Dynamical Quantum Geometry (DQG Programme)
aa r X i v : . [ g r- q c ] S e p Dynamical Quantum Geometry(DQG Programme)
Tim A. Koslowski [email protected]
Institut f¨ur Theoretische Physik und AstrophysikUniversit¨at W¨urzburgAm HublandD-97074 W¨urzburgEuropean UnionSeptember 24, 2018
Abstract
In this brief note (written as a lengthy letter), we describe the con-struction of a representation for the Weyl-algebra underlying Loop Quan-tum Geometry constructed from a diffeomorphism variant state, whichcorresponds to a ”condensate” of Loop Quantum Geometry, resembling astatic spatial geometry. We present the kinematical GNS-representationand the gauge- and diffeomorphism invariant Hilbert space representationand show that the expectation values of the geometric operators take es-sentialy classical values plus quantum corrections, which is similar to a”local condensate” of quantum geometry. We describe the idea for theconstruction of a scale dependent asymptotic map into a family of scaledependent lattice gauge theories, where scale separates the essential ge-ometry and a low energy effective theory, which is described as degreesof freedom in the lattice gauge theory. If this idea can be implementedthen it is likely to turn out that this Hilbert space contains in addition togravity also gauge coupled ”extra degrees of freedom”, which may not bedynamically irrelevant.
The algebra that underlies Loop Quantum Gravity is generated by ma-trix elements h e ( A ) IJ of SU (2)-holonomies along arbitrary piecewise analyt-ical curves e in the Cauchy surface Σ and the fluxes E i ( S ) of the conjugatedelectric fields through arbitrary piecewise analytical surfaces S . This algebracarries a canonical representation of the SU (2)-gauge transformations and thepiecewise analytical diffeomorphisms on Σ. A C ∗ -algebra version of this algebrawas introduced by Fleischhack [1]. This algebra is constructed as a subalge-bra of B ( H = L ( A , dµ AL )), the bounded operators on the Hilbert space ofw.r.t. the Ashtekar-Lewandowski measure square integrable functions on the1uantum configuration space A , generated by matrix elements of holonomiesand Weyl operators corresponding to pull-backs under translations on A , thatare generated by the exponential action of the fluxes on A . The gauge trans-formations and a subset of the homeomorphisms (for details see [2]) are thenimplemented as pull-backs under homeomorphisms on A , that preform the par-ticular transformations on A . The operators corresponding to the pull-backs areunitary due to the invariance of dµ AL under Weyl-, gauge- and diffeomorphism-transformations.It is the aim of this letter to construct a generalization of coherent states onthe simple harmonic oscillator for a finite momentum p o in the sense of a finitespatial geometry E o . These states will be outside the usual Hilbert space ofLoop Quantum Gravity, and we will construct their GNS-Hilbert space and theGNS-representation of the algebra of Quantum Geometry thereon. The VEVsof the harmonic oscillator are ω o ( U λ V µ ) = h Ω , U λ V µ Ω i = e − ( λc ) − iλµ − ( cµ )22 ,where U λ = e iλx and V µ = e iµp denote the Weyl-operators. The expectationvalues in the coherent state Ω α = e αa ∗ +¯ αa Ω, labeled by α = x o + ip o are for x o = 0: ω α ( U λ V µ ) = e icp o µ ω o ( U λ V µ ) . The generalization of this relation will turn out to be a state on the algebra ofQuantum Geometry from which we construct a GNS-representation. This is theconcern of the first part of this letter. After discussing some properties of thisrepresentation we conclude this letter with the introduction of a programme toconstruct an effective field theory for this representation, which is as of now anuncompleted programme, so we can only present the framework here.It turns out that Fleischhacks Weyl-algebra is larger than what is needed forsupporting Quantum Geometry: Quantum Geometry is provided by a familyof operators for lengths, areas and volumes of paths, surfaces and regions in Σ.The length operator is constructed from the volume operator using Thiemann’strick [3]. The area operator for a surface S is constructed [4] from flux operatorsas A ( S ) = lim ←P ( S ) X S p ∈P ( S ) q η ij E i ( S p ) E j ( S p ) , (1)where P ( S ) runs over the partitions of S , which also have to include 1- and0-dimensional quasi-surfaces for technical reasons. The resulting area operatorturns out to be a sum over vertex-Laplace operators A ( S ) Cyl α = P v ∈ S πγl P l P v p ∆ v,S,α Cyl α , where α is a graph that contains a vertex v for all (transver-sal) intersections with S . The observation that makes the restriction to a sub-algebra possible is that one can find a quantization of the volume of a regionconstructed from area operators, which is different from [5, 6], where an expres-sion of the form R R p det ( E ) is quantized. This suggests to consider the areaoperator as fundamental and some of the fluxes as composite operators.The volume of a classical region is (if the metric is suitably regular) the limitof Riemann sums of the volume of cells of coordinate volume that approacheszero. For suitably regular regions R , we can find a sequence of coordinate-2ubical complexes that approach the interior of R , such that the coordinatevolume of the cells approaches zero. The metric inside each cell can be assumedto be homogeneous, e.g. a ”volume average” over the cell. The volume of a cubein a homogeneous metric is however precisely the volume of a parallelepiped inEuclidean metric, after changing into the (in this case global) Riemann normalcoordinate system. So, given a three-dimensional generalization of Herons for-mula of the area of a parallelogram in terms of the three independent lengthof the parallelogram to the three-dimensional case using the six independentareas on a parallelepiped to express its volume. The construction of this ex-pression goes as follows: Using a rotation to a adapted coordinate system, theparallelepiped is spanned by ~a = ( a , , ~b = ( b , b ,
0) and ~c = ( c , c , c ), sothe volume is V = a b c . The system of equations involving the squares ofthe three surface areas A i and the squares of the areas of three diagonal crosssections B j is: A a = | ~b × ~c | = ( b c − b c ) + ` b + b ´ c A b = | ~a × ~c | = a ` c + c ´ A c = | ~a × ~b | = a b B a = | ( ~b + ~c ) × ~a | = a ` ( b + c ) + c ´ B b = | ( ~a + ~c ) × ~b | = ( b ( a + c ) − b c ) + ` b + b ´ c B c = | ( ~a + ~b ) × ~c | = ( b c − ( a + b ) c ) + ` ( a + b ) + b ´ c . (2) This coupled system of seven multi-linear equations can be solved for the volume V , which was aided by computer algebra: V o = ˛˛˛˛˛ a b c d A b ( b d A b − b c d A b A c +16 c A c ) + b d e A b ( b d A b − b c d A b A c +16 c A c ) − b d f A b ( b d A b − b c d A b A c +16 c A c ) − a c A c b d A b − b c d A b A c +16 c A c + b c e A c ( b d A b − b c d A b A c +16 c A c ) + b c d f A c ( b d A b − b c d A b A c +16 c A c )+ √ a b c d e A b − a b c d e A b A c + b c d e A b A c − b c d e f A b A c +4 b c d e f A c ( b d A b − b c d A b A c +16 c A c ) ˛˛˛˛˛ (3) where: a = ` − A b − ( A c − B a ) + 2 A b ( A c + B a ) ´ ` A a + ( A c − B b ) − A a ( A c + B b ) ´ b = q A b + ( A c − B a ) − A b ( A c + B a ) c = A b ` A b + ( A c − B a ) − A b ( A c + B a ) ´ d = ( A b + A c − B a ) ` A b + ( A c − B a ) − A b ( A c + B a ) ´ e = ` − A b − ( A c − B a ) + 2 A b ( A c + B a ) ´ ( A a + A b − B c ) f = ` − A b − ( A c − B a ) + 2 A b ( A c + B a ) ´ ` A a + ( A b − B c ) − A a ( A b + B c ) ´ We will modify the formula for V by multiplying it with a factor θ ( A a , ..., B c ) de-fined 1 if there is a non-degenerate parallelepiped with respective areas A a , ..., B c and 0 otherwise. The classically equivalent formula for the volume is then: V ( A a , ..., B c ) := V o ( A a , ..., B c ) θ ( A a , ..., B c ) . (4)3igure 1: This figure illustrates the relation between the ”used diagonal crosssections” and the ”used faces” in the moved diagonals. The numbers indicatethe pair ( i, j ) in the labeling of the face. The coordinate chart U is assumed tobe right-handed cartesian and the 1-direction is assumed to be going from leftto right.It turns out to be useful to not use the diagonals themselves, due to their in-tersection at the ”coordinate center of mass”, but to use isometric surfaces asindicated in figure 1, which we call ”moved diogonals”. The classical volumedoes not depend on the particular choice of sequence of cubical decomposi-tions, the quantum operator is however sensitive to it. Hence we provide aparticular choice and then remove the finger prints of this choice by averag-ing over a suitable set of classically equivalent choices. Let us for simplicityassume that R is contained in a single chart ( U, φ ), if not we have to usea partition of unity to achieve this. Then for all ǫ > R into coordinate cubes C U,ǫ~n = { φ − ( ~x ) : n i ǫ ≤ x i < ( n i + 1) ǫ } . Thosecubes that are subsets of R form a cubical decomposition L U,ǫ ( R ). Clearly V ( R ) = lim ǫ → P C ∈ L U,ǫ V ( A a ( C ) , ..., B c ( C )) converges to the classical volumeof R for all charts U . Moreover, removing any family S ǫ of subsets from L U,ǫ for which the coordinate volume vanishes as ǫ → V ( R ). To treat the surfaces democratically, we will insert for A i = ( A ( S fi ) + A ( S bi )), where S fi is the ”front” surface in i direction and S bi the respective ”back” surface.The strategy for the construction of the quantum operator is as follows:We construct an essentially self-adjoint volume operator on the spin networkfunctions using the limit of V ( R ) = lim ǫ → P C ∈ L U,ǫ V ( A a ( C ) , ..., B c ( C )) with V as in equation 4 together with a suitable average of L U,ǫ then define itsHermitian extension to H as the desired volume operator. We will in particularadapt the definition to the underlying graph: Given a spin network function ψ α depending on a graph α , there is a chart ( U, φ ) and a value for ǫ , s.t. eachcell contains at most one vertex, which is at the ”coordinate center of mass”of the respective cell C U,ǫ~n , and all cells that do not contain a vertex contain4t most one edge, which is furthermore required that its restriction to the cellis connected. Moreover, this chart can be chosen s.t. the edges of α penetratethe surfaces A a ( C ) , ..., B c ( C ) through its interior. We call one of these charts( U α , φ α ). Given such a chart, we can define a refinement process, by subdividingeach cell into 3 × × θ -factor that each cell, that does not containa vertex, will be assigned zero volume, because a single connected edge cannot penetrate the boundary of a cell in three independent points, hence theparallelepiped is degenerate. Since the volume functional is additive, we haveto only sum over the value of V for vertex containing cells. Each cell can betreated separately:The investigation of the vertex containing cells shows that the value of V depends on the choice of chart through the topological relations of the edgesadjacent to the vertex with the surfaces A a , ..., B c (i.e. which surfaces are pen-etrated by which set of edges). To remove this dependence we first notice thatdue to the restriction to piecewise analytic edges there is a value of ǫ such thatall edges are ”outgoing from the vertex v ”, but at the same time for any value ǫ > ϕ ( α, v ) such that the adjacent edges canbe mapped to penetrate any of the surfaces A a , ..., B c , and this diffeomorphismaffects only the cell and its next neighbors. Thus, for each topological rela-tion between the vertex-adjacent edges and surfaces A a , ..., B c there is a chart U ϕ ( α,v ) α s.t. this topological relation is satisfied in the corresponding decom-position. Taking one representative out of each of these classes and averagingthe volume functional over all possible topological relations removes the chartdependence and we arrive at a background independent volume functional. Wedenote these topological relations by T ( v ).The final volume operator is defined as the extension by density of the op-erator acting on spin network functions ψ α : V ( R ) ψ α := X v ∈ V ( α ∩ R ) |T ( v ) | X t ∈T ( v ) V ( A ( S tA a ) , ..., A ( S tB c )) ψ α , (5)where S tA a , ..., S tB c denotes the resp. surfaces that satisfy the topological relation t and A denotes the area operator. This operator is well defined because thespin network functions are simultaneous eigenfunctions of the area operators inour ”external” regularization and is gauge invariant by inspection.We are now able to define the restricted Weyl-algebra and we start witha more general consideration: Generalizing the notion of [1], we consider acompact Hausdorff space X and a regular Borel probability measure µ thereon.The integral kernels K f : x R dµ ( x ′ ) K ( x, x ′ ) f ( x ′ ) that leave C ( X ) invariantand are invertible contain a group of ”unitary” elements, i.e. those elements for5hich R dµ ( x )( Kf )( x )( Kf )( x ) = R dµ ( x ) f ( x ) f ( x ), denoted by U ( µ ). Thecommutative C ∗ -algebra C ( X ) acts as multiplication operators on L ( X , dµ ),whereas the elements of W act as convolutions. This canonical action is denotedby π o . Fixing a subgroup W of U ( µ ), one can define the *-algebra A o ( X , W ) offinite sums of ordered pairs: A o ( X , W ) ∋ a = n X i =1 f i ◦ w i , where f i ∈ C ( X ) and w i ∈ W . The involution is given by a ∗ = P ni =1 w − i ( f i ) w ∗ i ,where w ( f ) : x R dµ ( x ′ ) w ( x, x ′ ) f ( x ′ ). The canonical representation of this*-algebra on L ( X , dµ ) can be completed to a C ∗ -algebra, denoted by A ( X , W ),in the operator norm and || f ◦ || ≤ || f || ∞ and || ◦ w || = 1.The construction of unitaries follows from a slight modification of continu-ous measure generating systems in [1]: A Hermitian measure generatingsystem is a set E of continuous functions that is dense in C ( X ) as well as L ( X , dµ ) containing 1, such that all elements of E \ { } are perpendicular to 1in the inner product of L ( X , dµ ), together with a labeling set H , consisting of acommuting set Hermitian operators in B ( L ( X , dµ )) that have the elements of E as eigenfunctions and their eigenvalues distinguish the elements of E . Withoutloss of generality, we assume that h ∀ h ∈ H . For any analytic function f on the eigenvalues of the labeling set H ( E ) we define the linear map τ f e := e exp( if ( H ( e ))) , as the extension by density to L ( X , µ ), where H ( e ) denotes the set of eigenval-ues of e ∈ E . It turns out that if f ( H (1)) = 0 then τ f defines a unitary elementof B ( L ( X , dµ )).Since A o ( X , W ) is dense in A ( X , W ), we define a state on A o , check whetherit is bounded and extend it by density to A : Given a continuous morphism from W to U (1), we define a functional ω F,µ on A o by: ω F,µ n X i =1 f i w i ! := n X i =1 F ( w i ) Z dµ ( x ) f i ( x ) . (6)This functional is bounded | ω F,µ ( a ) | ≤ P ni =1 || f i || ∞ and linear as is checked byinsertion. Positivity can be shown using the group morphism property of F andthe invariance of µ under the action of W , such that ω F,ν ( a ∗ a ) = R dµ ( x ) P ni,j =1 ( F ( w i ) f i )( x )( F ( w j ) f j )( x )= R dµ ( x ) | P ni =1 f i ( x ) | ≥ . To calculate the GNS-representation constructed from ω F,µ it is useful to relatethis to the canonical representation, usually constructed from the state ω o ( a ) := P ni =1 R dµ ( x ) f i ( x ), using the map κ F : a n X i =1 ( F ( w i ) ∗ f i ) ◦ w i . (7)6t turns out that κ F is an automorphism of A o and that κ f ( a ) is in the Gel’fandideal I F of ω F,µ , if and only if a is in the Gel’fand ideal I o of ω o and thisrelation can be carried over to A due to the boundedness of κ F . It follows thata dense set of representatives R F of N F in A , where N F := A / I F , implies a set ofrepresentatives R o and vice versa. This allows us to use a continuous generatingsystem for the canonical representation as a dense set in the GNS-Hilbert spaceconstructed from ω F,µ . We notice that a weakly continuously represented 1-parameter group in the canonical representation implies an weakly continuousrepresentation of this 1-parameter group in the GNS-representation due to thegroup morphism property of F and due to the assumed continuity of F .Fleischhacks definition of the Weyl-algebra of Loop Quantum Gravity fitsprecisely into this setup: The compact Hausdorff space is space of general-ized connections A = lim α ( P α , SU (2)), where P α is the path-groupoid of thegraph α and the projective limit is taken over the inclusion as subgraphs. Thetranslations W i ( S ) on A generated by the fluxes E i ( S ) are special invertibleconvolution operators, that leave the Ashtekar-Lewandowski measure invariant.The definition [1] of A then coincides with our definition when taking X = A , µ = µ AL and W is the Weyl group generated by the W i ( S ) through arbi-trary quasi-surfaces. The canonical state ω o ( Cyl ◦ W ) = R dµ AL Cyl ( A ), where Cyl ∈ Cyl ( A ) and W is an element of the Weyl group. The extended analyticdiffeomorphisms leave the Ashtekar-Lewandowski measure invariant and hencethe pull-backs under these diffeomorphisms define unitary operators in B ( H ).The gauge-variant spin network functions are a continuous measure generat-ing system. A Hermitian labeling set for this generating system is constructedas follows: Given a map τ : Σ → SU (2), we can consider the fluxes ”parallel to τ ”, i.e. E || τ ( S ) := R S E i τ i . It turns out that the ”fluxes parallel to τ ” togetherwith the area operators form a Hermitian labeling set for the gauge variant spinnetwork functions.Let τ : Σ → SU (2) be normalized, i.e. k ij τ i τ j = 1 at each point. Let usdefine three families of Weyl-operators for each oriented surface through theiraction on gauge variant spin network functions ψ α : W AS ( f ) : T α AS,λ ( f ) ( T ) W + S ( f ) : T (Θ S,σ, exp( λ ( f ) τ ) ) ∗ TW − S ( f ) : T (Θ S,σ, exp( λ ( f ) τ ) ) ∗ T, (8)where W AS ( λ ) ψ α := e iλA S ψ α = P i a i T γ S ,i e iλA S ( ψ αS,i ) =: α AS,λ ( ψ α ) and Θ de-notes the respective element of Fleischacks translation group on A correspondingto the exponential action of the respective flux parallel to τ with the orienta-tion function σ on the quasi-surface S resp. the flipped orientation function σ . The extension to operators in B ( H ) by density defines unitary operators,which are precisely of the kind described above, because their integral kernelcan be obtained through exponentiating a function on the labeling set for which f (1) = 0. The group W τ generated by these operators contains area- andflux-Weyl-operators on less than two-dimensional quasi-surfaces, but they allflux-Weyl-operators are parallel to τ . The algebra that we consider is then the7lgebra in B ( H ) generated by the finite sums A τ ∋ a n X i =1 Cyl i W i , where Cyl i inCyl ( A ) and W i ∈ W τ . Let us fix a regular densitized inverse triad E o . The state ω E o on the finite sums is: ω E o ( a ) := n X i =1 F E o ( w i ) Z A dµ o ( A ) f ( A ) , (9)where F E o ( W S ) = 1 whenever S is contained in a less than two-dimensionalsubset of Σ and otherwise F E o : W AS ( λ ) exp( iλ R S | E o | ) W + S ( λ ) exp( iλ R S E o ) W − S ( λ ) exp( iλ R S E o ) . (10)Let us use the automorphism κ F to describe the GNS-representation in termsof the canonical representation, for which we have: ω o ( a ∗ a a ) = h η o ( a ) , π o ( a ) η o ( a ) i o η o ( a ) : A P i f i π o ( a ) φ : A P i f i ( A )( α w i ( φ ))( A ) h φ, φ ′ i o := R A φ ( A ) φ ′ ( A ) , Since ω E o ( κ F ( a )) = ω o ( a ), we see obtain the GNS-representation for ω E o im-mediately: η E o ( a ) = η o ( κ F ( a )) : A P i F ( w i ) f i π E o ( a ) φ = π o ( κ F ( a )) : A P i F ( w i ) f i ( A )( α w i ( φ ))( A ) h φ, φ ′ i E o = h φ, φ ′ i o := R A φ ( A ) φ ′ ( A ) . (11)The canonical action of the diffeomorphisms on the algebra elements moves thegraphs γ i and the quasi-surfaces S i around: α φ ( a ) = α φ ( X i f i,γ i w i,S i ) = X i f iφ ( γ i ) w i,φ ( S i ) . This implies for the transformation of the vacuum vector Ω E o of the GNS-representation: ω E o ( a ) = h U φ Ω E o , U φ π o ( a )Ω E o i E o = h U φ Ω E o , π o ( α φ ( a )) U φ Ω E o i E o = ω φE o ( α φ ( a )) , The state ω φE o is then determined to coincide with a
7→ h Ω φ ( E o ) , π φ ( E o ) ( a )Ω φ ( E o ) i ,and we thus have the relation for the vacuum vectors: U φ Ω E o = Ω φ ( E o ) . (12)8o be able to define a unitary action of the diffeomorphisms, we need to takethe direct sums over all E o in the diffeomorphism orbit of E o . Using the preciseanalogue of this calculation, we have for gauge transformations, which act ontriads as α Λ E = Λ − E Λ, the relation U Λ Ω E o = Ω α Λ ( E o ) . Thus, to have a Hilbert space representation that carries a unitray representa-tion of the diffeomorphisms and the gauge transformations, we should take thedirect sum over the entire geometric orbit of E o . There is one caveat concerningthe map τ : To be able to use group averaging, we have to have the fitting gaugetransformed flux-Weyl-operators available in each summand, thus we need totake the direct sum over the GNS-representations with gauge transformed τ -maps. The kinematical Hilbert space representation is thus the sum over theentire geometrical orbit G ( E o ) of E o :( H G o , π G o ,τ ) := ⊕ { E o : G ( E o )= G o } ( H E o , π E o ,τ G ) , (13)where τ G denotes the gauge transformed τ -map. The gauge-variant spin net-work functions are orthogonal in each of the E o -representations, because for anytwo gauge-variant spin network functions T , T we have: h T , T i E o = h π ( T )Ω E o , π ( T )Ω E o i E o = ω E o ( T ∗ T ) = R A dµ o ( A ) T ( A ) T ( A )= ω o ( T ∗ T ) = h T , T i o This implies the kinematical orthogonality for normalized spin network functions T , T : h T ◦ E o , T ◦ E o i G o = (cid:26) T = T ∧ E o = E o , (14)so a complete orthogonal set is labeled by a gauge variant spin network functionand a background geometry E in the geometrical orbit of E o .The orthogonality of the different summands in the direct sum allows us tosplit the group averaging into three parts: First we classify the gauge invariantcouplings between the spin network function and E , then we average over thegroup of transformations that affect the coupled spin network non-trivially andfinally we average over the quotient of the gauge-transformation group by thegroup that acts non-trivially on the gauge-invariant spin network function, whichwill evidently leave the invariant couplings between the spin network and thebackground invariant. The treatment of the diffeomorphism constraint will beanalogous:Let us start with the gauge-invariant couplings: The basic observation thatwe need is that for edges ( y, x ) , ( x, z ) with f ( y, x ) = x = i ( x, z ) objects of theform: O a := T r (cid:0) (( someth. )( A, E o ))( y, z )( h ( y,x ) ( A )) E oai ( x ) τ i ( h x,z ( A )) (cid:1) h e ( A )) n,m (cid:0) Λ − ( i ( e )) h e ( A )Λ( f ( e )) (cid:1) n,m and E o ( x ) Λ − ( x ) E o ( x )Λ( x ), but obviously not diffeomorphism invariant. Thisis however only a special case of the general picture: Any function F x ( E o ) builtfrom E o ( x ) that transforms under some representation of SU (2) can be gaugeinvariantly coupled to a spin network function T with vertex x by constructinga gauge-invariant intertwiner between the representation of F x ( E o ) and repre-sentations adjacent to x in T . Thus, given a gauge-variant spin network T γ ona graph γ , then we can couple it gauge invariantly to E o by assigning a function F v ( E o ) and an gauge-invariant intertwiner M v between the representation of F v ( E o ) and adjacent spins to each vertex v of γ . We call spin network func-tions with invariant couplings to the background gauge-invariantly coupled spinnetwork functions.Let us now average over all gauge transformations that act nontrivially on thegauge-invariantly coupled spin network function π ( T α ( A, E o ))Ω E o : The solution(to this group averaging over the finite number of copies of SU (2), one for eachnon-invariantly coupled vertex) is as in Loop Quantum Gravity given by theproduct states of traces over closed loops, with the addition that there may bevertices in these closed loops, which represent gauge-invariant couplings. So,basically the solution space to the Gauss-constraint is enlarged by spin-transferbetween the spin network function and the geometric background. We callthese solutions gauge-invariantly coupled gauge-invariant spin networkfunctions. Given a gauge-variant cylindrical function Cyl , we denote its groupaverage by G ( Cyl ) :=
R Q v ∈ var ( α ) dµ H ( g v ) Cyl ( ..., g − v h v ,v g v , ... ). Notice thatthe gauge-orbit of two distinct gauge-invariant couplings can yield the samegroup average, when the ”transferred spin” of the two couplings equal.Finally, we preform the group averaging over the quotient of the gauge trans-formations by the finite group that acts nontrivially on the gauge-invariantlycoupled spin network functions, which are precisely those gauge transforma-tions Λ( v ) = 1 SU (2) for all non-invariantly coupled vertices v ∈ var ( α ): Given agauge-invariantly coupled gauge-invariant spin network function π ( T α ( A, E o ))Ω E o ,these transformations act Λ : π ( T α ( A, E o ))Ω E o π ( T α ( A, E o ))Ω Λ − E o Λ . Withthese preparations we can calculate the effect of the gauge-rigging map η ( π ( T ( E o ))Ω E o ) : π ( T ′ ( E ′ o ))Ω E ′ o X Λ ∈ n G T,Eo h U Λ π ( T ( E o ))Ω E o , π ( T ′ ( E ′ o ))Ω E ′ o i G o , and upon preforming the aforementioned three-step we obtain the closed ex-pression for the gauge-invariant inner product: h η ( π ( T ( E o ))Ω E o ) , η ( π ( T ′ ( E ′ o ))Ω E ′ o ) i O ( E o ) := η ( π ( T ( E o ))Ω E o )[ π ( T ′ ( E ′ o ))Ω E ′ o ]= (cid:26) R A dµ o ( A ) G ( T ( E o , A )) G ( T ′ ( E o , A )) for E ′ o ∈ O ( E o )0 otherwise . (15)The gauge-invariant Hilbert-space is thus spanned by gauge-invariantly coupledgauge-invariant spin network functions, which are embedded into a gauge-orbitof a background E o . 10igure 2: An example of a solution to the Gauss- and difeomrophism constraint:The graph is embedded up to isometries of the background, contains additionalcouplings to the background (black dots) and disconnected regions of the graphare not physically disconnected, due to the occurrence of the background geom-etry.The precise same line of reasoning can be applied to solving the diffeomor-phism constraint: The diffeomorphism invariant couplings between the spinnetwork-functions and the background are gauge- and diffeomorphism-invariantcouplings between the pin network function and the background. The closedexpression for the gauge- and diffeomorphism-invariant inner product turns outto be: h η diff ( π ( T γ )Ω O ( E o ) ) , η diff ( π ( T ′ γ ′ )Ω O ( E ′ o ) ) i G o = (cid:26) P φ ∈ Sym ( O ( E o )) R A dµ o ( A ) T φ ( γ ) ( A ) T ′ γ ′ ( A ) for: O ( E ′ o ) ∈ G ( O ( E o ))0 otherwise , (16)where Sym ( O ( E o )) denotes the subgroup of the diffeomorphisms that containsthe symmetries of O ( E o ). So the gauge-and diffeomorphism invariant Hilbertspace consists of gauge-and diffeomorphism-invariantly coupled gauge-invariantspin networks, which are embedded into a geometry (modulus isometries of thisgeometry).Having a Hilbert-space, that is spanned by spin-networks that are embeddedinto a background geometry G o , one can consider the spin network as quantumfluctuations around this geometry. Let us make this statement more precise:Given a state ω on the algebra A τ of Quantum Geometry, we call a countableset of zero- and one-dimensional embedded piecewise analytic submanifolds ofΣ the excess E ( ω ), if the expectation values of any area- or volume-operatoron Σ \ E ( ω ) do not change upon removing a finite number of one-dimensionalsubmanifolds form Σ. Since one can reconstruct a classical geometry form theareas and volumes of the embedded two- and three-dimensional submanifoldsand since this geometry is invariant under the removal of a countable numberof lower dimensional submanifolds, we are able to to define the the essentialgeometry of a state ω as the geometry reconstructed from the expectationvalues of the area ω ( A ( S \ E ( ω ))) and volume operators ω ( V ( R \ E ( ω ))) on Σ.The essential geometry turns out to be a feature of the entire representation,because any sequence of cylindrical functions will be defined a countable set11f graphs and thus the removal of this set of graphs will remove the excessof any element of the GNS representation and the removed set consists of acountable number of zero- and one-dimensional submanifolds. It thus turns outthat the essential geometry of a state coincides with the essential geometry ofany other state in the same GNS representation and are thus an invariant of therepresentation. So one can calculate the essential geometry of any state fromthe essential geometry of the ground state, which does not have an effect (dueto the regularity assumption on E o ) and hence the essential geometry is simplyreconstructed from the vacuum expectation value of the respective geometricoperators.The essential expectation values for the area operators on a surface S areeasily calculated as h A ( S ) i E o = lim t → t (cid:0) ω E o ( W SA ( t )) − ω E o ( W SA ( − t )) (cid:1) = ∂∂t exp( tA E o ( S ) ) | t =0 = A E o ( S ) , so the essential expectation values of the area operators coincide with the clas-sical areas of S calculated in the geometry described through E o . Calculatinghigher derivatives reveals that there are no fluctuations in the essential expec-tation values for the area operators. Moreover it turns out that the action onthe ground state Ω E o of any two all area operators commute. This allows usto calculate the essential expectatioon values of the volume operator withoutfurther effort: The expectation values for the volume operator of a region is h V ( R ) i E o = V ( R ) = h Ω E o , lim ǫ → P C ∈ L U,ǫ V ( A a ( C ) , ..., B c ( C ))Ω E o i = V E o ( R ) , which is independent of the choice of chart ( U, φ ). Thus, the essential geometryturns out to be precisely the geometry that is described through the classicaldensitized inverse triad E o .Since the essential geometry can be recovered from any state in the GNS-representation and is fixed by the E o -geometry, we have a geometric backgroundin the E o -geometry that can be determined operationally, since the effect of astate can be determined operationally: Consider the following family of pairsof surfaces { ( S, S \ x ) : S ∈ S (Σ) , x ∈ S } , where S (Σ) denotes the set ofpiecewise analytical surfaces in Σ. Moreover, consider the set of pairs of regions { ( R, R \ x ) : R ∈ R (Σ) , x ∈ R } , where R (Σ) denotes the set of piecewiseanalytical regions in Σ. If the area- resp. volume- expectation values of anyof the pairs disagree, then x is in the effect of the state, so the effect of thestate can be determined operationally, which means that there is a measurablegeometric background that we can use to define essential distances, essentiallength of curves, essential areas, essential volumes and so on.Using the essential geometry one can in particular measure the length ofthe edges of a cubical decomposition of compact subsets C ⊂ Σ. Each cubicaldecomposition has a dual graph with (at most ) six-valent vertices. It turns out The vertices inside cells adjacent to the boundary of the compact region may have lessthan valence six.
Cub with refine-ments as morphisms. A refinement of a cubical decomposition D o ( C ) is a cubicaldecomposition D ( C ) that contains a (possibly trivial) decomposition for eachcell c ∈ D o ( C ). This category is partially ordered by the refinement property.Since there exists a cubical decomposition D ( C ) for any pair D ( C ) , D ( C )that is finer than these two, they are furthermore a projective family. We willfurthermore make use of the essential geometry and assume that the cubicaldecomposition has contains only edge-length between l o and 2 l o , meaning D i is”close to regular”.To each cubical decomposition D ( C ), there is an embedded dual graph Γ( D )with generally six-valent vertices, which is constructed as follows: the verticesof Γ are given by the ”coordinate center of mass” of the cells and the links ofΓ are given by the concatenation of the geodesics from the coordinate center ofmass of a cell to the coordinate center of mass of a joint face. (This constructionis possible due to the occurrence of the essential geometry.) One can define therefinement of a lattice Γ( D o ) to a finer lattice Γ( D ) through the the existenceof a pair D o , D , such that the cubical decompositions D ≥ D o are dual toΓ and Γ respectively. This turns the set of graphs that are dual to cubicaldecompositions into a category itself with refinements as morphisms.Given any graph Γ, we can consider the lattice gauge theory LGT (Γ). Thebasic idea to construct an effective quantum field theory rests on the commonbelief that a Quantum Gauge Field Theory is the limit of lattice size goingto zero of a lattice gauge theory if this limit exists. If this limit does notexist, then one can at least call the family of lattice gauge theories a familyof effective field theories. Thus, if one succeeds with the construction of afunctor that takes a cubical decomposition D ( C ) into a lattice gauge theory LGT (Γ( D ( C ))), that encodes ”all relevant degrees of freedom” , then one hasconstructed an effective field theory. Before we give a candidate construction forsuch a functor, we need to consider the construction of a contravariant functorthat assigns a noncommutative C ∗ -algebra (the quantum algebra of an SU (2)-lattice gauge theory plus possible extra degrees of freedom) and a Hilbert-spacerepresentation thereof (the canonical representation of this lattice gauge theoryon L ( SU (2) | Γ | , dµ H ) ⊕ H extra ):Let D o ≤ D be two cubical decompositions of a compact subset C ⊂ Σ.Thus, for each cell c ∈ D o there is a set of cells r ( c ) ∈ D that constitutea decomposition of c . Moreover, for each face f ∈ D o there is a set of faces r ( f ) ∈ D that constitute a decomposition of the face f . Given a lattice gaugetheory on Γ( D ), we map all degrees of freedom on the n links across the faces r ( f ) into the degrees of freedom on the link across f . Moreover, we map alldegrees od freedom residing on the vertices at the centers of the cells in r ( c ) intothe degrees of freedom on the vertex at the center of c . Since there is no boundon the number of cells in the refinement r ( c ) of a cell c and no bound on the Notice that the metric limit is taken here, which is not possible in Loop Quantum Gravity,where only a projective limit over all cubical decompositions would be meaningful. We will at first map all degrees of freedom into the effective field theory and then allowto ”forget irrelevant degrees of freedom”. r ( f ) of a face f , there is an infinite numberof degrees of freedom on each link and each vertex. We denote this map fromthe finer LGT (Γ f ) to the coarser lattice gauge theory LGT (Γ c ) by R fc . Theconsistency condition for the map R is: R ( R ( LGT (Γ ))) = R ( LGT (Γ )) , (17)whenever Γ ≥ Γ ≥ Γ . Thus, each vertex contains all the degrees of freedomof a lattice gauge theory on a lattice of arbitrary size and each link carriesan arbitrary number of copies of SU (2)-degrees of freedom. These degrees offreedom are naturally ordered by (1) the lattice size for the vertex degrees offreedom and (2) the number of copies of SU (2). This suggests the construction ofthe algebra of vertex-observables as the limit of the observable algebras of finitelattice gauge theories together with their canonical Hilbert space representationand similarly for the link degrees of freedom, which allows us to solve consistencycondition 17 in the obvious way of embedding a sequence of sublattices into asequence of lattices. There is evidence that the pull-back of lattice observablesunder this construction is a quantum embedding in the sense of [7], furnishingthe morphisms in the then defined category LGT of the lattice gauge theorieswith extra degrees of freedom.Now we have to specify the ”irrelevant degrees of freedom”. Given a physicalHamiltonian H phys. , we have to include all degrees of freedom that have ”ob-servable effects”, when the initial state is given by a state that does not containone of these degrees of freedom. This means a degree of freedom is irrelevant, ifthere are no lattice measurements available that can effectively distinguish be-tween the effect of the dynamical occurrence of a particular degree of freedomand its time evolution and an effective lattice state and it time evolution of theeffective lattice state. It is thus important to know the dynamics in order todetermine the ”relevance” of a degree of freedom, so we have to postpone thediscussion of ”relevance” until after the definition of a suitable dynamics. We do not yet claim that this construction, which we denote by F G o : Tri → LGT ,yields a functor, due to some seemingly natural yet unproven assumptions thatwe had to make. We are however confident that at most ”technical details” haveto be adjusted and the general picture will turn out unchanged.Let us now generalize the construction F G o to the vacuum state Ω E o of aGNS-summand, more precisely: given a cubical decomposition D ( C ), we wanta state Ω effE o on the lattice gauge theory LGT (Γ( D )), such that the deviationsbetween the expectation values are small: h Ω effE o , LattObs Ω effE o i = ω E o ( I ( LattObs )) + small corr., here I denotes the canonical embedding of a lattice observable into an elementof the algebra of quantum geometry, given by mapping link observables on thelattice into the respective holomomy observable along the embedding of this The usual effect is that ”heavy” and ”weakly coupled” extra degrees of freedom are”irrelevant”, where heavy is understood w.r.t. the lattice spacing l o , when both the mass andthe spacing are measured in natural units. D that is dual to themomentum. A particular choice for such observables is given by a momentumsqueezing | Ψ E o (Γ) i := i lim t → large | ψ Γ t (0 , E o ) i of Thiemanns coherent sates[8],which are proven to yield the correct expectation values [9]. We will assume,although we cannot prove it due to the lack of a physical Hamiltonian, thatthe extra vertex and link degrees of freedom are dynamically irrelevant, whichshould hold at least for classical static solutions E o of Einstein’s equations.Under these condition it turns out that the construction F E o yields a family oflattice states: {| Ψ E o (Γ) i : Γ( D ) } D∈ comp.cub.dec. (Σ) , (18)one for each cubical decomposition of a compact subset of Σ. Given any surface S , the family has the feature that the expectation values h Ψ E o (Γ) , A ( S )Ψ E o (Γ) i = A E o ( S ) + small corr., whenever Γ = Γ( D ) and S can be decomposed into faces in D . The analogousstatement holds for the expectation values of volumes of regions. To make aconnection with the F/LOST representation of the algebra of quantum geom-etry, let us consider the lattice states as states on the embedded lattice in theF/LOST-representation and put this observation on its head: Consider a (pro-jective) family of states { ψ E o , S , R } S , R indexed by finite sets of surfaces S andregions R in Σ in the F/LOST-representation, such that for any countable setof surfaces and regions the expectation values h ψ E o , S , R , A ( S ) ψ E o , S , R i = A E o ( S ) and h ψ E o , S , R , V ( R ) ψ E o , S , R i = V E o ( R ) , for all S ∈ S and all R ∈ R . This family is partially ordered (using the jointsubset relation) and projective, so one can heuristically consider the projectivelimit: Ψ E o := lim ← ( S , R ) ψ E o , S , R , (19)which does not exist in the F/LOST-representation. We can however definea state through the vacuum expectation values, which will then coincide with ω E o .This situation is reminiscent of a Bose condensate ground state of a freescalar field theory: Given a particular ground state density ρ o , one can considerthe thermodynamic limit Λ → ∞ of a family of grand canonical states ω Λ ,β,µ ( a ) | ρ ( β,µ )= ρ o := T r ( e − βH + µN a ) T r ( e − βH + µN ) V, where the inverse temperature β and the chemical potential µ are adjusted sothey yield the expectation value ρ o in an increasing region as Λ → ∞ . Generallythere is no element in the Hilbert space of the free theory that reproduces thislimit and one has to preform the GNS-construction w.r.t. the state definedthrough ω ρ o ( a ) := lim Λ →∞ ω Λ ,β,µ ( a ) | ρ ( β,µ )= ρ o . (20)15he rˆole of ρ o is similar to the rˆole of E o in the new representation of the algebraof quantum geometry, so one can view the state ω E o as a ”local condensateof quantum geometry”. The term ”local” is motivated by: Given any openset O ⊂ Σ (that is contained in a compact set) of arbitrarily small size onecan preform the limit construction Ψ E o := lim ← ( S , R ) ψ E o , S , R for surfaces andregions in O and one obtains that the projective limit does not exist in theF/LOST-representation, but there exists a state ω E o | O , which can be definedsimilar to equation 20.One can also draw a similarity between the GNS-Hilbert space constructedfrom the BEC ground state and the E o -GNS-Hilbert space: The GNS-constructionin the BEC-case yields a Hilbert space that contains states that have groundstate density ρ o everywhere, except for quantum fluctuations around the conden-sate that vanish at infinity. These fluctuations can be characterized in preciselythe way that the effect of a state was characterized and the ground state density ρ o has the rˆole of the essential geometry. So if we adopt the interpretation of ω E o as a state describing a local condensate of geometry, we are let to viewthe spin network functions as quantum fluctuations around the geometric con-densate. The interpretation of the F/LOST-ground state ω o is in light of theseconsiderations rather simple as the ω E o state with totally degenerate E o = 0.Let us now return to the construction F E o , which we want to generalize to anarbitrary state in the E o -GNS-Hilbert space, so we can find a family of latticegauge theory-states that describe the dynamically relevant degrees of freedom ofthe E o -state. Let us again assume that the geometric background correspondsto a classical static solution E o of Einstein’s equations and that this is suffi-cient for not producing extra lattice degrees of freedom. Let us consider a sate π ( Cyl α )Ω E o and a cubical decomposition D ( C ) of a compact subset C ⊂ Σ. Itis our aim to construct a state on the lattice gauge theory on Γ( D ( C )), suchthat there are no lattice measurements that deviate significantly from the corre-sponding embedded measurements in the GNS-representation. Since there areno restrictions on the cylindrical function Cyl or on its graph α , we cannot rulethe dynamical relevance of any part of it out, so we have to construct a statethat contains all degrees of freedom. A particular construction is:(1) Construct a state Ψ E o (Γ), that captures the essential geometry of π ( Cyl α )Ω E o precisely as previously done for Ω E o . Construct the effective state Cyl ′ throughthe multiplication operators: Cyl ′ := Ψ E o (Γ) Cyl α This procedure is supposed to absorb the E o -geometry inside the cell and let isreappear on scales larger than the cell.(2) Denote the restriction of Cyl ′ to a cell c ∈ D ( C ) by Cyl α | c . Use the smallestcubical lattice that supports a graph γ c that is topologically equivalent to α | c .Use φ c : γ c α | c and map the state φ ∗ c Cyl ′ | c into the vertex Hilbert spaceprecisely as prescribed in the construction F E o for lattice gauge theories onrefined lattices. If there is a set Γ c ( α ) of minimal lattices, then preform this16igure 3: Some of the ”extra vertex degrees of freedom” arising in our con-struction. Notice that these ”extra degrees of freedom” do not occur in a latticeversion of Ashtekar gravity. The relevance of the degrees of freedom is decided bythe dynamics. The usual argument from perturbatively renormizable QFT thatthe new degrees of freedom on a finer lattice effectively decouple from degrees offreedom on the coarser lattice is not obvious in our case due to the occurrence ofcouplings to the background, that may occur between extra degrees of freedomon the finer lattice in the E o -representation.construction for all minimal lattices γ ic ∈ Γ c ( α ) and construct1 | Γ c ( α ) | X γ ic ∈ Γ c ( α ) φ ∗ c,i Cyl ′ | c in the vertex Hilbert space.(3) The graph α will in general penetrate the face f ∈ D ( C ) n times and the n spin quantum numbers of the penetration are mapped into the link Hilbertspace.(4) This procedure is only well defined, if all vertices of α are in the interior of acell and if all edges penetrate the faces transversally. To resolve the ”degenerate”cases, we have to define for each node in D to belong to the inside of a link,for each link to belong to the inside of a face and for each face to belong tothe inside of a cell. Loosely speaking this procedure ”pinches α a bit, so thedegenerate topological relation is deformed into a general topological relation.”Let us briefly notice the extra effective vertex degrees of freedom that arisefrom this construction:(1) Couplings to the background (which are due to the enlargement of thesolution space of the Gauss-constraint in the E o -representation and not an effectof the construction F E o )(2) The topology-class of α | c and the cylindrical function of this topology class(3) High valent vertices (since the dual graph contains vertices of at most valencesix)(4) Knotting with the effective lattice state graphThe extra effective link degrees of freedom arising from this construction are:(1) multiple penetration of a face by possibly different edges(2) linking information of the penetrations and α | c for adjacent cells c The construction again yields a family of lattice states {| Ψ E o ( Cyl, Γ) i : Γ( D ) } D∈ comp.cub.dec. (Σ) , that describe the state π ( Cyl )Ω E o up to small corrections on the respectivelattices Γ( D ). With the interpretation of the states π ( Cyl )Ω E o as a conden-sate of quantum geometry, we can view the family | Ψ E o ( Cyl, Γ) i as states in17he F/LOST-representation in which the effect of the geometric condensate isintegrated out and absorbed into the state on scales larger than the latticespacing l o , that we assumed to be ”almost regular” from the onset. We canthus view our construction as a scale dependent (i.e. l o dependent) map thatmaps the essential geometry E o into an effective state in the ”fundamental”F/LOST-representation. We can also try to put this on its head and assert thata particular representative | Ψ E o ( Cyl, Γ) i in the F/LOST-representation is thetrue quantum state of our geometry and that the state π ( Cyl )Ω E o is only aneffective state that describes the ”smooth part” of the quantum geometry of | Ψ E o ( Cyl, Γ) i up to a scale l o given by the average lattice spacing of Γ. Theproblem with this interpretation is however, that there is no sense of ”nearness”in the F/LOST-representation.Let us now briefly discuss the possible dynamics for these states: Ideally onewould have a master constraint for the F/LOST-representation and induce onefor the E o -representation through requiring that the following diagram com-mutes: | ( E o ) i f F E o ( D f ) F/LOST + ˆ M → LGT f + ˆ M f ← E o − rep. + dynamics ց ↓ ւ| ( E o ) i c LGT c + ˆ M c F E o ( D c ) , where the diagram reads as follows: We start in the top line and add to theF/LOST-representation a lattice state | ( E o ) i f on a graph Γ( D f ), that is dual toa fine cubical decomposition D f . Applying an adaption of the construction F E o with E o = 0 to this leads to a lattice gauge theory on a fine lattice. We triedto construct F E o , such that the application of F E o to the E o -state on the left(using the same fine cubical decomposition D f ) yields the same lattice gaugetheory (with extra degrees of freedom). But a master constraint on the leftyields a constraint surface, that induces a constraint surface on the right.The lower line describes the analogous construction for a coarsening D c of D f . The nontrivial consistency condition is that the construction F between thelattice gauge theories yields the lattice gauge theory with extra degrees of free-dom on the coarser lattice. Since the dynamics of the F/LOST-representation isstill disputed, we may take a different route and forget about the F/LOST-sideof the diagram and try to ”invent” a consistent family of master constraints forthe lattice gauge theories with extra degrees of freedom in the middle.This letter is necessarily incomplete, particularly the proofs of our statementswhere not carried out in detail. The missing details about the volume operatorwill follow in [10], details about the algebra, the non-vacuum state and the GNS-representation will follow in [11], details about the construction of effective fieldtheories will follow in [12] and the precise mathematical formulation of thecategory LAT and the functor F E o as well as the possibility of the definitionof a consistent dynamics for these effective field theories are currently underinvestigation. 18 cknowledgements: One third of this work was supported by the Deutsche Forschungsgemeinschaft.Another third of this work profited from an invited visit to the Perimeter In-stitute. I am grateful for helpful discussions with Bianca Dittrich in particularand useful discussions with Lee Smolin, Laurent Freidel, Thorsten Ohl, Mar-tin Bojowald and Jonathan Engle. I am also thankful for my education at theBenedictine Abbey of M¨unsterschwarzach, which amongst many other usefulthings taught me austerity, which was necessary to complete this work.
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