Continuous formulation of the Loop Quantum Gravity phase space
aa r X i v : . [ g r- q c ] A p r Continuous formulation of the Loop Quantum Gravity phase space
Laurent Freidel, ∗ Marc Geiller, † and Jonathan Ziprick
1, 3, ‡ Perimeter Institute for Theoretical Physics31 Caroline St. N, N2L 2Y5, Waterloo ON, Canada Laboratoire APC – Astroparticule et CosmologieUniversit´e Paris Diderot Paris 7, 75013 Paris, France Department of Physics, University of WaterlooWaterloo, Ontario N2L 3G1, Canada
In this paper, we study the discrete classical phase space of loop gravity, which is ex-pressed in terms of the holonomy-flux variables, and show how it is related to the continuousphase space of general relativity. In particular, we prove an isomorphism between the loopgravity discrete phase space and the symplectic reduction of the continuous phase spacewith respect to a flatness constraint. This gives for the first time a precise relationshipbetween the continuum and holonomy-flux variables. In our construction the fluxes dependnot only on the three-geometry, but also explicitly on the connection, providing a naturalexplanation of their non-commutativity. It also clearly shows that the flux variables do notlabel a unique geometry, but rather a class of gauge-equivalent geometries. This allows us toresolve the tension between the loop gravity geometrical interpretation in terms of singulargeometry, and the spin foam interpretation in terms of piecewise flat geometry, since weestablish that both geometries belong to the same equivalence class. This finally gives us aclear understanding of the relationship between the piecewise flat spin foam geometries andRegge geometries, which are only piecewise-linear flat: While Regge geometry correspondsto metrics whose curvature is concentrated around straight edges, the loop gravity geometrycorrespond to metrics whose curvature is concentrated around not necessarily straight edges.
Introduction
The classical starting point of Loop Quantum Gravity (LQG) [1, 2] is a Hamiltonian formula-tion of general relativity in terms of first order connection and triad variables. The basic fieldsparametrizing the phase space are chosen to be the su (2)-valued Ashtekar-Barbero connection A [3], and its canonically conjugate densitized triad field E , both being defined over spatial hyper-surfaces foliating the spacetime manifold. The theory comes with a set of first class constraints,namely, the vector constraint generating diffeomorphisms of the spatial hypersurface, the scalarconstraint generating time reparametrizations, and the Gauss constraint generating internal SU(2)gauge transformations.As a first step towards the construction of the quantum theory, one defines a smearing ofthe classical Poisson algebra formed by the canonical pair ( A, E ) by introducing oriented graphs.Given a graph Γ embedded in the spatial manifold, the continuous variables A ( x ) and E ( x ) arereplaced by a pair ( h e , X e ) associated to each edge e . The variable h e ∈ SU(2) corresponds to theholonomy of the connection along the edge e , and X e ∈ su (2) represents the “electric” flux of thedensitized triad field across a surface dual to e . At the quantum level these new variables formthe so-called holonomy-flux algebra [5], which is a cornerstone of the entire construction of LQG. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] Note that there is an alternative approach where one takes the holonomy-flux variables associated to graphs asfundamental. The notion of a continuous spatial manifold is then seen as an emergent feature of the theory [4].
The Hilbert space H Γ of representations associated with this algebra is the so-called spin networkHilbert space. It captures only a finite number of degrees of freedom in the theory. One recoversthe continuous kinematical Hilbert space by taking the projective limit of graph Hilbert spaces H Γ . The main challenge is then to formulate a consistent and semi-classically meaningful versionof the Hamiltonian constraint acting on the spin network basis.In this construction, two very different procedures are realized at once. There is a discretizationprocedure in which the continuous fields are replaced by discrete holonomies and fluxes associatedwith graphs, and in the same stroke, these variables are promoted into quantum operators. Themain idea we want to take advantage of is that the processes of discretization and quantization aretotally independent. In this work we would like to disentangle these two steps. We propose to studyonly the process of discretization using graphs, without delving into the quantization of the theory.This means that we first associate to a given graph a finite-dimensional holonomy-flux phase spacegenerated by ( h e , X e ) ∈ T ∗ SU(2). The phase space of loop gravity on a graph is obtained as adirect product over the edges of SU(2) cotangent bundles. The main point of the present paper isto understand the exact relationship between this finite-dimensional discrete phase space, and thecontinuous phase space of gravity. We show explicitly that an element of the discrete phase spacerepresents a specific equivalence class of continuous geometries.The advantage of considering classical loop gravity is threefold. First, it provides a truncationof the classical phase space of gravity in terms of finite-dimensional holonomy-flux phase spaces,whose quantizations are given by spin network states. Second, it allows us to shed some light onthe geometrical interpretation of the holonomy-flux variables, and the type of geometry that theyrepresent. For instance, we will see that both the singular geometry of LQG and the piecewise flatgeometry of spin foam models are represented by the same flux data as two representatives of thesame equivalence class. As we will see in the end, our result also allows us to understand moreprecisely the relationship between the spin foam geometrical interpretation and Regge geometry.Namely, it shows that twisted geometries [6] described by fluxes can be understood as piecewiseflat geometries which are not necessarily piecewise-linear flat, as is the case for Regge geometry[7, 8]. Finally, this approach is designed to allow us to address at the classical level one of the mostchallenging questions of LQG: Is it possible to express, in the classical setting, the dynamics ofgeneral relativity in terms of a collection of truncated dynamics between finite-dimensional phasesspaces parametrized by holonomies and fluxes. In other words, can we capture the full dynamicsof gravity in terms of the holonomy-flux phase spaces if we simultaneously consider all graphs.If there is a clear positive answer to this question at the classical level, then the quantization ofloop gravity will be reduced to the treatment of quantization ambiguities in a finite-dimensionalsystem. If, on the other hand, we get a negative answer at the classical level, then no quantizationin terms of holonomy-flux variables can express the quantum gravitational dynamics. It is thereforeof utmost importance to eventually understand the classical dynamics of general relativity in termsof the holonomy-flux representation.Let us stress that the classical picture of the loop gravity phase space that we develop here is,when quantized, related to the picture first proposed by Bianchi in [9]. In this precursor work, it isargued that the spin network Hilbert space can be identified with the state space of a topologicaltheory on a flat manifold with defects. Our analysis makes the same type of identification at theclassical level and emphasizes the fact that the frame field determines only an equivalence classof geometries. The idea that the discrete data labels only an equivalence class of geometries hasalready been advocated in [10] on a general basis. Our approach gives a precise understanding ofwhich set or equivalence class of continuous geometries is represented by the discrete geometricaldata.We begin in section I by defining the continuous phase space of gravity in terms of the connectionand triad variables A and E , and recall some facts about the process of symplectic reduction. Insection II we introduce the discrete classical spin network phase space associated to a graph. Inparticular, we explain how to obtain the discrete data ( h e , X e ) starting from the continuous fields A and E , and showing the fluxes cannot depend only on E but need to involve the connectionin their definition. This construction explains why the flux variable carries information aboutboth intrinsic and extrinsic geometry, in agreement with what has been pointed out already in[6]. In section III, we prove that the discrete holonomy-flux phase space can be obtained as asymplectic reduction of the continuous phase space. This shows that the discrete data correspondsto an equivalence class of continuous three-geometries related by gauge transformations. In sectionIV we show that given a particular gauge choice, the discrete data can be used to reconstruct aconfiguration of the continuous fields. We will show in particular that it is possible to representa given equivalence class of geometries by either a singular gauge choice in agreement with theLQG interpretation of polymer geometry, or a flat gauge choice corresponding to the geometricalinterpretation of spin foams. Finally, in section V we discuss the notion of cylindrical consistencyand cylindrical operators, and explain how it is possible to relate operators constructed on thediscrete and the continuous phase spaces.Notations are such that µ, ν, . . . refer to spacetime indices, a, b, . . . to spatial indices, I, J, . . . toLorentzian indices, and i, j, . . . to su (2) indices. We will assume that the four-dimensional space-time manifold is topologically Σ × R , where Σ is a three-dimensional manifold without boundaries. I. CONTINUOUS PHASE SPACE OF GRAVITY
The loop approach to quantum gravity relies on the well-known idea that the phase space ofLorentzian or Riemannian general relativity can be parametrized in terms of an su (2)-valued con-nection one-form A ia and a densitized triad field ˜ E ai , both fields being defined over a base three-dimensional spacetime manifold Σ (which here we assume to be isomorphic to S ). The su (2)Ashtekar-Barbero connection A ia is related to the spacetime so (3 ,
1) spin connection ω IJµ and tothe geometrodynamical variables of the ADM phase space via A ia ≡ ǫ ijk ω jka + γω ia = Γ ia + γK ia , (1.1)where γ ∈ R − { } is the Barbero-Immirzi parameter, Γ ia is the Levi-Civita spin connection, and K ia the extrinsic curvature one-form. The densitized triad and the three-dimensional frame field e ia are related by ˜ E ai = 12 ǫ abc ǫ ijk e jb e kc , det( e ) e ia = 12 ǫ ijk ǫ abc ˜ E bj ˜ E ck . (1.2)These variables form the Poisson algebra (cid:8) A ia ( x ) , A jb ( y ) (cid:9) = (cid:8) ˜ E ai ( x ) , ˜ E bj ( y ) (cid:9) = 0 , (cid:8) A ia ( x ) , ˜ E bj ( y ) (cid:9) = γδ ij δ ba δ ( x − y ) . (1.3)The classical configuration space of the theory is the space A of smooth connections on Σ. Thephase space is the cotangent bundle P ≡ T ∗ A , and carries a natural symplectic potential. In thefollowing we will denote by E ab i (without tilde) the Lie algebra-valued two-form related to thedensitized vector ˜ E ia through E ab i ≡ ǫ abc ˜ E ci , E i ≡ E ab i d x a ∧ d x b . (1.4)The symplectic potential of the cotangent bundle is given byΘ = Z Σ E i ∧ δA i = Z Σ Tr ( E ∧ δA ) , (1.5)where we denote by Tr the natural metric on su (2) which is invariant under the adjoint actionAd SU(2) of the group. The phase space P also carries an action of the gauge group SU(2) andof spatial diffeomorphisms. In fact, since P is 18-dimensional at each point of Σ, the (first class)constraints of the canonical theory have to be taken into account in order to obtain the physicalphase space with 4 degrees of freedom at each point. This can be achieved through the process ofsymplectic (or Hamiltonian) reduction, which we now describe.Let P be a symplectic manifold, which is seen as the classical phase space of the theory ofinterest, and G a group of transformations. Suppose that the infinitesimal group transformationsare generated via Poisson bracket by a Hamiltonian H . Then the Marsden-Weinstein theorem[11–13] ensures that the symplectic reduction of P by the group G , denoted by the double quotient P (cid:12) G , is still a symplectic manifold and carries a unique symplectic form. The reduced phasespace is given by imposing the constraints and dividing the constraint surface by the action ofgauge transformations. This is written as P (cid:12) G ≡ H − (0) /G. (1.6)For notational simplicity, we will denote the group of transformations G and the associated Hamil-tonian H with the same letters. Note that the Marsden-Weinstein theorem is proven in general forfinite-dimensional phase spaces, but these methods are commonly extended to infinite-dimensionalphase spaces. See [14] for a symplectic reduction of P ≡ T ∗ A , and the first two chapters of [15]for a description of this method as commonly employed in physics.In the case of four-dimensional gravity, the physical phase space is obtained from the kinematical(unconstrained) phase space P by performing three symplectic reductions. The first one is definedwith respect to the group of SU(2) gauge transformations G ≡ C ∞ (cid:0) Σ , SU(2) (cid:1) . Since the actionof this gauge group on P is Hamiltonian, we can define the gauge-invariant phase space T ∗ A (cid:12) G .More precisely, the Hamiltonian generating these transformations is the smeared Gauss constraint: G ( α ) = Z Σ α i (d A E ) i = 0 , (1.7)where d A denotes the covariant differential and α is a Lie algebra-valued function. Its infinitesimalaction on the phase space variables is given by δ G α A = (cid:8) A, G ( α ) (cid:9) = d A α, δ G α E = (cid:8) E, G ( α ) (cid:9) = [ E, α ] . (1.8)The other relevant symplectic reduction is defined with respect to the group of spatial diffeomor-phisms, and enables one to construct the diffeomorphism-invariant phase space T ∗ A (cid:12) (cid:0) G ×
Diff(Σ) (cid:1) .Here, the action of the group of diffeomorphisms on the phase space variables is given by δ D ξ A = (cid:8) A, D ( ξ ) (cid:9) = L ξ A, δ D ξ E = (cid:8) E, D ( ξ ) (cid:9) = L ξ E, (1.9)where L ξ is the Lie derivative along the vector field ξ a . This group is generated through Poissonbrackets with the Hamiltonian D ( ξ ) = H ( ξ ) − G ( ξ a A a ) , with H ( ξ ) = Z Σ ξ a F iab E bi . (1.10)Finally, the physical phase space can be obtained from the gauge and diffeomorphism-invariantphase space by performing a symplectic reduction with respect to the scalar constraint. This latteris given by H ( N ) = Z Σ N E ai E bj p det( E ai ) (cid:16) ǫ ij k F kab − γ − σ ) K i [ a K jb ] (cid:17) = 0 , (1.11)where the smearing variable is the lapse function N , and σ = ∓ γ = ± i in the Lorentzian case,or ± II. SPIN NETWORK PHASE SPACE
In loop gravity, one does not work directly with the continuous kinematical Hilbert space, but in-stead with the projective limit of Hilbert spaces associated to embedded oriented graphs Γ [5, 16].The Hilbert space associated with one graph is the so-called spin network Hilbert space. It repre-sents a truncation of the full Hilbert space to a finite number of degrees of freedom. What we wouldlike to emphasize here is that spin network Hilbert spaces can be obtained as the quantization offinite-dimensional phase spaces associated to embedded oriented graphs Γ. Each of these truncatedphase spaces are spanned by a finite number of holonomies and fluxes which reproduce the Poissonalgebra of T ∗ SU(2). This fact has already been recognized in the literature [10] and is at the basisof most of the recent semi-classical analyses of LQG [17–19]. Our main point is that the process oftruncating the theory to a finite number of degrees of freedom and the process of quantizing thistruncated theory are separate constructions which have to be studied individually. Here we wouldlike to adopt the point of view that the continuous kinematical phase space P can be described asthe projective limit of phase spaces P Γ associated to embedded oriented graphs Γ. In particular,we would like to understand the relationship between these finite-dimensional phase spaces P Γ andthe continuous phase space variables.In [20] it has been shown how, for a given graph Γ, the graph phase space P Γ can be obtainedfrom the continuous phase space, and carries the Poisson structure of finite direct products of SU(2)cotangent bundles. It has furthermore been shown how the regulator corresponding to the graphcan be removed, thereby defining a continuum limit (via a suitable projective sequence) which leadsback to the original infinite-dimensional continuous phase space T ∗ A . While in the present workwe will recall some elements of this construction like the definition of the discrete spin networkphase spaces, our new message is to show how, without taking the continuum limit, it is possibleto understand the discrete holonomy and flux elements as labels for particular configurations onthe original phase space parametrized by continuous fields A ( x ) and E ( x ).An oriented graph Γ is defined as a one-cellular complex [21] consisting of a set E Γ of orientededges e (one-dimensional analytic submanifolds of Σ) and a set V Γ of vertices v . The end pointsof the oriented edges are the vertices, and we denote by s, t the two functions assigning a sourcevertex s ( e ) and a target vertex t ( e ) to each edge e . We also denote by e − the edge e with areverse orientation. The kinematical spin network phase space P Γ associated with such a graph isisomorphic to a direct product for each edge of SU(2) cotangent bundles : P Γ ≡ × e T ∗ SU(2) e . (2.1) Given a Lie group G , the group action on itself by left (or right) multiplication can be used to obtain an isomorphismof vector fields with the Lie algebra g , and to trivialize the cotangent bundle as T ∗ G = G × g ∗ [22]. Explicitly, this phase space is labeled by couples ( h e , X e ) ∈ SU(2) × su (2) of Lie group and Liealgebra elements for each edge e ∈ Γ. This data depends on a choice of orientation for each edge,and under an orientation reversal we have h e − = h − e , X e − = − h − e X e h e . (2.2)Since we have chosen here to trivialize T ∗ SU(2) with right-invariant vector fields, this last relationmeans that under orientation reversal of the edge we obtain the left-invariant ones. The variables( h e , X e ) satisfy the Poisson algebra (cid:8) X ie , X je ′ (cid:9) = δ ee ′ ǫ ijk X ke , (cid:8) X ie , h e ′ (cid:9) = − δ ee ′ τ i h e + δ ee ′− τ i h − e , (cid:8) h e , h e ′ (cid:9) = 0 , (2.3)where we have used notations such that X e ≡ X ie τ i . As shown in [23, 24], the symplectic potentialand symplectic two-form for this Poisson structure are given respectively byΘ Γ = X e Tr (cid:0) X e δh e h − e (cid:1) , Ω Γ = − dΘ Γ . (2.4)On the spin network phase space P Γ , we can define the action of the gauge group G Γ ≡ SU(2) | V Γ | at the vertices V Γ of the graph. Given an element g v ∈ SU(2), finite gauge transformations aregiven by g v ⊲ h e = g s ( e ) h e g − t ( e ) , g v ⊲ X e = g s ( e ) X e g − s ( e ) , (2.5)where s ( e ) (resp. t ( e )) denotes the starting (resp. terminal) vertex of e . This action on thevariables h e and X e is generated at each vertex by the Hamiltonian G v ≡ X e ∋ v X e = X e | s ( e )= v X e + X e | t ( e )= v X e − , (2.6)which can be understood as a discrete Gauss constraint. Since this action is Hamiltonian, we candefine the gauge-invariant phase space P G Γ = × e T ∗ SU(2) e (cid:12) SU(2) | V Γ | = G − v (0) / SU(2) | V Γ | (2.7)by symplectic reduction where, as explained above, the double quotient means imposing the Gaussconstraint at each vertex v and then dividing out the action of the SU(2) gauge transformation(2.5) that it generates.The question we would like to address is: What is the relationship between the continuousphase space P described in the previous section, and the spin network phase space P Γ ? Moreprecisely, we would like to know if it is possible to reconstruct from the discrete data P Γ a pointin the continuous phase space P ? In order to describe the relationship between the discrete andcontinuous data, we need a map from the continuous to the discrete phase space. We can thenstudy its kernel and see to what extent it can be inverted. This is the object of the next sections. In this work, we define τ i = − iσ i /
2, where σ i are the Pauli matrices. The su (2) commutation relations are thengiven by [ τ i , τ j ] = ǫ kij τ k , where ǫ kij is the completely antisymmetric Levi-Civita tensor. A. From continuous to discrete data
In order to construct the discrete data, let us first choose an embedding f Γ : Γ −→ Σ of the graph Γinto the spatial manifold Σ. Given this embedding, it is well understood in the discrete picture thatthe group elements h e represent holonomies of the Ashtekar-Barbero connection A ia along edges e . It is necessary to work with such objects because an important step toward the quantizationof the canonical theory is the smearing of the Poisson algebra (1.3). Since the connection A ia is aone-form, it is natural to smear it along paths e . Now we could just take the integral of A along e as a smearing but this will not respect the gauge transformations. What is needed is a smearingthat does intertwine the notion of continuous and discrete gauge transformations. It is well knownthat this is given by the notion of parallel transport along e , encoded in the holonomy h e ( A ) ≡ −→ exp Z e A = −→ exp Z e A ia ˙ e a τ i = −→ exp Z t ( e ) s ( e ) A ia ˙ e a τ i , (2.8)where ˙ e a denotes the tangent vector to the path and −→ exp denotes the path-ordered exponential.Let us recall some fundamental properties of the holonomy functional. The holonomy is invariantunder reparametrizations of the path e , and the holonomy of a path corresponding to a singlepoint is the identity. If we consider the composition e = e ◦ e of two paths which are such that s ( e ) = t ( e ), the holonomy satisfies h e = h e h e . (2.9)If we reverse the orientation of a path, we have h e − = h − e . (2.10)These properties come from the fact that the holonomy is a representation of the groupoid oforiented paths [25]. Under SU(2) gauge transformations, the holonomy transforms as g ⊲ h e = g s ( e ) h e g − t ( e ) , (2.11)which shows that the finite gauge transformation g ⊲A = gAg − + g d g − of the connection becomesa discrete gauge symmetry acting on the vertices defining the boundary of the edge e . Finally,under the action of a diffeomorphism Φ ∈ Diff(Σ), the holonomy transforms as h e (Φ ∗ A ) = h Φ( e ) ( A ) . (2.12)The exact meaning of “momentum” variable X e is less clear. Roughly speaking, we usually builda flux operator by smearing the field E ai along a surface F e dual to an edge e [2]. But if one wantsthis integrated flux to have a covariant behavior under gauge transformations, it is essential forthe integration along F e to involve some notion of parallel transport. Indeed, the naive definition¯ X e ( E ) = Z F e E ( x ) (2.13)of the flux is not covariant under gauge transformations, i.e.¯ X e ( g ⊲ E ) = ¯ X e (cid:0) gEg − (cid:1) = g s ( e ) ¯ X e ( E ) g − s ( e ) . (2.14)This is an important point which has often been ignored in the LQG literature, the only noticeableexceptions being [20, 26], and more recently [6, 27]. For the holonomy, the only reason we considerthe parallel transport operator instead of the simple integral of A along e is to have a discretizationcovariant under gauge transformation. It is as important to preserve this covariance for the flux asit is for the holonomy. Another drawback is that the non-covariant definition of the flux does notproduce the Poisson algebra given in (2.3), unless the intersection F e ∩ e between face and edgeis at the start point s ( e ) or terminal point t ( e ) of the edge. If we consider a face that intersectssomewhere in the middle of the edge, i.e. write the edge as e = e ◦ e and have the intersection F e ∩ e = s ( e ) = t ( e ), then we have (cid:8) ¯ X ie , h e ′ (cid:9) = − δ ee ′ h e τ i h e + δ ee ′− h − e τ i h − e , (2.15)which splits the holonomy in two.The way around this problem is to define a flux operator which also depends on the connectionthrough its holonomy. Given an oriented edge e ∈ Γ and a point u on this edge, we choose a surface F e intersecting e transversally at u = F e ∩ e . We also choose a set of paths π e assigning to anypoint x ∈ F e a unique path π e going from the source s ( e ) to x . Such a path starts at the sourcevertex of the edge e , goes along e until it reaches the intersection point u = F e ∩ e , and then goesfrom u to any point x ∈ F e while staying tangential to the surface F e . More precisely, we have π e : F e × [0 , −→ Σ such that π e ( x,
0) = s ( e ) and π e ( x,
1) = x . With the set of data ( F e , π e ), onecan define the flux operator X ( F e ,π e ) ( A, E ) ≡ Z F e h π e ( x ) E ( x ) h π e ( x ) − , (2.16)where h π e ( x ) ≡ −→ exp Z xs ( e ) A. (2.17)Notice that by definition, the source of the path π e is s ( e ), and its target is the point x ∈ F e .Therefore, under the gauge transformations g ⊲ E ( x ) = g ( x ) E ( x ) g ( x ) − , g ⊲ h π e ( x ) = g s ( e ) h π e ( x ) g ( x ) − , (2.18)the flux operator becomes X ( F e ,π e ) ( g ⊲ A, g ⊲ E ) = g s ( e ) X ( F e ,π e ) ( A, E ) g − s ( e ) , (2.19)which is in agreement with (2.5). The existence of a covariant transformation property is one ofthe main justifications for introducing the extra holonomy dependance in the definition of the fluxoperator. With the definition (2.16), the flux operator intertwines the continuous and discreteactions of the gauge group.From the definition of the paths π e we see that reversing the orientation of the edge givesa system of paths beginning at t ( e ) and ending at a point x ∈ F e , i.e. π e − ( x,
0) = t ( e ) and π e − ( x,
1) = x . This implies that π e − = e − ◦ π e , and therefore h π e − ( x ) = h − e h π e ( x ) . (2.20)Moreover, the surface F e possesses a reverse orientation F e − = − F e , and thus we have X ( F e − ,π e − ) = − h − e X ( F e ,π e ) h e , (2.21)which proves that our mapping is consistent with (2.2). Notice also that any two fluxes that differonly by the choice of surfaces are in the commutant of the holonomy algebra: (cid:8) X ( F ′ e ,π ′ e ) − X ( F e ,π e ) , h e (cid:9) = 0 , (2.22)where π e and π ′ e each follow the edge until the intersection points with their respective surfaces asdefined above. An important feature of the mapping that we have described is that it reproducesthe Poisson algebra (2.3), specifically the Poisson bracket between flux and holonomy. To showthis, let us use the notation R ( h π e ) ij E j ≡ ( h π e Eh − π e ) i in writing the flux. In the case where theflux and holonomy are associated to the same edge and have the same orientation, we have (cid:8) X i ( F e ,π e ) ( A, E ) , h e ( A ) (cid:9) = Z F e R ( h π e ) ij (cid:8) E j , h e ( A ) (cid:9) = − R ( h e ) ij h e τ j h e = − h e h − e τ i h e h e = − τ i h e , (2.23)where we are using the same notation as above when splitting the edge into e and e at the pointof intersection, and we have h π e = h e at the only point contributing to the integral in the firstline. Also notice that the SU(2) rotation R ( h π e ) ij acts on the basis elements τ i inversely to theway it acts on E i . A similar calculation with the inverse holonomy yields the second term shownin (2.3).Finally, we know that the requirement of consistency with the Jacobi identity imposes that thefluxes do not commute among each other. This property, which seems inconsistent if X e dependspurely on the (commuting) densitized triad field, is perfectly understandable if the flux dependsalso on the connection, and provides a natural explanation to the “mystery” behind the non-commutativity of the fluxes [28]. This is consistent with the understanding of the spin networkphase space in terms of twisted geometries [6], where it appears clearly that the flux operators alsocontain information about the holonomies, and cannot be thought of as being purely geometrical.In other words, the flux operators are not commuting because they capture information not onlyabout the intrinsic geometry, but also about the extrinsic curvature.The map that we have described depends on three types of data. It depends on a choice ofembedding f Γ of Γ into Σ, a choice of surface F e transverse to the edge e at u , and a choice of path π e going from s ( e ) to a point x ∈ F e . Once this data is given, we can construct a map I : P −→ P Γ ( A, E ) (cid:0) h e ( A ) , X ( F e ,π e ) ( A, E ) (cid:1) , (2.24)which has the key property of intertwining gauge transformations on the continuous and discretephase spaces, is compatible with the orientation reversal of the edges, and respects the Poissonstructure of T ∗ SU(2).
B. From discrete to continuous data
Now we would like to investigate to what extent it is possible to invert the map from continuousto discrete data I : P −→ P Γ . In other words, to what extent does the discrete data determine thecontinuous data? Can we reconstruct a unique representative of the continuous data starting fromthe discrete one, or describe a specific equivalence class?At first sight, this seems like an impossible task. Indeed, if one first focuses on the connection,0one needs to choose an embedding f Γ to construct the holonomies, so there is no way the discretegroup elements will determine the connection unless we know this embedding. Moreover, oneclearly sees that the flux operator is not uniquely defined by the electric field E . There are severalambiguities in its definition. There are many possible choices of surfaces F e that are transverseto the edge e , and also many possible paths that one can choose on F e . Different choices lead todifferent mappings from the continuous data to the discrete data. This means that giving a flux X ( F e ,π e ) (which we will call X e for simplicity) does not allow one to reconstruct a continuous field E , which constitutes a fundamental ambiguity. This state of affairs is fine if one treats the discretedata as some approximate description of continuous geometry which only takes physical meaning insome continuous limit. This is the usual point of view [10], and it implies that operators expressedin terms of the fluxes X e do not have a sharp semi-classical geometric interpretation.In this work we would like to be more ambitious and interpret the discrete data as potentialinitial value data for the continuous theory of gravity. The challenge is to show that one canreconstruct continuous fields ( A, E ) explicitly from the knowledge of the discrete data ( h e , X e ).How can this be possible in light of all the ambiguities that we have listed above? In order to makesome progress in this direction, let us first remark that there are configurations of fields for whichthe ambiguities disappear. This is the case in particular for a flat connection.Suppose that we focus on a region C v of simple topology (isomorphic to a three-ball) around avertex v ∈ C v , and that in this region the connection A is flat. In this case, the expression (2.16)for the flux becomes independent of the system of paths π e , since the flatness of the connectionimplies that there exists an SU(2) element a ( x ) such that A = a d a − and h π e ( x ) = a ( v ) a ( x ) − .Indeed, we have X ( F e ,π e ) = X e = a ( v ) (cid:18)Z F e a ( x ) − E ( x ) a ( x ) (cid:19) a ( v ) − , (2.25)and the dependence on the system of paths has disappeared. Moreover, one can see that the Gausslaw expresses the fact that X iF e = X iF ′ e , for if F e and F ′ e have the same oriented boundary, theirunion encloses a volume ∂C v and we have that:0 = Z C v a ( x ) − d A E ( x ) a ( x ) = Z C v d (cid:0) a ( x ) − E ( x ) a ( x ) (cid:1) = a ( v ) − (cid:0) X F e − X F ′ e (cid:1) a ( v ) . (2.26)In the next section, we are going to make this statement more precise, and study the case of apartially flat connection. III. PARTIALLY FLAT CONNECTION
In this section, we formulate and prove the equivalence between the continuous phase space ofpartially flat geometries and the discrete spin network phase space. In order to do so, we first needto introduce some notions of topology.
Definition 1.
A cellular decomposition ∆ of a space Σ is a decomposition of Σ as a disjoint union(partition) of open cells of varying dimension satisfying the following conditions:i) An n -dimensional open cell is a topological space which is homeomorphic to the n -dimensionalopen ball.ii) The boundary of the closure of an n -dimensional cell is contained in a finite union of cellsof lower dimension.The n -skeleton ∆ n of a cellular decomposition is the union of cells of dimension less than orequal to n . n -skeleton of a cellular decomposition is also a cellular decomposition. In particular,the one-skeleton ∆ of a cellular decomposition is a graph. Let us now suppose that we have agraph Γ embedded in Σ. We need to introduce the notion of a cellular decomposition dual to Γ. Definition 2.
A cellular decomposition ∆ of a three-dimensional space Σ is said to be dual to thegraph Γ if there is a one-to-one correspondence v C v between vertices of Γ and three-cells of ∆ , and a one-to-one correspondence e F e between edges of Γ and two-cells of ∆ , such that:i) There is a unique vertex v inside each three-cell C v .ii) The two-cells F e intersect Γ transversally at one point only, and the intersection belongs tothe interior of the edge e of Γ . In other words, a cellular decomposition dual to Γ is such that each vertex of Γ is dual to athree-cell, and each edge of Γ is dual to a two-cell. Finally, let us consider a pair (Γ , Γ ∗ ) of graphsembedded in Σ. Definition 3.
We say that an embedded graph Γ ∗ is dual to the embedded graph Γ (and vice versa),or that (Γ , Γ ∗ ) forms a pair of dual graphs, if there exists a cellular decomposition ∆ dual to Γ ,whose one-skeleton ∆ is Γ ∗ . From now on, we consider that (Γ , Γ ∗ ) is a pair of dual embedded graphs, and we denote by ∆the cellular decomposition dual to Γ with a one-skeleton ∆ given by Γ ∗ . Notice that if we takeany diffeomorphism Φ o on Σ which does not act on Γ ∗ or the vertices of Γ, we obtain an equivalent cellular decomposition Φ o (∆). Given such a pair of dual graphs, we are going to construct a certainphase space P Γ , Γ ∗ , and prove that it is the continuous analogue of the discrete spin network phasespace P Γ . In fact, we are going to show that there is a symplectomorphism between P Γ , Γ ∗ and P Γ . A. The reduced phase space P Γ , Γ ∗ To define the reduced phase space P Γ , Γ ∗ , we first construct a group F Γ ∗ × G Γ of gauge transforma-tions acting on P . For this, let us consider an infinite-dimensional Abelian group of transformations F Γ ∗ parametrized by Lie algebra-valued one-forms φ i ∈ Ω (cid:0) Σ , su (2) (cid:1) which have the property thatthey vanish on Γ ∗ : φ i ( x ) = 0 , ∀ x ∈ Γ ∗ . (3.1)This group action is Hamiltonian and generated by the curvature constraint F Γ ∗ ( φ ) = Z Σ φ i ∧ F i ( A ) , (3.2)whose action on the continuous phase space P is given by δ F Γ ∗ φ A = (cid:8) A, F Γ ∗ ( φ ) (cid:9) = 0 , δ F Γ ∗ φ E = (cid:8) E, F Γ ∗ ( φ ) (cid:9) = d A φ. (3.3)This constraint enforces the flatness of the connection outside of the one-skeleton graph Γ ∗ . SeeThe second group, G Γ , is the group of gauge transformations parametrized by Lie algebra-valuedfunctions α i ∈ Ω (cid:0) Σ , su (2) (cid:1) which have the property that they vanish on the vertices of Γ: α i ( x ) = 0 , ∀ x ∈ V Γ . (3.4) Since these diffeomorphisms vanish on Γ ∗ , the duality between edges and faces is preserved. G Γ ( α ) = Z Σ α i (d A E ) i , (3.5)whose infinitesimal action on the phase space variables is given by δ G Γ α A = (cid:8) A, G Γ ( α ) (cid:9) = d A α, δ G Γ α E = (cid:8) E, G Γ ( α ) (cid:9) = [ E, α ] . (3.6)From the various Poisson brackets (cid:8) G Γ ( α ) , G Γ ( α ′ ) } = G Γ ([ α, α ′ ]) , (3.7a) (cid:8) G Γ ( α ) , F Γ ∗ ( φ ) } = F Γ ∗ ([ α, φ ]) , (3.7b) (cid:8) F Γ ∗ ( φ ) , F Γ ∗ ( φ ′ ) (cid:9) = 0 , (3.7c)we see that the Hamiltonians (3.2) and (3.5) form a first class algebra.We are interested in the phase space obtained from P by symplectic reduction with respect to F Γ ∗ and G Γ , which we denote by P Γ , Γ ∗ ≡ T ∗ A (cid:12) (cid:0) F Γ ∗ × G Γ (cid:1) = C / (cid:0) F Γ ∗ × G Γ (cid:1) , (3.8)where C ≡ (cid:8) ( A, E ) ∈ T ∗ A| F ( A )( x ) = d A E ( y ) = 0 , ∀ x ∈ Σ \ Γ ∗ , ∀ y ∈ Σ \ V Γ (cid:9) (3.9)is the constrained space. This is the infinite-dimensional space of flat SU(2) connections on ˜Σ ≡ Σ \ Γ ∗ , and fluxes satisfying the Gauss law outside of V Γ . Once we divide this constrained spaceby the action of the two gauge groups introduced above, we obtain the finite-dimensional orbitspace P Γ , Γ ∗ [14]. We are going to prove that P Γ , Γ ∗ is the continuous analogue of the discrete spinnetwork phase space P Γ .Let us start by constructing a three-dimensional cellular decomposition of the region. Sincewe have chosen Γ ∗ to be the one-skeleton ∆ of the cellular decomposition ∆ of Σ, the cellulardecomposition of ˜Σ is simply given by ˜∆ ≡ ∆ \ ∆ . Explicitly, the decomposition ˜∆ can be writtenas ˜∆ = [ v C v [ e F e , (3.10)where C v are three-dimensional open cells labeled by the vertices v ∈ Γ, and F e are two-dimensionalopen cells labeled by the edges e ∈ Γ. We would like to solve the curvature constraint F ( A ) =d A A = 0 on ˜Σ and the Gauss constraint d A E = 0 on Σ \ V Γ . We start by solving them for eachthree-dimensional cell C v .To solve the curvature constraint, let us define on a three-cell C v a group-valued map a v ( x ) : C v −→ SU(2) as the path-ordered exponential a v ( x ) ≡ −→ exp Z vx A, (3.11)where the integration can be taken over any arbitrary path from the point x ∈ C v to the vertex v because the connection is flat and C v is simply connected. By construction, this map is such that3 a v ( v ) = 1. This allows us to reconstruct on C v the flat connection A as A ( x ) = a v ( x )d a − v ( x ) . (3.12)The second constraint to satisfy is the Gauss law outside of the vertex v which lies inside the cell C v . Because the connection is flat, the covariant derivative of the electric field E can be written asd A E = dE + (cid:2) a v d a − v , E (cid:3) = a v d (cid:0) a − v Ea v (cid:1) a − v = a v d X v a − v , (3.13)where we have introduced the Lie algebra-valued two-form field X v ( x ) ≡ a v ( x ) − E ( x ) a v ( x ) . (3.14)Therefore, we see that the Gauss law implies that the two-form X v is closed outside of v sinced X v ( x ) = a v ( x ) − d A E ( x ) a v ( x ) = 0 , ∀ x ∈ C v − { v } . (3.15)The electric field can now easily be reconstructed since we have E ( x ) = a v ( x ) X v ( x ) a v ( x ) − . (3.16)One can conclude that a general solution of the two constraints F ( A ) = d A A = 0 and d A E = 0 on C v and C v − { v } respectively, is given in terms of a Lie algebra-valued closed two-form X v and agroup element a v : C v −→ SU(2), the connection and flux fields being given by (3.12) and (3.16).Now we can extend this solution to the whole space ˜Σ by gluing consistently the solutions on eachcell. We have labeled the three-dimensional cells C v with vertices of the graph Γ. Consequently,the two-dimensional cells F e , labeled by edges e = ( v v ) of Γ connecting two vertices (such that s ( e ) = v and t ( e ) = v ), are obtained by intersecting two three-dimensional cells as F e = C v ∩ C v , (3.17)where the bar denotes the closure of the cell. We assume that the two-dimensional cells F e areoriented, and that their orientation is reversed when we change the orientation of the edge e .Demanding that the connection and flux fields be continuous across the two-dimensional cellsamounts to assuming that there exists, for each F e , an SU(2) element h e such that a v ( x ) = a v ( x ) h e , X v ( x ) = h − e X v ( x ) h e , (3.18)for x ∈ F e . Notice that the first equality can be written as h e ( A ) = a s ( e ) ( x ) − a t ( e ) ( x ) = −→ exp Z e A, (3.19)where x is any point on the two-cell F e , and once again the definition does not depend on x becausethe connection is flat. By construction, one can see that under an orientation reversal we have h e − = h − e .This construction shows that the constrained space C is isomorphic to the data ( a v , X v , h e ),subject to the conditions (3.18). We are now interested in the quotient of this constrained spaceby the gauge group F Γ ∗ × G Γ . Elements of this gauge group are pairs (cid:0) φ ( x ) , g o ( x ) (cid:1) , where φ isa Lie algebra-valued one-form which vanishes on Γ ∗ , and g o is an element of SU(2) (obtained byexponentiation of α ) fixed to the identity of the group at the vertices V Γ . The action of F Γ ∗ × G Γ on the pair ( A, E ) ∈ P translates on the constraint surface C into an action on the data ( a v , X v , h e )4given by a v ( x ) −→ g o ( x ) a v ( x ) , X v ( x ) −→ X v ( x ) + d (cid:0) a v ( x ) − φ ( x ) a v ( x ) (cid:1) , h e −→ h e . (3.20)Following (2.16), let us compute the flux X e across a surface dual to an edge e which is such that s ( e ) = v . It is given by X e = Z F e h π e ( x ) E ( x ) h π e ( x ) − = Z F e a v ( v ) a v ( x ) − E ( x ) a v ( x ) a v ( v ) − = Z F e X v , (3.21)where we have used the fact that a v ( v ) = 1. We see that the observables which are invariant underthis gauge transformation are simply given by the holonomies h e and the fluxes X e . B. The symplectomorphism between P Γ , Γ ∗ and P Γ Now we come to our main result, which is the symplectomorphism between the continuous phasespace P Γ , Γ ∗ and the discrete spin network phase space P Γ . Let us construct a map between theconstrained continuous data in C (see (3.9)) and discrete data on the spin network phase space P Γ ,and denote it by I : C −→ P Γ ( A, E ) (cid:0) h e ( A ) , X e ( A, E ) (cid:1) . (3.22)For this, we define for every three-cell C v a group-valued map a v : C v −→ SU(2) such that a v ( v ) = 1and a Lie algebra-valued two-form X v : C v −→ Ω (cid:0) C v , su (2) (cid:1) closed outside of the vertices of Γ.Given these fields, we can reconstruct on C v the connection and the two-form field using A ( x ) = a v ( x )d a v ( x ) − , E ( x ) = a v ( x ) X v ( x ) a v ( x ) − . (3.23)The map I is then defined by h e ( A ) ≡ −→ exp Z e A = a s ( e ) ( x ) − a t ( e ) ( x ) , (3.24a) X e ( A, E ) ≡ Z F e h π e ( x ) E ( x ) h π e ( x ) − = Z F e X s ( e ) ( x ) , (3.24b)where in the definition of h e , x is any point on the two-cell F e , and once again the definition doesnot depend on x because the connection is flat. To compute the holonomy h e , we have used thegroup elements a s ( e ) ( x ) and a t ( e ) ( x ) to define the connection on the two cells dual to the vertices s ( e ) and t ( e ) respectively.It is possible to use equation (3.24b) to write down the relationship between the discrete andcontinuous Gauss laws. We already know from (3.15) that the Gauss law is equivalent to therequirement that the two-form X v be closed outside of the vertex v . We can now write that Z C v a v ( x ) − d A E ( x ) a v ( x ) = Z C v d X v = Z ∪ e F e = ∂C v X s ( e ) = X e | s ( e )= v X e = G v , (3.25)which relates the continuous and discrete constraints. This shows that the violation of the contin-uous Gauss constraint is located at the vertices of Γ, and given by a distribution determined by5the discrete Gauss constraint: d A E ( x ) = X v ∈ V Γ G v δ ( x − v ) . (3.26)Since the map I is invariant under the gauge transformations F Γ ∗ × G Γ we can write it as amap [ I ] : P Γ , Γ ∗ −→ P Γ . We will now show that this map is not only invertible, but also a symplectomorphism.
Proposition 1.
The map [ I ] : P Γ , Γ ∗ −→ P Γ defined by (3.24) is a symplectomorphism, and isinvariant under the action of diffeomorphisms connected to the identity preserving Γ ∗ and the set V Γ of vertices of Γ . We are going to prove this proposition in the remainder of this work. Before doing so, let usstress that this result implies the existence of an inverse map which allows one to reconstruct fromthe discrete data an equivalence class [ A ( h e ) , E ( h e , X e )] of continuous configurations satisfying thecurvature and Gauss constraints (i.e. configurations in the constrained space C ). Explicitly, thisequivalence class is defined with respect to the equivalence relation( A, E ) ∼ (cid:0) g o ⊲ A, g − o ( E + d A φ ) g o (cid:1) , (3.27)where once again φ is a Lie algebra-valued one-form vanishing on Γ ∗ , and g o is an element of SU(2)fixed to the identity of the group at the vertices V Γ .Evidently, Proposition 1 implies a similar proposition for the gauge-invariant phase spaces.Indeed, if one defines P G Γ , Γ ∗ ≡ T ∗ A (cid:12) (cid:0) F Γ ∗ × G (cid:1) = C G / (cid:0) F Γ ∗ × G (cid:1) , (3.28)where C G ≡ (cid:8) ( A, E ) ∈ T ∗ A| F ( A )( x ) = d A E ( y ) = 0 , ∀ x ∈ Σ \ Γ ∗ , ∀ y ∈ Σ (cid:9) , (3.29)and G = C ∞ (cid:0) Σ , SU(2) (cid:1) is the group of full SU(2) gauge transformations, we have the symplec-tomorphism P G Γ , Γ ∗ = P G Γ between the continuous and discrete gauge-invariant phase spaces. Thisfollows directly from Proposition 1, and the fact that G = G Γ × G Γ , where G Γ is the group ofdiscrete gauge transformations acting at the vertices v ∈ V Γ only.Notice that when we act with the full group G of SU(2) transformations, the holonomies h e andthe fluxes X e clearly become gauge-covariant, i.e. satisfies I ( g ⊲ A, g ⊲ E ) = g ⊲ I ( A, E ). Indeed,since the group element g is not fixed to the identity at the vertices v anymore, we have g ⊲ a v ( x ) = g ( x ) a v ( x ) g ( v ) − , and therefore the definition (3.18) tells us that we have g ⊲ h e = g v h e g − v , where e is an edge of Γ connecting the vertices v and v . C. The symplectic structures
In this subsection we use the map (3.24) to prove the equivalence of the symplectic structures onthe continuous and discrete spaces P Γ , Γ ∗ and P Γ . We know that the spaces P and P Γ are symplecticmanifolds, their symplectic structures being given by (1.5) and (2.4) respectively. Since the space P Γ , Γ ∗ has been obtained from P by symplectic reduction, the Marsden-Weinstein theorem ensures6that it also carries a symplectic structure. We are now going to show that the symplectic structureson the spaces P Γ , Γ ∗ and P Γ are in fact identical.Let us start with the symplectic potential coming from the first order formulation of gravity. Itis given by Θ = 12 Z Σ Tr ( ⋆ ( e ∧ e ) ∧ δA ) = Z Σ Tr ( E ∧ δA ) , (3.30)where ⋆ denotes the Hodge duality map in the Lie algebra su (2). We first use the cellular decom-position ∆ to evaluate this symplectic potential on the set of partially flat connections and writeΘ = X v Z C v Tr (cid:0) E ∧ δ (cid:0) a v d a − v (cid:1)(cid:1) (3.31a)= X v Z C v Tr (cid:0) X v ∧ d (cid:0) δa − v a v (cid:1)(cid:1) (3.31b)= X v Z ∂C v Tr (cid:0) X v δa − v a v (cid:1) − X v G v δa − v a v ( v ) (3.31c)= X v Z ∂C v Tr (cid:0) X v δa − v a v (cid:1) , (3.31d)where we have used the identity δ (cid:0) a v d a − v (cid:1) = a v d (cid:0) δa − v a v (cid:1) a − v , the definition (3.14) of the two-form field X v , and the fact that d X v = G v δ ( x − v ) (see equation(3.26)). The last equality followsfrom the condition a v ( v ) = 1, which implies δa v ( v ) = 0. The summation over three-cells dual tothe vertices v can be rearranged as a sum over two-cells dual to the edges e , which givesΘ = X e Z F e h Tr (cid:16) X s ( e ) δa − s ( e ) a s ( e ) (cid:17) − Tr (cid:16) X t ( e ) δa − t ( e ) a t ( e ) (cid:17)i . (3.32)Now we can use the condition (3.18) of compatibility of the group elements a v across the edges torewrite the second term and obtainΘ = X e Z F e h Tr (cid:16) X s ( e ) δa − s ( e ) a s ( e ) (cid:17) − Tr (cid:16) h − e X s ( e ) h e δ (cid:16) h − e a − s ( e ) (cid:17) a s ( e ) h e (cid:17)i . (3.33)Finally, we can expand the last term to find the resultΘ = − X e Z F e Tr (cid:0) h − e X s ( e ) h e δh − e h e (cid:1) = X e Tr (cid:0) X e δh e h − e (cid:1) . (3.34)This is exactly the symplectic potential associated to | E Γ | copies of the cotangent bundle T ∗ SU(2).It shows that the symplectic structure of the spin network phase space is equivalent to that of firstorder gravity evaluated on the set of partially flat connections. In particular, since the symplecticforms are invertible by definition, this proves that the continuous phase space P Γ , Γ ∗ is indeedfinite-dimensional and isomorphic to P Γ .7 D. Action of diffeomorphisms
Now we prove the second point of Proposition 1, which concerns the invariance of the symplecto-morphism under a certain class of diffeomorphisms. The isomorphism I : P Γ , Γ ∗ −→ P Γ dependson a choice of cellular decomposition ∆ dual to Γ with one-skeleton ∆ = Γ ∗ . DiffeomorphismsΦ ∈ Diff(Σ) act naturally on the continuous phase space P Γ , Γ ∗ by A Φ ∗ A and E Φ ∗ E .Let us start by choosing a particular diffeomorphism Φ o which preserves the graph Γ ∗ and thevertices V Γ inside the cells C v , and is connected to the identity . Because the connection is flat on˜Σ, the holonomy h e ( A ) is independent of the choice of path between s ( e ) and t ( e ) as long as anytwo paths are in the same homotopy class of ˜Σ. The edges e and Φ o ( e ) are in the same homotopyclass if Φ o is connected to the identity and not moving Γ ∗ . Then it is clear that we have h e (Φ ∗ o A ) = h Φ o ( e ) ( A ) = h e ( A ) . (3.35)Similarly, the action of Φ o on the group element a v ( x ) maps it to a v (cid:0) Φ o ( x ) (cid:1) . This implies that thetwo-form X v defined by (3.14) satisfies X v (cid:0) Φ o ( x ) (cid:1) = Φ ∗ o X v ( x ). Recall from the definitions of thecellular decomposition that each face F e is bounded by links in the one-skeleton Γ ∗ . Now, sinceΦ o does not move the graph Γ ∗ , we have that ∂F e = ∂ (cid:0) Φ o ( F e ) (cid:1) ∈ Γ ∗ , and therefore F e ∪ Φ o ( F e )encloses a volume, which furthermore does not contain any vertices of Γ. Thus, by virtue of (2.26)and (3.21), we have that X e (Φ ∗ o A, Φ ∗ o E ) = X e ( A, E ) . (3.36)Relations (3.35) and (3.36) together show that I ◦ Φ o = I .We can give another very elegant proof of the invariance of the map I under the diffeomorphismsΦ o . For this, recall that given a vector field ξ a , a diffeomorphism acts on the connection like L ξ A = d( ι ξ A ) + ι ξ d A = ι ξ F + d A ( ι ξ A ) , (3.37)and on the electric field like L ξ E = d( ι ξ E ) + ι ξ d E = ι ξ d A E + d A ( ι ξ E ) + [ E, ι ξ A ] , (3.38)where ι denotes the interior product. Now, if the data ( A, E ) is on the constraint surface C , thecurvature vanishes outside of Γ ∗ , while d A E vanishes outside of the set V Γ of vertices. Therefore,if we consider a vector field ξ a which vanishes on Γ ∗ and on V Γ , we see that the action of dif-feomorphisms is a combination of flat transformations (3.3) and gauge transformations (3.6) withfield-dependent parameters of transformation, i.e. L ξ A = δ F Γ ∗ ι ξ E A + δ G Γ ι ξ A A, L ξ E = δ F Γ ∗ ι ξ E E + δ G Γ ι ξ A E. (3.39)We can write this more succinctly as simply L ξ = δ F Γ ∗ ι ξ E + δ G Γ ι ξ A . (3.40)Now, since the holonomy and flux variables are invariant under the flatness and gauge transforma-tions, such diffeomorphisms vanish on the variables ( h e , X e ). This means that there exists a smooth one-parameter family of diffeomorphism Φ t such that Φ t =0 = id andΦ t =1 = Φ o . IV. GAUGE CHOICES FOR THE ELECTRIC FIELD
Now that we have established the isomorphism between P Γ and the continuous phase space P Γ , Γ ∗ ,we have a correspondence between discrete geometries and an equivalence class of continuous ge-ometries related according to (3.27) by group gauge transformations and translations. Up to groupgauge transformations, the holonomy uniquely determines a choice of connection. For the E field,however, the story is different since even after we have performed a group gauge transformation,there is still a huge ambiguity coming from the transformation E → E + d A φ on the continuouselectric field determined by the fluxes. It is clear that in order to construct a continuous fieldconfiguration starting from the discrete data, one has to specify which continuous field represen-tative to pick in the particular equivalence class determined by the discrete data. In other words,a choice of a representative in this equivalence class is a choice of gauge. More precisely, we havethe following definition: Definition 4.
A choice of gauge is a map from the discrete data to the continuous phase space, T : P Γ −→ C ( h e , X e ) ( A, E ) , (4.1) which is the inverse of I in the sense that I ◦ T = id . (4.2) We say that a gauge fixing is diffeomorphism-covariant if Φ ∗ T is equal to the map T defined onthe graphs Φ(Γ) and
Φ(Γ ∗ ) , for any diffeomorphism Φ : Σ −→ Σ . In other words, choosing a gauge amounts to giving a prescription for reconstructing continuousfields A ( h e ) and E ( X e , h e ) starting from the discrete data, such that (4.2) holds, i.e. h e (cid:0) A ( h e ) (cid:1) = h e , X e (cid:0) A ( h e ) , E ( X e , h e ) (cid:1) = X e . (4.3)Note that a gauge fixing T is a right inverse for I , while the reverse is not true. The map T ◦ I isnot the identity, it just maps a continuous configuration (
A, E ) that solves the Gauss and curvatureconstraints into another gauge-equivalent configuration which satisfies the gauge choice.As we have already seen, at the continuous level a flat connection on ˜Σ is determined on everycell C v by a group element a v ( x ). Locally, it is always possible to perform a gauge transformationthat sends this element to the identity of the group, and thereby construct a trivial connection. Ifwe pick two neighboring cells C v and C v such that the vertices v and v bound the edge dual tothe face F e = C v ∩ C v , the relevant gauge-invariant information about the connection is encodedin the transition group element h e .For the electric field, there is more gauge freedom since the variable E can be acted upon by both F Γ ∗ and G Γ . Therefore, there is a priori a huge ambiguity in the choice of gauges that one can chooseto reconstruct the continuous data. This means that knowledge of the fluxes does not accuratelydetermine the geometry of space, but only a family of geometries that are gauge-equivalent undertranslations of the type E E + d A φ .However, there is a powerful way in which we can restrict the gauge choices that are avail-able. This can be done by asking that a gauge choice transforms covariantly under the actionof diffeomorphisms. A diffeomorphism Φ of Σ acts on the continuous data in the usual manner( A, E ) (cid:0) Φ ∗ A, Φ ∗ E (cid:1) . The same diffeomorphism also acts on the discrete data ( h e , X F e ) as (cid:0) h Φ( e ) , X Φ( F e ) (cid:1) . Note that here we have made explicit the fact that the flux field X e depends on9Γ ∗ via the choice of a surface F e whose boundary is supported on Γ ∗ . A gauge choice is said to becovariant if this action of the diffeomorphisms commutes with the gauge map T .If we restrict ourselves to gauge choices that are covariant under the action of diffeomorphisms,the ambiguity in the gauge choices is dramatically resolved, and there are only a few choicesavailable. In the following we present two such gauge choices . First, the singular gauge choice inwhich the electric field E vanishes outside of Γ, and then the flat gauge in which E is flat outsideof Γ ∗ . It is remarkable that these two gauge choices correspond to the two main interpretationsof the fluxes used in the literature. In loop quantum gravity one usually interprets the E field ashaving support only on Γ since the corresponding operator acting on a spin network state gives b E ( x ) | ψ i = 0 for x Γ. On the other hand, the spin foam literature usually interprets the E fieldas being flat outside of Γ ∗ . Our analysis shows that these two pictures are not contradictory, butthat they correspond to two different covariant gauge choices underlying the same discrete data.Now we want to emphasize that the restriction on the gauge choices coming from the require-ment of covariance under diffeomorphisms is the analog of the so-called uniqueness theorem of thequantum representation of the holonomy-flux algebra [29]. This theorem states that there is aunique diffeomorphism-covariant gauge choice, which corresponds to the singular gauge in which E has support on the graph Γ only and vanishes on Γ ∗ . In this singular gauge, which we refer toas the LQG gauge, the electric field E vanishes outside of the graph Γ dual to the triangulation ∆.This can be written as E | i = 0, where the vacuum state | i is the state of no geometry. Indeed,in LQG excitations of quantum geometry have support on the graph Γ only. Therefore, in all theregions of ∆ outside of Γ, there is simply no geometry, and the electric field vanishes. We are goingto give below an explicit construction of the continuous singular electric field.The key observation is that there is another legitimate choice of representative configuration inthe equivalence class (3.27) of continuous geometries which respects the diffeomorphism symmetry.As we already said, it is given by the flat gauge. At the quantum level, this corresponds to achoice of a vacuum state | F i in which the curvature vanishes. This corresponds to the flat, or spinfoam gauge, in which we have F ( A ) | F i = 0. This diffeomorphism-invariant vacuum is differentfrom the one singled out by the LOST theorem [29] (which obviously corresponds to the singulargauge) and it would be interesting to investigate further its properties. What should be noted isthat such a vacuum state appears naturally in our context and that it corresponds to the spinfoam description. It can be seen as the dual of the singular gauge, in the sense that it defines aflat geometry within the cells C v , with a non-vanishing electric field E on the dual graph Γ ∗ . Aswe will see in more detail, the availability of this gauge clearly shows that it is possible to define alocally flat geometry without necessarily having a triangulation with straight edges and flat faces.In Regge geometries [7], the extrinsic curvature is concentrated along the one-skeleton ∆ of thetriangulation, but in the present construction, the edges of Γ ∗ are not necessarily straight.Here we have drawn a parallel between a choice of gauge at the classical level and a choice of avacuum state at the quantum level. It would be interesting to develop this analogy further.In the remainder of this section, we are going to study in more detail the singular and flatgauges for the electric field. Our goal is to study the gauge freedom for the basic variables on thecontinuous phase space, and to construct explicitly the electric field as a functional of the discretevariables h e and X e . We conjecture these are the only two possible gauge choices, but a detailed investigation of this is still needed. A. Singular gauge
The singular gauge is a gauge in which the electric field E vanishes outside of the graph Γ. In thissection, we show by an explicit construction that it always possible to make such a gauge choice.More precisely, we construct explicitly continuous fields A ( h e ) and E S ( h e , X e ) which are such that E S ( x ) = 0 if x / ∈ Γ, and which satisfy the property I ( A, E S ) = ( h e , X e ) under the action of themap (3.24).In order to prove this, let us first introduce the following form: ω ( x, y ) ≡ ω i ( x − y ) ǫ ijk d x j ∧ d y k , with ω i ( x ) ≡ π x i | x | . (4.4)This object is a (1,1)-form, i.e. a one-form in x , and a one-form in y . This form satisfies a keyproperty, which is summarized in the following lemma. Lemma 1.
There exists an α ( x, y ) which is a (2,0)-form (i.e. a two-form in x and a zero-form in y ), such that d x ω ( x, y ) + d y α ( x, y ) = δ ( x, y ) , (4.5) where d x ≡ d x i ∂ x i , and δ ( x, y ) is the distributional (2,1)-form δ ( x, y ) = δ ( x − y ) ǫ ijk d x i ∧ d x j ∧ d y k (4.6) vanishing outside of x = y .Proof. First, it is straightforward to show that ∂ i ω i ( x ) = 0 for x = 0. Moreover, it is possible toshow by a direct computation in spherical coordinates that Z S ε ω i ( x ) ǫ ijk d x j ∧ d x k = 2 , (4.7)where S ε is a sphere of radius ε . Since this integral is also equal to2 Z B ε ∂ i ω i ( x )d x, (4.8)where B ε is the ball of radius ε , we obtain that ∂ i ω i ( x ) = δ ( x ). By a direct computation we cannow get that d x ω ( x, y ) = s k ∧ d y k (cid:0) ∂ i ω i (cid:1) ( x − y ) − s j ∧ d y k (cid:0) ∂ k ω j (cid:1) ( x − y ) , (4.9)with s i = 18 π ǫ ijk d x j ∧ d x k . (4.10)The lemma is therefore established by introducing α ( x, y ) ≡ ω i ( x − y ) s i . (cid:4) Given this lemma, it is now a straightforward task to construct a singular flux field. For this,we first construct a flat connection A on ˜Σ following the construction of subsection III A, and then1we define the singular flux field as E S ( x ) ≡ d A X e ∈ Γ h π e ( x ) − X e h π e ( x ) Z e ( y ) ω ( x, y ) ! . (4.11)The integral entering this definition is a one-dimensional integral over the edge e parametrized bythe variable y , which implies that the term inside the parenthesis is a one-form in x .The proof that this flux satisfies all the desired requirements is straightforward. First, it isobvious that the Gauss law d A E S = 0 is satisfied on Σ \ Γ ∗ since d A = F ( A ) = 0 on this space.Moreover, using the previous lemma and the definition of the holonomy, we can compute explicitlythe covariant derivative: E S ( x ) = X e h π e ( x ) − X e h π e ( x ) (cid:18) δ e ( x ) − α (cid:0) x, s ( e ) (cid:1) + α (cid:0) x, t ( e ) (cid:1)(cid:19) , (4.12)where δ e ( x ) ≡ Z e ( y ) δ ( x, y ) . (4.13)The last two terms in (4.12) can be reorganized in terms associated with the vertices to find E S ( x ) = X e h π e ( x ) − X e h π e ( x ) δ e ( x ) − X v α ( x, v ) h v ( x ) − X e | s ( e )= v X e − X e | t ( e )= v h − e X e h e h v ( x ) , (4.14)where h v ( x ) is the holonomy going from the vertex v to the point x . Now the last term vanishesdue to the discrete Gauss law (2.6). Therefore, we finally find that the singular electric field is E S ( x ) = X e h π e ( x ) − X e h π e ( x ) δ e ( x ) . (4.15)This electric field is obviously vanishing outside of Γ, and is such that X e ( A, E S ) = X e . It isinteresting to note that the integral of the two-form α ( x, y ) along S , Z S α ( x, y ) = 18 π Z S ω i ( x − y ) ǫ ijk d x j ∧ d x k , (4.16)is simply the solid angle of S as viewed from y divided by 4 π . B. Flat cell gauge
The flat cell gauge is a choice of electric field with vanishing intrinsic and extrinsic curvature withinthe cells, i.e. with the scalar curvature R ≡ d Γ Γ = 0 and the extrinsic curvature K = 0 in each cell C v . Note that R and K are not necessarily zero on the faces F e and their boundaries ∂F e . Thisgauge choice requires that we be within the SU(2)-invariant phase space.We are about to prove that it is always possible to find a gauge transformation, generated bythe flatness constraint, which takes an arbitrary electric field E ∈ P G Γ , Γ ∗ to a flat electric field ¯ E .In the following we assume the frame field e is invertible.Let us begin with two lemmas: Lemma 2.
Extrinsic curvature is zero if and only if the frame field is torsion-free. Proof.
Torsion is given by d A e = d e + [Γ + γK, e ] = γ [ K, e ] , (4.17)where by definition, the spin connection Γ is the solution to d e + [Γ , e ] = 0. This equation showsthat K = 0 implies d A e = 0. To show that the reverse is also true, we use (4.17) in index-form towrite 1 γ ǫ abc D b e ic = ǫ abc ǫ ijk K jb e kc = (det e ) K jb ( e ai e bj − e aj e bi ) , (4.18)where D a is the covariant exterior derivative in index notation and we used the identity ǫ abc ǫ ijk e kc ≡ (det e )( e ai e bj − e aj e bi ) in the second equality. Contracting both sides of this equation with e ia leadsto an equation on the trace of the extrinsic curvature: K aa = 12 γ (det e ) ǫ abc e ia D b e ci . (4.19)Now using (4.19) in (4.18), we find K ia = 1 γ (det e ) (cid:18) e ia e jd − e id e ja (cid:19) ǫ dbc D b e cj . (4.20)This shows that d A e = 0 implies K = 0 and establishes the proof. (cid:4) Lemma 3.
A flat connection, together with vanishing extrinsic curvature, imply that intrinsiccurvature is zero.Proof.
Using the definition A ≡ Γ + γK of the Ashtekar-Barbero connection, we can write itscurvature as F ( A ) = R + γ d Γ K + γ K, K ] . (4.21)Setting F ( A ) = 0 and K = 0 implies that R = 0. (cid:4) We showed previously (see (3.12) and (3.16)) that the gauge-invariant fields (
A, E ) are writtenin each cell C v in general as A = a v d a − v , E = a v d Z v a − v , (4.22)where we have used a Lie algebra-valued one-form Z v ∈ Ω (cid:0) C v , su (2) (cid:1) to write X v ( x ) = d Z v ( x ).Since a flat triad must be torsion-free by the above lemmas, we can similarly write a flat triad ingeneral as ¯ e = a v d z v a − v , (4.23)for some Lie algebra-valued function z v ∈ Ω (cid:0) C v , su (2) (cid:1) . The function z v provides a set of flatcoordinates in C v . Requiring the triad to be invertible places the following condition on z v : ǫ abc ǫ ijk ∂ a z iv ∂ b z jv ∂ c z kv > . (4.24)The electric field constructed from this triad is given by¯ E = a v [d z v , d z v ] a − v . (4.25)3Consider that we are given a pair ( a v , Z v ) defining an electric field E within a cell C v . Lookingat (3.3), we seek a gauge field φ v ∈ Ω (cid:0) C v , su (2) (cid:1) such thatd A φ v = ¯ E − E = a v ([d z v , d z v ] − d Z v ) a − v . (4.26)Using a − v d A φ v a v = d( a − v φ v a v ), we can solve for φ v to obtain φ v = a v ([ z v , d z v ] − Z v + d g v ) a − v , (4.27)for g v ∈ Ω (cid:0) C v , su (2) (cid:1) . For x ∈ Γ ∗ the value of g v ( x ) is fixed up to an overall constant by thecondition φ v ( x ) = 0: g v ( x ) = Z xs ( e ) ( Z v − [ z v , d z v ]) , (4.28)where the integration is along a link ℓ in the boundary ∂F e of the face F e .We have shown the existence of a gauge field φ v taking us from an arbitrary electric field E ∈ P G Γ , Γ ∗ to a flat electric field with vanishing intrinsic and extrinsic curvature in a cell. The nextquestion to ask is whether this choice unique. Since g v is fixed only on Γ ∗ (and even there only upto a constant), and it is not fixed in C v or the faces F e , there are many choice of φ v which give thetransformation E → ¯ E . Moreover, any z v satisfying (4.24) gives a flat, invertible triad, so there isnot even a unique choice of flat electric field. Therefore the transformation to the flat cell gauge isnot unique.Having found a gauge transformation to a flat electric field in a single cell, we now consider howthis transformation affects the geometry at cell boundaries when performing this transformationin all cells of the cellular decomposition. Consider a region C ≡ C v ∪ C v formed by two cellsand their boundaries. The requirement of continuity of E and ¯ e at the face F e = C v ∩ C v givesconditions at the face:lim y → x + d z ( y ) = lim y → x − h − e d z ( y ) h e , lim y → x + d Z ( x ) = lim y → x − h − e d Z ( x ) h e , (4.29)for x ∈ F e and y ∈ C , where y → x + is used to indicate that the coordinate y approaches x fromwithin C v , and y → x − indicates that y approaches x from within C v . Using these relations andrequiring φ ( x ) to vanish on the boundary of the face adds another condition:lim y → x + d g ( x ) = lim y → x + h − e d g ( x ) h e . (4.30)Together, these relations imply that φ ( x ) = φ ( x ), so that the gauge field is continuous acrossfaces.Let us look more closely at the extrinsic curvature on the face F e . In Lemma 2 we showed thatvanishing torsion implies zero extrinsic curvature. What is the torsion at the face in the flat cellgauge? To answer this question we zoom in to a small neighborhood of C which contains the faceso that we can define a local cartesian coordinate system where y is perpendicular to the face, y µ for µ = 1 , y = 0 on the face. In this neighborhood we definea one-form: d z ( y ) = d z ( y ) + Θ( y ) (cid:0) h − e d z ( y ) h e − d z ( y ) (cid:1) , (4.31)where Θ( y ) is a step function whose value is 0 for y ∈ C v , and 1 for y ∈ C v . The torsion is given4by: d A ¯ e ( y ) = d A (cid:0) a ( y )d z ( y ) a ( y ) − (cid:1) = a ( y )dd z ( y ) a ( y ) − . (4.32)Now, the right hand side of this equation is zero away from the face, but more scrutiny is requiredat the face. Since ∂ Θ( y ) = δ ( y ), we see explicitly that ∂ ∂ µ z ( y ) − ∂ µ ∂ z ( y ) = 0 when y = 0, andtherefore d A ¯ e = 0 at the face. Considering equation (4.20), this leads to a non-vanishing extrinsiccurvature at the faces of the cellular decomposition.Finally, we close this section with a reconstruction of the flux elements X e starting from theflat frame field ¯ e . The flux elements in this gauge are given by the simple form: X ie = 12 ǫ ijk Z F e h π e ¯ e j ∧ ¯ e k h − π e = 12 ǫ ijk Z F e d z jv ∧ d z kv = 12 ǫ ijk Z ∂F e z jv d z kv , (4.33)where we have used the fact that h π e = a − v . C. Regge geometries
The previous calculation shows that we can think of the phase space P Γ as the phase space ofpiecewise (metric) flat geometries on Σ \ Γ ∗ . Such geometries possess an invertible locally flatmetric, with extrinsic curvature concentrated on the faces of the cellular decomposition. Thisdescription is reminiscent of Regge geometries. However, it is known that the phase space of loopgravity is bigger than the phase space of Regge geometry [8]; Regge geometries appear only as aconstrained subset. This fact has triggered the search for the proper geometrical interpretation ofthe loop gravity phase space, for instance in terms of twisted geometries [6].We can now clearly understand the key difference between the phase space of loop gravity andthat of Regge geometries. In the flat gauge, the loop gravity phase space corresponds to a cellulardecomposition of the spatial manifold Σ where the extrinsic curvature is zero within each three-cell C v but non-zero on the faces F e . The faces do not need to be flat two-surfaces, and may bearbitrarily curved so long as they do not self-intersect and only intersect with other faces alongcommon boundaries. The difference between this setting and a Regge geometry is the arbitrarinessin the shape of the faces; the faces are all flat in a Regge geometry.In order to see how the loop gravity phase space (in the flat gauge) may be reduced to a Reggegeometry, we must ask how can the faces be made flat? A necessary condition for a face F e to beflat is that the boundary ∂F e is composed of flat links. Since Γ ∗ is the union of all face boundaries,Γ ∗ must consist entirely of flat links in order to obtain a Regge geometry.Let us go back to the formula for the fluxes that we have derived in the previous subsection: X ie = 12 ǫ ijk Z ∂F e z jv ∧ d z kv , (4.34)where z v is the flat coordinate in the cell C v . One sees that if the links ℓ ∈ ∂F e are chosen to beflat, then z v is linear and d z v is constant over ∂F e . This simplifies the expression drastically. Recallthat due to the Gauss law, the fluxes are independent of the choice of faces (2.26) for fixed Γ ∗ . Thismeans that (4.34) is independent of the choice of face, so that we obtain the same flux whether theface is chosen curved or flat, so long as the boundary of the face is composed of flat links. Indeed,a Regge geometry is given by a unique set of link lengths which can be reconstructed from thefluxes and dihedral angles between links, independently from the choice of faces. Imposing that Γ ∗ be composed entirely of flat links implies that the fluxes can be constructed, using (4.34), entirelyin terms of a discrete piecewise flat geometry `a la Regge.5In the twisted geometries construction [6] the geometry is seen as flat polyhedra glued togetheralong faces. While two faces that are glued together have the same area, they may generally havedifferent shapes. This means the metric is discontinuous across faces, although it is still possible todefine a spin connection [30]. The reduction to a Regge geometry is done using gluing constraints[8]. These constraints impose that the shapes match by enforcing that corresponding dihedralangles on the face boundaries agree.In our cellular decomposition there is only one face between neighboring cells, so there is nonotion of pairs of faces that must be made to fit together. Once the links of Γ ∗ are made flat,the gluing constraints are automatically satisfied by construction. This means that the set ofholonomies and fluxes on a graph can be implemented as a piecewise flat geometry on Σ \ Γ ∗ bymaking a particular gauge choice, and corresponds to a Regge geometry if we impose the additionalconstraint that the edges of Γ ∗ are straight with respect to the flat structure . The phase spaceof full loop gravity then corresponds to piecewise geometries where this additional restriction isnot imposed. In other words, the edges of Γ ∗ do not have to be flat when mapping from the loopgravity phase space to the continuous phase space using the flat cell gauge. D. Cotangent bundle
The result of our construction is that after a choice of gauge, we can express the elements of P Γ asa connection A and an su (2)-valued frame field e , which are solutions to F ( A )( x ) = 0 , d A e ( x ) = 0 , ∀ x ∈ Σ \ Γ ∗ . (4.35)Since δF ( A ) = d A δA , this is nothing but the cotangent bundle of the space of flat SU(2) connectionson Σ \ Γ ∗ . That is P Γ = T ∗ M Γ ∗ , (4.36)where M Γ ∗ denotes the moduli space of flat connections modulo gauge transformations. This meansthat at the quantum level we can represent the quantization of holonomies and fluxes in terms ofoperators acting on holonomies of flat connections. This interpretation has already proposed byBianchi in [9]. It is interesting to note that this is reminiscent of the geometry considered byHitchin in [31]. E. Diffeomorphisms and gauge choices
We have seen in subsection III D that diffeomorphisms Φ o connected to the identity that do notmove Γ ∗ and the vertices of Γ leave the construction of the holonomy-flux algebra invariant. Wehave also seen in the beginning of this section that the singular gauge and the flat gauge arediffeomorphism covariant. In general, the construction of h e and X F e depends both on Γ viathe choice of e , and on Γ ∗ via the choice of a two-cell F e . Now, because of the flatness of theconnection, the holonomy does not really depend on the choice of edge e , but solely on the choiceof the homotopy class of e , which itself is left unchanged by diffeomorphisms that are connected tothe identity. For the isomorphism between P G Γ , Γ ∗ and P G Γ , it is interesting to note that the choiceof the singular gauge is invariant under a diffeomorphism that does not move Γ, whereas the choiceof the flat gauge is invariant under diffeomorphisms that do not move Γ ∗ . Indeed, in the singular This means that d z v is constant on the edges of Γ ∗ . e ∈ Γ, and we have Φ ∗ E = E if Φ(Γ) = Γ.Moreover, under an infinitesimal diffeomorphism ξ , the flux becomes δ ξ X e = Z ∂F e ι ξ (cid:0) h π e ( x ) E ( x ) h π e ( x ) − (cid:1) , (4.37)where h π e ( x ) is again the holonomy going from the source vertex of the edge e to the point x in F e . We clearly see that this expression vanishes for all ξ when the electric field is in the singulargauge. In the flat gauge, the flux does not depend on Γ, and the construction is therefore invariantunder diffeomorphisms leaving Γ ∗ invariant.This shows that there is an interesting duality between the two gauges. While the singulargauge respects diffeomorphism invariance with respect to Γ, the flat one respects diffeomorphisminvariance with respect to Γ ∗ . V. CYLINDRICAL CONSISTENCY
An important property of operators in LQG is that of cylindrical consistency associated with aprojective family of graphs [16]. In a projective family of graphs we have an ordering such thatwe may write for any two graphs in the family that Γ < Γ if Γ contains all the edges of Γ inaddition to other edges. A cylindrically consistent function is such that the pull-back from P Γ to P Γ is identified with the function on the P Γ .In this section we give a proposal for extending the notion of cylindrical consistency to func-tionals O [ A, E ] of the continuous fields. We analyze to what extent the knowledge of a collectionof functions on P Γ for all Γ determines a continuous functional. Given a collection of functions O Γ ∈ P Γ , we now propose an extension of cylindrical consistency to continuous functionals. Definition 5.
Suppose that we are given a collection of functions O Γ ∈ P Γ . We say that sucha collection of functions is cylindrically consistent if there exists a continuous functional O [ A, E ] such that its restriction on the constraint surface C is equal to O Γ . That is O| C [ A, E ] = O Γ (cid:2) h e ( A ) , X e ( A, E ) (cid:3) . (5.1)The results presented in the previous sections show that such a continuous functional O [ A, E ]is characterized by the the following property:
Proposition 2. O [ A, E ] is a cylindrical functional if and only if its restriction to the constraintsurface C is invariant under the gauge group F Γ ∗ × G Γ for every pair of dual graphs (Γ , Γ ∗ ) . Indeed, suppose that we have a functional O [ A, E ] defined on the phase space P such that itsrestriction to the constraint surface C is then O| C [ A, E ], where the field configurations now satisfy F ( A ) = 0 outside of the dual graph Γ ∗ , and d A E = 0 outside of the vertices V Γ . O [ A, E ] is acylindrically consistent functional if and only if O| C (cid:2) g ⊲ A, ( φ, g ) ⊲ E (cid:3) = O| C [ A, E ] , (5.2)which necessarily implies that O [ A, E ] = O (cid:2) h e ( A ) , X e ( A, E ) (cid:3) .This proposition gives us a powerful criterion to check wether a continuous functional can berepresented as a collection of functions associated with P Γ . For instance, we can analyze the statusof geometrical functionals such as area and volume. We know that the continuous expression for7the area functional is A ( S ) = Z S q ˜ E ia ˜ E ai . (5.3)One can easily see that even when we restrict this functional to the constraint surface F ( A ) = 0outside Γ ∗ and d A E = 0, this functional is not invariant under the translations E E + d A φ .Therefore, this functional is not expressible purely in terms of holonomies and fluxes associatedwith the graph Γ. However, in loop quantum gravity, the area operator is expressed as an operatoracting on the graph Γ, and is the quantum version of a function of the fluxes : A LQG ( S ) = X e | e ∩ S =0 q X ie X ei . (5.4)Our proposition therefore shows that the LQG area operator does not come from the continuousarea functional. This means that we have A ( S ) | C − A LQG ( S ) = 0 . (5.5)So in that sense, the LQG operator is not a proper approximation of the continuous area functional.This is puzzling since the LQG area operator has been used extensively and derived in manyways. This result thus raises the question of the exact relationship between these two objects. Towhat extend does the LQG operator capture information about the continuous area functional?Now, since we have the exact relationship between the discrete and continuous phase spaces, wecan investigate this question a bit further.First, let us recall that the continuous and LQG areas are not unrelated. In fact, for any product h Γ of holonomies supported on the graph Γ, they satisfy (cid:8) A ( S ) | C − A LQG ( S ) , h Γ (cid:9) = 0 . (5.6)So even if A | C − A LQG does not vanish, it belongs to the commutant of the holonomy algebra.The second key remark is that if we have a non-gauge-invariant functional like A ( S ), we canpromote it to a gauge-invariant functional under F Γ ∗ by picking up a gauge. This can be done byworking with A T ( S ) ≡ A ( S ) (cid:0) E ( X e ) (cid:1) instead of A ( S )( E ), where T is a gauge choice as describedin section IV. Such a functional is by construction invariant under F Γ ∗ , since it depends only on thefluxes. Moreover, the difference between two functionals that differ by a choice of gauge belongsto the commutant of the holonomy algebra: (cid:8) A T ( S ) − A T ′ ( S ) , h Γ (cid:9) = 0 . (5.7)This implies that the LQG area operator is the quantization of the continuous area functionalwritten in a particular gauge, and as described in section IV, the interpretation of geometry inLQG is given by the singular gauge. This explains why it can be expressed purely in terms offluxes.So far in (5.4) we have considered the covariant flux (2.16) rather than the usual definition(2.13). Does this analysis hold for an area operator defined from the traditional definition of flux?In the singular gauge the electric field is given by (4.15), and the integral defining the covariantflux (2.16) receives a contribution only at the point of intersection between the surface S and the For the moment we shall use the covariant fluxes (2.16) in this definition, even though the traditional LQG areaoperator descends from a functional defined using (2.13). We shall comment more on this below. e . The dependence on h π is traced out in the definition (5.4) so that in the singular gauge, thearea functional is the same whether one uses the covariant flux or the usual definition. Therefore,the above analysis is valid for either form of the flux.Now, what is unclear is to what extent the knowledge of a function in a given gauge allowsreconstruction of the continuous functional. Also, if one chooses another gauge, like the flat gaugeof spin foam models, we are going to construct a different family of area functions associated withgraphs, which will differ from A LQG by an element of the commutant of the holonomy algebra. Itis not clear which family of operators (if any) we should use to capture in the most efficient wayinformation about the continuous volume operator.
Discussion and conclusion
In this paper, we have shown that the discrete phase space of loop gravity associated with agraph Γ can be interpreted as the symplectic reduction of the continuous phase space of gravitywith respect to a constraint imposing the flatness of the connection everywhere outside of the dualgraph Γ ∗ . This allows us to give a clear interpretation of the discrete flux variables as labelingan equivalence class of continuous geometries. The point of view that the discrete data representsa set of continuous geometries has already been advocated in [10]. Our approach gives a preciseunderstanding of which set or equivalence class of continuous geometries is represented by thediscrete geometrical data ( h e , X e ) on a graph. It provides a classical understanding of the work byBianchi [9], who showed that the spin network states can be understood as states of a topologicalfield theory living on the complement of the dual graph. It also allows us to reconcile the tensionbetween the loop quantum gravity picture, in which geometry is thought to be singular, and thespin foam picture, in which the geometry is understood as being locally flat. We now see thatboth interpretations are valid and correspond to different gauge choices in the equivalence class ofgeometries represented by the fluxes. It gives us a new understanding of the geometrical operatorsused in loop quantum gravity as gauged fixed operators, and allows us to investigate further therelationship between these operators and the continuous ones. Finally, it opens the way to aclassical formulation of loop gravity. We can now face the question of whether the dynamics ofclassical general relativity can be formulated in terms of these variables. We plan to come back tothis issue of defining a loop classical gravity in the future. Acknowledgments
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