Propagation in Polymer Parameterised Field Theory
aa r X i v : . [ g r- q c ] S e p Propagation in Polymer Parameterised Field Theory
Madhavan Varadarajan ∗ Raman Research Institute, Bangalore 560080, India
The Hamiltonian constraint operator in Loop Quantum Gravity acts ultralocally.Smolin has argued that this ultralocality seems incompatible with the existenceof a quantum dynamics which propagates perturbations between macroscopicallyseperated regions of quantum geometry. We present evidence to the contrary withinan LQG type ‘polymer’ quantization of two dimensional Parameterised Field Theory(PFT). PFT is a generally covariant reformulation of free field propagation on flatspacetime. We show explicitly that while, as in LQG, the Hamiltonian constraintoperator in PFT acts ultralocally, states in the joint kernel of the Hamiltonian anddiffeomorphism constraints of PFT necessarily describe propagation effects. Theparticular structure of the finite triangulation Hamiltonian constraint operator playsa crucial role, as does the necessity of imposing (the continuum limit of) its kinematicadjoint as a constraint. Propagation is seen as a property encoded by physical statesin the kernel of the constraints rather than that of repeated actions of the finitetriangulation Hamiltonian constraint on kinematic states. The analysis yields robuststructural lessons for putative constructions of the Hamiltonian constraint in LQGfor which ultralocal action co-exists with a description of propagation effects byphysical states.
I. INTRODUCTION
A key open problem in Loop Quantum Gravity is a satisfactory definition of the Hamil-tonian constraint operator which generates its quantum dynamics. A mathematically welldefined procedure to construct this operator using canonical quantization techniques wasdeveloped by Thiemann [1] . While signifiant progress has been made since Thiemann’spioneering work the construction still yields an operator which is far from unique. Thereforeit is desireable to subject candidate operators to further requirements so as to cut down onthe quantization choices responsible for this non-uniqueness. For example, in [4] a Hilbertspace is constructed which supports the action of the operator so that the operator may beconfronted with adjointness properties. Another example is of recent work which attemptsto impose the requirement that the candidate operator yield an anomaly free algebra ofconstraints so as to impose spacetime covariance in the quantum theory [5–7].Here we are interested in yet another requirement, namely that the quantum dynamicsdescribe the propagation of perturbations from one part of the quantum geometry to another.This requirement was articulated several years ago by Smolin [8]. In his work, Smolin alsooffers a critique of Thiemann’s general construction and concludes that such a constructionmethod yields a constraint action which is ‘too local’ to allow for propagation effects in the ∗ Electronic address: [email protected] This work itself was based on cumulative progress achieved by many workers; for a detailed bibliographysee, for e.g. the review article [2] and the book [3] quantum theory. More in detail, recall that quantum states in LQG are superpositions of‘spin network’ states labelled by graphs. Each such state describes 1d excitations of spatialgeometry along the edges of the graph which underlies the state. All known versions ofthe Hamiltonian constraint which derive from the general Thiemann procedure act only atvertices of the graph and the action at one vertex is independent of the action at neighbouringvertices. Further, repeated actions of the Hamiltonian constraint create a finer and finernested structure about each vertex. Thus, modifications of the graph structure wrought bythe action of the Hamiltonian constraint in the immediate vicinity of one vertex do not seemto propagate to other vertices. Since such propagation is thought of as the non-perturbativeseed for graviton propagation in semiclassical states in LQG, one would like the quantumdynamics to allow for such propagation. Since the issue of the (semi)classical limit of LQG isstill open, Smolin’s criticisms are based, at least partly, on physical intuition. Nevertheless,these criticisms seem compelling and this article seeks to address them.Given the complications of full blown gravity and the open issues within LQG, we focuson Smolin’s criticisms in the context of a generally covariant field theoretic toy model whichhas already proven to be extremely useful in addressing other issues concerned with theHamiltonian constraint. The model is that of 1+1 Parameterised Field Theory (PFT) whichis a generally covariant reformulation of free massless scalar field theory on 1+1 Minkwoskispacetime in which, in addition to the scalar field, the Minkwoskian coordinates are treatedas dynamical variables to be varied in the action. Since on the one hand the classicaltheory describes scalar wave propagation , and on the other, a complete LQG type ‘polymer’quantization exists for the model [9, 10, 13] this model serves as an ideal testing ground for ananalyis and possible resolution of Smolin’s criticisms. By virtue of its general covariance, thedynamics of PFT is driven by Hamiltonian and diffeomorphism constraints. The kinematicHilbert space is spanned by ‘charge network states’. Each such state is labelled by a 1dgraph on the Cauchy slice, the edges of the graph being colored by integer value chargesassociated with quantum excitations of the Minkowsian coordinates and the scalar field. Thephysical Hilbert space can be constructed by group averaging techniques [11] and, despitebeing a rigorous quantization of continuum scalar field theory, these physical states describequantum scalar field excitations on discrete spacetime. More in detail, a superselectionsector of such states exists which describes quantum scalar field excitations propagating on adiscrete spacetime lattice , the lattice spacing being governed by the analog of the Barbero-Immirzi parameter [12]. For reasons explained in [10] we call this sector the ‘finest lattice’sector. The Hamiltonian constraint operator can also be constructed following the broadprocedure introduced by Thiemann and quantization choices can be made in such a way thatits action is an infinitesmal version of the finite transformations used in the group averagingprocedure referred to above (see [13] and Section IV). The Hamiltonian constraint operatorso constructed acts ultralocally. By this we mean that, as in LQG, the finite triangulationHamiltonian constraint acts on vertices of kinematical states, its action at a vertex is onlysensitive to structure in a small neighbourhood of that vertex and its action at one vertexis independent of that at other vertices. As a result, its repeated action at a vertex does Recall that the Thiemann procedure introduces a set of finer and finer triangulations of the Cauchy slice,chooses at each triangulation an approximant to the constraint constructed from the basic holonomy-fluxfunctions of the theory so that at infinitely fine triangulation the approximant becomes exact, replaces theclassical functions by quantum operators, and, finally, defines the resulting operator in a limit of infinitelyfine triangulation. not lead by itself to any propagation for reasons similar to that articulated by Smolin in thecontext of LQG.Nevertheless, as we show in this work, despite this ultralocal action, the joint kernel ofthe Hamiltonian constraint operator and the diffeomorphism constraint operator necessarilycontains the finest lattice sector of physical states which do provide a description of prop-agation effects. Thus our work shows that propagation is not to be seen as a property ofrepeated actions of the (finite triangulation) Hamiltonian constraint but rather as a (logi-cally independent) property encoded in physical states which are in the joint kernel of theHamiltonian and diffeomorphism constraints. With this shift in focus to the structure ofphysical states in the kernel of the constraints, the two key features which enable propaga-tion effects turn out to be(i) the structure of the Hamiltonian constraint at finite triangulation which is quite differentfrom the structure of Thiemann’s Hamiltonian constraint despite their shared property ofultralocality.(ii) the imposition of the continuum limit of the finite triangulation Hamiltonian constraintas well as the continuum limit of the adjoint of this finite triangulation constraint, as operatorconstraints for physical states.We believe that the emphasis on propagation as a property encoded in physical statestogether with the general structural lessons learnt from PFT are robust and applicable toLQG and offer a way out of Smolin’s criticisms. In other words, while Smolin’s criticismsseem to hold for Thiemann’s choice of Hamiltonian constraint and while this choice leads toan ultralocal action, it does not follow that the same criticisms need hold for other choiceseven if these choices also lead to an ultralocal action. The obstacle to propagation is then not ultralocality. Hence, while the Thiemann procedure does seem to lead to ultralocalconstraint action, we are optimistic that there exist choices of constraints constructed viathe general Thiemann procedure which co-exist with physical states describing propagationeffects. The layout of this paper is as follows. In section II we provide a brief review of classicalPFT and its polymer quantization wherein physical states are constructed through groupaveraging techniques. The interested reader may consult [9, 10] for details of the formal-ism. In section III we focus on a particular physically relevant superselection sector of theHilbert space known as the ‘finest lattice sector’. We review the pictorial interpretation ofkinematic and group averaged physical states in this sector in terms of discrete slices on aspacetime lattice carrying quantum matter excitations. In section IV we review the actionof the Hamiltonian constraint from [13] and show that its action is ultralocal and that itsrepeated action on a finest lattice charge net does not cause long range propagation exactlyas anticipated by Smolin. In the pictorial interpretation of section III this repeated actionfails to evolve the discrete Cauchy slice and its data beyond a single lattice step. It turnsout that the key to long range propagation is getting beyond one lattice step to two lat- Preliminary work [14] on the ‘ U (1) ’ model [6, 7, 15] indicates that the notion of physical states should befurther restricted by demanding that they lie not only in the kernel of the Hamiltonian and diffeomorphismconstraints but also in that (of certain combinations of the Hamiltonian and) the ‘electric’ diffeomorphismconstraints, the latter being obtained by smearing the diffemorphism constraint by triad dependent shifts[6]. It is not clear to us if this further restriction in the context of Thiemann’s specific choices in [1] sufficesto yield physical propagation effects or if a different choice of the Hamiltonian constraint is required inaddition to this restriction; at the moment we believe the latter is more likely. tice steps. In section V we show how elements in the joint kernel of the Hamiltonian anddiffeomorphism constraints encode evolution beyond a single lattice step to a second latticestep. We are able to isolate the key structural properties ( see (i) and (ii) above) of theHamiltonian constraint responsible for this. This is the main result of this paper. Once wehave demonstrated evolution beyond a single lattice step to two lattice steps, a technicalproof can be constructed to show that long range evolution is encoded in physical states.We relegate this proof to the appendix as it relies on certain detailed technicalities discussedin [9, 10] for which familiarity is assumed. Section VI contains a discussion of our results inthe context of LQG. II. REVIEW OF POLYMER PFTA. Classical Theory
The action for a free scalar field f on a fixed flat 2 dimensional spacetime ( M, η AB ) interms of global inertial coordinates X A , A = 0 , S [ f ] = − Z M d Xη AB ∂ A f ∂ B f, (2.1)where the Minkowski metric in inertial coordinates, η AB , is diagonal with entries ( − , f = f + ( X + T )+ f − ( T − X ) where f ± are arbitrary functions of their ‘light cone’ arguments.Due to this functional dependence f + describes left moving modes and f − , right movingmodes on the flat spacetime.If, in the action (2.1), we use coordinates x α , α = 0 , X A are ‘parameterized’by x α , X A = X A ( x α )), we have S [ f ] = − Z M d x √ ηη αβ ∂ α f ∂ β f, (2.2)where η αβ = η AB ∂ α X A ∂ β X B and η denotes the determinant of η αβ . The action for PFT isobtained by considering the right hand side of (2.2) as a functional, not only of f , but alsoof X A ( x ) i.e. X A ( x ) are considered as 2 new scalar fields to be varied in the action so that η αβ is considered to be a functional of X A ( x )). Thus S P F T [ f, X A ] = − Z M d x p η ( X ) η αβ ( X ) ∂ α f ∂ β f. (2.3)Note that S P F T is a diffeomorphism invariant functional of the scalar fields f ( x ) , X A ( x ).Variation of f yields the equation of motion ∂ α ( √ ηη αβ ∂ β f ) = 0, which is just the flat space-time equation η AB ∂ A ∂ B f = 0 written in the coordinates x α . On varying X A , one obtainsequations which are satisfied if η AB ∂ A ∂ B f = 0. This implies that X A ( x ) are undeterminedfunctions (subject to the condition that the determinant of ∂ α X A is non- vanishing). This2 functions- worth of gauge is a reflection of the 2 dimensional diffeomorphism invariance of S P F T . Clearly the dynamical content of S P F T is the same as that of S ; it is only that thediffeomorphism invariance of S P F T naturally allows a description of the standard free fielddynamics dictated by S on arbitrary foliations of the fixed flat spacetime.The spacetime is assumed to be of topology Σ × R . A 1+1 Hamiltonian de-composition yields a phase space coordinatized by the canonically conjugate pairs( T ( x ) , Π T ( x )) , ( X ( x ) , Π X ( x )) , ( f ( x ) , π f ( x )) where x coordinatizes the 1 dimensional t =constant Cauchy slice Σ. For each value of x , the functions ( T ( x ) , X ( x )) locate a pointin flat spacetime by virtue of their interpretation as Minkowskian coordinates so that as x varies, ( T ( x ) , X ( x )) describe an embedding of the Cauchy slice coordinatized by x intothe flat spacetime. As a result we refer to the sector of phase space coordinatized by( T ( x ) , Π T ( x )) , ( X ( x ) , Π X ( x )) as the embedding sector. A canonical transformation can bemade into ‘left and right moving’ embedding variables ( X + , Π + ) , ( X , Π − ) with X ± = T ± X and Π ± their conjugate momenta. It is also useful to transform to the variables Y ± = π f + f ′ in the matter sector. It can be checked that the ‘+’ and ‘-’ variables Poisson commute witheach other.The dynamics is generated by a pair of constraints H ± ( x ) = [ Π ± ( x ) X ± ′ ( x ) ±
14 ( Y ± ( x )) ] . (2.4)These constraints are of density weight two. In 1 spatial dimension their transformationproperties under coordinate transformations are identical to those of spatial covector fields.Integrating them against multipliers N ± , which can therefore be thought of as spatial vec-tor fields, one finds that the integrated ‘+’ (respectively ‘-’) constraint generates spatialdiffeomorphisms on the ‘+’ (respectively ‘-’) variables while keeping the ‘-’ (respectively‘+’) variables untouched. Thus, PFT dynamics can be thought of as the action of two independent spatial diffeomorphisms Φ + , Φ − on the ‘+’ and ‘-’ sectors of the phase space.Instead of the H ± constraints we may consider the constraints C diff = H + + H − (2.5)and C ham = H + − H − . (2.6)It can be checked that the diffeomorphism constraint C diff generates spatial diffeomorphismson the Cauchy data ( X − , Π − , X + , Π + , Y + , Y − ) whereas the (density weight two) Hamilto-nian constraint C ham generates evolution of this data along the unit timelike normal n α tothe slice [16] (recall that the phase space data ( X + , X − ) define an embedded Cauchy slicein Minkowski spacetime; n α is the unit timelike normal to this slice with respect to the flatspacetime metric).In terms of the finite transformations Φ + , Φ − , it follows from (2.5), (2.6) that finitespatial diffeomorphisms generated by C diff correspond to the choice Φ + = Φ − whereas finitetransformations generated by the Hamiltonian constraint C ham correspond to the choiceΦ + = (Φ − ) − .The relation between evolution on phase space data generated by the constraints and freefield evolution of f on flat spacetime can be seen as follows [16]. The constraints ( C diff , C ham or, equivalently, H + , H − ) generate transformations of the phase space data ( X ± ( x ) , Y ± ( x ))to new data ( X ± new ( x ) , Y ± new ( x )). The phase space data ( X + ( x ) , X − ( x )) define an embeddedslice in flat spacetime. Initial data on this slice for evolution via the scalar wave equation canbe given in terms of left and right moving values of the scalar field on the slice (or, ignoringissues of zero modes, the values of their derivatives). As discussed in [16] the relationshipbetween the phase space data Y ± and these derivatives is given by ∂f∂X ± | X ± = X ± ( x ) = Y ± ( x ) X ±′ ( x ) .The transformed embedding data X ± new then define an ‘evolved’ slice in flat spacetime withmatter phase space data X ±′ new ( x ) ∂f∂X ± | X ± = X ± new ( x ) = Y ± new ( x ) where f is the restriction to thenew slice of the solution to the wave equation with initial data ∂f∂X ± on the old slice.As in the previous works [9, 10, 13] we shall restrict attention to a flat spacetime ( M, η )with cylindrical topology S × R . We denote the length of the T = 0 circle in the flatspacetime ( M, η ) by L . The cylindrical topology of M implies that any Cauchy slice Σ iscircular. Certain subtelities related to the use of a single angular inertial coordinate X onthe flat spacetime M as well as a single angular coordinate x on the Cauchy slice Σ arise butthese subtelities and their ramifications constitute technical details which may be ignoredin as much as the key arguements in this work are concerned. The interested reader mayconsult [16] and section IIC of [9] for an account of these subtelities in classical theory. B. Quantum Theory
We shall concentrate mainly on the embedding sector in this brief review. For furtherdetails regarding the matter sector please see [10, 13]. We shall mention, but not explainin any detail, the subtelities in the quantum theory concerning the circular spatial topologybecause such details will only serve to distract from the main arguments in subsequentsections. The interested reader may consult [10] for such details.
1. Kinematics
The embedding sector Hilbert space is a tensor product of ‘+’ and ‘-’ sectors. On the ‘+’sector the operator correspondents of functions on the ‘-’ sector of phase space act triviallyand vice versa.The ‘+’ embedding sector is spanned by an orthonormal basis of charge network stateseach of which is denoted by | γ + , ~k + i where γ + is a graph i.e. a set of edges which coverthe circle with each edge e labelled by a ‘charge’ k + e , the collection of such charges forall the edges in the graph being denoted by ~k + . The + embedding sector Hilbert spaceprovides a representation of the Poisson bracket algebra between the classical ‘holonomyfunctions’ exp i ( P e ( k + e R e Π + )), and the embedding coordinate X + ( x ). In this representationthe embedding momenta are polymerized so that the holonomy functions are well definedoperators but the embedding momenta themselves are not. The embedding coordinateoperator ˆ X + ( x ) is well defined and the charge net states are eigen states of this operator.In particular the action of the embedding coordinate operators ˆ X + ( x ) on a charge networkstate | γ + , ~k + i when the argument x lies in the interior of an edge e of γ + is:ˆ X + ( x ) | γ, ~k + i = ~ k + e | γ, ~k + i (2.7)Identical results hold for + → − . The tensor product states | γ + , ~k + i ⊗ | γ − , ~k − i form a basisof the embedding sector Hilbert space and are referred to as embedding charge networkstates. By going to a graph finer than γ + , γ − , each such state can be equally well denotedby | γ, ~k + , ~k − i where each edge e of the graph γ is labelled by a pair of charges ( k + e , k − e ) andthe collection of these charges is denoted by ( ~k + , ~k − ). Such a state is an eigen ket of boththe ˆ X + and ˆ X − operators. Similar to (2.7) the action of ˆ X ± ( x ) on the charge net | γ, ~k + , ~k − i when x is in the interior of an edge e of γ is:ˆ X ± ( x ) | γ, ~k + , ~k − i = ~ k ± e | γ, ~k + , ~k − i (2.8)The charges are chosen to be integer valued multiples of a dimensionful parameter a ~ so that ~ k ± e ∈ Z a (2.9)where a is a Barbero- Immirzi like parameter. In the context of the circular spatial topologyrelevant to this work, we restrict attenion, as in [10], to a value of a such that La = N, (2.10) N being a positive integer.The matter sector Hilbert space is also a tensor product of ‘+’ and ‘-’ sectors. On the‘+’ sector the field Y + is polymerised and on the ‘-’ sector the Y − field is polymerised.Thus taken together, neither Y + nor Y − (and hence neither π f nor f ) exist as well de-fined operators. The ‘+’ sector provides a representation of matter holonomy functionsexp i ( P e l + e R e Y + ) on a basis of ‘+’ matter charge nets, each such charge net denoted by | γ + , ~l + i in obvious notation. The matter charges ~l + on each such charge net are real and aresubject to the following restriction [10]: for every pair of edges edges e, e ′ ∈ γ + , the mattercharge difference l + e − l + e ′ is an integer valued multiple of a dimensionful parameter ǫ . The‘-’ sector structure is identical. The parameter ǫ along with the parameter ‘ a ’ for the embed-ding sector constitute ‘Barbero-Immirzi’ parameters and label inequivalent representations.The tensor product states | γ + , ~l + i ⊗ | γ − , ~l − i form an orthonormal basis of the matter Hilbertspace and are referred to as matter charge net states. By going to a fine enough graph anysuch state can be denoted, in notation similar to that for embedding states, as | γ, ~l + , ~l − i .The tensor product of the matter and embedding Hilbert spaces yields the kinematicHilbert space H kin for PFT. This Hilbert space is spanned by charge net states each ofwhich is a tensor product of a matter charge net and an embedding charge net. By going toa fine enough graph underlying the matter and embedding charge nets we may denote sucha tensor product state by | γ, ~k + , ~k − , ~l + , ~l − i .Since the ‘+’ and ‘-’ sectors are independent, we also have the tensor product decompo-sition: | γ, ~k + , ~k − , ~l + , ~l − i = | γ + , ~k + , ~l + i ⊗ | γ − , ~k − , ~l − i (2.11)where | γ ± , ~k ± , ~l ± i is itself a product of a ‘ ± ’ embedding charge network and a ‘ ± ’ mattercharge network. We do not impose the vanishing zero constraint [9, 10]. We remark further on the zero mode issue insection VI.
2. Gauge transformations generated by the constraints.
Recall that the finite transformations generated by the constraints H + , H − correspond to apair of diffeomorphisms Φ + , Φ − . The quantum kinematics supports a unitary representationof these diffeomorphisms by the unitary operators ˆ U + (Φ + ) , ˆ U − (Φ − ). The operator ˆ U + (Φ + )acts on a ‘+’ charge network state | γ + , ~k + , ~l + i by moving the graph and its colored edges bythe diffeomorphism Φ + while acting as identity on ‘-’ charge network states, and a similaraction holds for + → − . We denote this action byˆ U ± (Φ ± ) | γ ± , ~k ± , ~l ± i =: | γ ± , Φ ± , ~k ± Φ ± , ~l ± Φ ± i (2.12)The action of finite gauge transformations on a charge net state | γ, ~k + , ~k − , ~l + , ~l − i can thenbe deduced from equation (2.11):ˆ U + (Φ + ) ˆ U − (Φ − ) | γ, ~k + , ~k − , ~l + , ~l − i = | γ + , Φ + , ~k +Φ + , ~l +Φ + i ⊗ | γ − , Φ − , ~k − Φ − , ~l − Φ − i . (2.13)By going to a graph finer than γ + , Φ + , γ , Φ − the right hand side can again be written as achargenet labelled by a single graph with each edge labelled by a set of 4 charges namelythe ‘+’ and ‘-’ embedding and matter charge labels.
3. Group Averaging
Physical states can be constructed by group averaging [11] over the action of all Φ + , Φ − .From [10] we have that the group average of any charge net | γ, ~k + , ~k − , ~l + , ~l − i := | γ + , ~k + , ~l + i⊗| γ − , ~k − , ~l − i is the distribution Ψ given by:Ψ := X (Φ + , Φ − ) ∈ Orbit h γ, ~k + , ~k − , ~l + , ~l − | ˆ U † + (Φ + ) ˆ U †− (Φ − )= X (Φ + , Φ − ) ∈ Orbit h γ + , Φ + , ~k +Φ + , ~l +Φ + | ⊗ h γ − , Φ − , ~k − Φ − , ~l − Φ − | (2.14)where Orbit comprises of gauge transformations such that for each distinct chargenet whichis gauge related to | γ, ~k + , ~k − , ~l + , ~l − i there is a unique element (Φ + , Φ − ) in Orbit which maps | γ, ~k + , ~k − , ~l + , ~l − i to this gauge related image. In other words, the sum is over all distinctgauge related chargenets. We shall ignore issues of ambiguities in the group averagingprocedure (see for example [9, 10, 17]) as this will not be important for the arguements inthis paper. While the matter charges are unaffected, each time an edge moves past the point x = 0 ≡ x = 2 π itsembedding charge labels are augmented by factors of L where L is the length of the T =constant circlesin flat spacetime. This is one of the subtelities arising from circular spatial topology. III. THE FINEST LATTICE SECTOR AND ITS PICTORIALREPRESENTATION
Since the embedding charges are eigen values of the embedding coordinate operators,we can associate an embedding charge net with a discrete slice of Minkowski spacetime asfollows. For every edge e the pair ~ k + e , ~ k − e specifies the point X + = ~ k + e , X − = ~ k − e inflat spacetime. The set of such points for all edgewise pairs of eigen values then defines aset of points in the flat spacetime which we may refer to as a discrete slice. It turns out[10, 13] that there exists a superselected sector (with respect to all gauge transformationstogether with a complete set of Dirac observables) of states with the following ‘finest lattice’property.Consider a lightcone lattice of spacing a in the flat spacetime so that X + , X − ∈ Z a onthis lattice. We shall say that a pair of points are nearest neighbours if they are either lightlike seperated and one lattice spacing away from each other or if they are spacelike seperatedand located at a spatial distance a (as measured by the flat spacetime metric) from eachother. Thus each point on the lattice has 6 nearest neighbours, 4 lightlike and 2 spacelike. Next, consider the set of edgewise pairs of embedding charges for any charge net in thesuperselected sector under discussion. Plot these as a set of points in flat spacetime in themanner described above. Then it turns out that these points fall on the spacetime lattice X + , X − ∈ Z a in such a way that points obtained from adjacent edges are nearest neighbours.Further, by virtue of the minimal spacing of the eigen values of the X ± operators (2.8)- (2.9),flat spacetime point sets defined by any chargenet in the kinematic Hilbert space cannotfall on any finer lattice. Finally, for any charge net in this sector (see (i) below), the mattercharges are distributed on the underlying graph in a coarser manner than the embeddingcharges. Hence with each pair of embedding charges which specify a point in flat spacetime,we can uniquely associate a pair of matter charges. This means that we can label each pointon the discrete slice in flat spacetime by a pair of matter charges so that the chargenet canbe interpreted as a specification of quantum matter on a discrete ‘Cauchy’ slice, this slicesatisfying the ‘nearest neighbour’ property on the finest lattice allowed by the spectrum(2.8),(2.9). No other chargenet outside this superselection sector admits this interpretation.Hence this sector is called the ‘finest lattice’ sector.Technically, chargenets in this sector are specified as follows [13]. A chargenet | γ, ~k + , ~k − , ~l + , ~l − i = | γ + , ~k + , ~l + i ⊗ | γ − , ~k − , ~l − i belongs to the finest lattice sector iff (i)- (iii)below hold:(i) its matter chargenet labels are ‘coarser’ than the embedding ones so that each pair ofsuccessive edges of the coarsest graph γ coarse ± underlying | γ ± , ~k ± , ~l ± i is necessarily labelledby distinct pairs of ± embedding charges but not necessarily distinct pairs of ± mattercharges.(ii) its embedding charges on the coarsest graph γ coarse ± underlying | γ ± , ~k ± , ~l ± i satisfy ~ k ± e ′± − ~ k ± e ± = a where e ′± , e ± are adjacent edges in γ coarse ± such that e ′± lies to the right of e ± in the coordinatization x (i.e.the edges are located such that given any point p ± in the There are also 2 ‘timelike’ neighbours. It turns out that these are not relevant to our discussion. Hencewe exclude them from our definition of nearest neighbours. e ± with coordinate x = x ± and any point p ′± in the interior of e ′± with coordinate x = x ′± , we have that x ′± > x ± ).Further, if the difference in the embedding charge value on the last edge and the firstedge of γ coarse ± is ± L ~ , the matter charge values on these edges are identical. (iii) there are N + ‘+’ and N − ‘-’ distinct embedding charges with N + such that either N + = N + 1 or N + = N and N − such that either N − = N + 1 or N − = N where N isdefined in equation (2.10). Property (i) ensures that we may think of the matter charges l + e , l − e as sitting on thelattice point ~ k + e , ~ k − e . Property (ii) is the technical formulation of the ‘nearest neighbour’condition. Property (iii) turns out to be necessary for the consistency of the ‘discrete Cauchyslice’ interpretation in the context of circular spatial topology. To summarise: The kinematiccharge nets of polymer PFT in this sector can be interpreted as describing quantum matterdegrees of freedom on discrete Cauchy slices which fall on a (light cone) lattice in Minkowskispacetime. For the remainder of this work we shall focus, for concreteness, exclusively onstates in this finest lattice sector.Let us consider the action of a gauge transformation Φ + on a finest lattice state | γ, ~k + , ~k − , ~l + , ~l − i . It moves the ‘+’ charged edges along the circle relative to the fixed‘-’ charged edges. Consider a fine enough graph which underlies the new set of ‘+’ edgestogether with the old ‘-’ edges. On this graph the set of edgewise pairs of charges is dif-ferent from the set corresponding to the original charge net. This new set defines a new discrete Cauchy slice and matter data on this slice. Thus, the matter data propagate fromone discrete Cauchy slice to another. A similar picture ensues for the action of a gaugetransformation Φ − and one explicitly sees how, just as in classical theory, the finite gaugetransformations generated by the constraints propagate matter from one slice to another. Itturns out (see [10] for details) that the considerations of section II B 2 in conjunction withFootnote 5 ensure that, for appropriate choices of gauge transformations, the initial and finaldiscrete slices can be ‘macroscopically’ seperated (i.e. by arbitarily many lattice points) intime so that such gauge transformations implement long range propagation .From the discussion in section II B 3 a physical state obtained by group averaging overa charge net | γ, ~k + , ~k − , ~l + , ~l − i can be written as a sum over all distinct charge net statesobtained from this one by action of all possible finite gauge transformations. As assertedin [10], if we plot the lattice points in flat spacetime associated with each of these statestogether with their matter charge labelling, one obtains a single discrete spacetime latticewith uniquely specified matter charges at each lattice point. This specification is consistent in the sense that if a single lattice point derives from the same pair of embedding chargesarising from different states in the sum, the matter charge labels for this point provided bythese different states are the same. In other words the discrete Cauchy slices with matterdata which occur as summands in (2.14) stack up consistently to give a single discretespacetime with quantum matter at each spacetime point. Putting this picture together withthat of the action of finite gauge transformations discussed above it follows that any physical This additional restriction on the matter charges is due to subtelities connected with circular spatialtopology (see Footnote 5). We assume that
N >> edges of a chargenet define points in flat spacetime. If two successive edges define light like relatedpoints we refer to the vertex of the chargenet at which these edges intersect as a null vertex.If the points are spacelike related we refer to the corresponding vertex as a spacelike vertex.Note that from (ii) above successive edges can never define flat spacetime points which aretimelike seperated.
IV. ULTRALOCALITY OF THE HAMILTONIAN CONSTRAINT ANDSMOLIN’S CRITICISM
In the section II B 3 we showed that physical states could be constructed by group aver-aging over the finite transformations generated by the ‘light cone’ constraints H + , H − . SinceSmolin’s criticisms apply to the LQG formalism wherein finite spatial diffeomorphisms areaveraged over and the Hamiltonian constraint is constructed via Thiemann’s procedure, wenow turn our attention to a similar treatment for PFT in terms of its diffeomorphism andHamiltonian constraints, C diff (2.5) and C ham (2.6)First consider, as in LQG, the finite transformations generated by C diff . Recall from thediscussion after (2.5), (2.6) that these transformations correspond to the case Φ + = Φ − .Indeed, setting Φ + = Φ − = Φ in equation (2.13)we see that the associated unitary transfor-mations ˆ U + (Φ) ˆ U − (Φ) drag both the ‘+’ and ‘-’ chargenets by the same diffeomorphism Φ,which of course, just corresponds to the action of a spatial diffeomorphism :ˆ U diff (Φ) | γ, ~k + , ~k − , ~l + , ~l − i = ˆ U + (Φ) ˆ U − (Φ) | γ, ~k + , ~k − , ~l + , ~l − i = | γ + , Φ , ~k +Φ , ~l +Φ i ⊗ | γ − , Φ , ~k − Φ , ~l − Φ i . (4.1)By going to a finer graph that γ + , Φ , γ − , Φ , one can denote the tensor product chargenet inthe last line of the above equation by a single chargenet with edges labelled by a quadrupleof (2 embedding and 2 matter) charges. Since both the ‘+’ and ‘-’ labels are dragged aroundby the same diffeomorphism, it is straightforward to check that ˆ U diff (Φ) | γ, ~k + , ~k − , ~l + , ~l − i and | γ, ~k + , ~k − , ~l + , ~l − i define the same discrete Cauchy slice in flat spacetime with the samematter data. This is as it should be because the flat spacetime picture encodes the relation between the embedding and matter excitations and this relation is diffeomorphism invariant.Averaging over diffeomorphisms can then be done [13]. Specialising (2.14) to the caseΦ + = Φ − = Φ, and ignoring group averaging ambiguities, we have thatΨ diff := X Φ ∈ Orbit diff h γ, ~k + , ~k − , ~l + , ~l − | ˆ U † + (Φ) ˆ U †− (Φ)= X Φ ∈ Orbit diff h γ + , Φ , ~k +Φ , ~l +Φ | ⊗ h γ − , Φ , ~k − Φ , ~l − Φ | (4.2)where Orbit diff consists of elements which uniquely take the charge net being averaged over This turns out to be true despite the added subtelities alluded to in Footnote 5 [9, 10]. diff is a sum over distinctdiffeomorphic images of the chargenet being averaged so that if a certain bra is in the sum,then all its diffeomorphic images are also in the sum.Next consider the Hamiltonian constraint C ham (2.6). Recall from discussion after (2.5)-(2.6) that the finite transformations generated by C ham correspond to the case Φ + = (Φ − ) − .From Reference [13], it follows that with particular quantization choices at finite triangu-lation, the finite triangulation approximant to the Hamiltonian constraint C ham ( N ) smearedwith lapse N can be written asˆ C ham,δ ( N ) | γ, ~k + , ~k − , ~l + , ~l − i = − i ~ X v N ( v ) ˆ U + (Φ δ,v ) ˆ U − (Φ − δ,v ) − δ | γ, ~k + , ~k − , ~l + , ~l − i (4.3)= − i ~ X v N ( v ) | γ + , Φ δ , ~k +Φ δ , ~l +Φ δ i ⊗ | γ − , Φ − δ , ~k − Φ − δ , ~l − Φ − δ i (4.4)The sum in (4.4) is over ‘non-trivial’ vertices of the (finest lattice) charge net | γ, ~k + , ~k − , ~l + , ~l − i . By a ‘non-trivial’ vertex we mean a point on γ at which the ingoingand outgoing edges carry non-identical charges. Since the charge net is a finest latticecharge net, this implies that at least one of the incoming embedding charges differs fromits outgoing counterpart. Φ δ,v is a small diffeomorphism around the non-trivial vertex v which moves v to its right by an amount δ as measured by the coordinate x , where ‘right’means direction of increasing x . In addition Φ δ,v is identity outside a region of size of order δ around v and Φ δ,v is such that its inverse Φ − δ,v moves v to its left by an amount δ . Thusgiven a chargenet, the Hamiltonian constraint acts only at its non-trivial vertices and movesthe ‘+’ part of the charge net in the vicinity of the vertex v to the right and the ‘-’ partof the chargenet in the vicinity of the vertex v to the left. For small enough δ it is easyto see that the diffeomorphism class of the chargenet deformed in this manner remains thesame as δ →
0. Finally note that we may extend the sum in (4.4) to include any numberof ‘trivial’ vertices of γ for which the incoming and outgoing charges are identical because,as is straightforward to verify, on any such vertex v , the operator ˆ U + (Φ δ,v ) ˆ U − (Φ − δ,v ) acts asthe identity on the chargenet so that these vertices do not yield non-zero contributions. Itis convenient to define: ˆ U ham,δ,v := ˆ U + (Φ δ,v ) ˆ U − (Φ − δ,v ) . (4.5)so that ˆ C ham,δ ( N ) | γ, ~k + , ~k − , ~l + , ~l − i = − i ~ X v N ( v ) ˆ U ham,δ,v − δ | γ, ~k + , ~k − , ~l + , ~l − i (4.6)where it is understood that the sum ranges over all non-trivial vertices but can also includeany number of trivial vertices as well.Next, we note that choices could equally well be made in the Thiemann procedure such We shall comment more on these choices in Section VI. We discuss this further in Section VI. C ham,δ ( N ) | γ, ~k + , ~k − , ~l + , ~l − i = i ~ X v N ( v ) ˆ U † ham,δ,v − δ | γ, ~k + , ~k − , ~l + , ~l − i (4.7)where, from equation (4.5), we have thatˆ U † ham,δ,v = ˆ U †− (Φ − δ,v ) ˆ U † + (Φ δ,v ) = ˆ U − (Φ δ,v ) ˆ U + (Φ − δ,v ) = ˆ U + (Φ − δ,v ) ˆ U − (Φ δ,v ) (4.8)The action of the finite triangulation Hamiltonian constraint (4.7) would then be to movethe ‘+’ part of the charge net in the vicinity of the vertex v to the left and the ‘-’ partof the chargenet in the vicinity of the vertex to the right . Clearly the actions (4.6), (4.7)are ultralocal in the sense of Smolin. More in detail, these actions are only in the vicinityof vertices of the chargenet being acted upon and the action at one vertex is completelyindependent of the other. We now analyse repeated actions of the Hamiltonian constraintand show that they do not lead to propagation.We shall focus first on the action of ˆ U ham,δ,v at a vertex v with incoming embeddingcharges ( k +1 , k − ) and outgoing charges ( k +2 , k − ). The action bifurcates the vertex, givingrise to two new vertices v , v with v to the left of v and one new edge connecting thesevertices so that the sequence of charges from left to right is now ( k +1 , k − ) , ( k +1 , k − ) , ( k +2 , k − ).Thus the sequence of charges in the vicinity of v changes as:( k +1 , k − ) , ( k +2 , k − ) → ( k +1 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) (4.9)The following three cases are of interest (see the end of section III for the terminology usedin (ii),(iii) below):(i) The original vertex v is trivial: In this case the charge net is unchanged.(ii)The vertex v is null so that k +1 = k +2 or k − = k − : In this case the new charge sequence isequivalent to the one on the original charge net so that the new chargenet is diffeomorphicto the old one. It follows that the new charge net defines exactly the same points in flatspacetime as the old charge net.(iii)The vertex v is spacelike so that k +1 = k +2 and k − = k − . In this case we see that thenew vertices v and v are null. Further from (ii), section III and (4.9) it follows that thenew chargenet represents ‘one lattice step’ of evolution with respect to the original chargenet.Exactly the same conclusions ensue for the action (4.7). Note that the matter chargesare just dragged along together with the embedding charges by the actions (4.6),(4.7). Fromthe above discussion we see that if v is trivial or null the Hamiltonian constraint actions(4.6),(4.7) do not change the flat spacetime points (and the matter data thereon) associatedwith the chargenet so that there is no evolution in the flat spacetime. If v is spacelike,it follows from (iii) above that the action of the Hamiltonian constraint (4.6),(4.7) evolvesmatter data by one lattice step. Note however that this action replaces the spacelike vertex v of the charge net by a pair of null vertices v and v . From (ii) above it follows that further actions of the Hamiltonian constraint (whether (4.6) or (4.7)) at these new vertices of thechargenet do not evolve the discrete Cauchy slice any further.Applying these results to all the vertices of the chargenet we conclude that repeatedactions of the Hamiltonian constraint ( say n actions (4.6) followed by m actions (4.7)followed by n of (4.6) and so on) cannot evolve the discrete Cauchy slice with quantum4matter any further than one lattice step away in flat spacetime. It is in this precise sensethat the Hamiltonian constraint does not generate long range propagation and it is in thisprecise sense that Smolin’s criticism is formulated in the case of PFT.In order to understand exactly why repeated actions of the Hamiltonian constraint fail togenerate long range propagation, it is appropriate to compare the actions (4.6), (4.7) withthat of finite gauge transformations which, as explained in section III and [10], do describelong range propagation. Let us consider 3 successive edges e , e , e of the charge net so thatthe embedding charge sequence is now ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ). Let us call the succesivevertices u and v so that u is the ‘3-1’ vertex and v , as before, is the ‘1-2’ vertex and let usassume that these vertices are spacelike. Now let us consider a sequence of actions of ‘-’diffeomorphisms each of which stretches the ‘-’ labels to the left and each of which is identityto the left of e and to the right of e . In visualising this action recall that the charge networkhas a ‘ + ⊗− product structure (2.11). With an appropriate choice of sequence of such ‘-’diffeomorphisms, we obtain the following sequence of embedding charges:( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) → ( k +3 , k − ) , ( k +1 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) → (4.10)( k +3 , k − ) , ( k +3 , k − ) , ( k +1 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) → ( k +3 , k − ) , ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) (4.11) → ( k +3 , k − ) , ( k +3 , k − ) , ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) (4.12)Thus, at the end of this sequence of actions, the ‘-’ edge with charge ‘ k − ’ has moved leftwardso as to overlap the k +3 edge yielding the point ( ~ k +3 , ~ k − ) which is awayfrom points on the original slice. In order for this to happen in a continuous manner asdepicted in (4.10)- (4.12), it is necessary to have the intermediate step (4.11) where the k − charge completely displaces the k − charge from the ‘+’ edge with label k +1 .In contrast, since the action of the Hamiltonian constraint (4.6),(4.7) is for sufficientlysmall δ , this action even if repeated, can never completely erase any of the original pairs ofedge labels. In particular even after repeated actions of the Hamiltonian constraint, there isalways an edge with the label ( k +1 , k − ). This survival of the original edge labels is directlytied to the ultralocality of the action of the Hamiltonian constraint: since its action onlymodifies the structure in a small enough neighbourhood of a vertex, each one of theoriginal edges always has a part which is not affected by this action. Viewed in this manner,it is the inability of repeated actions of the Hamiltonian constraint to erase such originalpairs of labels which obstructs long range propagation.As mentioned in section I, in the next section we offer a new perspective on propagationand show how this obstruction is sidestepped. Once this obstruction is sidestepped it is notdifficult to prove that long range propagation ensues. Since we believe that this proof isjust an added layer of polymer PFT technicalities, we relegate this proof to the appendixand concentrate in the main body of the paper on the key lesson of this paper, namely theevasion of the obstacle described above and the robust structural reasons for this evasion. V. PROPAGATION
As shown in the last section and as anticipated by Smolin, repeated ultralocal actions ofthe finite triangulation Hamiltonian constraint on a kinematic state do not propagate quan- The precise notion of ‘small enough’ is defined in L1 , section V distributions which may be expressed as formal sums over kinematic states. Hence a putative solution toboth the Hamiltonian and the diffeomorphism constraints must also be a distributional sum.Consider such a solution and let a finest lattice charge net describing some discrete Cauchyslice with quantum matter be a summand in the sum which represents this solution. Recall,from section III that this slice (together with its quantum matter) evolves under the actionof the finite gauge transformations (Φ + , Φ − ). If the finest lattice charge net correspondingto any finite evolution of this discrete slice with quantum matter is also a summand in thesum which represents this solution then we shall say that the solution encodes propagationeffects. In the previous section we isolated the key obstruction to long range evolution by repeatedactions of the Hamiltonian constraint. We now show how our new formulation of propagationovercomes this obstruction. Specifically, as in section IV, let us consider 3 successive edgesof a finest lattice charge net with embedding charge sequence ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − )with succesive vertices u and v so that u is the ‘3-1’ vertex and v is the ‘1-2’ vertex. Re-call from section IV that repeated actions of the finite triangulation Hamiltonian constrainton this charge net are unable to produce the charge net with embedding charge sequence( k +3 , k − ) , ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) (see equation (4.11)), this inability being the key ob-struction to the generation of long range evolution through such repeated actions. We nowshow that if a physical state in the kernel of the diffeomorphism and Hamiltonian constraintshas the original ‘( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − )’ chargenet as a summand, it necessarily has achargenet with the desired ( k +3 , k − ) , ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) sequence. We proceed asfollows.Let Ψ be a distribution represented by a sum of ‘bra’ states. We shall say that a chargenet | γ, ~k + , ~k − , ~l + , ~l − i is in Ψ iff the bra h γ, ~k + , ~k − , ~l + , ~l − | is a summand in the sum over braswhich represents Ψ. Next recall that Ψ is a solution to the continuum limit of the finitetriangulation Hamiltonian constraint (4.6) iff for every | γ, ~k + , ~k − , ~l + , ~l − i we have thatlim δ → Ψ( − i ~ X v N ( v ) ˆ U ham,δ,v − δ | γ, ~k + , ~k − , ~l + , ~l − i ) = 0 . (5.1)Now suppose that Ψ is a solution to the Hamiltonian constraint (5.1) and that | γ, ~k + , ~k − , ~l + , ~l − i is in Ψ. Then it must be the case that for all sufficiently small δ thatˆ U ham,δ,v | γ, ~k + , ~k − , ~l + , ~l − i is also in Ψ (else the nontrivial contribution from the ‘ ’ term in(5.1) will not be cancelled). A similar arguementation leads to the converse namely that As seen in section III this statement of propagation holds for physical states obtained by group averaging.So we could as well have formulated propagation as the condition that the kernel of the diffeomorphismand Hamiltonian constraints be identical to the physical state space obtained by group averaging. Wechoose to formulate the statement in the way we have done because such a formulation generalises moreeasily to the case of LQG where we do not have the possibility of group averaging over the transformationsgenerated by all the constraints . U ham,δ,v | γ, ~k + , ~k − , ~l + , ~l − i is in Ψ for all sufficiently small δ , then | γ, ~k + , ~k − , ~l + , ~l − i is also in Ψ.Similarly, Ψ is a solution to the continuum limit of the finite triangulation Hamiltonianconstraint(4.7) iff for every | γ, ~k + , ~k − , ~l + , ~l − i we have thatlim δ → Ψ( i ~ X v N ( v ) ˆ U † ham,δ,v − δ | γ, ~k + , ~k − , ~l + , ~l − i ) = 0 (5.2)Similar arguments imply that for Ψ which is a solution to the Hamiltonian constraint (5.2),iff | γ, ~k + , ~k − , ~l + , ~l − i is in Ψ then for all for sufficiently small δ , ˆ U † ham,δ,v | γ, ~k + , ~k − , ~l + , ~l − i is inΨ. It is also useful for what follows to recall the following:(a) the action of ˆ U ham,δ,v on any charge net | γ, ~k + , ~k − , ~l + , ~l − i is such that for small enough δ > U ham,δ,v | γ, ~k + , ~k − , ~l + , ~l − i are in the same diffeomorphism class,(b) a similar statement holds for ˆ U † ham,δ,v ,(c) Ψ is a solution to the diffeomorphism constraints iff it is a linear combination of statesof the form (4.2).From the above discussion, it is straightforward to check that the following Lemma L1 holds. L1 : Let Ψ be a solution to the diffeomorphism constraint, the Hamiltonian constraint in(5.1) and the Hamiltonian constraint in (5.2). Then the following statements hold:(i) The charge net | γ, ~k + , ~k − , ~l + , ~l − i is in Ψ iff all diffeomorphic images of | γ, ~k + , ~k − , ~l + , ~l − i are in Ψ,(ii)The charge net | γ, ~k + , ~k − , ~l + , ~l − i is in Ψ iff for all sufficiently small δ > U ham,δ,v | γ, ~k + , ~k − , ~l + , ~l − i is in Ψ,(iii) The charge net | γ, ~k + , ~k − , ~l + , ~l − i is in Ψ iff for all sufficiently small δ > U † ham,δ,v | γ, ~k + , ~k − , ~l + , ~l − i is in Ψ,where in (ii) (respectively (iii)) ‘sufficiently small’ means ‘sufficiently small that thediffeomorphism class of ˆ U ham,δ,v | γ, ~k + , ~k − , ~l + , ~l − i (respectively ˆ U † ham,δ,v | γ, ~k + , ~k − , ~l + , ~l − i )remains the same for all such δ . We are now ready to state and prove our desired result.
Proposition : Let Ψ be a solution to the diffeomorphism constraint, the Hamiltonianconstraint (5.1) and the Hamiltonian constraint (5.2). Let | γ, ~k + , ~k − , ~l + , ~l − i be a finestlattice chargenet in Ψ. Let | γ, ~k + , ~k − , ~l + , ~l − i have 3 consecutive edges with embedding This characterization of ‘sufficiently small’ follows from (a)-(c) and the fact that Ψ in L1 is, in particular,a solution to the diffeomorphism constraint. The proof below applies independent whether the 2 vertices associated with these edges are spacelike or k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ). Then the chargenet with these 3 edges replaced by 4successive edges with the embedding charge sequence ( k +3 , k − ) , ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − )is necessarily in Ψ. Proof:
Denote the ‘1-2’ vertex of | γ, ~k + , ~k − , ~l + , ~l − i by u and the ‘2-3’ vertex by v . There are3 steps to the proof: Step 1 : Act with ˆ U ham,δ ,v on | γ, ~k + , ~k − , ~l + , ~l − i at its ‘1-2’ vertex v . We obtain the chargenetwith the above sequence replaced by( k +3 , k − ) , ( k +1 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) . (5.3) L1 (ii) implies the chargenet so obtained is also in Ψ (we have implicitly chosen δ smallenough that L1 (ii) applies). Step 2:
Act on the chargenet obtained at the end of Step 1 with ˆ U ham,δ ,u on its ‘3-1’ vertex u to get the sequence ( k +3 , k − ) , ( k +3 , k − ) , ( k +1 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) (5.4) L1 (ii) implies the chargenet so obtained is also in Ψ (similar to Step 1, we have implicitlychosen δ small enough that L1 (ii) applies). Step 3:
Consider the (desired) chargenet with sequence( k +3 , k − ) , ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) (5.5)and call the vertex at the intersection of the ( k +3 , k − ) and ( k +1 , k − ) edges as w . Act onthis chargenet by ˆ U † ham,δ,w for sufficiently small δ in the sense of (b) above. It followsthat for every such δ >
0, this action yields a chargenet with the sequence (5.4) which isdiffeomorphic to the particular chargenet obtained at the end of Step 2. Since the latterchargenet is in Ψ, L1 (i) implies that the chargenets, obtained by the action of ˆ U † ham,δ,w forall sufficiently small δ on the desired chargenet with sequence (5.5), are in Ψ. The (converseof) L1 (iii) then implies that the desired chargenet is also in Ψ. This completes the proof.Note that the first two steps of the proof involve actions by the Hamiltonian constraint onthe chargenet in question. Hence, from section IV these steps by themselves are incapableof generating the desired result. It is Step 3 which is the key step. The success of thisstep hinges on the imposition, as a constraint, of the ‘kinematic adjoint’ (4.7),(5.2) of theconstraint (4.6),(5.1) together with its ‘ ˆ U † −
1’ structure.Finally, we also note that, acting by ˆ U ham,δ,w on the chargenet (5.5) with sequence( k +3 , k − ) , ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) (5.6)at its vertex w (where the ( k +3 , k − ) and ( k +1 , k − ) edges meet) we obtain a chargenet withsequence ( k +3 , k − ) , ( k +3 , k − ) , ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ) . (5.7)If in the original chargnet the vertices u, v are spacelike, it is easy to see that the point not. ~ k +3 , ~ k − ) represents a 2 lattice displacement from the original set of points corresponding to( ~ k +3 , ~ k − ) , ( ~ k +1 , ~ k − ) , ( ~ k +2 , ~ k − ). Thus as indicated in section I we are able to demonstrateevolution beyond 1 lattice displacement to 2 lattice displacements. As mentioned at the endof section IV, the demonstration of long range evolution is relegated to the Appendix. VI. DISCUSSION
Before we proceed to more general remarks, we comment on the derivation of the con-straint action (4.7) from choices of finite triangulation approximants to the local fields whichcomprise the Hamiltonian constraint. The reader not interested in fine technical details ofour prior work [13] may skip the next two paragraphs and go on to peruse the more generalremarks.First consider the derivation of the action (4.6). In this regard Reference [13] provides adetailed derivation of finite triangulation approximants to H + , H − and it is from these thatthe approximant to the Hamiltonian constraint operator (equation (112),[13]) is obtained.It is straightforward to see that this is the same as (4.6) albeit in a slightly different no-tation. Recall that in order to obtained the desired constraint operator action (112),[13],the embedding sector approximants to the local fields in the constraint are constructedstraightforwardly in [13], first as appropriate classical approximants involving state depen-dent charges, and then as operators. Also recall that Reference [13] is unable derive mattersector approximants in this way. More in detail, that work is unable to construct classi-cal approximants to the local matter fields in the Hamiltonian constraint such that theirreplacement by operators is consistent with the desired action (112) of [13] . Hence anindirect appeal to the Hamiltonian vector fields of the matter part of the constraints is made[13] and this constitutes a slight departure from the strict ‘Thiemann-like’ prescription.Next consider the derivation of the adjoint action (4.7). It turns out that the action (4.7)requires a slightly different choice of approximants to local fields in the constraints. It is onceagain straighforward to construct the desired embedding momenta approximants . How-ever, for the matter sector one needs to again consider Hamiltonian vector fields. While wehave not done this in detail, we anticipate that the considerations of Section VB,[13] may bemimicked with slightly different choices so as to obtain an action which contributes appropri-ately to the Hamiltonian constraint approximant so as to obtain (4.7). More generally, ourcurrent viewpoint is that it is the Hamiltonian vector field structure of the constraint ratherthan the constraint itself which is primary and that this Hamiltonian vector field structureis what one should import in a suitable fashion into quantum theory even if one is unable toprovide concrete classical approximants to the constraint itself. From this viewpoint one candirectly posit the actions (4.6), (4.7) as approximants to the infinitesmal transformationsgenerated by the constraint (2.6) without unduly worrying about classical approximants tothe constraint itself.Finally, in this work we have not imposed the zero mode constraint [9, 10] as we feel We think that the underlying reason for this inability is that, as mentioned in II B 1, there is no ‘unpoly-merised’ matter variable. See the discussion immediately after (47),[13]. There is a typographical error in the choice of approximantfor the ‘+’ embedding momenta which is claimed to lead to the leftward displacement of the ‘+’ vertex:the subscript on the embedding holonomy should be △ − △ . δ , we expect it to commute with the continuum limit action of(4.6),(4.7). Hence we expect that it shouldnt matter whether we find the kernel of theHamiltonian and diffeomorphism constraints first and then group average these solutionsover the zero mode constraint or whether we first solve the zero mode constraint by groupaveraging, define the Hamiltonian and diffeomorphism constraints on the resulting statesand then find the kernel of the Hamiltonian and diffeomorphism constraints. Verifying theabove expectations, while straightforward, lies outside the scope of this work. Incidentally,we believe that in hindsight, the treatment of the zero mode in [13] was too perfunctory,that the arguements there should be seen as arguements prior to the implementation ofthe zero mode constraint and that a proper treatment of the zero mode constraint shouldexplicitly verify our expectations as stated above. This concludes our discussion of finetechnical matters in polymer PFT.We now discuss the key structural lessons from this work for LQG. Let us refer to thenew charge nets obtained by the action of a constraint on a given chargenet as ‘children’of this ‘parent’ chargenet. In this language Smolin’s general considerations imply, correctly,that such children and their descendants do not encode long range propagation. However,given the structure of the constraints (4.6)- (4.7), Lemma L1 implies that if a parent is inΨ so are its children, and, conversely if any child is in Ψ then so are all its parents. It isthe converse statement which provides the key ingredient of ‘non-unique parentage’ in thecrucial Step 3 of our proof in the previous section.From a general point of view what Step 3 effectively achieves is the merging of twovertices of a ‘child’ into a single vertex of a ‘parent’ (the single vertex being w in our proof).Note that this merging cannot be achieved by the action (4.6) nor by its kinematic adjointaction (4.7) on the child. This is apparent from the considerations of section III which applyequally to both actions, the key point being that these actions are only defined as finitetriangulation constraint actions for sufficiently small δ . It is this caveat of “ for sufficientlysmall δ ” that leads to ultralocality and the impossibility of merging vertices of the child byaction on the child. Rather, the merging is achieved by seperating vertices of the parentvia the action of the kinematic adjoint and using the structure of the constraint action interms of the difference of a unitary operator and the identity to conclude that the existenceof children in a physical state impy the existence of all their possible ancestors. Once the 2vertices of the child have been effectively merged through this structure to yield the desiredparent, a suitably chosen action of the Hamiltonian constraint (see the last paragraph of V)on this parent creates a different child and the sequence ‘parent → child → different parent → different child’ constitutes an evolution path to a final discrete Cauchy slice two latticespacings away. This sequence may be viewed as the propagation of a perturbation (namelythe k − charge together with the associated ‘-’ matter charges) ‘leftward’ along the chargenet.From the above discussion it is apparent that the key structures responsible for propaga-tion are exactly (i) and (ii) of section I and that propagation should be viewed as encodedin the structure of physical states rather than as a property of repeated actions of the finitetriangulation Hamiltonian constraint on kinematical states. To conclude, while we do expect0the general Thiemann procedure to yield a Hamiltonian constraint with ultralocal action,we are optmistic that the structural lessons arising out of this work can be imported in asuitable way to LQG so as to restrict the choice of this ultralocal action in such a way thatphysical states in the kernel of the Hamiltonian (and diffeomorphism) constraints do encodepropagation effects. Acknowledgements
I thank Lee Smolin for raising the general issues dealt with in this work during the Loops13 conference. This work was motivated by a conversation with Michael Reisenberger duringthat conference and I am very grateful for his remarks in the course of that conversation.
Appendix A: Proof of long range propagation
In this appendix we assume familiarity with the contents of [9, 10, 13]. We shall also set ~ = 1 by a suitable choice of units.Recall the following from [10]. The diffeomorphisms of the circle, Φ ± , admit periodicextensions to the real line, also denoted by Φ ± . As in [10], denote the set ( γ ± , ~k ± , ~l ± ) by s ± and the corresponding states | γ ± , ~k ± , ~l ± i by | s ± i with | s + i⊗| s − i := | s + , s − i . s ± is referred toas a charge network or charge network label and | s ± i as a charge network state. Whereas s ± is defined on [0 , π ] its extension s ± ext is defined on the entire real line by periodic extensionof the graph γ ± , its edges and its matter charge labels and quasi-periodic extension (withappropriate augmentation by factors of ± L ) of its embedding charge labels. The periodicextensions of Φ ± have a well defined action on s ± ext . The state defined by the restriction ofthe image of this action to the interval [0 , π ] coincides with that obtained by the action ofˆ U ± (Φ ± ) on | s ± i [9, 10] so that :ˆ U ± (Φ ± ) | s ± , i = | s ± Φ ± i , s ± Φ ± := Φ ± ( s ± ext ) | [0 , π ] . (A1)Clearly, the action of Φ ± on any interval of coordinate length 2 π determines its actioneverywhere on the real line by virtue of the periodicity of this action. Similarly the restrictionof s ± ext to any interval of coordinate length 2 π determines s ± ext everywhere on the real line.Since ˆ U ham,δ,v , ˆ U † ham,δ,v are constructed from ˆ U ± ( φ δ,v ) , ˆ U †± ( φ δ,v ) it follows that the actionof the finite triangulation Hamiltonian constraint on any state | s + , s − i is determined by theaction of the diffeomorphisms Φ ± = φ δ,v (and their inverses) on the restriction of s ± ext to anyinterval of coordinate length 2 π .More in detail, for some real y , consider the interval [ y, y + 2 π ]. Φ ± maps this interval tothe interval [ y ± , y ± +2 π ] where we have set Φ ± ( y ) =: y ± . Consider the restriction, s ± ext | [ y,y +2 π ] of s ± ext to the interval [ y, y +2 π ]. Clearly Φ ± has a natural action on s ± ext | [ y,y +2 π ] (it maps everyedge e of the graph underlying s ± ext | [ y,y +2 π ] into its image Φ ± ( e ) in [ y ± , y ± + 2 π ], with Φ ± ( e )being colored by the same charges as e ). Denote the resulting charge net on [ y ± , y ± + 2 π ] byΦ ± ( s ± ext | [ y,y +2 π ] ). It is straightforward to check that Φ ± ( s ± ext | [ y,y +2 π ] ) is just the restriction ofΦ ± ( s ext ) to the interval [ y ± , y ± +2 π ]. It then follows that the extension, (Φ ± ( s ± ext | [ y,y +2 π ] )) ext , See Footnote 3 in this regard. ± ( s ± ext | [ y,y +2 π ] ) to the real line is just Φ ± ( s ext ). Finally, the restriction ofthis extension to [0 , π ] is the just the charge net s ± Φ ± i.e.(Φ ± ( s ± ext | [ y,y +2 π ] ) ext ) | [0 , π ] = s ± Φ ± . (A2)Since ˆ U ± (Φ ± ) | s ± i = | s ± Φ ± i , the content of this paragraph is just a transcription to mathe-matical notation of what we said in words in the previous paragraph.The above discussion implies that L1 , Section V may be rephrased as follows in thenotation used above (this rephrasing, while cumbersome and seemingly roundabout, isuseful for our purposes in this appendix). L2 : Let Ψ be a solution to the diffeomorphism constraint, the Hamiltonian constraint in(5.1) and the Hamiltonian constraint in (5.2). Let | s + , s − i be a finest lattice chargenetstate. The charge net state | s + , s − i is in Ψ iff the set of charge net states {| t + , t − i} is in Ψwhere the elements | t + , t − i of this set are defined by any of (i)- (iii) below, with y any fixedreal number and v ∈ [ y, y + 2 π ]:(i) ( φ ( t ± ext ) | [ y,y +2 π ] ) ext | [0 , π ] = s ± for any diffeomorphism φ .(ii) t + = ( φ δ,v ( s + ext | [ y,y +2 π ] )) ext | [0 , π ] , t − = ( φ − δ,v ( s − ext | [ y,y +2 π ] )) ext | [0 , π ] for all sufficientlysmall δ .(iii) t + = ( φ − δ,v ( s + ext | [ y,y +2 π ] )) ext | [0 , π ] , t − = ( φ δ,v ( s − ext | [ y,y +2 π ])) ext | [0 , π ] for all sufficientlysmall δ .where in (ii)-(iii) ‘sufficiently small’ means ‘sufficiently small that the diffeomorphism classof | t + , t − i does not change. Next note that similar to (2.11) we may consider a fine enough graph which underliesboth s + and s − and whose edges carry both + and − charges. Let us denote the resultinglabel set ( γ, ~k + , ~k − , ~l + , ~l − ) by s and set | s i := | s + , s − i . (A3)We may also define the extended label s ext by a periodic extension γ ext of γ , a periodicextension of the matter charge labels and appropriate (quasi)periodic extensions of theembedding charges. Clearly, γ ext constitutes a fine enough graph which underlies both s + ext and s − ext , and s ext accomodates both the + and − charge labels of s + ext and s − ext .The Proposition of Section V can then be rephrased as: P1 : Let Ψ be a solution to the diffeomorphism constraint, the Hamiltonian constraint (5.1)and the Hamiltonian constraint (5.2). Let | s i = | s + , s − i be a finest lattice chargenet inΨ. Let y be some real number. Let the restriction of s ext to the interval [ y, y + 2 π ] be s ext | [ y,y +2 π ] . Let the graph underlying s ext | [ y,y +2 π ] have 3 consecutive edges e , e , e with Recall that φ δ,v is such that φ δ,v moves v to the right by a coordinate distance δ , its inverse φ − δ,v moves v to the left by a distance δ and φ δ,v is identity outside an interval of size of order δ about v . e , e , e ⊆ [ y, y + 2 π ] with embedding charges ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ). Consider thecharge net state | s ′ i = | s ′ + , s ′− i such that s ′ ext | [ y,y +2 π ] agrees with s ext | [ y,y +2 π ] except thatthese 3 edges of the latter are replaced in the former by 4 successive edges e ′ , e ′ , e ′ , e ′ withthe embedding charge sequence ( k +3 , k − ) , ( k +3 , k − ) , ( k +1 , k − ) , ( k +2 , k − ). Then | s ′ i is necessarily in Ψ.It is straightforward to check that P1 can be proved along the lines of the proof of theProposition of section VNext, let s, y, s ′ , Ψ be as in P1 . Clearly, there exists a diffeomorphism φ which is identityto the right of e and the left of e in [ y, y + 2 π ] and which maps s ′ to some s ′′ such thatin s ′′ ext | [ y,y +2 π ] we have that φ ( e ′ ) = e , φ ( e ′ ) = e , φ ( e ′ ∪ e ′ ) = e . Thus s ′′ ext | [ y,y +2 π ] agreeswith s ext | [ y,y +2 π ] outside the interval φ ( e ′ ) ∪ e . L2 (i) then implies that the followingcorollary to P1 holds: C1 : Given s, y, s ′ , s ′′ as above, | s ′′ i is also in Ψ.With these results in place we now show that if a finest lattice charge net state | s i is inany diffeomorphism invariant solution Ψ to the Hamiltonian constraints (5.1) and (5.2), thenall states related to | s i by the action of finite gauge transformations generated by H + , H − are also in Ψ. It then follows from the notion of propagation introduced in section V thatΨ encodes long range propagation effects. We proceed by proving the following lemmas L3 - L7 .In the proof of L3 below we shall use the following notion of ‘ sequence of ‘-’ embeddingcharges ’. Let s = ( s + , s − ) be a finest lattice chargenet. Let s − ext be the extension of s − andconsider the restriction s − ext | [ y,y +2 π ] to some 2 π interval [ y, y + 2 π ]. Consider any fine enoughgraph underlying the charges on s − ext | [ y,y +2 π ] . Let the edges of this graph be e I , I = 1 , .., B where e J is to the right of e I for J > I . Let the ‘-’ embedding charge on e I be k − I . Thenthe ordered set of ‘-’ charges ( k − , k − , .., k − B ) is referred to as the sequence of ‘-’ embeddingcharges associated with s ext | [ y,y +2 π ] . The finest lattice property implies that this sequenceis non-increasing and that for the coarsest graph underlying s − ext | [ y,y +2 π ] this sequence isstrictly decreasing. Thus, depending on the fineness of the graph, there may be severalinstances of a number of successive entries in the sequence being identical. In the proofof L3 below we shall refer to such sequences directly without explicitly constructing thegraphs which define them; however it is to be understood that such graphs exist (as thereader may verify, it is straightforward to show their existence). L3 Let | s i = | s + , s − i be a finest lattice state which is in a solution Ψ to the diffeomorphismconstraint and the Hamiltonian constraints (5.1) and (5.2). Let s ′ + = s + . Let s ′− be suchthat the set of its matter charge labels, ordered edgewise from left to right is identicalto the corresponding set for s − and such that the set of its embedding charge labels,ordered from left to right are obtained by decreasing each of the elements of the correspond-ing set for s − by M L for some arbitrary positive integer M . Then | s ′ i = | s ′ + , s ′− i is also in Ψ. Proof : As in Footnote 8, we shall assume
N >>
4. Let the coarsest graph underlying s haveedges e I , I = 1 , .., A with embedding charges ( k + I , k − I ) with e I to the left of e J for J > I .Clearly
A >>
4. Also note that the chargenets in L2 , P1 , C1 are related by the action ofgauge transformations so that these chargenets are all in the finest lattice sector and respect3properties (i)- (iii), section III. We shall implicitly use this fact repeatedly in what follows.Consider s ext and let e be the edge (of the coarsest graph underlying s ext ) in the interval[ − π,
0] starting at some point y ∈ [ − π,
0] and ending at the orgin. In the next 3 para-graphs we will repeatedly apply C1 to appropriately chosen triplets of edges in the interval[ y , y + 2 π ], this being the interval spanned by e I , I = 0 , .., A − C1 applied to the 3 edges e , e , e implies that we can displace the ‘-’ charge k − on theedge e by extending the coloring k − to e so that e , e are colored by k − . Denote theresulting extended charge net by s ext so that its restriction to [0 , π ] is the charge net s .Clearly the ‘-’ embedding charge on the last edge of s (in the interval [0 , π ]) is now k − − L (see [9, 10] and Footnote 5). Let e (1)0 be the edge in s ext which ends at x = 0. The edge e (1)0 corresponds to the edge φ ( e ′ ) and e to φ ( e ′ ∪ e ′ ) in the remarks before C1 . It followsthat e (1)0 is contained in e . Hence, denoting the left end point of e (1)0 by y (1) , we have that0 > y (1)0 > y .Iterate this process to spread k − leftwards in s at the cost of k − as follows. First notethat the graph which underlies s can be chosen such that its first 3 edges coincide with e , e , e . From C1 we can spread k − to e . From P1 and C1 note that in the resultingchargenet the edges e , e , e can still be taken to be the first 3 edges with e colored by k − and e , e by k − . Next consider the extension of this chargenet and the edges e (1)0 , e , e therein. Using C1 we can move k − from e to e . This yields the desired charge net s withfirst edge colored by k − and last edge by k − − L . Denote the edge in s ext ending at x = 0by e (2)0 and let its left end point be y (2)0 . Here e (2)0 corresponds to φ ( e ′ ) and e (1)0 to φ ( e ′ ∪ e ′ )in the remarks before C1 . It follows that C1 implies that 0 > y (2)0 > y (1) > y .Clearly, we can iterate this process such that after A − s A − withfirst charge k − A − , last charge k − A − − L and the edge in s A − ext ending at the origin with leftendpoint y ( A − such that 0 > y ( A − > y ( A − ... > y .Next, consider the interval [ y ( A − , y ( A − + 2 π ]. In this paragraph we shall repeatedlyapply C1 to appropriately chosen triplets of edges in this interval. Note that the sequence of‘-’ charges in s A − ext restricted to this interval reads ( k − A − , k − A − , k − A , k − − L, k − − L, .., k − A − − L ). Repeated application of C1 to appropriate edges in this interval results in the spread of k − A leftwards till x = 0 resulting in the charge net s A − . The sequence of charges in s A − ext restricted to this interval reads, as before, ( k − A − , k − A − , k − A , k − − L, k − − L, .., k − A − − L )except that now the edge ending at x = 0 has charge k − A − and the edge starting at x = 0has charge k − A . Finally, we repeatedly apply C1 to appropriate edges in this interval so asto spread k − − L leftwards till the origin to yield the charge net s A . The sequence of ‘-’charges in the restriction of s A ext to this interval is unaltered but now the edge starting outfrom the origin has charge k − − L and the edge ending at the origin has charge k − A . Thisimplies that the charge sequence in s A in the interval [0 , π ] reads( k − − L, k − − L, .., k − A − − L, k − A − − L, k − A − − L, k − A − L ).Finally taking s A as the initial charge net and repeating the above procedure M timesleads to the desired result. Note N1 : Interchanging the role of ˆ U δ,v and ˆ U † δ,v in the proof of the Proposition of SectionV and in that of P1 , C1 results in the rightward movement of the ‘-’ charges. Withthis modification considerations identical to L3 lead to the ‘-’ embedding charges beingaugmented by factors of + M L . Similarly, it is straightforward to see that the Propositionof Section V as well as P1 , C1 and L3 above can be modified to reflect leftward and4rightward movement of the ‘+’ charges with augmentation of the ‘+’ charges by factors of ± M L . The fact that we are dealing with finest lattice charge nets satisfying (ii) of sectionIII immeditiately implies that the matter charges are dragged along with the embeddingcharges yielding the desired charge configurations. L4 Let | s i = | s + , s − i be a finest lattice state with N ± edges as in (iii) of Section III. Let | s i be in a solution Ψ to the diffeomorphism constraint and the Hamiltonian constraints(5.1) and (5.2). Then there exists | s ′ i = | s ′ + , s ′− i in Ψ such that s ′± have N + 1 dis-tinct embedding charges i.e. the coarsest graphs γ ′± underlying s ′ ± have N ± = N + 1 edges. Proof : The proof consist of the following 2 steps.
Step 1 : If N − = N + 1 proceed to Step 2. If not then we have N − = N . From(ii), (iii) Section III, this means that the embedding charge sequence on s − is of the form k − , k − , ..., k − N with k − − k − N = L − a . Application of ˆ U δ,v =0 , for small enough δ , to | s i dragspart of the first edge of s − leftwards. The rightmost charge of resulting chargenet s is then,by (quasi)periodicity, k − − L and the left most charge is still k − so that now N − = N + 1.By L2 , the resulting chargenet state | s i is in Ψ. Step 2 : In the charge net state | s i at the end of Step 1, if N + = N + 1 we are done.If N + = N , an application of ˆ U δ ′ ,v =0 for small enough δ ′ to | s i leads to the rightwarddragging of part of the first edge of s +1 while maintaining N − = N + 1. It is then easy tosee that the resulting charge net has N + = N + 1. Further the resulting charge net is in Ψby L2 .This completes the proof. L5 Let | s i , | s ′ i be finest lattice charge nets with | s ′ i in Ψ ′ where Ψ ′ is a solution to thediffeomorphism constraint and the Hamiltonian constraints (5.1), (5.2). Let s ± , s ′± have N + 1 distinct ‘ ± ’ embedding charges. Then there exist | t ′ i in Ψ ′ such that s ± , t ′± haveidentical sets of ± embedding charge labels. Proof : Let us number the edges (and their embedding charge labels) in s − , s ′− from 1,..N+1as we proceed rightwards on the coarsest graphs underlying these charge nets. Thus the leftmost charge in s − is k − . By the finest lattice property there exists a unique charge label k ′− m in s ′− with the property that k ′− m is the largest ‘-’ embedding charge in s ′− such that k ′− m − k − is an integer multiple ‘ M − ’ times L . If m = 1 then we can apply L3 to construct t − such that its ‘-’ embedding charge set agrees with that of s − .If m = 1, we apply C1 repeatedly to s ′ so as to spread k ′− m leftwards in a manner identicalto that employed in the proof of L3 . As a result, on the resulting chargenet k ′− m becomesthe first charge. Since k ′− m = k ′− m − , it is easily verified that this charge net has N distinct‘-’ charges. It is straightforward to verify that an application of ˆ U δ,v =0 for small enough δ moves the first edge of the resulting chargenet slightly to the left so as to change N − to N + 1 while maintaining the ‘+’ embedding charge set. Call the chargenet so obtained as s ′ . Next, we can identify the unique smallest ‘+’ embedding charge on s ′ which differs from k +1 by an integer multiple M + of L . If this charge colors the first edge of s ′ we are done. This rules out m = N + 1 s ′ by application of N1 so that it becomesthe first ‘+’ embedding charge on the resulting chargenet. An application of ˆ U δ ′ ,v =0 forsmall enough δ ′ on this chargenet with N + = N ensures that the resulting chargenet s ′ has N + = N + 1 while maintaining N − = N + 1. As a result s ′ ± has the same ± embeddingchargesets as s modulo factors of M ± L .Finally we can apply L3 , N1 to obtain the desired charge net t ′ . From L1 , L3 , N1 itfollows that | t ′ i is in Ψ ′ . L6 Let | s i , | s ′ i be finest lattice charge nets with | s ′ i in Ψ ′ where Ψ ′ is a solution to the diffeo-morphism constraint and the Hamiltonian constraints (5.1), (5.2). Let s ± , s ′± have N +1 ‘ ± ’distinct embedding charges and let the sets of these ± embedding charge labels be identical.Then there exists | r ′ i in Ψ ′ such that the embedding charge nets underlying r ′ , s are identical. Proof : By an appropriate choice of spatial diffeomorphism φ such that φ = in a smallneighbourhood of x = 0 (and hence x = 2 π ), we can arrange for t = s ′ φ to be such that its‘+’ embedding charge net matches with that of s . Consider the coarsest graphs underlying s − , t − . Let their first edges be e , f . If f = e , we can proceed to a comparision of thesecond edges of these charge nets. If f is longer than e we can stretch the second edge f of t (with charge k − ) leftwards by repeated applications of C1 so that on the (coarsestgraph underlying the) resulting charge net t − , the first edge is also e . If f is shorter that e then we can stretch f rightwards by repeated applications of N1 so that in the coarsestgraph underlying the resulting chargenet t − , the first edge is again e . Next, we compare the second edges on s − , t − . If they are unequal, we can use C1 , N1 to stretch the 3rd edge of t − leftward or the second edge rightward so as to generate t − onwhich the first two edges match those of s − . Clearly iterations of this procedure ensure that r ′− := t − N − agrees with s − N − while maintaining r ′ + = t + = s + . From C1 , N1 it followsthat we have constructed the desired | r ′ i in Ψ ′ . L7 :Let | s ′ i be a finest lattice charge net in Ψ ′ where Ψ ′ is a solution to the diffeomorphismconstraints and the Hamiltonian constraints (5.1), (5.2). Let s be related to s ′ by a finitegauge transformation Φ + , Φ − . Then | s i is also in Ψ ′ . Proof : Since s is gauge related to s ′ , it is also a finest lattice state. The proof of L4 impliesthat there exists | s i which is obtained by the action of some ˆ U δ,v =0 , ˆ U δ ′ ,v =0 on | s i for smallenough δ, δ ′ , such that s has N + 1 ‘+’ and N + 1 ‘-’ embedding charges. From L2 it followsthat if | s i is in any solution to the diffeomorphism constraint and to (5.1), (5.2), then | s i must be in the same solution.Next, L4 implies that there exists s ′ with N + 1 ‘+’ and N + 1 ‘-’ embedding charges suchthat | s ′ i is in Ψ ′ and L5 implies that there exists | s ′ i in Ψ ′ such that s ′ , s have identicalsets of embedding charges. From L6 , there exists | s ′ i in Ψ ′ such that s ′ , s have identicalembedding charge networks.Next, it is straightforward to check that, as asserted in [10], if two finest lattice statesare gauge related and have the same embedding charge networks, they must have the samematter charge networks. Since the transformations relating s to s and s ′ to s ′ i , i = 1 , , Note that in applying C1 , N1 . we are free to choose an appropriately fine graph which underlies t . 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