( g ( m ) e ) defined in (2.28). Eq. (4.3) relates the expectation value of ˆ O to matrix elements ofeach individual ˆ O i . Thus we are motivated to study matrix elements of polynomial of holonomiesand flux. To begin with, we study the matrix element of the holonomy and flux. For the flux operator p αs ( e ), we have (cid:104) ψ g e | p αs | ψ g (cid:48) e (cid:105) = it dd (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 (cid:104) ψ g e | ψ e − (cid:15)τα g (cid:48) e (cid:105) = it dd (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 √ πe t/ t / ζ e e ζ et sinh( ζ e ) + O ( t ∞ ) (4.5)where we used the fact that χ j ( g (cid:48) e h − e e − (cid:15)τ α ) = χ j ( e − (cid:15)τ α g (cid:48) e h − e ) as well as Eq. (2.25), and ζ e isdefined by cosh( ζ e ) = 12 tr( e − (cid:15)τ α g (cid:48) e g † e ) . (4.6)By this definition, it has d ζ e d (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 = −
12 tr( τ α g (cid:48) e g † e )sinh( ζ (0) e ) (4.7)where ζ is given by 2 cosh( ζ (0) e ) = tr( g (cid:48) e g † e ), i.e. ζ (0) e = ζ e | (cid:15) =0 . Combining the results of (4.5) and(4.7), we finally have (cid:104) ψ g e | p αs | ψ g (cid:48) e (cid:105) = i (cid:104) ψ g e | ψ g (cid:48) e (cid:105) tr( τ α g (cid:48) e g † e )sinh( ζ (0) e ) (cid:32) − ζ (0) e + (coth( ζ (0) e ) − ζ (0) e ) t (cid:33) + O ( t ∞ ) . (4.8)Doing the same for p αt ( e ), one finally has (cid:104) ψ g e | p αt | ψ g (cid:48) e (cid:105) = − it dd (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 (cid:104) ψ g e | ψ g (cid:48) e e (cid:15)τα (cid:105) = i (cid:104) ψ g e | ψ g (cid:48) e (cid:105) tr( g † e g (cid:48) e τ α )sinh( ζ (0) e ) (cid:32) ζ (0) e − (coth( ζ (0) e ) − ζ (0) e ) t (cid:33) + O ( t ∞ ) . (4.9)– 25 –or the holonomy operator D ιab ( h e ), it is sufficient to consider the case of ι = 1 /
2. By using(C.8), we have (cid:104) ψ g e | D ab ( h e ) | ψ g (cid:48) e (cid:105) = (cid:88) j (cid:88) j (cid:48) = j ± / d j d j (cid:48) e − t ( j ( j +1)+( j (cid:48) ( j (cid:48) +1) ) (cid:88) m,n D jmn ( g e ) D j (cid:48) m + a,n + b ( g (cid:48) e )( − m + a − n − b × (cid:18) j (cid:48) jb − n − b n (cid:19) (cid:18) j (cid:48) ja − m − a m (cid:19) . (4.10)According to Eq. (A.21), the last equation can be simplified to (cid:104) ψ g e | D ab ( h e ) | ψ g (cid:48) e (cid:105) = (cid:88) j e − t ( j ( j +1)+(( j +1 / j +3 / (cid:88) m,n D jnm ( g † e ) (cid:18) [ ˆ∆ , D ab ( g (cid:48) e )] + ( j + 14 ) D ab ( g (cid:48) e ) (cid:19) D jmn ( g (cid:48) e ) − (cid:88) j e − t ( j ( j +1)+(( j − / ) (cid:88) m,n D jnm ( g † e ) (cid:18) [ ˆ∆ , D ab ( g (cid:48) e )] − ( j + 34 ) D ab ( g (cid:48) e ) (cid:19) D jmn ( g (cid:48) e ) (4.11)where ˆ∆ is defined in (A.19). By definition[ ˆ∆ , D ab ( g (cid:48) e )] = 34 D ab ( g (cid:48) e ) + 2 iD ab ( τ k g (cid:48) e ) ˆ X k , (4.12)which leads to (cid:104) ψ g e | D ab ( h e ) | ψ g (cid:48) e (cid:105) = (cid:88) j e − t ( j ( j +1)+(( j +1 / j +3 / (cid:16) − D ab ( τ k g (cid:48) e ) χ j ( τ k g (cid:48) e g † e ) + ( j + 1) D ab ( g (cid:48) e ) χ j ( g (cid:48) e g † e ) (cid:17) − (cid:88) j e − t ( j ( j +1)+(( j − / ) (cid:16) − D ab ( τ k g (cid:48) e ) χ j ( τ k g (cid:48) e g † e ) − jD ab ( g (cid:48) e ) χ j ( g (cid:48) e g † e ) (cid:17) . (4.13)Let 2 cosh( ζ (0) e ) := tr( g (cid:48) e g † e ) and 2 cosh( ζ ke ) := tr( e (cid:15)τ k g (cid:48) e g † e ). We have (cid:104) ψ g e | D ab ( h e ) | ψ g (cid:48) e (cid:105) = dd (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 (cid:88) j e − t ( j ( j +1)+(( j +1 / j +3 / (cid:32) − D ab ( τ k g (cid:48) e ) sinh( d j ζ ke )sinh( ζ ke ) + ( j + 1) D ab ( g (cid:48) e ) sinh( d j ζ (0) e )sinh( ζ (0) e ) (cid:33) − dd (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 (cid:88) j e − t ( j ( j +1)+(( j − / ) (cid:32) − D ab ( τ k g (cid:48) e ) sinh( d j ζ ke )sinh( ζ ke ) − j sinh( d j ζ (0) e )sinh( ζ (0) e ) (cid:33) . (4.14)By denoting n = d j , these two sums over j can be combined as (cid:104) ψ g e | D ab ( h e ) | ψ g (cid:48) e (cid:105) = dd (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 (cid:88) n ∈ Z e − t (2 n ( n +1) − (cid:32) D ab ( τ k g (cid:48) e ) sinh( nζ ke )sinh( ζ ke ) + n + 12 D ab ( g (cid:48) e ) sinh( nζ (0) e )sinh( ζ (0) e ) (cid:33) (4.15)to which the Poisson summation formula can be applied. Thus we have (cid:104) ψ g e | D ab ( h e ) | ψ g (cid:48) e (cid:105) =2 D ab ( τ k g (cid:48) e ) dd (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 √ π (cid:16) e ζ ke − (cid:17) e ( ζke )2 t + t − ζke √ t sinh( ζ ke )+ D ab ( g (cid:48) e ) √ πe ( ζ (0) e )2 t + t − ζ (0) e (cid:16) − te ζ (0) e + t + 4 (cid:16) e ζ (0) e + 1 (cid:17) ζ (0) e (cid:17) t / sinh( ζ (0) e ) + O ( t ∞ ) . (4.16)– 26 –ince d ζ ke d (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 = 12 tr( τ k g (cid:48) e g † e )sinh( ζ (0) e ) (4.17)we finally have (cid:104) ψ g e | D ab ( h e ) | ψ g (cid:48) e (cid:105) = (cid:104) ψ g e | ψ g (cid:48) e (cid:105) τ k g (cid:48) e g † e ) D ab ( τ k g (cid:48) e ) (cid:16) e ζ (0) e + 1 (cid:17) ζ (0) e e ζ (0) e + D ab ( g (cid:48) e )2 ζ (0) e e − ζ (0) e e − t (cid:18)(cid:16) e ζ (0) e + 1 (cid:17) ζ (0) e + ( − e ζ (0) e + 1) t (cid:19) + O ( t ∞ ) . (4.18)where Eq. (2.25) is used.In summary, Eqs. (4.8), (4.9) and (4.18) tell that the matrix elements of the fluxes and holo-momies are of a form described below (cid:104) ψ g e | ˆ O i | ψ g (cid:48) e (cid:105) = (cid:104) ψ g e | ψ g (cid:48) e (cid:105) ( E ( g e , g (cid:48) e ) + tE ( g e , g (cid:48) e ) + O ( t ∞ )) . (4.19)Recalling Eq. (4.3), we are going to consider integrals containing (cid:104) ψ ge | ψ g (cid:48) e (cid:105)(cid:107) ψ ge (cid:107)(cid:107) ψ g (cid:48) e (cid:107) . These integrals canbe analyzed with the generalized stationary phase approximation guided by H¨ormander’s theorem7.7.5 in [60]. Theorem 4.1.
Let K be a compact subset in R n , X be an open neighborhood of K, and k be apositive integer. If (1) the complex functions u ∈ C k ( K ) , f ∈ C k +1 ( X ) and (cid:61) ( f ) ≥ in X, with (cid:61) ( f ) being the imaginary part of f ; (2) there is a unique point x ∈ K satisfying (cid:61) ( f ( x )) = 0 , f (cid:48) ( x ) = 0 , and det( f (cid:48)(cid:48) ( x )) (cid:54) = 0 ( f (cid:48)(cid:48) denotes the Hessian matrix), f (cid:48) (cid:54) = 0 in K \ { x } then we havethe following estimation: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) K u ( x ) e iλf ( x ) dx − e iλf ( x ) (cid:20) det (cid:18) λf (cid:48)(cid:48) ( x )2 πi (cid:19)(cid:21) − k − (cid:88) s =0 (cid:18) λ (cid:19) s L s u ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) λ (cid:19) k (cid:88) | α |≤ k sup | D α u | . (4.20) Here the constant C is bounded when f stays in a bounded set in C k +1 ( X ) . We have used thestandard multi-index notation α = (cid:104) α , ..., α n (cid:105) and D α = ( − i ) α ∂ | α | ∂x α ...∂x α n n , where | α | = n (cid:88) i =1 α i (4.21) L s u ( x ) denotes the following operation on u: L s u ( x ) = i − s (cid:88) l − m = s (cid:88) l ≥ m ( − l − l l ! m ! n (cid:88) a,b =1 H − ab ( x ) ∂ ∂x a ∂x b l (cid:0) g mx u (cid:1) ( x ) , (4.22) where H ( x ) = f (cid:48)(cid:48) ( x ) denotes the Hessian matrix and the function g x ( x ) is given by g x ( x ) = f ( x ) − f ( x ) − H ab ( x ) ( x − x ) a ( x − x ) b (4.23) such that g x ( x ) = g (cid:48) x ( x ) = g (cid:48)(cid:48) x ( x ) = 0 . For each s, L s is a differential operator of order 2sacting on u ( x ) . – 27 – .2 Analysis of integrals containing (cid:104) ψ ge | ψ g (cid:48) e (cid:105)(cid:107) ψ ge (cid:107)(cid:107) ψ g (cid:48) e (cid:107) Define G ( (cid:126)p (1) , (cid:126)θ (1) , (cid:126)p (2) , (cid:126)θ (2) ) := (cid:104) ψ g (1) | ψ g (2) (cid:105)(cid:107) ψ g (1) (cid:107)(cid:107) ψ g (2) (cid:107) (4.24)with g ( k ) parameterized by g ( k ) = e i(cid:126)p ( k ) · (cid:126)τ e (cid:126)θ ( k ) · (cid:126)τ , k = 1 , . (4.25)According to Eq. (2.25) and Eq. (2.26), G ( (cid:126)p (1) , (cid:126)θ (1) , (cid:126)p (2) , (cid:126)θ (2) ) reads G ( (cid:126)p (1) , (cid:126)θ (1) , (cid:126)p (2) , (cid:126)θ (2) ) = ζ (cid:112) sinh( p (1) ) sinh( p (2) ) (cid:112) p (1) p (2) sinh( ζ ) e − − ζ p (1))2+( p (2))22 t (4.26)where p ( i ) = (cid:112) (cid:126)p ( i ) · (cid:126)p ( i ) and ζ is given by2 cosh( ζ ) = tr( g † g ) . (4.27)Denote S ( (cid:126)p (1) , (cid:126)θ (1) , (cid:126)p (2) , (cid:126)θ (2) ) := − ζ + ( p (1) ) + ( p (2) ) . We first claim that
Lemma 4.1. (cid:60) ( S ( (cid:126)p (1) , (cid:126)θ (1) , (cid:126)p (2) , (cid:126)θ (2) )) , the real part of S ( (cid:126)p (1) , (cid:126)θ (1) , (cid:126)p (2) , (cid:126)θ (2) ) , is non-negative andvanishes iff (cid:126)p (2) = (cid:126)p (1) and (cid:126)θ (2) = (cid:126)θ (1) . To prove this lemma, let us introduce the following proposition which is given in [19] and alsoproven in Appendix D.
Proposition 2.
Let g = e i(cid:126)p · (cid:126)τ e (cid:126)θ · (cid:126)τ . Considering ζ = s + iφ ∈ C with s ∈ R and φ ∈ [0 , π ] determinedby cosh( ζ ) = tr( g ) , we have, with denoting p := √ (cid:126)p · (cid:126)p , δ = p − s + φ ≥ where the equality occurs iff θ := (cid:112) (cid:126)θ · (cid:126)θ = 0 . Thanks to this proposition, we prove Lemma 4.1 as follows.
Proof of Lemma 4.1. g † g can be decomposed as g † g = e i(cid:126)x · (cid:126)τ e (cid:126)y · (cid:126)τ . Denote x = √ (cid:126)x · (cid:126)x and y = √ (cid:126)y · (cid:126)y . Then (cid:60) ( S ( (cid:126)p, (cid:126)θ, (cid:126)p, (cid:126)θ )) = 2 δ − x p (1) ) + ( p (2) ) . (4.29)where δ := x / − (cid:60) ( ζ ) is non-negative according to the proposition 2. Thus we only need to provethat − x / p (1) ) + ( p (2) ) ≥ x ) = tr( g † g g † g ), which leads to2 cosh( x ) = tr( e i(cid:126)p (1) · (cid:126)τ e i(cid:126)p (2) · (cid:126)τ ) . (4.30)Since e i(cid:126)µ · (cid:126)τ = cosh( µ ) I + 2 i (cid:126)µ · (cid:126)τµ sinh( µ ) with µ = √ (cid:126)µ · (cid:126)µ , we havecosh( x ) = 1 − β p (1) − p (2) ) + 1 + β p (1) + p (2) ) ≤ cosh( p (1) + p (2) ) (4.31)– 28 –here β = (cid:126)p (1) · (cid:126)p (2) p (1) p (2) ∈ [ − ,
1] and cosh( p (1) + p (2) ) ≥ cosh( p (1) − p (2) ) is used. Moreover, because of (cid:112) p (1) ) + 2( p (2) ) ≥ p (1) + p (2) ≥
0, it hascosh( (cid:113) p (1) ) + 2( p (2) ) ) ≥ cosh( p (1) + p (2) ) . (4.32)Combining the results of (4.31) and (4.32), one finally have − x p (1) ) + ( p (2) ) ≥ (cid:126)p (1) = (cid:126)p (2) .In summary we have (cid:60) ( S ( (cid:126)p, (cid:126)θ, (cid:126)p, (cid:126)θ )) ≥ (cid:60) ( S ( (cid:126)p, (cid:126)θ, (cid:126)p, (cid:126)θ )) = 0 only if (cid:126)p (1) = (cid:126)p (2) and δ = 0 which means (cid:126)θ (1) = (cid:126)θ (2) .It turns out below that the integrand of Eq. (4.3) consists of a Gaussian-like function e − t ( S ( (cid:126)p,(cid:126)θ,(cid:126)p (1) ,(cid:126)θ (1) )+ S ( (cid:126)p (1) ,(cid:126)θ (1) ,(cid:126)p (2) ,(cid:126)θ (2) )+ ··· + S ( (cid:126)p ( k ) ,(cid:126)θ ( k ) ,(cid:126)p,(cid:126)θ ) ) . (4.35)Lemma 4.1 suggests us to do the stationary phase approximation analysis at (cid:126)p ( i ) = (cid:126)p and (cid:126)θ ( i ) = (cid:126)θ .Notice that (cid:126)p and (cid:126)θ are given to parameterize g as g := e i(cid:126)p · (cid:126)τ e (cid:126)θ · (cid:126)τ , and g labels the coherent state | ψ g (cid:105) with respect to which the expectation value of ˆ O is computed. Thus, rather than consideringall values of (cid:126)p and (cid:126)θ , it is sufficient to set (cid:126)p o = (0 , , p ) , (cid:126)θ o = (0 , , θ ) (4.36)according to (3.10). Denote f k ; (cid:126)p,(cid:126)θ ( (cid:126)p (1) , (cid:126)θ (1) , · · · , (cid:126)p ( k ) , (cid:126)θ ( k ) ) ≡ S ( (cid:126)p, (cid:126)θ, (cid:126)p (1) , (cid:126)θ (1) ) + S ( (cid:126)p (1) , (cid:126)θ (1) , (cid:126)p (2) , (cid:126)θ (2) ) + · · · + S ( (cid:126)p ( k ) , (cid:126)θ ( k ) , (cid:126)p, (cid:126)θ ) , (4.37)we have the following result: Theorem 4.2. (i) (cid:60) ( f k ; (cid:126)p,(cid:126)θ ) ≥ and the equality occurs only when all g ( i ) coincide, namely (cid:126)p ( i ) = (cid:126)p and (cid:126)θ ( i ) = (cid:126)θ .(ii) At (cid:126)p ( i ) = (cid:126)p o = (0 , , p ) and (cid:126)θ ( i ) = (cid:126)θ o = (0 , , θ ) , it has ∇ (cid:126)p ( i ) f k ; (cid:126)p o ,(cid:126)θ o = 0 = ∇ (cid:126)θ ( i ) f k ; (cid:126)p o ,(cid:126)θ o , ∀ i = 1 , · · · , k. (4.38) (iii) The Hessian matrix f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o of f k ; (cid:126)p o ,(cid:126)θ o at (cid:126)p ( i ) = (cid:126)p o = (0 , , p ) and (cid:126)θ ( i ) = (cid:126)θ o = (0 , , θ ) isnon-degenerate with the determinant det (cid:16) f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o (cid:17) (cid:12)(cid:12)(cid:12) (cid:126)p ( i ) = (cid:126)p o ,(cid:126)θ ( i ) = (cid:126)θ o = (cid:32) (cid:0) θ (cid:1) θ (cid:33) k (4.39) Proof.
The first statement is true by using Lemma 4.1. For the second statement, let us consider S ( (cid:126)p (1) , (cid:126)θ (1) , (cid:126)p (2) , (cid:126)θ (2) ) = − ζ + ( (cid:126)p (1) ) + ( (cid:126)p (2) ) (4.40)where ζ is given by cosh( ζ ) = 12 tr( e (cid:126)θ (2) · (cid:126)τ e − (cid:126)θ (1) · (cid:126)τ e i(cid:126)p (1) · (cid:126)τ e i(cid:126)p (2) · (cid:126)τ ) . (4.41)– 29 –hen it has that ∂ cosh( ζ ) ∂p (1) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o = ∂ cosh( ζ ) ∂p (2) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o = δ j, p ) ∂ cosh( ζ ) ∂θ (1) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o = − ∂ cosh( ζ ) ∂θ (2) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o = iδ j, p ) (4.42)where the subscript (cid:126)p o , (cid:126)θ o indicates to take values at (cid:126)p (1) = (cid:126)p o = (cid:126)p (2) and (cid:126)θ (1) = (cid:126)θ o = (cid:126)θ (2) .According to Eq. (4.42), we have ∇ (cid:126)p (1) S ( (cid:126)p (1) , (cid:126)θ (1) , (cid:126)p (2) , (cid:126)θ (2) ) (cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o = 0 = ∇ (cid:126)p (2) S ( (cid:126)p (1) , (cid:126)θ (1) , (cid:126)p (2) , (cid:126)θ (2) ) (cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o (4.43)and, thus, ∇ (cid:126)p ( i ) f k ; (cid:126)p o ,(cid:126)θ o (cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o = 0 for all i = 1 , , · · · , k . For ∇ (cid:126)θ ( i ) f k ; (cid:126)p o ,(cid:126)θ o (cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o , the second equationin Eq. (4.42) gives us ∇ (cid:126)θ ( i ) f k ; (cid:126)p o ,(cid:126)θ o (cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o = ∇ (cid:126)θ ( i ) S ( (cid:126)p ( i − , (cid:126)θ ( i − , (cid:126)p ( i ) , (cid:126)θ ( i ) ) (cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o + ∇ (cid:126)θ ( i ) S ( (cid:126)p ( i ) , (cid:126)θ ( i ) , (cid:126)p ( i +1) , (cid:126)θ ( i +1) ) (cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o = 0 , (4.44)which completes the proof of the second statement.For the last statement, using the conclusion of the second statement, we can immediately getthat ∂f k ; (cid:126)p o ,(cid:126)θ o ∂x ( i ) m ∂y ( j ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o = 0 , | i − j | > x ( i ) m and y ( j ) n represent (cid:126)θ ( i ) m or (cid:126)p ( i ) m . Therefore, if we order the arguments (cid:126)p ( i ) , (cid:126)θ ( i ) as p (1)1 , p (1)2 , p (1)3 , θ (1)1 , θ (1)2 , θ (1)3 , p (2)1 , p (2)2 , p (2)3 , θ (2)1 , θ (2)2 , θ (2)3 , · · · . (4.46)to arrange the matrix elements of f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o ( (cid:126)p o , (cid:126)θ o ) ≡ f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o (cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o , the resulting matrix is block-tridiagonal matrix. Moreover, since all of the p -arguments, as well as the θ -arguments, in f k ; (cid:126)p o ,(cid:126)θ o are symmetric, we conclude that ∂f k ; (cid:126)p o ,(cid:126)θ o ∂x ( i ) m ∂y ( j ) l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o = ∂f k ; (cid:126)p o ,(cid:126)θ o ∂x ( i (cid:48) ) m ∂y ( j (cid:48) ) l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)p o ,(cid:126)θ o . (4.47)Consequently f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o ( (cid:126)p o , (cid:126)θ o ) takes the form f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o ( (cid:126)p o , (cid:126)θ o ) = A B · · · B T A B · · · B T A B · · · · · · B T A B · · · B T A (4.48)where A and B are 6 × A and B can be calculated by considering the casewith k = 2, which gives us A = ( p ) p − ( θ ) tanh ( p ) θ − θ ) tanh ( p ) θ ( p ) p θ ) tanh ( p ) θ − ( θ ) tanh ( p ) θ
00 0 2 0 0 0 − ( θ ) tanh ( p ) θ θ ) tanh ( p ) θ − p (cos( θ ) −
1) coth( p ) θ − θ ) tanh ( p ) θ − ( θ ) tanh ( p ) θ − p (cos( θ ) −
1) coth( p ) θ
00 0 0 0 0 2 – 30 –nd B = − ( p ) p ( θ ) tanh ( p ) + i sin( θ ) θ sin( θ ) tanh ( p ) + i (cos( θ ) − θ − ( p ) p − i cos( θ ) − sin( θ ) tanh ( p ) + iθ ( θ ) tanh ( p ) + i sin( θ ) θ
00 0 − i ( θ ) tanh ( p ) − i sin( θ ) θ i ( cos( θ )+ i sin( θ ) tanh ( p ) − ) θ p (cos( θ ) −
1) coth( p ) θ − ip (cos( θ ) − θ − i cos( θ )+sin( θ ) tanh ( p ) + iθ ( θ ) tanh ( p ) − i sin( θ ) θ ip (cos( θ ) − θ p (cos( θ ) −
1) coth( p ) θ
00 0 − i − . With the expression of A and B , it can be verified that BA − B T = B T A − B = 0 . (4.49)To calculate det( f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o ( (cid:126)p o , (cid:126)θ o )), we define matrices of ˜ B , ˜ C and D of dimensions 6 × k − k − × k − × k −
1) respectively such that f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o ( (cid:126)p o , (cid:126)θ o ) = (cid:18) A ˜ B ˜ C D (cid:19) . (4.50)Then by using the property of the Schur complement, it hasdet( f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o ( (cid:126)p o , (cid:126)θ o )) = det( A ) det( D − ˜ CA − ˜ B ) = det( A ) det( D ) (4.51)where we used ˜ CA − ˜ B = 0 because of Eq. (4.49) and that f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o is block-tridiagonal matrix.Because D is the Hessian matrix f (cid:48)(cid:48) k − (cid:126)p o ,(cid:126)θ o ( (cid:126)p o , (cid:126)θ o ), we finally havedet( f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o ( (cid:126)p o , (cid:126)θ o )) = det( A ) k = (cid:32) ( θ ) θ (cid:33) k . (4.52)By these results, the stationary phase approximation method introduced in Theorem 4.1 canbe applied to calculate the integral (4.3). Assigning to each edge e a complex number w e = p e − iθ e , we have the coherent state | Ψ (cid:126)w (cid:105) := (cid:79) e ∈ E ( γ ) | ψ w e (cid:105) . (4.53)For the operator ˆ O in Eq. (4.2), we state the following theorem Theorem 4.3.
Consider an operator ˆ O = (cid:81) ki =1 ˆ O i . Assume that, for each operator ˆ O i , its matrixelements (cid:104) Ψ (cid:126)g (1) | ˆ O i | Ψ (cid:126)g (2) (cid:105) take the following form (cid:104) Ψ (cid:126)g (1) | ˆ O i | Ψ (cid:126)g (2) (cid:105) = (cid:104) Ψ (cid:126)g (1) | Ψ (cid:126)g (2) (cid:105) (cid:16) E ( i )0 ( (cid:126)g (1) , (cid:126)g (2) ) + tE ( i )1 ( (cid:126)g (1) , (cid:126)g (2) ) + O ( t ∞ ) (cid:17) . (4.54) Let N be the number of operators ˆ O m ∈ { ˆ O i } ki =1 such that (cid:104) Ψ (cid:126)w | ˆ O m | Ψ (cid:126)w (cid:105)(cid:104) Ψ (cid:126)w | Ψ (cid:126)w (cid:105) = O ( t ) , (4.55)– 31 – here the O ( t ) term vanishes on the RHS. Then the expectation value of ˆ O with respect to thecoherent state | Ψ (cid:126)w (cid:105) satisfies (cid:104) Ψ (cid:126)w | ˆ O | Ψ (cid:126)w (cid:105)(cid:104) Ψ (cid:126)w | Ψ (cid:126)w (cid:105) = O ( t n ) , with n ≥ (cid:98) N + 12 (cid:99) (4.56) where (cid:98) x (cid:99) is the largest integer no larger than x .Proof. As in Eq. (4.3), it has (cid:104) Ψ (cid:126)w | ˆ O | Ψ (cid:126)w (cid:105)(cid:104) Ψ (cid:126)w | Ψ (cid:126)w (cid:105) = (cid:90) k − (cid:89) j =1 (cid:89) e ∈ E ( γ ) πt d µ H ( u ( j ) e )d (cid:126)p ( j ) e k (cid:89) i =1 (cid:104) Ψ (cid:126)g ( i − | ˆ O i | Ψ (cid:126)g ( i ) (cid:105)(cid:107) Ψ (cid:126)g ( i − (cid:107)(cid:107) Ψ (cid:126)g ( i ) (cid:107) (4.57)where we denoted | Ψ (cid:126)g (0) (cid:105) = | Ψ (cid:126)g ( k ) (cid:105) := | Ψ (cid:126)w (cid:105) , applied Eq. (2.28) and used the decomposition g ( i ) e = e i(cid:126)p ( i ) e · (cid:126)τ e (cid:126)θ ( i ) e · (cid:126)τ = e i(cid:126)p ( i ) e · (cid:126)τ u ( i ) e . (4.58)By the assumption, we have (cid:104) Ψ (cid:126)g ( i − | ˆ O i | Ψ (cid:126)g ( i ) (cid:105)(cid:107) Ψ (cid:126)g ( i − (cid:107)(cid:107) Ψ (cid:126)g ( i ) (cid:107) = (cid:104) Ψ (cid:126)g ( i − | Ψ (cid:126)g ( i ) (cid:105)(cid:107) Ψ (cid:126)g ( i − (cid:107)(cid:107) Ψ (cid:126)g ( i ) (cid:107) (cid:16) E ( i )0 ( (cid:126)g ( i − , (cid:126)g ( i ) ) + tE ( i )1 ( (cid:126)g ( i − , (cid:126)g ( i ) ) + O ( t ∞ ) (cid:17) . (4.59)Thus Eq. (4.57) takes the form (cid:104) Ψ (cid:126)w | ˆ O | Ψ (cid:126)w (cid:105)(cid:104) Ψ (cid:126)w | Ψ (cid:126)w (cid:105) = (cid:90) k − (cid:89) j =1 (cid:89) e ∈ E ( γ ) πt d µ H ( u ( j ) e )d (cid:126)p ( j ) e k (cid:89) i =1 (cid:104) Ψ (cid:126)g ( i − | Ψ (cid:126)g ( i ) (cid:105)(cid:107) Ψ (cid:126)g ( i − (cid:107)(cid:107) Ψ (cid:126)g ( i ) (cid:107) k (cid:89) l =1 E ( l ) ( (cid:126)g ( l − , (cid:126)g ( l ) ) (4.60)with P ( { (cid:126)g ( i ) } ki =1 ) denoting the function E ( l ) ( (cid:126)g ( l − , (cid:126)g ( l ) ) := (cid:16) E ( l )0 ( (cid:126)g ( l − , (cid:126)g ( l ) ) + tE ( l )1 ( (cid:126)g ( l − , (cid:126)g ( l ) ) + O ( t ∞ ) (cid:17) . (4.61)Eq. (4.60) can be analyzed with the stationary phase approximation according to Theorem 4.2. Itshould be noticed that the Haar measure d µ H can be expressed asd µ H ( u ) = sin (cid:16) (cid:112) (cid:126)θ · (cid:126)θ (cid:17) π ( (cid:126)θ · (cid:126)θ ) d (cid:126)θ (4.62)where u ∈ SU(2) is coordinatized as u = e (cid:126)θ · (cid:126)τ and d (cid:126)θ is the Lebesgue measure on R . Substitutingthe last equation into Eq. (4.60) and applying Eq. (4.20) as well as Eq. (4.39), we finally obtain (cid:104) Ψ (cid:126)w | ˆ O | Ψ (cid:126)w (cid:105)(cid:104) Ψ (cid:126)w | Ψ (cid:126)w (cid:105) = (cid:89) e ∈ E ( γ ) (cid:18) θ e sin ( θ e / (cid:19) k − l − (cid:88) s =0 (2 t ) s D ( s ) (cid:12)(cid:12)(cid:12) g ( k ) e = e iweτ , ∀ k,e + O ( t l ) (4.63)where D ( s ) takes the form D ( s ) = ( − s (cid:88) l − j = s (cid:88) l ≥ j ( − l − l l ! j ! n (cid:88) a,b =1 H − ab ( x ) ∂ ∂x a ∂x b l (cid:32) G ( j ) k (cid:89) i =1 E ( i ) (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( l ) e = e iweτ , ∀ l,e (4.64)with G ( j ) being defined as the following, G ( j ) (cid:16) { (cid:126)g ( i ) } k − i = k (cid:17) = (cid:89) e ∈ E ( γ ) k − (cid:89) n =1 sin ( | (cid:126)θ ( n ) e | / | (cid:126)θ ( n ) e | (cid:89) e ∈ E ( γ ) k − (cid:89) n =0 ζ ( n,n +1) e (cid:113) sinh( p ( n ) e ) sinh( p ( n +1) e ) (cid:113) p ( n ) e p ( n +1) e sinh( ζ ( n,n +1) e ) g x ( { (cid:126)g ( i ) } k − i =1 ) j . (4.65)– 32 –ere g x is some function defined by applying (4.23) to the current case.If the leading order term of D ( s ) is claimed to be O ( t d s ), then each term in the summation over s of Eq. (4.63) is O ( t s + d s ). Moreover, for each l in Eq. (4.64), the derivative acting on G ( j ) (cid:81) E ( i ) is of order 2 l in total. Because of the properties given by Eq.(4.23), the non-vanishing result appearswhen there are at least 3 j derivatives acting on G ( j ) , which indicates that the order of derivativethat acts on the term (cid:81) E ( i ) in Eq. (4.64) is 2 l − j . Because 2 l − j = s and l = s + j , there areat most 2 s − j derivatives acting on (cid:81) E ( i ) . Further, since j ≥
0, the maximum order of derivativeacting on the term (cid:81) E ( i ) is 2 s , which only occurs for l = s . Then, let us count the leading orderof D ( s ) for a given s . According to the expression of D ( s ) , once it is evaluated at the critical pointgiven by g ( k ) e = e iω e τ , those E ( m ) contributed by operators ˆ O m satisfying (4.55) will inevitablyincrease the power of its leading order term if they are not acted by any derivative operators. Fora fixed s there are at least n s of these “non-acted” terms with n s = ( N − s ) + | N − s | . (4.66)Therefore, (cid:104) Ψ (cid:126)w | ˆ O | Ψ (cid:126)w (cid:105)(cid:104) Ψ (cid:126)w | Ψ (cid:126)w (cid:105) is of order of t n with n ≥ min s ∈ Z + ( s + n s ) = (cid:98) N + 12 (cid:99) . (4.67)Because of the vanishing leading-order term of (cid:104) ˆ Q (cid:104) ˆ Q (cid:105) − (cid:105) , it can be regarded as operator ˆ O m satisfying (4.55). Thus, Theorem 4.3 is applied to count the power of t for the term including (cid:16) ˆ Q (cid:104) ˆ Q (cid:105) − (cid:17) k in (cid:104) (cid:92) H [ N ] (cid:105) . Moreover, in order to apply Theorem 4.3, matrix elements of ˆ O i have to becomputable. We have to factorize ˆ Q (cid:104) ˆ Q (cid:105) − (cid:16) ˆ Q (cid:104) ˆ Q (cid:105) + 1 (cid:17) (cid:16) ˆ Q (cid:104) ˆ Q (cid:105) − (cid:17) because every matrix elementof ˆ Q is a polynomial of matrix elements of the flux operators, while that of ˆ Q is not.Because the expectation values of ˆ p ± s ( e ), ˆ p ± t ( e ) and D ιab ( h e ) ( a (cid:54) = b ) with respect to Ψ (cid:126)ω vanish, each of them can also be considered as operator ˆ O m in (4.55). Therefore, this theorem canbe applied to study the leading order of monomial of holonomies and fluxes. Let us use ˆ p β ( e ) todenote either ˆ p βs ( e ) or ˆ p βt ( e ), and use M to denote the monomial of holonomies and fluxes. Let N ± be the number of ˆ p ± ( e ) respectively and, M + (respectively M − ) be the number of D −
12 12 ( h e )(respectively D , − ( h e )) in M . According to our analysis above, the expectation value of M withrespect to the coherent state | ψ z e (cid:105) with z e ∈ C is non-vanishing if m (cid:88) i =1 β i + k (cid:88) j =1 ( b j − a j ) = 0 . (4.68)Hence, we have N + + M + = N − + M − . (4.69)Therefore, this theorem gives us that the leading order the expectation value (cid:104)M(cid:105) z e is O ( t N + + M + )or higher. We have more discussions on this case. Since the matrix elements of ˆ p α ( e ) and D ab ( h e )are computable, the results on the leading order of M can be calculated more concretely. Theresult is summarized as the following theorem. Theorem 4.4.
Given M an arbitrary monomial of holonomies and fluxes. Let M (cid:48) be the operatorresulting from M by deleting all factors ˆ p ( e ) and D aa ( h e ) . Denote the number of ˆ p s ( e ) and ˆ p t ( e )– 33 – n M as N ,s and N ,t respectively, and the number of D ( h e ) and D − − ( h e ) as M and M − respectively. Then the leading order of (cid:104)M(cid:105) z e is exactly O ( t M + + N + ) if and only if the leadingorder of (cid:104)M (cid:48) (cid:105) z e is exactly O ( t M + + N + ) . Moreover, for the case when (cid:104)M(cid:105) z e and (cid:104)M (cid:48) (cid:105) z e are exactly O ( t M + + N + ) , it has (cid:104)M(cid:105) z e ∼ = ( (cid:104) ˆ p s ( e ) (cid:105) z e ) N ,s ( (cid:104) ˆ p t ( e ) (cid:105) z e ) N ,t ( (cid:104) D ( h e ) (cid:105) z e ) M , + ( (cid:104) D − − ( h e ) (cid:105) z e ) M , − (cid:104)M (cid:48) (cid:105) z e (4.70) where ∼ = means the leading-order terms, i.e. the O ( t M + + N + ) terms, of the left and right hand sidesare equal to each other. The proof of this theorem is divided into several lemmas which can be verified by straightforwardcalculations.At first, denote ˆ p ( e ), D ( h e ) and D − − ( h e ) by ˆ O d a with a = 1 , , p ± ( e ), D − ( h e ) or D −
12 12 ( h e ) by ˆ O nd a with a = 1 , , , O ia take the form (cid:104) ψ g (1) | ˆ O ia | ψ g (2) (cid:105) = (cid:104) ψ g (1) | ψ g (2) (cid:105) E ia ;0 ( g (1) , g (2) ) + O ( t ) , ∀ i = d , nd . (4.71)Then by formulae listed in Appendix E, we obtain: Lemma 4.2.
Given g ( i ) = e i(cid:126)p ( i ) · (cid:126)τ e i(cid:126)θ ( i ) · (cid:126)τ , we have ( ∂ x ( j ) k E d a ;0 )( e iwτ , e iwτ ) = 0 , j = 1 , , and k = 1 , ∂ x ( j )3 E nd a ;0 )( e iwτ , e iwτ ) = 0 , j = 1 , for all w = p − iθ ∈ C , where x ( i ) j denotes p ( i ) j or θ ( i ) j . Moreover, consider the matrix A and B inEq. (4.48) . E nd a ;0 satisfies that ∇ T(cid:126)x (1) E nd a ;0 ( e iwτ k , e iwτ ) A − B = 0 = ∇ T(cid:126)x (2) E nd a ;0 ( e iwτ k , e iwτ ) A − B T , ∇ T(cid:126)x (1) E nd a ;0 ( e iwτ k , e iwτ ) A − B T = −∇ T(cid:126)x (1) E nd a ;0 ( e iwτ k , e iwτ ) , ∇ T(cid:126)x (2) E nd a ;0 ( e iwτ k , e iwτ ) A − B = −∇ T(cid:126)x (2) E nd a ;0 ( e iwτ k , e iwτ ) ,B T A − ∇ (cid:126)x (1) E nd a ;0 ( e iwτ k , e iwτ ) = 0 = BA − ∇ (cid:126)x (2) E nd a ;0 ( e iwτ k , e iwτ ) ,BA − ∇ (cid:126)x (1) E nd a ;0 ( e iwτ k , e iwτ ) = −∇ (cid:126)x (1) E nd a ;0 ( e iwτ k , e iwτ ) ,B T A − ∇ (cid:126)x (2) E nd a ;0 ( e iwτ k , e iwτ ) = −∇ (cid:126)x (2) E nd a ;0 ( e iwτ k , e iwτ ) , (4.73) where ∇ (cid:126)x ( i ) = ( ∂ (cid:126)p ( i )1 , ∂ (cid:126)p ( i )2 , ∂ (cid:126)p ( i )3 , ∂ (cid:126)θ ( i )1 , ∂ (cid:126)θ ( i )2 , ∂ (cid:126)θ ( i )3 ) T and ∇ T(cid:126)x ( i ) is its transpose. The second lemma is on the inverse of the Hessian matrix (4.48). Recalling Eq. (4.50), one hasthat the inverse of f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o , denoted as H − k,(cid:126)p o ,(cid:126)θ o , is H − k,(cid:126)p o ,(cid:126)θ o = (cid:18) A − + A − ˜ B ( D − ˜ CA − ˜ B ) − ˜ CA − − A − ˜ B ( D − ˜ CA − ˜ B ) − − ( D − ˜ CA − ˜ B ) − ˜ CA − ( D − ˜ CA − ˜ B ) − (cid:19) . (4.74)Since ˜ CA − ˜ B = 0 and D = f (cid:48)(cid:48) k − (cid:126)p o ,(cid:126)θ o , we have H − k,(cid:126)p o ,(cid:126)θ o = (cid:32) A − + A − ˜ BH − k − ,(cid:126)p o ,(cid:126)θ o ˜ CA − − A − ˜ BH − k − ,(cid:126)p o ,(cid:126)θ o − H − k − ,(cid:126)p o ,(cid:126)θ o ˜ CA − H − k − ,(cid:126)p o ,(cid:126)θ o (cid:33) . (4.75)– 34 –or k = 1, H − ,(cid:126)p o ,(cid:126)θ o = A − . Thus one has that ˜ BH − ,(cid:126)p o ,(cid:126)θ o ˜ C = 0 and H − ,(cid:126)p o ,(cid:126)θ o = (cid:32) A − − A − ˜ BH − ,(cid:126)p o ,(cid:126)θ o − H − ,(cid:126)p o ,(cid:126)θ o ˜ CA − H − ,(cid:126)p o ,(cid:126)θ o (cid:33) . Doing this successively, one has ˜ BH − k − ,(cid:126)p o ,(cid:126)θ o ˜ C = 0, and H − k,(cid:126)p o ,(cid:126)θ o = (cid:32) A − − A − ˜ BH − k − ,(cid:126)p o ,(cid:126)θ o − H − k − ,(cid:126)p o ,(cid:126)θ o ˜ CA − H − k − ,(cid:126)p o ,(cid:126)θ o (cid:33) . (4.76)Finally, H − k ; (cid:126)p o ,(cid:126)θ o can be obtained with this recurrence relation and the initial data H − ,(cid:126)p o ,(cid:126)θ o = A − .The result is as follows: Lemma 4.3. H − k ; (cid:126)p o ,(cid:126)θ o satisfies that ( H − k ; (cid:126)p o ,(cid:126)θ o ) mn = ( − | m − n | A − ( BA − ) | m − n | , m < nA − , m = n ( − | m − n | A − ( B T A − ) | m − n | , m > n (4.77) where H − k,(cid:126)p o ,(cid:126)θ o is arranged as a block matrix as f (cid:48)(cid:48) k ; (cid:126)p o ,(cid:126)θ o in (4.48) ], with ( H − k ; (cid:126)p o ,(cid:126)θ o ) mn as a block. Now the theorem (4.4) can be proven.
Proof of Theorem 4.4.
For convenience, we define s o = M + + N + . By Theorem 4.3, (cid:104)M(cid:105) z e is oforder t s o or higher. Adopting the result from equation (4.63), O ( t s o ) only occurs when s = s o and D ( s o ) is of O ( t ).Set s = s o in the definition (4.64) of D ( s ) . For given l and j , if there is one E ( m ) taking theform of E nd a ;0 + O ( t ) not being acted by derivatives, then the eventual evaluation at the critical pointwill vanish. Moreover, in D ( s o ) , on one hand it contains 2 s o of E nd a ;0 + O ( t ), and on the other handthe maximum order of derivative that can act on (cid:81) E ( i ) is 2 s o , which only occurs when l = s o .Therefore, only when l = s o and m = 0, all E nd a ;0 + O ( t ) are acted by derivatives. Finally, D ( s o ) = (cid:32) sin ( θ ) θ (cid:33) |M|− − s o s o ! |M|− (cid:88) a,b =1 H − ab ∂ ∂x a ∂x b s o |M|− (cid:89) i =1 E ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( l ) = e iwτ , ∀ l , (4.78)where |M| denotes the number of factors in the monomial M .To calculate (4.78), we employ the notion introduced in Lemma 4.2 and 4.3 to treat H − as ablock matrix. Then (cid:80) |M|− a,b =1 H − ab ∂ ∂x a ∂x b is rewritten as |M|− (cid:88) a,b =1 H − ab ∂ ∂x a ∂x b = |M|− (cid:88) m,n =1 ∇ T(cid:126)x ( m ) (cid:16) ˆ H − |M|− (cid:126)p o ,(cid:126)θ o (cid:17) mn ∇ (cid:126)x ( n ) . (4.79)We then expand (cid:104)(cid:80) |M|− m,n =1 ∇ T(cid:126)x ( m ) (cid:16) ˆ H − |M|− (cid:126)p o ,(cid:126)θ o (cid:17) mn ∇ (cid:126)x ( n ) (cid:105) s o and let each individual term of theexpansion act on (cid:81) |M|− i =1 E ( i ) . In each individual term of the expansion, it contains derivative withrespect to certain (cid:126)x ( q ) . Because each E nd a ;0 + O ( t ) only depends on certain (cid:126)x ( q ) , we only considerthe case when all derivatives are paired with all E nd a ;0 + O ( t ) with the same argument. For the otherterms of the expansion, they give vanishing results because of the evaluation at the critical point.– 35 –he procedure mentioned above is equivalent to the follows. We first partition these E nd a ;0 + O ( t )into ordering pairs. Denote all possibilities of the partition as P . Given a pair ( E nd a m ;0 ( (cid:126)x ( m − , (cid:126)x ( m ) )+ O ( t ) , E nd a n ;0 ( (cid:126)x ( n − , (cid:126)x ( n ) ) + O ( t )) in a partition p ∈ P . It can be acted by ∇ T(cid:126)x ( m ) ( H − M− (cid:126)p o ,(cid:126)θ o ) mn ∇ (cid:126)x ( n ) , (4.80) ∇ T(cid:126)x ( m − ( H − M− (cid:126)p o ,(cid:126)θ o ) m − ,n ∇ (cid:126)x ( n ) , (4.81) ∇ T(cid:126)x ( m ) ( H − M− (cid:126)p o ,(cid:126)θ o ) m,n − ∇ (cid:126)x ( n − , (4.82) ∇ T(cid:126)x ( m − ( H − M− (cid:126)p o ,(cid:126)θ o ) m − ,n − ∇ (cid:126)x ( n − (4.83)According to Lemmas 4.3 and 4.2,(1) If m < n , only the operator (4.82) gives non-vanishing results which reads ∇ T(cid:126)x ( m ) E nd a m ;0 ( (cid:126)x ( m − , (cid:126)x ( m ) )( H − M− (cid:126)p o ,(cid:126)θ o ) m,n − ∇ (cid:126)x ( n − E nd a n ;0 ( (cid:126)x ( n − , (cid:126)x ( n ) )= ∇ T(cid:126)x ( m ) E nd a m ;0 ( (cid:126)x ( m − , (cid:126)x ( m ) ) A − ∇ (cid:126)x ( n − E nd a n ;0 ( (cid:126)x ( n − , (cid:126)x ( n ) ) (4.84)(2) If m > n , only the operator (4.81) gives non-vanishing results which reads ∇ T(cid:126)x ( m − E nd a m ;0 ( (cid:126)x ( m − , (cid:126)x ( m ) )( H − M− (cid:126)p o ,(cid:126)θ o ) m − ,n ∇ (cid:126)x ( n ) E nd a n ;0 ( (cid:126)x ( n − , (cid:126)x ( n ) )= ∇ T(cid:126)x ( m − E nd a m ;0 ( (cid:126)x ( m − , (cid:126)x ( m ) ) A − ∇ (cid:126)x ( n ) E nd a n ( (cid:126)x ( n − , (cid:126)x ( n ) )= ∇ T(cid:126)x ( n ) E nd a n ;0 ( (cid:126)x ( n − , (cid:126)x ( n ) ) A − ∇ (cid:126)x ( m − E nd a m ;0 ( (cid:126)x ( m − , (cid:126)x ( m ) ) , (4.85)where in the last step we used A = A T .It should be reminded that an evaluation at the critical point has been done in Eqs.(4.84) and(4.85).According to the results in Eqs. (4.84) and (4.85), rather than partitioning the E nd a ;0 + O ( t ) into ordering pairs, we can identify the partitions p and p if p can be the same as p by reorderingeach of its pairs. The set with this identification will be denoted by ˜ P . Then we finally have |M|− (cid:88) m,n =1 ∇ T(cid:126)x ( m ) (cid:16) ˆ H − |M|− (cid:126)p o ,(cid:126)θ o (cid:17) mn ∇ (cid:126)x ( n ) s o |M|− (cid:89) i =1 E ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( l ) = e iwτ , ∀ l =2 s o (cid:88) p ∈ ˜ P (cid:89) ( m,n ) ∈ p with m T (cid:111) F where F is the isotropic subgroup and T is the translation subgroup. Denote the subgroup of T (cid:111) F preserving γ by S γ . A classical state g is said to be symmetric with respect to S γ if s ∗ g := g ◦ s isidentical with g up to a gauge transformation s ( ∀ s ∈ S γ ). According to this definition, classicallysymmetric states g are of the form g : e (cid:55)→ g e = n e e izτ n − e (5.1)with n e ∈ SU(2) satisfying n e τ n − e = (cid:126)n e · (cid:126)τ . (5.2)In the last equation, (cid:126)n e is the unit vector pointing to direction of edge e . Then, for each s = ( t, f ) ∈ T (cid:111) F , it can be verified that g ◦ s = Ad f ◦ g (5.3)where Ad f ◦ g ( e ) = f g ( e ) f − for all e ∈ E ( γ ). Given ˆ F e as a polynomial of fluxes and holonomies on e . For s = ( t, f ) ∈ T (cid:111) F , Eq. (5.3) resultsin (cid:104) ψ g s ( e ) | ˆ F s ( e ) | ψ g s ( e ) (cid:105) = (cid:104) ψ fg e f − | ˆ F e | ψ fg e f − (cid:105) = f (cid:46) (cid:104) ψ g e | ˆ F e | ψ g e (cid:105) (5.4)where f (cid:46) (cid:104) ψ g e | ˆ F e | ψ g e (cid:105) denotes the gauge transformation of the expectation value (cid:104) ψ g e | ˆ F e | ψ g e (cid:105) andwe used Eq. (3.10) to derive the last equality.To expand the expectation value of ˆ H E and ˆ H L to order O ( t ), one needs to replace the operatorˆ V v by ˆ V ( v ) GT defined in (2.29). Then the Euclidean part (cid:92) H E [ N ] is rewritten in terms of (there is nosummation over I, J, K here)ˆ H ( n ) E ( v ; e I , e J , e K ) = 1 iβa t (cid:15) IJK tr( h α IJ [ h e K , ˆ Q nv ] h − e K ) , (5.5)– 37 –nd the Lorentzian part, in terms ofˆ H ( (cid:126)k ) L ( v ; v , v , v , v ; e I , e J , e K )= − iβ a t (cid:15) IJK tr([ h e I , [ ˆ Q k v , ˆ H ( k ) E ( v )]] h − e I [ h e J , [ ˆ Q k v , ˆ H ( k ) E ( v )]] h − e J [ h e K , ˆ Q k v ] h − e K ) (5.6)with (cid:126)k = ( k , k , k , k , k ). Defineˆ H ( n ) E ( v ) = (cid:88) e I ,e J ,e K ˆ H ( n ) E ( v ; e I , e J , e K ) (5.7)and ˆ H ( (cid:126)k ) L ( v ) = (cid:88) v ,v ,v ,v ,e I ,e J ,e K ˆ H ( (cid:126)k ) L ( v ; v , v , v , v ; e I , e J , e K ) . (5.8)The Euclidean and Lorentzian parts, with the replacement ˆ V v → ˆ V ( n ) GT truncated at a finite n , arelinear combinations of ˆ H ( n ) E ( v ) and ˆ H ( (cid:126)k ) L ( v ) with various n and (cid:126)k respectively. According to Eq. (5.4) and the gauge invariance of ˆ H ( n ) E ( v ; e I , e J , e K ), one realizes the followingsymmetry (cid:104) ˆ H ( n ) E ( v ; e I , e J , e K ) (cid:105) = (cid:104) ˆ H ( n ) E ( s ( v ); s ( e I ) , s ( e J ) , s ( e K )) (cid:105) , (5.9)where (cid:104)·(cid:105) denotes the expectation value with respect to the cosmological coherent state given by(5.1), and s = ( t, f ) is a symmetry of the graph.By this relation, Eq. (5.7) is simplified asˆ H ( n ) E ( v ) = 24( ˆ H ( n ) E ( v ; e + x , e + y , e + z ) + ˆ H ( n ) E ( v ; e + x , e + y , e − z )) (5.10)where the prefactor 24 is deduced by the fact that there are totally 48 terms in the RHS of (5.7).Moreover, [ h e ± z , ˆ Q nv ] h − e ± z appearing in ˆ H ( n ) E ( v ; e + x , e + y , e ± z ) potentially relates ˆ H ( n ) E ( v ; e + x , e + y , e + z )with ˆ H ( n ) E ( v ; e + x , e + y , e − z ).Concisely,[ h e ± z , ˆ Q nv ] h − e ± z = n (cid:88) l =1 (cid:88) P l (cid:16) ∓ it ( βa ) (cid:17) l ˆ Q p v (cid:15) α β γ ˆ X α ˆ Y β τ γ ˆ Q p v (cid:15) α β γ ˆ X α ˆ Y β τ γ · · · ˆ Q p l v (cid:15) α l β l γ l ˆ X α l ˆ Y β l τ γ l ˆ Q p l +1 v (5.11)where the edges e ± z are oriented so that s ( e + z ) = v = s ( e − z ), ˆ X α = ˆ p αs ( e (cid:48) ) − ˆ p αt ( e (cid:48) ) and ˆ Y α =ˆ p αs ( e (cid:48)(cid:48) ) − ˆ p αt ( e (cid:48)(cid:48) ), and P = { p , p , · · · , p l +1 } with p i ∈ Z , p i ≥ (cid:80) l +1 i =1 p i = 2 n − l . Substitutingthe last equation into the expression of ˆ H ( n ) E ( v ), one has thatˆ H ( n ) E ( v ; e + x , e + y , e + z ) + ˆ H ( n ) E ( v ; e + x , e + y , e − z ) = 2 ˆ˜ H ( n ) E ( v ; e + x , e + y , e + z ) (5.12)where ˆ˜ H ( n ) E ( v ; e + x , e + y , e + z ) is the operator ˆ H ( n ) E ( v ; e + x , e + y , e + z ) with applying the following replacement[ h e + z , ˆ Q nv ] h − e + z → (cid:88) l is odd (cid:88) P l (cid:16) − it ( βa ) (cid:17) l ˆ Q p v (cid:15) α β γ ˆ X α ˆ Y β τ γ ˆ Q p v (cid:15) α β γ ˆ X α ˆ Y β τ γ · · · ˆ Q p l v (cid:15) α l β l γ l ˆ X α l ˆ Y β l τ γ l ˆ Q p l +1 v . (5.13) A general equation can be obtained analogously if the holonomy h e ± z is replaced by a holonomy along otheredges. In the following context, Eq. (5.11) will be usually referred as this general equation. – 38 –y Eq. (5.10), ˆ H ( n ) E ( v ) becomes ˆ H ( n ) E ( v ) = 48 ˆ˜ H ( n ) E ( v ; e + x , e + y , e + z ). Thus, when we calculate theexpectation value of the Euclidean part, it is only necessary to consider ˆ˜ H ( n ) E ( v ; e + x , e + y , e + z ) ratherthan ˆ H ( n ) E ( v ; e + x , e + y , e ± z ).Further, according to Eq. (5.11), the Euclidean Hamiltonian is of the formˆ H ( n ) E ( v ; e I , e J , e K ) = (cid:15) IJK tr( h α IJ τ α ) ˆ O α , where ˆ O α is a polynomial of fluxes. Then the fact tr( hτ α ) = − tr( h − τ α ) givesˆ H ( n ) E ( v ; e I , e J , e K ) = ˆ H ( n ) E ( v ; e J , e I , e K ) . (5.14)In summary, originally there are totally 48 terms for every ˆ H ( n ) E ( v ) in Eq. (5.7). However,thanks to the symmetries discussed in this section, we have ˆ H ( n ) E ( v ) = 48 ˆ˜ H ( n ) E ( v ; e + x , e + y , e + z ), whichmeans that only the expectation value of ˆ˜ H ( n ) E ( v ; e + x , e + y , e + z ) is necessary to be computed. Considering a list of vertices and edges ( v ; v , v , v , v ; e I , e J , e K ) with e I , e J and e K being outgoingfrom v , we have that ( e I , e J , e K ) is either left-handed or right-handed. Thus, there exists a rotation f which leaves v invariant such that ( v ; f ( v ) , f ( v ) , f ( v ) , f ( v ); f ( e I ) , f ( e J ) , f ( e K )) is either( v ; f ( v ) , f ( v ) , f ( v ) , f ( v ); e + x , e + y , e + z )or ( v ; f ( v ) , f ( v ) , f ( v ) , f ( v ); e + x , e + y , e − z ) . Therefore, (5.8) is simplified toˆ H ( (cid:126)k ) L ( v ) = 24 (cid:88) v ,v ,v ,v (cid:16) ˆ H ( (cid:126)k ) L ( v ; v , v , v , v ; e + x , e + y , e + z ) + ˆ H ( (cid:126)k ) L ( v ; v , v , v , v ; e + x , e + y , e − z ) (cid:17) (5.15)Moreover, since the term [ h e + z , ˆ Q nv ] h − e + z appears in ˆ H ( (cid:126)k ) L too, Eq. (5.11) can be applied again tosimplify Eq. (5.15) to obtain the following expressionˆ H ( (cid:126)k ) L ( v ) = 48 (cid:88) v ,v ,v ,v ˆ˜ H ( (cid:126)k ) L ( v ; v , v , v , v ; e + x , e + y , e + z ) , (5.16)where this ˆ˜ H ( (cid:126)k ) L operator is given by ˆ H ( (cid:126)k ) L ( v ; v , v , v , v ; e + x , e + y , e + z ) with the replacement (5.13).As a consequence, it is only necessary to compute the expectation value ofˆ˜ H ( (cid:126)k ) L ( v ; v , v , v , v ; e + x , e + y , e + z )for different vertices v , v , v and v .The above discussion simplifies the computation of the Lorentzian part. However, more symme-tries are required in order to reduce the computation time to an acceptable level. For this purpose,let us firstly look at the term [ h e [ ˆ Q mv , h − e ] , ˆ Q v ] which is from the commutator between the volumeand the Euclidean part. We obtain the following proposition which can be proven by Eq. (5.11)directly. Proposition 3. Given an edge e with the source s ( e ) and the target t ( e ) , [ h e [ ˆ Q ms ( e ) , h − e ] , ˆ Q v ] = 0 for all v (cid:54) = s ( e ) . – 39 –ith this proposition, we consider the the commutator [ ˆ Q kv , ˆ H ( n ) E ( v ; e I , e J , e K )] which definesthe operator ˆ K as ˆ K = 1 it [ ˆ V , ˆ H E ] . (5.17)By definition, we have [ ˆ Q kv , ˆ H ( n ) E ( v ; e I , e J , e K )] = 2 iβa t ( ˆ K + ˆ K ) (5.18)with K := (cid:15) IJK tr([ˆˆ Q kv , h α IJ ][ h e K , ˆ Q nv ] h − e K )ˆ K := (cid:15) IJK tr( h α IJ (cid:2) ˆ Q kv , [ h e K , ˆ Q nv ] h − e K (cid:3) ) . (5.19)The classical analogy of Eq. (5.17) is K = { V, H E } . (5.20)Substituting the expression of H E , one has K = { V, H E } = (cid:90) d x { V, F iab ( x ) } (cid:15) ijk E ai E bj (cid:112) det( E ) . (5.21)According to Eq. (5.21), only the Poisson bracket between volume V and the curvature F iab isinvolved in the classical expression of K . In the quantum theory, F iab is quantized to a holonomyalong some loop α IJ . Thus, comparing to Eq. (5.18), the operator ˆ K corresponds to the RHS ofEq. (5.21), while ˆ K gives an extra term in ˆ K . According to Proposition 3, this extra term ˆ K vanishes unless v = v = s ( e K ) at which Eq. (5.11) can be applied to cancel the holonomies insidethe commutators of ˆ K . Then ˆ K is simplified to the following formtr( h α IJ [ ˆ Q kv , polynomial of only fluxes]) . Therefore, it is because of the non-commutativity between the flux operators that the operator ˆ K appears in ˆ K . Note that the existence of ˆ K does not affect the continuum limit of lim t → (cid:104) (cid:92) H [ N ] (cid:105) (the classical limit of (cid:104) (cid:92) H [ N ] (cid:105) reduces to the classical continuum expression of H [ N ] when the sizesof lattice edges are neglected [23]).By Eq. (5.13), ˆ K can be simplified asˆ K = (cid:15) IJK (cid:88) p + p =2 n − (2 k ) − it ( βa ) h α IJ τ γ ) ˆ Q p v (cid:2) ˆ Q v , (cid:15) α β γ ˆ X α I ˆ X β J (cid:3) ˆ Q k − p v + (cid:15) IJK (cid:88) p + p =2 n − k (2 k − − it ( βa ) h α IJ τ γ ) ˆ Q p v (cid:2) ˆ Q v , (cid:2) ˆ Q v , (cid:15) α β γ ˆ X α I ˆ X β J (cid:3)(cid:3) ˆ Q k − p v + O ( t ) (5.22)where ˆ X αI = ˆ p αs ( e I ) − ˆ p αt ( e I ) and the conclusion that ˆ K (cid:54) = 0 if v = v ≡ v is used. For the firstterm, we have up to O ( t )first term = (cid:15) IJK (2 n )(2 k ) − it ( βa ) h α IJ τ γ ) ˆ Q nv (cid:2) ˆ Q v , (cid:15) α β γ ˆ X α I ˆ X β J (cid:3) ˆ Q k − v − (2 k ) 2 n (2 n + 1)2 − it ( βa ) h α IJ τ γ ) ˆ Q n − v (cid:2) ˆ Q v , (cid:2) ˆ Q v , (cid:15) α β γ ˆ X α I ˆ X β J (cid:3)(cid:3) ˆ Q k − v (5.23)For the second term, up to O ( t ) we havesecond term = (cid:15) IJK (2 n ) 2 k (2 k − − it ( βa ) h α IJ τ γ ) ˆ Q n − v (cid:2) ˆ Q v , (cid:2) ˆ Q v , (cid:15) α β γ ˆ X α I ˆ X β J (cid:3)(cid:3) ˆ Q k − v (5.24)– 40 –inally, ˆ K isˆ K = (cid:15) IJK (2 n )(2 k ) − it ( βa ) h α IJ τ γ ) ˆ Q nv (cid:2) ˆ Q v , (cid:15) α β γ ˆ X α I ˆ X β J (cid:3) ˆ Q k − v + (cid:15) IJK (2 k )(2 n )(2 k − n − − it ( βa ) h α IJ τ γ ) ˆ Q n − v (cid:2) ˆ Q v , (cid:2) ˆ Q v , (cid:15) α β γ ˆ X α I ˆ X β J (cid:3)(cid:3) ˆ Q k − v + O ( t ) (5.25)Because of the commutators between fluxes operators,[ˆ p α ( e ) , ˆ p β ( e )] = t ( − γ ε − γαβ ˆ p γ ( e ) =: tC αβγ ˆ p γ ( e ) (5.26)with ε − , , = 1 and p α ( e ) denoting p αt ( e ) or p αs ( e ), one obtains the following[ˆ p αs ( e + ) + s ˆ p αt ( e − ) , ˆ p βs ( e + ) + s ˆ p βt ( e − )] = tC αβγ (ˆ p γs ( e + ) + s s ˆ p γt ( e − )) , (5.27)with s , s = ± 1. Substituting Eq. (5.27) into Eq. (5.25), we express ˆ K , as well as ˆ K , as apolynomial of h e and p αs ( e + ) ± p αt ( e − ).Moreover, thanks to the above results, ˆ H ( (cid:126)k ) L ( v ) in Eq. (5.16) can finally be simplified to be interms of Ct tr( h e + x F h − e + x h e + y F h − e + y G ) (5.28)where C is some constant of order t or higher, F i is some monomials of holonomies and ( p αs ( e + ) ± p αt ( e − )) and G is a monomial of ( p αs ( e + ) − p αt ( e − )).The results in Sec. 4.3 can be used to reduce the computational complexity too. To use theseresults, one needs to apply the basic commutation relations (2.10) to simplify the Hamiltonianoperator such that the operators after the simplification are written in terms of C ˆ P with C beingsome constant of order t or higher, and ˆ P being some monomial of holonomies and fluxes.In order to achieve so, one needs to permute h e + x and ˆ F , as well as h e + y and ˆ F , in Eq. (5.28)with applying Eq. (3.14). Take the permutation of h e + x and ˆ F as an example: Implementingthe results of Eq. (3.14), one substitutes ˆ O by h e + x , and ˆ O i k by ˆ p αs ( e + x ) and/or ˆ p αt ( e + x ). One ofmany these substitutions inevitably generates some special terms in which the commutators onlycontain ˆ p αs ( e + x ). The computation of these commutators with (2.10) will lead to results that areproportional to h e + x . After substituting these permuted results into Eq. (5.28), this h e + x eventuallycancels with h − e + x . (similar to Eq. (5.11)). One can apply the same mechanism to permute h e + y andˆ F . Let us collect these special terms coming from permuting h e + x and ˆ F as well as permuting h e + y and ˆ F . Denote the partial sum of these special terms in ˆ H ( k ) L by alt ˆ H ( (cid:126)k ) L . Because of thecancellation between holonomies and their inverses, alt ˆ H ( (cid:126)k ) L no longer depends on h e + x and h e + y .It turns out that alt ˆ H ( (cid:126)k ) L possesses more symmetries which will be discussed shortly below. Thesespecial terms can be equivalently selected by considering only the non-commutativity between h e + x and p αs ( e + x ) but ignoring the non-commutativity between h e + x and p αt ( e + x ). That isthe special terms of h e + x ˆ F h − e + x = h s + x ˆ F h − s + x (5.29)where s + x is the segment within e + x and does not contain the target t ( e + x ). Because of the afore-mentioned cancellation between the holonomies and their inverses, it is remarkable that the lengthof the segment does not cause any ambiguity. Concretely, Eq. (5.29) results in alt ˆ H ( (cid:126)k ) L ( v ; v , v , v , v ; e I , e J , e K )= − iβ a t (cid:15) IJK tr (cid:16) [ h s I , [ ˆ Q k v , ˆ H ( k ) E ( v )]] h − s I [ h s J , [ ˆ Q k v , ˆ H ( k ) E ( v )]] h − s J [ h s K , ˆ Q k v ] h − s K (cid:17) . (5.30)– 41 –t is interesting that the RHS could be understood as that from an alternative definition of theLorentzian part, alt (cid:92) H L [ N ] = − iβ a t (cid:88) v N ( v ) (cid:88) s I ,s J ,s K at v ε IJK tr (cid:16) [ h s I , [ ˆ V , ˆ H E ]] h − s I [ h s J , [ ˆ V , ˆ H E ]] h − s J [ h s K , ˆ V v ] h − s K (cid:17) . (5.31)in which all edges e I , e J , e K are replaced by their corresponding segments s I , s J , s K with s I ⊂ e I .Indeed, alt (cid:92) H L [ N ] is obtained by an alternative regularization/quantization of the Hamiltonian, i.e.via the following replacement { K, ˙ e a A a ( x ) } → − κ ( i (cid:126) β ) [ h s e , [ ˆ V , ˆ H E ]] h − s e , where the holonomy along the segment s e ⊂ e instead of the entire edge e is used. Here, ˙ e a denotethe vector tangent to e .Collect the terms in ˆ H ( k ) L other than the special terms discussed above, and denote their sumby extra ˆ H ( (cid:126)k ) L , namely extr ˆ H ( (cid:126)k ) L ( v ; v , v , v , v ; e I , e J , e K )= ˆ H ( (cid:126)k ) L ( v ; v , v , v , v ; e I , e J , e K ) − alt ˆ H ( (cid:126)k ) L ( v ; v , v , v , v ; e I , e J , e K ) . (5.32)The operators alt ˆ H ( (cid:126)k ) L and extr ˆ H ( (cid:126)k ) L are dealt with separately in our algorithm.For alt ˆ H ( (cid:126)k ) L , the simplification procedures discussed above result in Ct h s + x F h − s + x h s + y F h − s + y G , (5.33)instead of Eq. (5.28). Since [ h s I , ˆ p αt ( e I )] = 0, we can simplify these terms with h s ± m (cid:89) i =1 ( σ + i p α i s ( e + ) + σ − i p α i t ( e − )) h − s ± = (cid:88) I ( it ) |I| (cid:89) i/ ∈I ( σ + i p α i s ( e + ) + σ − i p α i t ( e − )) (cid:89) j ∈I σ ± j τ α j (5.34)where I is a subsets of { , , · · · , m } with its length denoted by |I| , s + and s − are segments of e + and ( e − ) − respectively with e ± oriented such that e + and ( e − ) − are both outgoing, σ + i = 1 and σ − i = ± 1. The summation over e I , e J and e K in Eq. (5.8) motivates us to compute the following h s + m (cid:89) i =1 ( σ + i p α i s ( e + ) + σ − i p α i t ( e − )) h − s + − h s − m (cid:89) i =1 ( σ + i p α i s ( e + ) + σ − i p α i t ( e − )) h − s − . (5.35)Then, one can apply the aforementioned replacement h s + m (cid:89) i =1 ( σ + i p α i s ( e + ) + σ − i p α i t ( e − )) h − s + → (cid:88) J ( it ) |J | (cid:89) i/ ∈J ( σ + i p α i s ( e + ) + σ − i p α i t ( e − )) (cid:89) j ∈J τ α j (5.36)with J ⊂ { , , · · · , m } such that (cid:89) j ∈J σ − j = − . (5.37)Substitute Eq. (5.36) into (5.33), one cancels the prefactor 1 /t , simplifying (5.33) to be inthe form of C ˆ P with some constant C of order t or higher. Hence the results in Sec. 4.3 can beapplied. – 42 –urther, alt ˆ H ( (cid:126)k ) L brings the following symmetries. Consider a π -rotation which transforms either e + x to e − x or e + y to e − y . Denote it by s . Moreover, to indicate the dependence of F i on vertices andedges, we will rewrite F i in Eq. (5.33) as F i ( v, e ). Then Eq. (5.34) tells us h s ( s + x ) F ( s ( v ) , s ( e )) h − s ( s + x ) h s ( s + y ) F ( s ( v ) , s ( e )) h − s ( s + y ) G ( s ( v ) , s ( e ))= h s + x F ( s ( v ) , s ( e )) h − s + x h s + y F ( s ( v ) , s ( e )) h − s + y G ( s ( v ) , s ( e )) . (5.38)Furthermore, with recalling Eq. (5.4), we obtain the following equation (cid:104) h s + x F ( v, e ) h − s + x h s + y F ( v, e ) h − s + y G ( v, e ) (cid:105) = (cid:104) h s + x F ( s ( v ) , s ( e )) h − s + x h s + y F ( s ( v ) , s ( e )) h − s + y G ( s ( v ) , s ( e )) (cid:105) , (5.39)which reduces the number of contributing vertices and edges in the computation of alt ˆ H ( (cid:126)k ) L .For the operator extr ˆ H ( (cid:126)k ) L , Eq. (5.34) can no longer be applied. Therefore, the symmetry impliedby (5.39) is not manifested. To reduce the complexity of the computation, the following strategyis proposed. Consider a rotation, denoted by r , about the axis (1 / √ , / √ , 0) for π radians whichexchanges the x - and y -axes, and flips the z -axis. We obtainˆ˜ H ( (cid:126)k ) L ( v ; v , v , v , v ; e + x , e + y , e + z ) = ˆ˜ H ( (cid:126)k ) L ( v ; r ( v ) , r ( v ) , r ( v ) , r ( v ); e + y , e + x , e + z ) (5.40)in which we use the definition of ˆ˜ H ( (cid:126)k ) L in (5.16) and assume r ( v ) = v without loss of generality.Then consider the operatorˆ F ( v ; v , v , v , v ; e + x , e + y , e + z ):= ˆ H ( (cid:126)k ) L ( v ; v , v , v , v ; e + x , e + y , e + z ) + ˆ H ( (cid:126)k ) L ( v ; r ( v ) , r ( v ) , r ( v ) , r ( v ); e + x , e + y , e + z ) (5.41)According to (5.40), ˆ F isˆ F ( v ; v , v , v , v ; e + x , e + y , e + z )= − tr([ h e + y , [ ˆ Q k t ( v ) , ˆ H ( k ) E ( t ( v ))]] h − e + y [ h e + x , [ ˆ Q k v , ˆ H ( k ) E ( t ( v ))]] h − e + x [ h e + z , ˆ Q k v ] h − e + z )+ tr([ h e + x , [ ˆ Q k t ( v ) , ˆ H ( k ) E ( t ( v ))]] h − e + x [ h e + y , [ ˆ Q k t ( v ) , ˆ H ( k ) E ( t ( v ))]] h − e + y [ h e + z , ˆ Q k v ] h − e + z ) (5.42)up to an overall factor. For clarity, we will denoteˆ X ≡ [ h e + x , [ ˆ Q k v , ˆ H ( k ) E ( t ( v ))]] h − e + x ˆ Y ≡ [ h e + y , [ ˆ Q k t ( v ) , ˆ H ( k ) E ( t ( v ))]] h − e + y ˆ Z ≡ [ h e + z , ˆ Q k v ] h − e + z . (5.43)Because of the holonomies therein, they are all operator-valued matrices whose entries, thus, willbe denoted as ˆ X ˜ a ˜ b , ˆ Y ˜ a ˜ b and ˆ Z ˜ a ˜ b respectively. Then, one hasˆ F = ˆ X ˜ a ˜ b ˆ Y ˜ b ˜ c ˆ Z ˜ c ˜ a − ˆ Y ˜ a ˜ b ˆ X ˜ b ˜ c ˆ Z ˜ c ˜ a = ˆ Y ˜ b ˜ c ˆ X ˜ a ˜ b ˆ Z ˜ c ˜ a − ˆ Y ˜ a ˜ b ˆ X ˜ b ˜ c ˆ Z ˜ c ˜ a + [ ˆ X ˜ a ˜ b , ˆ Y ˜ b ˜ c ] ˆ Z ˜ c ˜ a . (5.44)According to Eq. (3.10), we first compute the expectation values of the operators with respectto coherent states labeled by e iz e τ , and then gauge transform the results correspondingly. Hence,applying this procedure to the the first two terms of Eq. (5.44), one finally simplifies the subtractionof their expectation values as (cid:104) ˆ Y ˜ b ˜ c ˆ X ˜ a ˜ b ˆ Z ˜ c ˜ a − ˆ Y ˜ a ˜ b ˆ X ˜ b ˜ c ˆ Z ˜ c ˜ a (cid:105) { g e } = ( C abcdef − C (cid:48) abcdef ) (cid:104) ˆ Y ab ˆ X cd ˆ Z ef (cid:105) { z e } (5.45)– 43 –here C abcdef and C (cid:48) abcdef are two sets of constant coefficients produced by the gauge-transformation.To explicitly compute ∆ C := C abcdef − C (cid:48) abcdef , let us use O i to denote expectation values ofpolynomials of fluxes and holonomies with respect to coherent states labeled by e iz e τ . Then ˆ H ( (cid:126)k ) L and the corresponding results of ∆ C take the following forms, which are discussed case by case.(1) If ˆ H ( (cid:126)k ) L is expressed in terms of the form(1.1) ( n x O τ α n − x ) ˜ a ˜ b ( n y O τ β n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ;(1.2) ( n x O τ α n − x ) ˜ a ˜ b ( n y O τ β τ β n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ;(1.3) ( n x O τ α τ α n − x ) ˜ a ˜ b ( n y O τ β n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ,then ∆ C is defined as the following,∆ C =tr ( n − x n y ) m y (cid:89) β =1 τ β ( n − y n z ) τ γ ( n − z n x ) (cid:32) m x (cid:89) α =1 τ α (cid:33) − tr ( n − y n x ) (cid:32) m x (cid:89) α =1 τ α (cid:33) ( n − x n z ) τ γ ( n − z n y ) m y (cid:89) β =1 τ β where ( m y , m x ) = (1 , , 1) and (1 , 2) for the cases (1.1), (1.2) and (1.3) respectively.(2) If ˆ H ( (cid:126)k ) L is expressed in terms of the form(2.1) ( n x O h e + x τ α h − e + x n − x ) ˜ a ˜ b ( n y O h e + y τ β h − e + y n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ;(2.2) ( n x O h e + x τ α h − e + x n − x ) ˜ a ˜ b ( n y O h e + y τ β τ β h − e + y n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ;(2.3) ( n x O h e + x τ α τ α h − e + x n − x ) ˜ a ˜ b ( n y O h e + y τ β h − e + y n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ,then ∆ C is defined as the following,∆ C = m y (cid:89) β =1 τ β bc (cid:32) m x (cid:89) α =1 τ α (cid:33) da (cid:110) ( n − x n y ) ab [( n − y n z ) τ γ ( n − z n x )] cd − ( n − y n x ) cd [( n − x n z ) τ γ ( n − z n y ) ab (cid:111) (3) If ˆ H ( (cid:126)k ) L is expressed in terms of the form(3.1) ( n x O h e + x τ α h − e + x n − x ) ˜ a ˜ b ( n y O τ β h e + y τ β h − e + y n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ,then ∆ C is defined as the following,∆ C = ( τ β ) bc ( τ α ) da (cid:110) [ n − x n y τ β ] ab [( n − y n z ) τ γ ( n − z n x )] cd − [( n − x n z ) τ γ n − z n y τ β ] ab [( n − y n x )] cd (cid:111) (4) If ˆ H ( (cid:126)k ) L is expressed in terms of the form(4.1) ( n x O τ α h e + x τ α h − e + x n − x ) ˜ a ˜ b ( n y O h e + y τ β h − e + y n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ,then ∆ C is defined as the following,∆ C = ( τ α ) da ( τ β ) bc (cid:110) [( n − x n y ) ab [( n − y n z τ γ n − z n x τ α ] cd − [( n − x ) n z ) τ γ ( n − z n y )] ab [( n − y n x ) τ α ] cd (cid:111) (5) If ˆ H ( (cid:126)k ) L is expressed in terms of the form– 44 –5.1) ( n x O h e + x τ α h − e + x n − x ) ˜ a ˜ b ( n y O τ β n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ;(5.2) ( n x O h e + x τ α h − e + x n − x ) ˜ a ˜ b ( n y O τ β τ β n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ;(5.3) ( n x O h e + x τ α τ α h − e + x n − x ) ˜ a ˜ b ( n y O τ β n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ,then ∆ C is defined as the following,∆ C = (cid:32) m x (cid:89) α =1 τ α (cid:33) ba (cid:110) [( n − x n y ) (cid:0) m y (cid:89) β =1 τ β (cid:1) ( n − y n z ) τ γ ( n − z n x )] ab − [( n − x n z ) τ γ ( n − z n y )( m y (cid:89) β =1 τ β )( n − y n x )] ab (cid:111) (6) If ˆ H ( (cid:126)k ) L is expressed in terms of the form(6.1) ( n x O τ α n − x ) ˜ a ˜ b ( n y O h e + y τ β h − e + y n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ;(6.2) ( n x O τ α n − x ) ˜ a ˜ b ( n y O h e + y τ β τ β h − e + y n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ;(6.3) ( n x O τ α τ α n − x ) ˜ a ˜ b ( n y O h e + y τ β h − e + y n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ,then ∆ C is defined as the following,∆ C = m y (cid:89) β =1 τ β ba (cid:110) [( n − y n z ) τ γ ( n − z n x )( m x (cid:89) α =1 τ α )( n − x n y )] ab − [( n − y n x )( m x (cid:89) α =1 τ α )( n − x n z ) τ γ ( n − z n y )] ab (cid:111) (7) If ˆ H ( (cid:126)k ) L is expressed in terms of the form(7.1) ( n x O τ α n − x ) ˜ a ˜ b ( n y O τ β h e + y τ β h − e + y n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a ,then ∆ C is defined as the following,∆ C = ( τ β ) ba (cid:8) [( n − y n z ) τ γ ( n − z n x ) τ α ( n − x n y ) τ β ] ab − [( n − y n x ) τ α ( n − x n z ) τ γ ( n − z n y ) τ β ] ab (cid:9) (8) If ˆ H ( (cid:126)k ) L is expressed in terms of the form(8.1) ( n x O τ α h e + x τ α h − e + x n − x ) ˜ a ˜ b ( n y O τ β n − y ) ˜ b ˜ c ( n z O τ γ n − z ) ˜ c ˜ a the subtraction of the coefficients are of the form∆ C = ( τ α ) ba (cid:110) [( n − x n y ) τ β ( n − y n z ) τ γ ( n − z n x ) τ α ] ab − [( n − x n z ) τ γ ( n − z n y ) τ β n − y n x τ α ] ab (cid:111) . Finally, let us consider the last term in (5.44). Since [ ˆ X ˜ a ˜ b , ˆ Y ˜ b ˜ c ] itself is of O ( t ), it is onlynecessary to compute the leading-order expectation value of this term. Thus, expectation value of[ ˆ X ˜ a ˜ b , ˆ Y ˜ b ˜ c ] ˆ Z ˜ c ˜ a term can be sufficiently computed by applying Theorem 4.4.In summary, let N be the original number of terms in alt ˆ H ( k ) L . The symmetries implied by Eqs.(5.12) and (5.14) reduces this number to N / , where the power 2 is from the fact that alt ˆ H ( k ) L consists of two ˆ H E . Then Eq. (5.16) reduces this number further to N / (4 × N × × N . Let N be the original number of terms in extr ˆ H ( k ) L . The symmetries implied by Eqs. (5.12) and(5.14) are used at first to reduce this number to N / . Then Eq. (5.16) reduces it to N / (4 × N × × N . The word “about” is because there exists cases with ˆ X = ˆ Y in Eq. (5.44). – 45 – .2 Introducing the algorithm In order to demonstrate the idea of our algorithm, some simple examples are used in this section.All of the cases illustrated by these examples finally occur in our computation.Roughly speaking, the computation is divided into two steps. The first is to simplify the oper-ator with applying the commutation relations (2.10) and the second is to compute the expectationvalues of the simplified operator.In the first step, the non-commutative multiplications between operator make a non-trivialsimplification. Because of the operator ˆ Q , we actually need to deal withˆ P α ( v, i ) = ˆ p αs ( e + v ) − ˆ p αt ( e − v )where e ± v are the edges along the i th direction satisfying s ( e + v ) = v = t ( e − v ). In the computation,we mainly need to deal with the commutators between ˆ P α ( v, i ) and holonomy. Thus let us considerthe example[ m (cid:89) j =1 ˆ P α j ( v, i ) , h e + v ] = it m (cid:88) k =1 τ α k h e + v (cid:89) j (cid:54) = k ˆ P α j ( v, i ) + ( it ) (cid:88) k 1) ˆ P β ( v, 2) ˆ P γ ( v, m occuring in ˆ Q mv can be denotedas ( ˆ P α ( v, 1) ˆ P β ( v, 2) ˆ P γ ( v, m = P ( v, , I (1 ,m ) ) P ( v, , I (2 ,m ) ) P ( v, , I (3 ,m ) ) . Similarly, let us define WDt( I ( i,k ) ) := τ α k τ α k − · · · τ α (5.48)where I ( i,k ) = { α , · · · , α k } . With these notions, Eq. (5.46) can be written as[ P ( v, i, I ( i,m ) ) , h e + v ] =( it ) m (cid:88) k =1 WDt(˜ I ( i, k ) h e + v P ( v, i, I ( i,m − k )+ ( it ) (cid:88) k 3, which leads to n ≤ H ( n ) E . A very similardiscussion for ˆ H ( (cid:126)k ) L can be done, which gives us that, for (cid:126)k = ( k , k , · · · , k ), | k + k − | + ( k + k − | k + k − | + ( k + k − k ≤ . (5.58)– 48 – Quantum correction in the expectation value The resulting expectation value of the Hamiltonian with unit lapse (cid:100) H [1] = (cid:100) H E + (cid:0) β (cid:1) (cid:99) H L atcoherent states with cosmological data ( η < (cid:104) (cid:100) H E (cid:105) = 6 a (cid:112) − βη sin ( ξ ) − at (cid:115) − βη sin (cid:18) ξ (cid:19) cos (cid:18) ξ (cid:19) (cid:40) cos (cid:18) ξ (cid:19) (cid:104) η + 8 η (4 cosh( η ) − η ) − (cid:105) − iη sin (cid:18) ξ (cid:19) (cid:41) + O ( t ) , (6.1) (cid:104) (cid:99) H L (cid:105) = − a √− βη sin ( ξ ) cos ( ξ ) β − at − βη ) / (cid:40) (cid:0) − η (cid:1) cos(6 ξ )+ 4 iη (4838 sin( ξ ) − ξ ) + 4685 sin(3 ξ ) − ξ ) − 105 sin(5 ξ ))+ 2( − η (492 η + 11 i )) cos( ξ ) − − 789 + 4 η (305 η + 18 i )) cos(2 ξ )+ (4413 − η (928 η + 49 i )) cos(3 ξ ) + 8( − η (4192 η − i )) cos(4 ξ )+ ( − − η + 5 i ) η ) cos(5 ξ ) − η coth( η ) (cid:104) 536 cos( ξ ) + 1731 cos(2 ξ )+ 1524 cos(3 ξ ) − ξ ) + 116 cos(5 ξ ) + 117 cos(6 ξ ) + 37292 (cid:105) + 8 η csch( η ) (cid:104) 130 cos( ξ ) + 918 cos(2 ξ ) + 801 cos(3 ξ ) − ξ )+ 125 cos(5 ξ ) + 58 cos(6 ξ ) + 16362 (cid:105) + 8(1436 + η ( − η + 25 i )) (cid:41) + O ( t ) . (6.2)According to (5.32), (cid:104) (cid:99) H L (cid:105) = (cid:104) extr (cid:99) H L (cid:105) + (cid:104) alt (cid:99) H L (cid:105) , in which (cid:104) alt (cid:99) H L (cid:105) is, (cid:104) alt (cid:99) H L (cid:105) = − a √− βη sin (2 ξ )8 β − at − βη ) / (cid:40) (cid:0) η − (cid:1) cos( ξ ) + (cid:0) − η (cid:1) cos(2 ξ )+ 4 (cid:0) − η (cid:1) cos(3 ξ ) + (cid:0) η − (cid:1) cos(4 ξ ) − η − η (cid:104) coth( η )( − (262 cos( ξ ) − ξ ) + 65 cos(3 ξ ) − 380 cos(4 ξ ) + 366))) − csch( η )( − 292 cos( ξ ) + 268 cos(2 ξ ) + 420 cos(3 ξ ) − ξ ) + 1799) − i (6 sin( ξ ) − 10 sin(2 ξ ) + 6 sin(3 ξ ) − ξ )) (cid:105) + 241 (cid:41) + O ( t ) , (6.3)– 49 –nd (cid:104) extr (cid:99) H L (cid:105) is (cid:104) extr (cid:99) H L (cid:105) = − a √− βη sin (2 ξ )8 β + 3 at − βη ) / (cid:40) − (cid:0) η + 72 iη − (cid:1) cos(2 ξ )+ 2 (cid:0) − η (cid:1) cos(6 ξ ) + 4 iη (cid:104) ξ ) − ξ ) + 3149 sin(3 ξ ) − ξ ) + 3 sin(5 ξ )) (cid:105) + 2( − η (284 η + 11 i )) cos( ξ )+ (2653 − η (96 η + 49 i )) cos(3 ξ ) + 56( − 211 + η (464 η − i )) cos(4 ξ )+ ( − − η + 5 i ) η ) cos(5 ξ ) − η coth( η ) (cid:104) ξ ) + 627 cos(2 ξ ) − 36 cos(3 ξ ) − ξ ) + 116 cos(5 ξ ) + 117 cos(6 ξ ) + 28508 (cid:105) + 8 η csch( η ) (cid:104) 714 cos( ξ ) + 382 cos(2 ξ ) − 39 cos(3 ξ ) − ξ ) + 125 cos(5 ξ ) + 58 cos(6 ξ ) + 12764 (cid:105) + 8 η ( − η + 25 i ) + 9560 (cid:41) + O ( t ) . (6.4)When expressing (cid:104) (cid:100) H [1] (cid:105) to be (cid:104) (cid:100) H [1] (cid:105) = H + tH + O ( t ), we notice that the O ( t )-term H contains η in its denominator and, thus, is divergent if η → 0. This feature is implied by the factthat if ˆ Q v (cid:105) → 0, ˆ V ( v ) GT is divergent. This is because when if η → (cid:104) ˆ Q v (cid:105) → (cid:104) ˆ Q v (cid:105) ∼ | η | .Hence, the expansion of (cid:104) (cid:100) H [1] (cid:105) becomes invalid when η is too small. More precisely, expressing (cid:104) (cid:100) H [1] (cid:105) as (cid:104) (cid:100) H [1] (cid:105) = (cid:112) | η | (cid:2) f + ( t/η ) f + O ( t ) (cid:3) , we get that f is independent of η , and f is regularat η → 0. Thus, it is concluded that our expansion is valid when η (cid:29) t . This aspect is importantfor a new improvement of cosmological effective dynamics derived from the full LQG [61]. Theexpansion is valid for large | η | , because f is regular at | η | → ∞ .Consider the reduced-phase-space LQG of gravity coupled to Gaussian dust. Then, the rela-tional evolution with respect to the dust time T will be generated by the physical Hamiltonian (cid:98) H = (cid:16) (cid:100) H [1] + (cid:100) H [1] † (cid:17) . Its coherent state expectation value reads (cid:104) (cid:98) H (cid:105) = (cid:60) ( (cid:104) (cid:100) H [1] (cid:105) ) ≡ H + t (cid:101) H + O ( t ) , (cid:101) H = (cid:60) ( H ) . (6.5)In order to demonstrate the physical application and effects from the O ( (cid:126) ) correction we adoptthe proposal in [16] as follows. Firstly, we view (cid:104) (cid:98) H (cid:105) as the effective Hamiltonian on the 2-dimensionalphase space P cos of homogeneous and isotropic cosmology. Then, one can verify that η = µ P βa and ξ = µβK where µ is the coordinate length of e ∈ E ( γ ), P is the square of the scale factor and K is the extrinsic curvature. Thus, the Poisson bracket between ξ and η reads { η, ξ } = κ a . With thisPoisson bracket, the Hamiltonian time evolution on P cos generated by (cid:104) (cid:98) H (cid:105) is computable. Thenumerical result is shown in Fig. 1, which respectively depicts the dynamics of the spatial volumegoverned by H , (cid:104) (cid:98) H (cid:105) = H + t (cid:101) H and the classical FLRW Hamiltonian H cl = − aβ − / √− η ξ .In the example shown in Fig. 1, a relatively small t = 10 − is chosen to display the effects ofthe next-to-leading-order term. The coincided initial data of η, dη/dT are chosen to be at T = 0 forall of the three cases. Since η | T =0 gives a large spatial volume, T = 0 is in the low-energy-densityregime. As shown in Fig.1, toward T < 0, the evolution with respect to H + t (cid:101) H is similar asthat corresponding to H , where the latter one gives the µ -scheme effective dynamics. Both of theeffective Hamiltonian H + t (cid:101) H and H resolve the big-bang singularity by “a bounce”. Moreover,the dynamics of H and H cl at late time T > t (cid:101) H in H + t (cid:101) H behaves like an additional matter distribution with negative energy density of O ( t ), which causes the universe to re-collapse and have another bounce at very late time. It shouldbe noted that the time of re-collapse is extremely late because of the tiny value of t .– 50 – - Figure 1 . Evolution of the spatial volume generated by (cid:104) (cid:98) H (cid:105) = H + t (cid:101) H in (6.5) (black curve), H (reddashed curve), and H cl (black circles). The coincided initial data of η, dη/dT are chosen to be at T = 0 forthese 3 cases. The parameters are set to be a = 1, t = 10 − , and β = 0 . Figure 2 . Plots of the critical density ρ c of the dynamics governed by H (dashed curve) and H + t (cid:101) H with t = 0 . 01 (solid curve). The parameters are set as a = 1 and β = 0 . For the critical density ρ c plotted in Fig.2, it also receives correction from t (cid:101) H at the T < µ -scheme dynamics governed by H reads ρ c = a κβ ( β +1) | η b | with | η b | being the value of | η | at the bounce. As shown in Fig.2, instead ofblowing up for small | η b | , the corrected ρ c from (cid:104) (cid:98) H (cid:105) = H + t (cid:101) H is bounded from above for smallvalues of | η b | . In an optimistic viewpoint, this correction of ρ c might hint that the correction fromhigher-order term in t could potentially flatten the dependence of η b in ρ c . This flattened behaviorof ρ c is also supported by the current model of ¯ µ -scheme effective dynamics with complete quantumcorrections, which is considered as an important feature of ¯ µ -scheme Loop Quantum Cosmology.However, by recalling the fact that our expansion in t requires η (cid:29) t , one realizes that the small | η b | regime, where the correction of ρ c becomes significant, mostly violates this requirement (see– 51 –g.2). Hence the quantum dynamics near the bounce is still an open problem from this point ofview.It should be emphasized that the proposal [16] adopted here for studying the effective dynamicsis not as rigorous as the path integral formula (1.2). As argued in Section 1, the O ( t ) correctionin (cid:104) (cid:100) H [1] (cid:105) is only a partial contribution in the quantum effective action that ultimately determiningthe quantum effect in the dynamics. Hence, the cosmological dynamics plotted in Fig.1 only shows O ( t ) correction in (cid:104) (cid:100) H [1] (cid:105) from one of the three O ( t ) contributions in Γ, and is not yet a rigorousprediction from the principle of LQG. In this paper, we developed an algorithm to overcome the complexity of computing the expectationvalue of LQG Hamiltonian operator (cid:92) H [ N ]. With this algorithm, the O ( (cid:126) ) correction in the expecta-tion value (cid:104) (cid:92) H [ N ] (cid:105) at the coherent state peaked at the homogeneous and isotropic data of cosmologyis computed. In the current work, there are several perspectives which should be addressed in thefuture:The first one is to complete the computation of the quantum effective action Γ mentioned inSection 1. After completing of this current work, the only missing ingredient in the O ( (cid:126) ) terms ofΓ is the “1-loop determinant” det( H ). Therefore, a research to be carried out immediately is tocompute this correction of det( H ) at the homogeneous and isotropic background. Once we obtainall of the O ( (cid:126) ) contribution to Γ, the variation of Γ should give the quantum corrected effectiveequations which will demonstrate the quantum correction to the cosmological model implied byLQG.The next step of generalizing our computation is to study the expectation values of (cid:104) (cid:92) H [ N ] (cid:105) with respect to the coherent states peaked at cosmological perturbations. The semiclassical limitsof the expectation value and the cosmological perturbation theory from the path integral (1.2)have been studied in [47]. Thus, it is interesting to study the O ( (cid:96) p ) correction to the cosmologicalperturbation theory.Finally, the computation of the quantum correction in (cid:104) (cid:92) H [ N ] (cid:105) should also be extended to themodel of gravity coupled to standard matter fields. The contributions of matter fields to (cid:92) H [ N ]have been studied in [62, 63]. Since the matter parts in (cid:92) H [ N ] is much simpler than the Lorentzianpart in (cid:92) H [ N ], the computation of their expectation values should not be hard. Study of the mattercontributions and their quantum corrections is a project currently undergoing [64]. Acknowledgements M.H. acknowledges Andrea Dapor, Klaus Liegener, and Hongguang Liu for various discussionsmotivating this work. M.H. receives support from the National Science Foundation through grantPHY-1912278. C. Z. acknowledges the support by the Polish Narodowe Centrum Nauki, Grant No.2018/30/Q/ST2/00811 A SL( , C ) and SU(2) groups Let σ i with i = 1 , , τ k := − iσ k / 2. Define (cid:126)θ = ( θ sin( ψ ) cos( φ ) , θ sin( ψ ) sin( φ ) , θ cos( ψ )) . (A.1) h ∈ SU(2) can be coordinatized as h = e (cid:126)θ · (cid:126)τ = (cid:18) cos (cid:0) θ (cid:1) − i sin (cid:0) θ (cid:1) cos( ψ ) − i sin (cid:0) θ (cid:1) sin( ψ ) e − iφ − i sin (cid:0) θ (cid:1) sin( ψ ) e iφ cos (cid:0) θ (cid:1) + i sin (cid:0) θ (cid:1) cos( ψ ) (cid:19) (A.2)– 52 –ith θ, φ ∈ (0 , π ) and ψ ∈ (0 , π ). The Haar measure then isd µ H ( (cid:126)θ ) = 14 π sin ( θ ψ )d θ d ψ d φ. (A.3)Moreover, by defining (cid:126)p = ( p sin( α ) cos( β ) , p sin( α ) sin( β ) , p cos( α )) (A.4)with p > α ∈ (0 , π ) and β ∈ (0 , π ), g ∈ SL(2 , C ) can be parameterized as g ( (cid:126)p, (cid:126)θ ) = e i(cid:126)p · (cid:126)τ e (cid:126)θ · (cid:126)τ =: e i(cid:126)p · (cid:126)τ h ( (cid:126)θ ) . (A.5)For each p > 0, there exists u ± (cid:126)p ∈ SU(2) such that( u ± (cid:126)p ) − (cid:126)p · (cid:126)τ ( u ± (cid:126)p ) = ± pτ . (A.6)Note that u ± (cid:126)p is determined by Eq. (A.6) up to a right transformation by e ατ . Namely u ± (cid:126)p e ατ forall α ∈ R are solution to Eq. (A.6) provided u ± (cid:126)p does. Moreover, u ± (cid:126)p has the relation u ± (cid:126)p = u ∓− (cid:126)p . (A.7)Let us denote η ≡ ± p and u ≡ u ± (cid:126)p for convenience. Then g ( (cid:126)p, (cid:126)θ ) = u e iητ u − h ( (cid:126)θ ) . (A.8)For u − h ( (cid:126)θ ) ∈ SU(2), decomposing it as u − h ( (cid:126)θ ) = e − ξτ n − , (A.9)one get g ( (cid:126)p, (cid:126)θ ) = ue i ( η + iξ ) τ n − . (A.10)Note that Eq. (A.9) determines n up to a right transformation by e ατ as that for u ± (cid:126)p . n satisfies n ( ητ ) n − = h ( (cid:126)θ )( (cid:126)p · (cid:126)τ ) h ( (cid:126)θ ) − . (A.11)The Wigner 3- j symbol (cid:18) j j j m m m (cid:19) is an SU(2)-invariant tensor, namely D j n m ( h ) D j n m ( h ) D j n m ( h ) (cid:18) j j j m m m (cid:19) = (cid:18) j j j n n n (cid:19) , ∀ h ∈ SU(2) . (A.12)Define τ α with α = − , , τ ± = ∓ τ ± τ √ , τ = τ . (A.13)We obtain the j -representation of τ α in terms of the 3 j symbol, according to the Wigner–Eckarttheorem, as D (cid:48) jmm (cid:48) ( τ α ) = iw j (cid:15) jnm (cid:18) j j n m (cid:48) α (cid:19) (A.14)where w j = (cid:112) j ( j + 1)(2 j + 1) and (cid:15) jnm = ( − j + m δ ( n, − m ) is the 2- j symbol. The 2- j symbol isalso SU(2) invariant. By 3- j symbol and (cid:15) jnm , any SU(2)-intertwiner can be constructed as ι ( k k · · · k n − ) m m m ··· m n = (cid:18) j j k m m l (cid:19) (cid:15) k l l (cid:48) (cid:18) k j k l (cid:48) m l (cid:19) (cid:15) k l l (cid:48) · · · (cid:15) k n − l n − l (cid:48) n − (cid:18) k n − j n − j n l (cid:48) n − m n − m n (cid:19) . (A.15)– 53 –oreover, the Clebsch-Gordan coefficients relates to 3 j -symbol as (cid:104) j m j m | JM (cid:105) =( − j √ J + 1 (cid:15) JMN (cid:18) J j j N m m (cid:19) =( − j − j − J √ J + 1 (cid:18) j j Jm m N (cid:19) (cid:15) JNM (A.16)The facts that D jmn ( e ipτ ) = e pm δ mn and (cid:18) j j j m m m (cid:19) ∝ δ ( m + m + m , 0) lead to D j n m ( e ipτ ) D j n m ( e ipτ ) D j n m ( e ipτ ) (cid:18) j j j m m m (cid:19) = (cid:18) j j j n n n (cid:19) . (A.17)Thus Eq. (A.12) holds for all g ∈ SL(2 , C ) which can be concluded with applying Eq. (A.8). Extendthe right invariant vector field ˆ R i on SU(2) to SL(2 , C ) to define an operator ˆ X i asˆ X i D jmn ( g ) = i dd (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 D jmn ( e (cid:15)τ i g ) . (A.18)Then ˆ∆ := ˆ X i ˆ X i takes D jmn ( g ) as its eigenstate with eigenvalue j ( j + 1), i.e.,ˆ∆ D jmn ( g ) = j ( j + 1) D jmn ( g ) . (A.19)The commutator [ ˆ∆ , D ab ( g )] is[ ˆ∆ , D ab ( g )] D jmn ( g )= (cid:88) J = j ± / ( J ( J + 1) − j ( j + 1)) d J ( − m + a − n − b (cid:18) / j Ja m − m − a (cid:19) (cid:18) / j Jb n − n − b (cid:19) D Jm + a,n + b ( g )(A.20)Therefore, we have (cid:18) [ ˆ∆ , D ab ( g )] + ( j + 14 ) D ab ( g ) (cid:19) D jmn ( g )= d j +1 / d j ( − m + a − n − b (cid:18) / j j + 1 / a m − m − a (cid:19) (cid:18) / j j + 1 / b n − n − b (cid:19) D j +1 / m + a,n + b ( g ) (cid:18) [ ˆ∆ , D ab ( g )] − ( j + 34 ) D ab ( g ) (cid:19) D jmn ( g )= − d j d j − / ( − m + a − n − b (cid:18) / j j − / a m − m − a (cid:19) (cid:18) / j j − / b n − n − b (cid:19) D j − / m + a,n + b ( g ) . (A.21) B the Clebsch-Gordan coefficients with negative parameters Given j , m , j , m and m = m + m , the Clebsch-Gordan coefficients (cid:104) j m j m | pm (cid:105) ≡ (cid:20) j j pm m m (cid:21) for various p satisfy the difference equation [65] A ( p + 1) (cid:20) j j p + 1 m m m (cid:21) + A ( p ) (cid:20) j j p − m m m (cid:21) + ( A ( p ) − m + m ) (cid:20) j j pm m m (cid:21) = 0 (B.1)– 54 –here max( | m | , | j − j | ) ≤ p ≤ j + j and A ( p ) = 1 p (cid:115) − ( p − ξ )( p − ξ )( p − ξ )4 p − ,A ( p ) = ξ ξ ξ p ( p + 1) (B.2)with ξ = j − j , ξ = j + j + 1 , ξ = m. With the initial data (cid:20) j j j + j m m m (cid:21) = (cid:115) (2 j )! (2 j )! ( j + j − m )! ( j + j + m )!(2 j + 2 j )! ( j − m )! ( j + m )! ( j − m )! ( j + m )! (B.3)the Clebsch-Gordan coefficients for other values of p are computable with Eq. (B.1).In order to extend the Clebsch-Gordan coefficients to negative parameters, we define a function C ( x ) := (cid:115) (2 j )! Γ(2 x + 1)Γ( x + j − m + 1)Γ( x + j + m + 1)( j − m )! ( j + m )!Γ(2 x + 2 j + 1)Γ( x − m + 1)Γ( x + m + 1) , (B.4)with which (cid:20) j j j + j m m m (cid:21) = C ( j ) . (B.5)It is remarkable that C ( x ) is well-defined not only for positive x such that x − m ∈ Z , x ≥ | m | (B.6)but also for negative x satisfying x − m ∈ N , x ≤ min( −| m | , − j − | m | − 1) (B.7)where those gamma functions with negative integers is understood asΓ( − m ) · · · Γ( − m k )Γ( − n ) · · · Γ( − n k ) = lim z → Γ( z − m ) · · · Γ( z − m k )Γ( z − n ) · · · Γ( z − n k ) = ( − n + ··· + n k − m −···− m k n ! · · · n k ! m ! · · · m k ! . (B.8)By definition, the Clebsch-Gordan coefficients (cid:20) j j j + j − ιm m m (cid:21) is obtained by applying therecurrence relation (B.1) successively for ι steps with the initial data C ( j , j + j ). Then, wedefined (cid:20) − j j − j + j − ιm m m (cid:21) , the Clebsch-Gordan coefficients with negative parameters, as theresult by applying the recurrence relation (cid:20) − j j − q − m m m (cid:21) = − A ( − q ) (cid:18) ˜ A ( − q + 1) (cid:20) − j j − q + 1 m m m (cid:21) + ( ˜ A ( − q ) − m + m ) (cid:20) − j j − qm m m (cid:21)(cid:19) (B.9)with the initial data C ( − j ), where˜ A ( q ) = A ( q ) (cid:12)(cid:12) j →− j , ˜ A ( q ) = A ( q ) (cid:12)(cid:12) j →− j . (B.10)This definition extended the Clebsch-Gordan coefficients to negative parameters. It guaranteesthat, (cid:20) − j j − j + j − ιm m m (cid:21) = (cid:20) j j j + j − ιm m m (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) j →− j . (B.11)– 55 –y definition, it has C ( − j ) = (cid:115) (2 j − j − j )! ( j − m − j + m − j − j − j − m − j − j + m − j − m )! ( j + m )!=( − j + m (cid:20) j − j j − j − m m m (cid:21) . (B.12)Moreover, (cid:20) j − j j − j + ι − m m m (cid:21) can also be obtained by applying successively the recurrencerelation (cid:20) j j qm m m (cid:21) = − B ( q ) (cid:18) B ( q − (cid:20) j j q − m m m (cid:21) + ( B ( q − − m + m ) (cid:20) j j q − m m m (cid:21)(cid:19) (B.13)with the initial data (cid:20) j j j − j − m m m (cid:21) , where B ( q ) = A ( q ) (cid:12)(cid:12) j → j − , B ( q ) := A ( q ) (cid:12)(cid:12)(cid:12) j → j − . (B.14)Furthermore, it can be verified that B ( q ) = − ˜ A ( − q ) , B ( q − 1) = ˜ A ( − q ) . (B.15)Therefore, according to Eqs. (B.12), (B.9) and (B.13), we finally have (cid:20) − j j − j + j − ιm m m (cid:21) = ( − j + m − ι (cid:20) j − j j − j + ι − m m m (cid:21) , (B.16)namely (cid:20) − j j − j + ∆ m m m (cid:21) = ( − ∆+ m (cid:20) j − j j − − ∆ m m m (cid:21) . (B.17) C Graphical method The 2- j symbol is graphically represented as (cid:15) jmn = ( − j + n δ ( m, − n ) = j At first, for θ = 0, it has p = s/ φ = 0, which lead to that δ = 0 . (D.1)Then, for p = 0, it has s = 0 and φ = θ/ 2. Thus one has δ = θ / ≥ θ = 0.Finally, consider the case when θ (cid:54) = 0 and p (cid:54) = 0. Using e (cid:126)µ · (cid:126)τ = cos( | (cid:126)µ | ) + 2 (cid:126)µ · (cid:126)τ | (cid:126)µ | sin( | (cid:126)µ | ), we havethat 12 tr( g ) = cosh( p θ − ix sinh( p θ s ) cos( φ ) + i sinh( s ) sin( φ ) = cosh( s + iφ ) (D.3)where we denoted x := (cid:126)θ · (cid:126)p/ ( pθ ). By realizing that s = ± p/ φ = θ/ x = ± 1, weare motivated to study the monotonicity of s , φ and, thus, δ with respect to x for fixed values of p and θ .Denote R := cosh( p θ B ( x ) := 1 + cosh ( p ( θ ( p ( θ x . Since sin ( φ ) + cos ( φ ) = 1 = cosh ( s ) − sinh ( x ), we finally obtain from Eq. (D.3) thatcosh( s ) = 1 √ (cid:113) B ( x ) + (cid:112) B ( x ) − R cos( φ ) = sgn(cos( θ )) √ (cid:113) B ( x ) − (cid:112) B ( x ) − R (D.4)with which one gets d s d x = coth( s )2 (cid:112) B ( x ) − R d B ( x )d x d φ d x = cot( φ )2 (cid:112) B ( x ) − R d B ( x )d x . (D.5)Therefore d δ d x = ( φ cot( φ ) − s coth( s )) (cid:112) B ( x ) − R d B ( x )d x =2 ( φ cot( φ ) − s coth( s )) (cid:112) B ( x ) − R sinh ( p ( θ x (D.6)Because φ cot( φ ) ≤ φ ∈ [0 , π ] while s coth( s ) ≥ 1, we conclude that δ ≥ δ | x = ± = θ / > derivative of the matrix element of p i ( e ) and D ab ( h e ) Note that the results show below ignore the terms of order t and higher. We denote ∇ (cid:126)x ( i ) := (cid:0) ∇ (cid:126)p ( i ) , ∇ (cid:126)θ ( i ) (cid:1) and x ∗ is the complex conjugate of x . ∇ (cid:126)x (1) (cid:104) ψ g (1) | ˆ p s ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:32) − , i tanh (cid:16) p (cid:17) , , − ip sin( θ )csch( p )2 θ , ip sin (cid:0) θ (cid:1) csch( p ) θ , (cid:33) T = (cid:18) ∇ (cid:126)x (2) (cid:104) ψ g (1) | ˆ p s ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) (cid:19) ∗ , ∇ (cid:126)x (1) (cid:104) ψ g (1) | ˆ p s ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:32) − i tanh (cid:16) p (cid:17) , − , , − ip sin (cid:0) θ (cid:1) csch( p ) θ , − ip sin( θ )csch( p )2 θ , (cid:33) T = (cid:18) ∇ (cid:126)x (2) (cid:104) ψ g (1) | ˆ p s ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) (cid:19) ∗ , ∇ (cid:126)x (1) (cid:104) ψ g (1) | ˆ p s ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:18) , , − , , , − i (cid:19) T = (cid:18) ∇ (cid:126)x (2) (cid:104) ψ g (1) | ˆ p s ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) (cid:19) ∗ . (E.1) ∇ (cid:126)x (1) (cid:104) ψ g (1) | ˆ p t ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:32) e iθ + e p − iθ e p + 2 , ie − iθ (cid:0) e p − e iθ (cid:1) e p + 1) , , e − iθ (cid:0) − e iθ (cid:1) p (cid:0) e p + e iθ (cid:1) θ ( e p − ,ie − iθ (cid:0) − e iθ (cid:1) p (cid:0) e p − e iθ (cid:1) θ ( e p − , (cid:33) T = (cid:18) ∇ (cid:126)x (2) (cid:104) ψ g (1) | ˆ p t ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) (cid:19) ∗ , ∇ (cid:126)x (1) (cid:104) ψ g (1) | ˆ p t ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:32) ie − iθ (cid:0) − e p + e iθ (cid:1) e p + 1) , e iθ + e p − iθ e p + 2 , , ie − iθ (cid:0) − e iθ (cid:1) p (cid:0) − e p + e iθ (cid:1) θ ( e p − ,e − iθ (cid:0) − e iθ (cid:1) p (cid:0) e p + e iθ (cid:1) θ ( e p − , (cid:33) T = (cid:18) ∇ (cid:126)x (2) (cid:104) ψ g (1) | ˆ p t ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) (cid:19) ∗ , ∇ (cid:126)x (1) (cid:104) ψ g (1) | ˆ p t ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:18) , , , , , i (cid:19) T = (cid:18) ∇ (cid:126)x (2) (cid:104) ψ g (1) | ˆ p t ( e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) (cid:19) ∗ . (E.2)– 59 – (cid:126)x (1) (cid:104) ψ g (1) | D ( h e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:18) , , − e − iθ , , , − ie − iθ (cid:19) T , ∇ (cid:126)x (1) (cid:104) ψ g (1) | D − ( h e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:32) − e iθ tanh (cid:0) p (cid:1) p , ie iθ tanh (cid:0) p (cid:1) p , , − i sin (cid:0) θ (cid:1) θ + θe p , − sin (cid:0) θ (cid:1) θ + θe p , (cid:33) T , ∇ (cid:126)x (1) (cid:104) ψ g (1) | D − 12 12 ( h e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:32) − e − iθ tanh (cid:0) p (cid:1) p , − ie − iθ tanh (cid:0) p (cid:1) p , , − ie p sin (cid:0) θ (cid:1) θ + θe p , e p sin (cid:0) θ (cid:1) θ + θe p , (cid:33) T , ∇ (cid:126)x (1) (cid:104) ψ g (1) | D ( h e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:18) , , e iθ , , , ie iθ (cid:19) T , ∇ (cid:126)x (2) (cid:104) ψ g (1) | D ( h e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:18) , , e − iθ , , , − ie − iθ (cid:19) T , ∇ (cid:126)x (2) (cid:104) ψ g (1) | D − ( h e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:32) e iθ tanh (cid:0) p (cid:1) p , − ie iθ tanh (cid:0) p (cid:1) p , , − ie p sin (cid:0) θ (cid:1) θ + θe p , − e p sin (cid:0) θ (cid:1) θ + θe p , (cid:33) T , ∇ (cid:126)x (2) (cid:104) ψ g (1) | D − 12 12 ( h e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:32) e − iθ tanh (cid:0) p (cid:1) p , ie − iθ tanh (cid:0) p (cid:1) p , , − i sin (cid:0) θ (cid:1) θ + θe p , sin (cid:0) θ (cid:1) θ + θe p , (cid:33) T , ∇ (cid:126)x (2) (cid:104) ψ g (1) | D − − ( h e ) | ψ g (2) (cid:105)(cid:104) ψ g (1) | ψ g (2) (cid:105) = (cid:18) , , − e iθ , , , ie iθ (cid:19) T . 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