Thermally correlated states in Loop Quantum Gravity
TThermally correlated states in Loop Quantum Gravity
Goffredo Chirco , , Carlo Rovelli , , Paola Ruggiero , Aix Marseille Universit´e, CNRS, CPT, UMR 7332, 13288 Marseille, France. Universit´e de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France. Dipartimento di Fisica dell’Universit`a di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy. SISSA, Via Bonomea 265, 34136 Trieste, Italy. (Dated: March 3, 2017)We study a class of loop-quantum-gravity states characterized by (ultra-local) thermal correlationsthat reproduce some features of the ultraviolet structure of the perturbative quantum field theoryvacuum. In particular, they satisfy an analog of the Bisognano-Wichmann theorem. These statesare peaked on the intrinsic geometry and admit a semiclassical interpretation. We study how thecorrelations extend on the spin network beyond the ultra local limit.
I. INTRODUCTION
The Bisognano–Wichmann theorem [1] states that therestriction of the vacuum of a Lorentz-invariant quantumfield theory to the algebra of field operators with supporton the Rindler wedge x > | t | is a KMS (Kubo-Martin-Schwinger) state, that is, a thermal equilibrium state [2],with inverse temperature 2 π , with respect to the flowgenerated by the boost operator K in the x direction [3].That is, it is described by the density matrix ρ ∝ e − πK . (1)This fact is at the root of the thermal aspects of quantumfield theory, such as the Unruh effect [4]. On a curvedspacetime, quantum fields mimic flat space properties lo-cally, and (a local version of) (1) can be argued to un-derpin the thermal properties of black holes [4–10]. This effect derives from the quantum correlations inthe field. This is particularly clear by considering statesat fixed time. Eq. (1) can be obtained –at least formally–by tracing the state on the t = 0 surface over the degreesof freedom with support on x <
0. The resulting state isnot pure because of the field correlations across x = 0. Ingeneral, we say that a state on a 3d spatial surface Σ hasthe Bisognano–Wichmann property if in any sufficientsmall patch of Σ a version of (1) holds locally for any2d surface S , after tracing over the degrees of freedomon one side of S . (More precision below.) This propertycaptures aspects of the field’s local correlations.This is of interest in quantum gravity for the follow-ing reason. The full background-independent nonpertur-bative theory must include states yielding conventionalphysics at low energy, including quantum field correla-tions. But the ultraviolet structure of these correlationswhich characterises theories defined on a background ge-ometry (cid:104) | φ ( x ) φ ( y ) | (cid:105) ∼ | x − y | , (2) Hawking’s black hole temperature is precisely equal to the Unruhtemperature observed by a stationary observer near the horizon,red-shifted from this observer’s location to infinity; see [11]. does not remain true in a quantum gravity theory (suchas loop quantum gravity) where the Planck scale is aphysical cut-off and there is no background metric defin-ing the distance on the right hand side of this equation.Thus, (2) is not useful for characterising semiclassicalstates. On the other hand, as we shall see, (1) makessense naturally in the theory. And it better hold true forsemiclassical states, for these to yield the expected lowenergy phenomenology. Here we explore the possibil-ity of using (1) as a (partial) characterisation of “good”semiclassical states in quantum gravity.A similar suggestion has been recently put forwardby two papers. In [10], Bianchi and Myers have sug-gested Bisognano–Wichmann-like correlations to charac-terize semiclassical states in any nonperturbative quan-tum theory of spacetime. The smooth structure of space-time geometry at the classical level may be intimately re-lated to the structure of correlations of the quantum grav-itational state. A similar perspective has received atten-tion in string theory, in the context of the gauge/gravityduality, where the entanglement of the boundary gaugefield degrees of freedom has been associated to the con-nectivity of the bulk space-time dual [17–19]. In [20], theBisognano–Wichmann property has been suggested as apossible replacement, in a background independent con-text, of the Hadamard condition that characterises the“good” states in quantum field theory on curved space.In loop quantum gravity, semiclassical states have beenstudied extensively [21–25]. Today we know how to writestates where the expectation value of the the gravitationalfield appropriately matches a given smooth geometry.However, little is known so far about states where alsothe fluctuations of the gravitational field, and especiallythe nonlocal correlations, match the ones of conventionalfield theory. Here we construct and study states witha Bisognano–Wichmann-like property, as a step in thisdirection.Notice that the main hypotheses of the Bisognano–Wichmann theorem are positivity of energy and Lorentz Entanglement entropy due to short-scale quantum correlationshas been studied in loop quantum gravity, especially in the con-text of black holes thermodynamics [12–16]. a r X i v : . [ g r- q c ] M a r l S T n T n S FIG. 1. A facet l separating two cells (a source cell S anda target cell T ) punctured by the link l that joins the twocorresponding nodes. invariance. The last is a dynamical property in thesense that a boost generates the change of a state froma given (spacelike) plane to a boosted one. Thereforethe Bisognano–Wichmann property captures aspects ofa state’s evolution. As we will see, this is reflected inthe states we define below: their definition depends onthe (covariant [25–30]) definition of the loop quantumdynamics. Therefore they can also be viewed as a steptowards fully physical dynamical quantum gravity states.Section II recalls the covariant definition of loop quan-tum gravity states. In section III we define the thermallycorrelated link state, the fundamental brick of our con-struction. In Section III C we study the semiclassicalproperties of this state. In Section IV we construct thethermally correlated SU (2) spin network state. Section Vshows how local correlations ‘propagate’ along the spinnetwork. Results are summarised and discussed in thelast section. II. LORENTZ COVARIANT LQG STATES
We start introducing the conventional loop quantumgravity space state, but using the SL (2 , C ) covariant lan-guage [31–33] adapted for what follows.Consider an oriented three dimensional space-like hy-persurface Σ embedded in a four dimensional space-timemanifold M . Fix an oriented cellular decomposition ofΣ. We call n (for “node”) the cells and l (for “link”) thefacets separating two adjacent cells. Fix a dual graph Γ,with a node n in each cell and a link l connecting thenodes of neighbouring cells. We use the same notation, n and l for the nodes and links of the graph and the dualcells and facets of the triangulation. Each link l is ori-ented: we call n S (for “source”) and n T (for “target”) itsinitial and final nodes. The corresponding facet is equallyoriented and separates a “source” cell S from a “target”cell T . See Figure 1.Consider the spin connection ω of the Cartan formula-tion of general relativity, restricted to Σ. This describesaspects of the gravitational field on Σ. A quantum statecan be expressed as a functional of ω . In particular, a quantum state on Γ is defined to be a (cylindrical ) func-tion Ψ[ ω ] = ψ ( g l [ ω ]) of the holonomy g l [ ω ] ∈ SL (2 , C )of the spin connection along the L links l of the graph.These states, we assume, can be expanded in matrix el-ements of unitary representations of SL (2 , C ).The SL (2 , C ) generators J IJl = − J JIl , I, J = 0 , ..., l play the role of the basicobservables of the theory. They are the quantum oper-ators representing the momentum conjugate to the spinconnection, which on Σ is proportional to the Plebanskitwo-form e I ∧ e J , where e I is Cartan’s tetrad one-form.More precisely, they are determined by the flux of thisquantity across the facet lJ IJl, S ∼ (cid:90) l e I ∧ e J , (3)parallel transported to the source node n S of l . The op-erator J IJl, S acts on a function of g l as the left-invariantvector field. It is important for what follows to observethat the right -invariant vector ˜ J IJl, T , related to J IJl, S by thetransformation defined by g l (in the adjoint representa-tion), is a distinct operator. It represents the same flux,but parallel transported to the target node n T of the link l , namely in the frame of the adjacent cell. The relationbetween the two operators depends on the spin connec-tion along the link.It is convenient to pick the time gauge, which ties thenormal to Σ to a direction t in the internal Minkowskispace. Then J IJl splits into rotation generators (cid:126)L andboost generators (cid:126)K . It is easy to see that (in a locallyflat context [34]) the first is a vector normal to the facet l , with length proportional to the area of the facet [30].The unitary representations of SL (2 , C ) are labelled bya positive real number p and non negative half-integer k [35]. At each node, the vector t determines a subgroup SU (2) ⊂ SL (2 , C ) that leaves it invariant. The Hilbertspace H ( p,k ) that carries the ( p, k ) representation decom-poses into irreducible representations of the subgroup asfollows H ( p,k ) = ⊕ ∞ j = k H j , (4)where H j is the (finite dimensional) SU (2) representationof spin j . Therefore H ( p,k ) admits a basis | ( p, k ); j, m (cid:105) ,called the canonical basis, obtained by diagonalizing thetotal angular momentum L and the L z = (cid:126)L · (cid:126)z compo-nent of the SU (2) subgroup. The statesΨ p l k l j l m l j (cid:48) l m (cid:48) l [ ω ] = ⊗ l D ( p l ,k l ) j l m l ,j (cid:48) l m (cid:48) l ( g l [ ω ]) , (5)where D ( p l ,k l ) are the representations matrices of the SL (2 , C ) unitary representations, span the space of thestates on Γ. A cylindrical function on an infinite dimensional space is a func-tion depending on only on a finite number of coordinates on thisspace.
Within H ( p,k ) the physical subspace of the theory isdetermined (in a given Lorentz frame) by the linear sim-plicity condition (cid:126)K = γ(cid:126)L (6)satisfied, in general relativity, by the momentum conju-gate to the spin connection. Here γ ∈ R + is the Barbero-Immirzi parameter [27, 30]. This relation determines arestriction on the set of the unitary representations andpicks a subspace within each representation. Indeed, therelation (6) is weakly (in matrix elements) true [36] whenrestricted to states of the form | p, k ; j, m (cid:105) = | γ ( j + 1) , j ; j, m (cid:105) . (7)Accordingly, the physical subspace is formed by thestates of the formΨ j l m l m (cid:48) l [ ω ] = ⊗ l D ( γ ( j l +1) ,j l ) j l m l ,j l m (cid:48) l ( g l [ ω ]) . (8)This state space is naturally isomorphic to the space L [ SU (2)] L , the conventional (non gauge invariant)Hilbert space of loop quantum gravity on the graph Γ.The isomorphism maps (8) into ψ ( h l ) = ⊗ l D ( j ) m l ,m (cid:48) l ( h l ) (9)where h l ∈ SU (2) and D j ( h ) are Wigner matrices, andis determined by the injection Y γ : H j → H ( γ ( j +1) ,j ) (10) | j, m (cid:105) (cid:55)→ | ( γ ( j + 1) , j ); j, m (cid:105) .L [ SU (2)] L is a Hilbert space and this isomorphism en-dows the physical state space with the scalar productneeded to define a quantum theory.Let us now see how local gauge invariance affects thisconstruction. In the Cartan formulation, general rela-tivity is invariant under local SL (2 , C ) gauge transfor-mations. Of these, only the Lorentz transformation Λ n at the nodes n of Γ affect the states on Γ (because onlythese affect the holonomies g l [ ω ]). Consider first gaugetransformations where Λ n are rotations. These do not af-fect the local frame at each node, and transform physicalstates into themselves. The states invariant under thesetransformation are the well known spin network states ψ ( h l ) = (cid:79) l D j l ( h l ) · (cid:79) n ι n (11)where ι n is an SU (2) intertwiner at the node n and thecontraction is determined by the structure of the graph.These gauge invariant states form the Hilbert space H Γ H Γ = L [ SU (2) L /SU (2) N ]the standard loop quantum gravity state space on agraph. The states in this space have a direct interpre-tation as quantum geometry of the spatial section Σ ofspace-time. More interesting are the Lorentz transformation thatare not rotations. These act on the SL (2 , C ) states,changing (rotating) the class of physical states. Say t is a vector in the Minkowski representation, left invari-ant by SU (2); a generic Lorentz transformation boosts t into Λ t , which stabilises a different SU (2) subgroup,which in turn defines a different class of physical states.Therefore the spin network formalism is invariant underlocal rotations but is covariant under boosts. See [33] fora full discussion.We are interested in the structure of correlations ofthese states. In other words. we are interested in theway different regions of a spin network can be correlatedto one another. III. THERMAL LINK STATESA. Bisognano–Wichmann property on a single link
Let us begin by focusing on a single link, and disre-garding, for now, gauge invariance. The states on a sin-gle link are given by functions ψ ( g ) on SL (2 , C ) satisfyingthe simplicity constraint, that is, linear combinations ofthe states of the formΨ jmm (cid:48) ( g ) ≡ (cid:104) g | jmm (cid:48) (cid:105) = D ( γ ( j +1) ,j ) jm,jm (cid:48) ( g ) . (12)The operator (cid:126)L S acts on this state as the generator ofrotations on the first index (cid:104) jmm (cid:48) | (cid:126)L S | jm (cid:48)(cid:48) m (cid:48)(cid:48)(cid:48) (cid:105) = (cid:126)τ jmm (cid:48)(cid:48) (13)where (cid:126)τ j is the generator of rotations in the spin j rep-resentation of SU (2) and summation over related indicesis understood. The operator (cid:126)L T acts on this state as thegenerator of rotations on the second index (cid:104) jmm (cid:48) | (cid:126)L T | jm (cid:48)(cid:48) m (cid:48)(cid:48)(cid:48) (cid:105) = (cid:126)τ jm (cid:48) m (cid:48)(cid:48)(cid:48) . (14)The two boost generators, (cid:126)K S and (cid:126)K T restricted to thisspace, have the same matrix elements, multiplied by γ .This set of operators splits naturally into two groups: (cid:126)L S and (cid:126)K S act on the first magnetic index and represent ob-servables living on the cell on the source side of the facet;while (cid:126)L T and (cid:126)K T act on the second magnetic index andrepresent operators living on the cell on the target side ofthe facet. Recall that they represent quantities paralleltransported to two different nodes, and their differencemeasures the connection along the link. We can thereforesplit the observables into two groups, associated to thetwo cells on opposite sides of the facet l .All operators considered here are diagonal in j (theboost operator mixes different j sectors of the same SL (2 , C ) irreducible, but not different SL (2 , C ) irre-ducibles, of course; also, states with different j belongto different irreducibles, because k = j ). It is thereforeconvenient to work at fixed quantum number j , namelyon a L eigenspace (clearly | (cid:126)L S | = | (cid:126)L T | ). This spacehas the structure H = H S j ⊗ H T j . (15)Given a state in this subspace, we can trace on one factorand define a density matrix over the other. Explicitly,tracing on the target factor, a state of the form | ψ (cid:105) = (cid:88) mn c mn | jmn (cid:105) (16)gives the density matrix ρ = T r T | ψ (cid:105)(cid:104) ψ | ≡ (cid:88) n c nm c nm (cid:48) | j, m (cid:105)(cid:104) j, m (cid:48) | (17)on H S j . Since the restriction of (cid:126)K to H j is given by γ(cid:126)L ,because of the simplicity conditions, we can define thedensity matrix e − π (cid:126)K · (cid:126)z = (cid:88) m e − πγm | j, m (cid:105)(cid:104) j, m | . (18)where here | j, m (cid:105) is a basis of eigenstates of (cid:126)L · (cid:126)z . Wenow have the language for the following definition. Wesay that a link state ψ with spin j has the Bisognano–Wichmann property if there is a (cid:126)n such thattr T [ | ψ (cid:105)(cid:104) ψ | ] = e − π (cid:126)K S · (cid:126)n . (19) and there is a (cid:126)n (cid:48) such thattr S [ | ψ (cid:105)(cid:104) ψ | ] = e − π (cid:126)K T · (cid:126)n (cid:48) . (20)Armed with this definition, let us now see what are thestates with this property. B. States
We want to find a class of states {| ψ (cid:105)} satisfying (19)and (20).For a given (cid:126)z , we set (cid:126)n = (cid:126)n (cid:48) ≡ (cid:126)z . Sandwiching (19)between eigenstates of (cid:126)K S · (cid:126)z gives (cid:104) jm | tr T [ | ψ (cid:105)(cid:104) ψ | ] | jm (cid:48) (cid:105) = e − γπm δ mm (cid:48) (21)Using (17), this reads (cid:88) n c nm c nm (cid:48) = e − γπm δ mm (cid:48) . (22)Let Λ be the diagonal matrix with entries e − γπm and c be the matrix with matrix elements c nm . Then the lastequation can be written in the form cc † = ΛΛ † (23)or equivalently ( c Λ − )( c Λ − ) † = I , which is solved for any unitary matrix U by c = Λ U (24)In components, our coefficients read c mn = e − πγm U mn (25)Moreover, recall that the definition of the Bisognano–Wichmann property demands the state to be thermalwhen traced on either side. Repeating the above deriva-tion with source and target swapped yields c mn = V mn e − πγn (26)with V mn also a unitary matrix.Now, for two generic directions (cid:126)n , (cid:126)n (cid:48) , it is easy to showthat equations (25) and (26) generalize to c mn = (cid:88) k D ( (cid:126)n ) mk e − πγk U kn (27) c mn = (cid:88) k V mk e − πγk D † ( (cid:126)n (cid:48) ) kn (28)where D ( (cid:126)n ) and D ( (cid:126)n (cid:48) ) are the Wigner matrices (in therepresentation j ) corrisponding to the SU (2) elementsrotating the (cid:126)z into the (cid:126)n and the (cid:126)n (cid:48) axes, respectively.A wide class of states satisfying both (27) and (28) isgiven by | ψ (cid:105) = (cid:88) mnl D ( (cid:126)n ) lm e − πγm D † ( (cid:126)n (cid:48) ) mn | j, l, n (cid:105) (29)Note that the effect of the Wigner matrices on the ba-sis states | j, l, n (cid:105) = | j, l (cid:105) ⊗ | j, n (cid:105) † is simply to transformthe L z eigenbasis | j, l (cid:105) into the eigenbasis | j, l (cid:105) (cid:126)n of (cid:126)L · (cid:126)n for a generic vector (cid:126)n . Therefore this class of (spin j )states labelled by two arbitrary vectors and the SU (2)representation j , that satisfy the Bisognano–Wichmannproperty, has the compelling from | ψ j(cid:126)n(cid:126)n (cid:48) (cid:105) = (cid:88) m e − πγm | j, m (cid:105) (cid:126)n ⊗ | j, m (cid:105) (cid:126)n (cid:48) (30)These states are not normalised. Their norm is easilycomputed; it is the square root of N j = (cid:104) ψ j(cid:126)n(cid:126)n (cid:48) | ψ j(cid:126)n(cid:126)n (cid:48) (cid:105) = j (cid:88) k = − j e − πγk (31)These are the Bisognano–Wichmann link states. C. Semiclassicality
Before extending the Bisognano–Wichmann states tothe full graph, let us study their properties. First ofall, we have defined states at fixed spin j . Therefore weexpect the corresponding conjugate momentum, namelythe extrinsic curvature at the facet, to be fuzzy. Weleave open, for the moment, the task of combining thesestates into extrinsic [30] semiclassical states, and we con-centrate on the properties of the intrinsic geometry theydefine.For this, we estimate the mean value and the dispersionof the geometrical operators on the states (30). To beginwith, consider the case with (cid:126)n = (cid:126)n (cid:48) = (cid:126)z . Choosing thebasis that diagonalises L z we have immediately (cid:104) (cid:126)L S (cid:105) ≡ (cid:104) ψ j(cid:126)z(cid:126)z | (cid:126)L S | ψ j(cid:126)z(cid:126)z (cid:105) = (cid:88) m e − γπm (cid:104) jm | L x L y L z | jm (cid:105) , (32)where, we recall, L x L y L z | jm (cid:48) (cid:105) S = ( c | j, m (cid:48) − (cid:105) S + c | j, m (cid:48) + 1 (cid:105) S ) i ( c | j, m (cid:48) − (cid:105) S − c | j, m (cid:48) + 1 (cid:105) S ) m (cid:48) | jm (cid:48) (cid:105) S the coefficients c , c being defined by c = (cid:104) j, m − | L x − iL y | j, m (cid:105) = (cid:112) ( j + m )( j − m + 1) c = (cid:104) j, m + 1 | L x + iL y | j, m (cid:105) = (cid:112) ( j + m + 1)( j − m ) . Easily, (cid:104) (cid:126)L S (cid:105) = j (cid:88) m = − j e − πγm m = j (cid:88) m = − j e − πγm m (cid:126)z (33)The mean value, properly normalized, reads (cid:126)L S ≡ (cid:104) (cid:126)L S (cid:105)N j = (cid:80) jm = − j e − πγm m (cid:80) jm = − j e − πγm (cid:126)z. (34)The vector operator points in the direction identified bythe state and, for large j , we have: (cid:126)L S ( − j ) j →∞ −−−→ (cid:126)L S ∼ − j + O ( j ). The correction is actually aconstant given by: (cid:126)L S ∼ − j + 12( e πγ −
1) (35)In order to understand if the state become sharp for large j , we look to the relative dispersion of (cid:126)L S in the planeorthogonal to the direction identified by the mean value.We saw above that (cid:104) L x (cid:105) = (cid:104) L y (cid:105) = 0 , (36)while, due to symmetry, we can write (cid:104) L x (cid:105) = (cid:104) L y (cid:105) = 12 (cid:104) ( L x + L y ) (cid:105) = 12 (cid:104) ( (cid:126)L − L z ) (cid:105) . (37) The relative dispersions of the components L x , L y (thescale parameter is chosen to be the norm square of thevector itself) is given by σ ( L x ) (cid:104) (cid:126)L (cid:105) = σ ( L y ) (cid:104) (cid:126)L (cid:105) = 12 (cid:104) ( (cid:126)L − L z ) (cid:105)(cid:104) (cid:126)L (cid:105) . (38)The expectation value of (cid:126)L is j ( j + 1), while for L z weget L z = (cid:80) jm = − j m e − γπm (cid:80) jm = − j e − γπm . (39)The spread is then given by σ ( L y ) (cid:104) (cid:126)L (cid:105) = σ ( L x ) (cid:104) (cid:126)L (cid:105) = 12 j ( j + 1) − (cid:80) jm = − j m e − γπm (cid:80) jm = − j e − γπm j ( j + 1) , which goes to zero in the limit j → ∞ .It is easy to generalize this result for a generic direc-tion: the mean value of (cid:126)L S on | ψ j(cid:126)n(cid:126)n (cid:48) (cid:105) is given by (cid:126)L S = 1 N j (cid:32)(cid:88) m e − πm m (cid:33) (cid:126)n. (41)And the mean value of the right invariant vector field (cid:126)L T is (cid:126)L T = 1 N j (cid:32)(cid:88) n e − πn n (cid:33) (cid:126)n (cid:48) . (42)with the same relative dispersion as above. D. Overcomplete basis
Finally an important property of the Bisognano–Wichmann link states is that they form an overcompletebasis for each j , in the Hilbert space H j ⊗ H ∗ j . The res-olution of the identity is I j = d j (4 π ) N j (cid:90) S d (cid:126)n (cid:90) S d (cid:126)n (cid:48) | ψ j(cid:126)n(cid:126)n (cid:48) (cid:105)(cid:104) ψ j(cid:126)n(cid:126)n (cid:48) | (43)The integration is over the two-sphere of the normalizedvectors (cid:126)n, (cid:126)n (cid:48) , with the standard R measure restrictedto the unit sphere. This property is crucial: it indicatesthat every state can be expressed as a superposition ofstates with semiclassical labels. The proof of (43) isgiven in Appendix A. The calculation is easy to perform if one notes that all the quan-tities appearing are all of the form: j (cid:88) m = − j m n e − πm = d n dα n j (cid:88) m = − j e − αm | α =2 π (40)The only sum one needs to perform is then (cid:80) jm = − j e − αm whichcan be split into two geometric sums. IV. THERMALLY CORRELATED SPINNETWORK STATES
So far we have studied single link states. We now moveto states defined on the full graph. The first step for thisis to combine Bisognano–Wichmann states associated tothe links that join on a single node n . To this aim, we sim-ply take the tensor product of a Bisognano–Wichmannlink state per each of the links meeting at n and projecton the SU (2) gauge invariant subspace. The projectionis performed by integrating over the local gauge group SU (2), | Ψ ( n ) j l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105) = (cid:90) dh (cid:79) l ∈ n D ( h ) | ψ j l (cid:126)n l (cid:126)n (cid:48) l (cid:105) . (44)The Bisognano–Wichmann graph state is then deter-mined by a spin associated to each link and two vectors (cid:126)n l and (cid:126)n (cid:48) l associated, respectively, to the source and thetarget of each link. The resulting gauge invariant stateis | Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105) = (cid:90) (cid:89) n dh n (cid:79) l ≡(cid:104) n l ,n (cid:48) l (cid:105) D j l ( h n l ) D j l † ( h n (cid:48) l ) | ψ j l (cid:126)n l (cid:126)n (cid:48) l (cid:105) (45)where we identify each link with the two node at its end-points, l ≡ (cid:104) n l , n (cid:48) l (cid:105) .These are the Bisognano–Wichmann states on thegraph.In the Schr¨odinger representation, namely on the groupelement basis, they readΨ j l ,(cid:126)n l ,(cid:126)n (cid:48) l ( U l ) = (cid:104) U l | Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105) = (cid:90) (cid:89) n dh n (46) (cid:89) l ≡(cid:104) n l ,n (cid:48) l (cid:105) tr j l [ D ( U l ) D ( h n l ) D ( (cid:126)n l ) e − πL z D † ( (cid:126)n (cid:48) l ) D † ( h n (cid:48) l )] . These states resemble the common intrinsic Livine-Speziale states on the graph, but there is a crucial differ-ence. The space of the states with fixed spin is the tensorproduct of one intertwined space per node, that is H j l = (cid:79) n H n (47)where the intertwined space H n of the node n is the SU (2) invariant part of the tensor product of the rep-resentation spaces associated to the spins of the linksjoining in n : H n = Inv SU (2) [ (cid:79) l H j l ] . where the product in l runs over the links joining in n . The Livine-Speziale states | j l , (cid:126)n l , (cid:126)n (cid:48) l (cid:105) are tensor stateswith respect to this decomposition | j l , (cid:126)n l , (cid:126)n (cid:48) l (cid:105) = (cid:79) n | ι nj l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105) (48) where ι nj l ,(cid:126)n l ,(cid:126)n (cid:48) l is the Livine-Speziale intertwiner. On thecontrary, the Bisognano–Wichmann states do not fac-torise. To see this, it is sufficient to consider the densitymatrix of the state defined in (44) and reduce it to theintertwined space H n , by tracing over the external rep-resentation spaces of the links. A straightforward calcu-lation shows this to be ρ = tr[ | Ψ ( n ) j l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105)(cid:104) Ψ ( n ) j l ,(cid:126)n l ,(cid:126)n (cid:48) l | ] == (cid:79) l ∈ n (cid:90) dh (cid:90) d ˜ h [ D ( h ) e − πγ(cid:126)L l · (cid:126)n l D † (˜ h )]where the tensor product is on the links that join at thenode n and for simplicity we have assumed the node to bethe source of these all. One may notice in this expressionthat [ D ( h ) e − π (cid:80) l (cid:126)L l · (cid:126)n l D † (˜ h )] does not act as a rotationof the vectors (cid:126)n l , since the adjoint rapresentation actswith the same group element, whereas here we have twodifferent SU (2) elements ( h, ˜ h ). This density matrix ingeneral is not pure.This indicates that the Bisognano–Wichmann statescarry nontrivial quantum correlations between differentnodes. In Appendix C we compute the correlations be-tween two operators in a Bisognano–Wichmann state ona simple graph (the dipole graph), to verify explicitlythat they are indeed non-vanishing (see also Appendix Bfor details on the observables).This is the main property we were seeking. V. LONG DISTANCE CORRELATIONS
The Bisognano–Wichmann states defined in the previ-ous section have non trivial quantum correlations acrossadjacent nodes. Do they also have correlations betweennodes that are not adjacent? Here we show that theanswer is yes and we give some preliminary elements ofanalysis of these correlations.The simplest spin network we can use to try to addressthese questions, is an open spin network composed by achain of N nodes, each pair sharing a single link. We startby writing the explicit form of the state for the specialcase N = 2 to understand the structure of the state itself.The non-gauge invariant state on the two node graph isgiven by | ˜Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105) = (cid:88) { k l } e − πγ (cid:80) l k l | j , k , (cid:126)n (cid:105) ×× | j , k (cid:126)n (cid:48) (cid:105) † || j l , k l , (cid:126)n ( (cid:48) ) l (cid:105) ( † ) i (cid:54) =4 gathering together the external half links into the espres-sion || j l , k l , (cid:126)n ( (cid:48) ) l (cid:105) ( † ) l (cid:54) =4 . The projection to the gauge invari-ant subspace is given by | Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105) == (cid:88) { k l } e − πγ (cid:80) l k l (cid:88) αβ φ α ( j k , (cid:126)n ) ×× φ ∗ β ( j , k (cid:126)n (cid:48) ) | ι α (cid:105)| ι β (cid:105) † || j l , k l , (cid:126)n ( (cid:48) ) l (cid:105) ( † ) l (cid:54) =4 where the projector operator and the φ α,β coefficientsare those defined in Appendix C. The generalization toa chain of N nodes is straightforward.We have computed numerically the correlations on achain of N = 7 nodes, fixing all j l = 1 /
2. We havecomputed the following quantity (cid:104) P (0) i P (0) j (cid:105) − (cid:104) P (0) i (cid:105)(cid:104) P (0) j (cid:105) (49)for i, j = 1 , · · · , N , where P (0) i = | ι (cid:105)(cid:104) ι | is the projec-tor on the first element of the recoupling basis on the i -th node. The results of the numerical calculation aredisplayed in Figure 2.The correlations that we find can be interpreted as theresult the interplay between the thermal correlations onthe single links, which correlate any two adjacent nodes,and the effect of gauge-invariance at nodes, which tiesthe links in quadruples and allows for the propagation ofthe those thermal correlations among far nodes. VI. SUMMARY AND DISCUSSION
We have defined a family of states in loop quan-tum gravity which are peaked on an (intrinsic) geom-etry and have non trivial correlations between distinctnodes. These correlations are such that tracing on onenode yields, on a neighbouring node, a thermal statewith respect to a flow related, via the simplicity con-ditions, to the boost generator. We have called thesestates Bisognano–Wichmann states, and this feature theBisognano–Wichmann property. Correlations extend tonon-neighbouring nodes.We list in the following a number of questions whichwe think deserve to be investigated. • We have investigated in this paper states at fixed j . Extrinsic coherent states obtained relaxing thesharpness condition on j are of course interestingfor physics. • The boost operator (cid:126)K is the generator of internal boosts in the full covariant theory. In a gauge fixedformalism, as is implicitly the loop formalism whichis formulated in the time gauge, this is related toa physical boost. The situation is analogous to therotations of the tetrad in general relativity: if wedescribe a measuring apparatus gauge fixing thetetrad to its axes, then a rotation of the tetradhas the physical interpretation of a relative rotation D N = N1 - N210 - - < P H N1 L P H N2 L > c FIG. 2. Fit of the correlation function ( (cid:104) P ( N P ( N (cid:105) c = (cid:104) P ( N P ( N (cid:105) − (cid:104) P ( N (cid:105)(cid:104) P ( N (cid:105) ) as a function of the dis-tance between nodes (∆ N = N − N f (∆ N ) = a exp( − b ∆ N ). Fit results: a = 0 , b = 5 , between the apparatus and its exterior. Similarly,the boost generator (cid:126)K can also be interpreted asthe generator of physical boosts: it evolves a stateon a surface to the state on a boosted surface, whichis to say to the surfaces of a boosted observer [37]. • In the context of QFT, the restriction of the fieldvacuum state to the right Rindler wedge, automat-ically gives a restriction to the positive eigenvaluesof the boost generator. In closer analogy, perhapsthe restriction of states to those where (cid:126)K · (cid:126)n haspositive eigenvalues is of physical interest. In par-ticular, this aspect could turn out to be crucial forthe normalizability of the states once the sum overthe SU (2) representations ( j ) is considered. Weleave this question open. • The spin-networks Hilbert space is the same asin a conventional SU (2) lattice Yang-Mills theory.In lattice gauge theory, nonlocal correlations havebeen studied by Donnelly in [38]. See also [39].The construction here is related to these analyses,and the precise relation deserves to be better un-derstood. • In the Bisognano–Wichmann states, the thermalcorrelations get mixed up by gauge invariance. Thedensity matrix on a single node Hilbert space ob-tained by tracing the state on the rest of the nodesis not thermal, because the correlations defined onthe links get mixed up by the gauge at the node.This is not in contradiction with the Bisognano–Wichmann property, which refers to a single surface(a single link), but deserves better understanding. • The 2 π in the Bisognano–Wichmann temperatureis related to the Minkowski geometry and its com-plex extension, as well as the corner terms of theaction on the spitting surface [40–42]. The role ofthese in the loop dynamics is strictly connected tothe Bisognano–Wichmann property and is a tanta-lising issue, which still deserves clarification. ACKNOWLEDGEMENTS
Thanks to Aldo Riello and Hal Haggard for many ex-changes. PR thanks the support of the Della Riccia Foun-dation.
Appendix A: Resolution of identity
Here we derive explicitly the resolution of the identitygiven in the text. Starting from equation (43), we have I j = 1 N j d j (4 π ) (cid:90) S d (cid:126)n (cid:90) S d (cid:126)n (cid:48) | ψ ( (cid:126)n,(cid:126)n (cid:48) ) j (cid:105)(cid:104) ψ ( (cid:126)n,(cid:126)n (cid:48) ) j | = (A1)= 1 N j d j (4 π ) (cid:90) S d (cid:126)n (cid:90) S d (cid:126)n (cid:48) (cid:88) k | j, k (cid:105) S D ( (cid:126)n ) e − πγk ×× D † ( (cid:126)n (cid:48) ) | j, k (cid:105) † S (cid:88) l | j, l (cid:105) T D ( (cid:126)n (cid:48) ) e − πγl D † ( (cid:126)n ) | j, l (cid:105) † T = 1 N j d j (4 π ) (cid:90) S d (cid:126)n (cid:90) S d (cid:126)n (cid:48) (cid:88) k,α,β | j, α (cid:105) S D ( (cid:126)n ) α,k e − πγk D † ( (cid:126)n (cid:48) ) k,β | j, β (cid:105) † T (cid:88) l, ˜ α, ˜ β | j, ˜ β (cid:105) T D ( (cid:126)n (cid:48) ) ˜ β,l e − πγl D † ( (cid:126)n ) l, ˜ α | j, ˜ α (cid:105) † S . Rearranging factors, I j = 1 N j d j (cid:88) k,α,β (cid:88) l, ˜ α, ˜ β | j, α (cid:105) S (cid:104) j, ˜ α | S | j, ˜ β (cid:105) T (cid:104) j, β | T e − πγ ( k + l ) (cid:90) S d (cid:126)n π D ( (cid:126)n ) α,k D † ( (cid:126)n ) l, ˜ α (cid:90) S d (cid:126)n (cid:48) π D † ( (cid:126)n (cid:48) ) k,β D ( (cid:126)n (cid:48) ) ˜ β,l = 1 N j d j (cid:88) k,α,β (cid:88) l, ˜ α, ˜ β | j, α (cid:105) S (cid:104) j, ˜ α | S | j, ˜ β (cid:105) T (cid:104) j, β | T ×× e − πγ ( k + l ) δ α, ˜ α δ k,l d j δ β, ˜ β δ k,l d j = 1 N j (cid:88) k,α,β | j, α (cid:105)(cid:104) j, α || j, β (cid:105)(cid:104) j, β | e − πk = (cid:88) α | j, α (cid:105)(cid:104) j, α | × (cid:88) β | j, β (cid:105)(cid:104) j, β | × (cid:80) k e − πk (cid:80) k e − πk = I S ⊗ I T (A2)which is the identity in H j ⊗ H ∗ j . Appendix B: Node observables
To study correlations we need an observable to probethem. A good example of observable is the scalar product( (cid:126)L ( a ) · (cid:126)L ( b ) ) A (where a and b are two links that meet atthe node A ). Its geometrical interpretation is a measure l a l b n A ~L ( a ) ~L ( b ) FIG. 3. Thetrahedron cell dual to the node A. The two links l a and l b meet at the node A . The observable given by thescalar product ( (cid:126)L ( a ) · (cid:126)L ( b ) ) A is a measure of the dihedral angle(shaded above) between the two facets of the cell A . of the dihedral angle between the two facets of the cell A .Recall that the area of these facets is | L a | and | L b | . Herewe assume for simplicity that all nodes are four-valent.See Figure 3.This observable is diagonal in the appropriate recou-pling basis, | ι α (cid:105) ≡ {| j · · · j , ι α (cid:105)} , labelled by the spinnumber α of the “virtual link” associated to the node[30]. This can be seen explicitly by looking at the opera-tor ( (cid:126)L ( a ) + (cid:126)L ( b ) ) first: consider the explicit form of theintertwiner state | ι α (cid:105) = | j · · · j , ι α (cid:105) = (B1)= (cid:88) k ,k ,k ,k ι k k k k α | j k (cid:105)| j k (cid:105)| j k (cid:105)| j k (cid:105) = (cid:88) k ,k ,k ,k m ι k k m ι k k m | j k (cid:105)| j k (cid:105)| j k (cid:105)| j k (cid:105) . Acting with the operator ( (cid:126)L ( a ) + (cid:126)L ( b ) ), where a = 1 , b = 2we have( L i (1) + L i (2) ) | j k (cid:105)| j k (cid:105) = [ J i ( j ) + J i ( j ) ] | j k (cid:105)| j k (cid:105) (B2)where J i are the generators of SU (2) in the rapresen-tation j . If we now use this for the full state (B1), wehave( L i (1) + L i (2) ) | ι α (cid:105) = (B3)= (cid:88) k ,k ,k ,k m ι k k m ι mk k [ J i ( j ) + J i ( j ) ] × (B4) × | j k (cid:105)| j k (cid:105)| j k (cid:105)| j k (cid:105) = (cid:88) k ,k ,k ,k m ι k k m [ − J i ( α ) ] ι mk k | j k (cid:105)| j k (cid:105)| j k (cid:105)| j k (cid:105) = (cid:88) k ,k ,k ,k m, ˜ m ι k k ˜ m [ − J ( α ) ] i ˜ mm ι mk k | j k (cid:105)| j k (cid:105)| j k (cid:105)| j k (cid:105) where we used the definition of the intertwiner, D ( j ) m n D ( j ) m n D ( j ) m n ι n n n = ι m m m to get( J ( j ) + J ( j ) + J ( j ) ) ι n n n = 0 ⇒ ( J ( j ) + J ( j ) ) ι n n n = − J ( j ) ι n n n Finally, applying the same operator a second time, weobtain( L i (1) + L i (2) )( L i (1) + L i (2) ) | ι α (cid:105) == (cid:88) k ,k ,k ,k m ι k k m [ − J i ( α ) ] ι mk k | j k (cid:105)| j k (cid:105)| j k (cid:105)| j k (cid:105) = α ( α + 1) (cid:88) k ,k ,k ,k m ι k k m ι mk k | j k (cid:105)| j k (cid:105)| j k (cid:105)| j k (cid:105) = α ( α + 1) | ι α (cid:105) since [ J i ( α ) ] is the Casimir operator. Analogously, theoperator ( L i ( a ) L i ( b ) ) will be diagonal on this basis, as L i (1) L i (2) = 12 [( L i (1) + L i (2) ) − ( L i (1) ) − ( L i (2) ) ](B5)with eigenvalues given by C α = α ( α + 1) − j ( j + 1) − j ( j + 1) . (B6)In the recoupling basis, the operator takes the form L i (1) L i (2) = (cid:88) α C α | ι α (cid:105)(cid:104) ι α | . (B7) Appendix C: Correlation in dipole graph
Here we show that the correlations between nodes arein fact non vanishing, by providing a detailed example fora simple graph. We consider the dipole graph ∆ ∗ : twofour-valent nodes, A and B , sharing four links (Figure4). We consider two operators acting on the two nodes:( (cid:126)L (1) S · (cid:126)L (2) S ) A and ( (cid:126)L (3) T · (cid:126)L (4) T ) B , with (cid:126)L S and (cid:126)L T beingrespectively the left and right invariant vector fields. Wewant to measure the correlation between the two nodes.We expand the state in the appropriate recoupling ba-sis, where the chosen operators are diagonal: • | ι α (cid:105) ≡ | j , j , j , j ι α (cid:105) s.t. ι k k k k α = ι k k a ι k k a • | ι β (cid:105) ≡ | j , j , j , j ι β (cid:105) s.t. ι k k k k β = ι k k b ι k k b Instead of integrating over the group for each node, weimpose the gauge invariance through the projectors P A = (cid:88) α | ι α (cid:105)(cid:104) ι α | and P B = (cid:88) β | ι β (cid:105)(cid:104) ι β | . (C1)The non-gauge invariant state is given by | ˜Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105) == (cid:88) { k l } e − πγ (cid:80) l k l | j , k , (cid:126)n (cid:105) S | j , k (cid:126)n (cid:48) (cid:105) † T n A n B ~L (1) S ~L (2) S ~L (3) T ~L (4) T l l l l FIG. 4. Dipole graph ∆ ∗ : two four-valent nodes, A and B ,sharing four links. We consider two operators acting on thetwo nodes: ( (cid:126)L (1) S · (cid:126)L (2) S ) A and ( (cid:126)L (3) T · (cid:126)L (4) T ) B Projecting with (C1), we get its gauge invariant version | Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105) = (cid:88) { k l } e − πγ (cid:80) l k l ×× (cid:88) αβ φ α ( j k , (cid:126)n ) φ ∗ β ( j , k (cid:126)n (cid:48) ) | ι α (cid:105)| ι β (cid:105) † where we used the definition: φ α ( j k , (cid:126)n ) = (cid:104) ι α | j , k , (cid:126)n (cid:105) (C2) φ β ( j , k (cid:126)n (cid:48) ) = (cid:104) ι β | j , k , (cid:126)n (cid:48) (cid:105) We can write these coefficients explicitly. Consider firstthe case in which (cid:126)n i ≡ (cid:126)z . In this case we have (cid:104) ι α | j , k (cid:105) = ι k k k k α (C3)whose components can be calculated using the decom-position of this invariant tensor with { j } symbols. Ifinstead we keep generic directions (cid:126)n i , we need to takeinto account a rotation matrix for each link: (cid:104) ι α | j , k , (cid:126)n (cid:105) = (cid:88) D j l k ( (cid:126)n ) D j l k ( (cid:126)n ) ×× D j l k ( (cid:126)n ) D j l k ( (cid:126)n ) ι l l l l α Now we can take the expectation value of the operators.We obtain (cid:104) Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l | ( (cid:126)L (1) S · (cid:126)L (2) S ) A ( (cid:126)L (3) T · (cid:126)L (4) T ) B | Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105) == (cid:88) { k l } , { ˜ k l } e − πγ (cid:80) l ( k l +˜ k l ) ×× (cid:88) α, ˜ α φ { k l ,(cid:126)n l } α φ ∗{ ˜ k l (cid:126)n l } ˜ α (cid:104) ι ˜ α | ( (cid:126)L (1) S · (cid:126)L (2) S ) A | ι α (cid:105) ×× (cid:88) β, ˜ β φ ∗{ k l (cid:126)n (cid:48) l } β φ { ˜ k l ,(cid:126)n (cid:48) l } ˜ β (cid:104) ι ˜ β | ( (cid:126)L (3) T · (cid:126)L (4) T ) B | ι β (cid:105) (cid:88) { k l } , { ˜ k l } e − πγ (cid:80) l ( k l +˜ k l ) (cid:88) α, ˜ α φ { k l ,(cid:126)n l } α φ ∗{ ˜ k l ,(cid:126)n l } ˜ α C α δ α, ˜ α ×× (cid:88) β, ˜ β φ ∗{ k l ,(cid:126)n (cid:48) l } β φ { ˜ k l ,(cid:126)n (cid:48) l } ˜ β C β δ β, ˜ β = (cid:88) { k l } , { ˜ k l } e − πγ (cid:80) l ( k l +˜ k l ) (cid:88) α, ˜ α φ { k l ,(cid:126)n l } α φ ∗{ ˜ k l ,(cid:126)n l } α C α ×× (cid:88) β, ˜ β φ ∗{ k l ,(cid:126)n (cid:48) l } β φ { ˜ k l ,(cid:126)n (cid:48) l } β C β Similarly, (cid:104) Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l | ( (cid:126)L (1) S · (cid:126)L (2) S ) A | Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105) == (cid:88) { k l } , { ˜ k l } e − πγ (cid:80) l ( k l +˜ k l ) (cid:88) α, ˜ α φ { k l ,(cid:126)n l } α φ ∗{ ˜ k l ,(cid:126)n l } α C α ×× (cid:88) β, ˜ β φ ∗{ k l ,(cid:126)n (cid:48) l } β φ { ˜ k l ,(cid:126)n (cid:48) l } β and (cid:104) Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l | ( (cid:126)L (3) T · (cid:126)L (4) T ) B | Ψ j l ,(cid:126)n l ,(cid:126)n (cid:48) l (cid:105) == (cid:88) { k l } , { ˜ k l } e − πγ (cid:80) l ( k l +˜ k l ) (cid:88) α, ˜ α φ { k l ,(cid:126)n l } α φ ∗{ ˜ k l ,(cid:126)n l } α ×× (cid:88) β, ˜ β φ ∗{ k l ,(cid:126)n (cid:48) l } β φ { ˜ k l ,(cid:126)n (cid:48) l } β C β . We want to prove the following inequivalence (cid:104) ( (cid:126)L (1) S · (cid:126)L (2) S ) A ( (cid:126)L (3) T · (cid:126)L (4) T ) B (cid:105) (cid:54) = (cid:104) ( (cid:126)L (1) S · (cid:126)L (2) S ) A (cid:105)(cid:104) ( (cid:126)L (3) T · (cid:126)L (4) T ) B (cid:105) . Since we know the explicit form of the coefficients, wecan verify this statement in an explicit example. Forsemplicity, let us fix all j l = 1 / {| ι (cid:105) , | ι (cid:105)} , and consider (cid:126)n l = (cid:126)n (cid:48) l = (cid:126)z , for each l . We find (cid:104) ( (cid:126)L (1) S · (cid:126)L (2) S ) A ( (cid:126)L (3) T · (cid:126)L (4) T ) B (cid:105) = 54 (C4) (cid:104) ( (cid:126)L (1) S · (cid:126)L (2) S ) A (cid:105) = (cid:104) ( (cid:126)L (3) T · (cid:126)L (4) T ) B (cid:105) = −
12 (C5)which implies (cid:104) ( (cid:126)L (1) S · (cid:126)L (2) S ) A ( (cid:126)L (3) T · (cid:126)L (4) T ) B (cid:105) − (C6) −(cid:104) ( (cid:126)L (1) S · (cid:126)L (2) S ) A (cid:105)(cid:104) ( (cid:126)L (3) T · (cid:126)L (4) T ) B (cid:105) = 34 (cid:54) = 0The conclusion is that the Bisognano–Wichmann stateshave correlations between neighbouring nodes. [1] J. J. Bisognano and E. H. Wichmann, “On the Dual-ity Condition for Quantum Fields,” J. Math. Phys. (1976) 303–321.[2] R. Haag, N. M. Hugenholtz, and M. Winnink, “Onthe equilibrium states in quantum statistical mechan-ics,” Comm. Math. Phys. (1967). 215–236.[3] R. Haag, Local Quantum Physics: Fields, Particles, Al-gebras . Springer, 1996.[4] W. G. Unruh, “Notes on black hole evaporation,”
Phys.Rev. D (1976) 870.[5] L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, “Quan-tum source of entropy for black holes,” Phys. Rev. D (1986) 373–383.[6] M. Srdnicki, “Entropy and Area,” Phys. Rev. Lett. (1993) 666–669.[7] T. Jacobson and R. Parentani, “Horizon entropy,” Found. Phys. (2003) 323–348, arXiv:0302099 .[8] S. N. Solodukhin, “Entanglement entropy of black holes,” Living Rev. Relativity (2011) 8.[9] E. Bianchi, “Black hole entropy from graviton entangle-ment,” arXiv:1211.0522 .[10] E. Bianchi and R. C. Myers, “On the Architecture ofSpacetime Geometry,” arXiv:1212.5183 .[11] E. Frodden, A. Ghosh, and A. Perez, “A local first lawfor black hole thermodynamics,” arXiv:1110.4055 .[12] A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov,“Quantum geometry and black hole entropy,” Phys. Rev. Lett. (1998) 904–907, arXiv:9710007 .[13] E. Bianchi, “Entropy of Non-Extremal Black Holes fromLoop Gravity,” arXiv:1204.5122 .[14] E. Frodden, A. Ghosh, and A. Perez, “Black hole entropyin LQG: Recent developments,” AIP Conf. Proc. (2011) 100–115.[15] J. F. Barbero G., J. Lewandowski, and E. J. S. Vil-lasenor, “Quantum isolated horizons and black hole en-tropy,” arXiv:1203.0174 .[16] A. Ghosh, K. Noui, and A. Perez, “Statistics, hologra-phy, and black hole entropy in loop quantum gravity,” arXiv:1309.4563 .[17] M. Van Raamsdonk, “Building up spacetime with quan-tum entanglement,” arXiv:1005.3035 .[18] B. Czech, J. L. Karczmarek, F. Nogueira, and M. VanRaamsdonk, “The Gravity Dual of a Density Matrix,” arXiv:1204.1330 .[19] B. Czech, J. L. Karczmarek, F. Nogueira, andM. Van Raamsdonk, “Rindler Quantum Gravity,” arXiv:1206.1323 .[20] G. Chirco, H. M. Haggard, A. Riello, and C. Rovelli,“Spacetime thermodynamics without hidden degrees offreedom,” arXiv:1401.5262 .[21] T. Thiemann, “Complexifier coherent states for quantumgeneral relativity,”
Class. Quant. Grav. (2006) 2063–2118, arXiv:0206037 .[22] E. R. Livine and S. Speziale, “Physical boundary state for the quantum tetrahedron,” Class. Quant. Grav. (2008) 85003, arXiv:0711.2455 .[23] E. Bianchi, E. Magliaro, and C. Perini, “Coher-ent spin-networks,” Phys. Rev. D (2010) 24012, arXiv:0912.4054 .[24] E. Bianchi, P. Don`a, and S. Speziale, “Polyhedra inloop quantum gravity,” Phys. Rev. D (2011) 44035, arXiv:1009.3402 .[25] A. Ashtekar, Quantum Gravity and Quantum Cosmology , Lecture Notes in Physics . Springer Berlin Heidelberg,Berlin, Heidelberg, (2013).[26] L. Freidel and K. Krasnov, “A New Spin Foam Modelfor 4d Gravity,”
Class. Quant. Grav. (2008) 125018, arXiv:0708.1595 .[27] J. Engle, E. Livine, R. Pereira, and C. Rovelli, “LQGvertex with finite Immirzi parameter,” Nucl. Phys. B (2008) 136–149, arXiv:0711.0146 .[28] W. Kaminski, M. Kisielowski, and J. Lewandowski,“Spin-Foams for All Loop Quantum Gravity,”
Class.Quant. Grav. (2010) 95006, arXiv:0909.0939 .[29] C. Rovelli, “Zakopane lectures on loop gravity,” PoS ( QG -QG) (2011) 3, arXiv:1102.3660 .[30] C. Rovelli and F. Vidotto, Introduction to covariant loopquantum gravity . Cambridge University Press, to ap-pear., 2015.[31] S. Alexandrov and E. R. Livine, “SU(2) loop quantumgravity seen from covariant theory,”
Phys. Rev. D (2003) 44009, arXiv:0209105 .[32] M. Dupuis and E. R. Livine, “Lifting SU(2) Spin Net-works to Projected Spin Networks,” Phys. Rev. D (2010) 64044.[33] C. Rovelli and S. Speziale, “Lorentz covariance of loopquantum gravity,” Phys. Rev. D (2011) 104029, arXiv:1012.1739 .[34] B. Dittrich and M. Geiller, “A new vacuum for LoopQuantum Gravity,” arXiv:1401.6441 .[35] W. Ruhl, The Lorentz group and harmonic analysis .W.A. Benjamin, Inc, New York, 1970.[36] Y. Ding and C. Rovelli, “The volume operator in co-variant quantum gravity,”
Class. Quant. Grav. (2010)165003, arXiv:0911.0543 .[37] C. Rovelli, “What is observable in classical and quantumgravity?,” Class. Quant. Grav. (1991) 297–316.[38] W. Donnelly, “Decomposition of entanglement entropy inlattice gauge theory,” Physical Review D (Apr., 2012)085004, arXiv:1109.0036 .[39] P. Buividovich and M. Polikarpov, “Entanglement en-tropy in gauge theories and the holographic principle forelectric strings,” Physics Letters B (Dec., 2008) 141–145, arXiv:0806.3376 .[40] G. Hayward, “Gravitational action for spacetimes withnonsmooth boundaries,”
Physical Review D (Apr.,1993) 3275–3280.[41] Y. Neiman, “Parity and reality properties of theEPRL spinfoam,” Class.Quant.Grav. (2012) 65008, arXiv:1109.3946 .[42] E. Bianchi and W. Wieland, “Horizon energy as the boostboundary term in general relativity and loop gravity,” arXiv:1205.5325arXiv:1205.5325