Coherent states, quantum gravity and the Born-Oppenheimer approximation, III: Applications to loop quantum gravity
aa r X i v : . [ m a t h - ph ] A p r Coherent states, quantum gravity and the Born-Oppenheimerapproximation, III: Applications to loop quantum gravity
Alexander Stottmeister a) and Thomas Thiemann b) Institut für Quantengravitation, Lehrstuhl für Theoretische Physik III,Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7/B2, D-91058 Erlangen,Germany
In this article, the third of three, we analyse how the Weyl quantisation for compact Liegroups presented in the second article of this series fits with the projective-phase spacestructure of loop quantum gravity-type models. Thus, the proposed Weyl quantisationmay serve as the main mathematical tool to implement the program of space adiabaticperturbation theory in such models. As we already argued in our first article, space adia-batic perturbation theory offers an ideal framework to overcome the obstacles that hinderthe direct implementation of the conventional Born-Oppenheimer approach in the canonicalformulation of loop quantum gravity.
CONTENTS
I. Introduction II. Loop quantum gravity and phase space quantisation
III. Conclusions and perspectives Acknowledgments IV. Bibliography I. INTRODUCTION
In our previous articles in this series , we pointed out the need for a Weyl quantisation for modelsof loop quantum gravity-type to realise the (time-dependent) Born-Oppenheimer approximation formulti-scale quantum dynamical systems along the lines of space adiabatic perturbation theory . Inthe second article of this series, we introduced a Weyl quantisation for compact Lie groups anddeveloped the basis for an associated calculus of Paley-Wiener-Schwartz symbols, which allowedus to tackle the “problem of non-commutative fast-slow coupling” (originally pointed out in thecontext of loop quantum gravity ).But, if we intend to use the Born-Oppenheimer approach to extract a continuum limes in the slow a) Electronic mail: [email protected] b) Electronic mail: [email protected] (gravitational) sector (cf. ), there is a second obstacle. The latter can be addressed in terms ofcompatibility conditions of the Weyl quantisation with the projective limit structures involved inthe construction of the models à la loop quantum gravity. We expect, that such compatibilityconditions, in addition to a selection of admissible observables, play a major role in the possibleextraction of quantum field theory on curved spacetimes from loop quantum gravity (with matter).This said, it is the primary objective of the present article to investigate the possibility of formulatinga Weyl quantisation suitable for loop quantum gravity-type models.Before we come to the main part of the article, let us briefly outline its structure and content:The main section II is devoted to applications of the (abstract) methods introduced in the previousarticle . In the first subsection II A, we apply the global and local Weyl quantisations for compactLie groups to the basic building blocks of loop quantum gravity-type models, T ∗ G , G a compactLie group. Moreover, we show how and to what extent compatibility with the projective limit,Γ = lim ←− l ∈ L Γ l , Γ l ∼ = T ∗ G n l (cf. ), of finite dimensional truncations of the gravitational phasespace, Γ = | Λ | T ∗ A P (in Ashtekar-Barbero variables), can be achieved, thus, allowing for a genuineWeyl quantisation of loop quantum gravity-type models. In the course of this analysis, we discovercertain subtle differences between the phase space quantisation and the quantisation in terms of theholonomy-flux algebra, that was so far only noticed in recent work by Lanéry and Thiemann . Inrespect of the Born-Oppenheimer approximation, the main difference of our approach to previousones (notably ) is that we aim, already from the beginning, for a technical setup, which is able todeal with full loop quantum gravity (in its common realisations).In subsection II B, we analyse the possibility to define a “non-commutative phase space” by meansof the inductive family of quantum algebras that is obtained from the Weyl quantisation of theprojective family of (truncated) phase spaces. We also comment on the dual notion of projectivefamilies of (algebraic) state spaces (cp. ).In the last subsection II C of the main part, we explain the behaviour of gauge transformation w.r.t.the Weyl quantisation and the projective/inductive limit structures.Finally, we present some concluding remarks and perspectives in section III. II. LOOP QUANTUM GRAVITY AND PHASE SPACE QUANTISATION
While the previous articles were of a rather general mathematical character, the present sectionis devoted to discussing applications of the outlined framework to models of a loop quantum gravity-type (cf. for an application to spin systems). We show how the transformation group C ∗ -algebra C ( G ) ⋊ L G makes a, quite natural, appearance in the phase space quantisation of loop quantumgravity type models that are based on a gauge theory with compact (Lie) structure group G , anddiscretisations w.r.t. graphs (cf. and references therein). Furthermore, we discuss how Weyl andKohn-Nirenberg quantisation enter the picture. To begin with, we recall some basic notions fromloop quantum gravity. We follow closely , although we refine certain aspects of the presentation:Loop quantum gravity is based on a Hamiltonian formulation of general relativity in terms ofa constrained Yang-Mills-type theory, i.e. in a field theoretic description the phase space of theclassical theory is given by the (densitiesed) cotangent bundle | Λ | T ∗ A P to the space of connections A P on a given (right, semi-analytic ) principal G -bundle P π → Σ, where Σ is the spatial manifoldin a 3+1-splitting of a (globally hyperbolic) spacetime M ∼ = R × Σ. In general relativity, we have G = SU(2) , Spin , or central quotients of these groups, but for most of what follows we only needto assume that G is a compact Lie group.The basic variables, the theory is phrased in, are the Ashtekar-Barbero connection A ∈ A P andits conjugate momentum E ∈ Γ (cid:0) T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ) (cid:1) . Strictly speaking, we further require E to be non-degenerate as a (densitiesed) section of the bundle of linear operators L(Ad(P) , T Σ). Ingeneral relativity, the existence of E is ensured by the triviality of the orthogonal frame bundle P SO (Σ). This mathematical setup also appears to be valid in the context of the new variablesproposed in . Here, Ad ∗ (P) := P × Ad ∗ g ∗ and | Λ | (Σ) denotes the bundle of 1-densities on Σ.Since A P is an affine space modelled on Ω (Ad(P)) := Γ( T ∗ Σ ⊗ Ad(P)), Ad(P) := P × Ad g , thefollowing Poisson structure { E ai ( x ) , A jb ( y ) } = δ ab δ ji δ ( x, y ) (2.1)is meaningful in local coordinates φ : U ⊂ Σ → V ⊂ R subordinate to a local trivialisation ψ : P | U → U × G , i.e. (( φ ◦ ψ ) − ) ∗ A | P | U = A jb dx b ⊗ τ j , (2.2)( φ ◦ ψ ) ∗ E | P | U = E ai ∂∂x a ⊗ τ ∗ i . Here, { τ j } j is a basis of g and { τ ∗ i } i its dual in g ∗ .The variables ( A, E ) are directly related to the Arnowitt-Deser-Misner variables ( q, P ). Namely, E ai is a densitiesed triad for the spatial metric q ab E ai E bj = det( q ) δ ij , and A ia = Γ ia + K ia is built outof the Levi-Civita connection Γ of the spatial metric q and the extrinsic curvature K determinedby the momentum P .What makes the variables ( A, E ) special, is that they allow to carry out a canonical quantisation ofgeneral relativity, i.e. loop quantum gravity (cf. for general accounts on the topic). Especially, itis possible to construct mathematically well-defined operators for all constraints acting in a suitableHilbert space within this approach, most prominently the Wheeler-DeWitt constraint (cf. ). A. Projective-phase space structure and Weyl quantisation
The canonical quantisation of Γ := | Λ | T ∗ A P , adapted optimally to our framework, starts fromthe following functionals of the basic variables ( A, E ): Definition II.1:
Let γ ∈ Γ sa , ↑ be an oriented, embedded, semi-analytic, compactly supported, finite graph in Σ , P γ an oriented, semi-analytic, polyhedronal decomposition of Σ dual to γ , and Π γ a system of oriented,semi-analytic paths adapted to γ and P γ , i.e. for every edge e ∈ E ( γ ) there exists a unique face S e of P γ having unique transversal intersection x e = e ∩ S e in an interior point, and a collectionof paths { ρ e ( x ) | x ∈ S e } ⊂ Π γ from any x ∈ S e to x e . Moreover, S e carries a compatibleorientation of its normal bundle, i.e. aligned with the orientation of the edge e . As usual, the edges(connected, oriented semi-analytic submanifolds, possibly with two-point boundary) and vertices(boundary points of edges) of a graph γ are denoted by e ∈ E ( γ ) , respectively, v ∈ V ( γ ) . An edgewill be treated as an embedded submanifold e : [0 , → Σ . The orientationsThe triple l = ( γ, P γ , Π γ ) is called a finite, oriented, semi-analytic structured graph in Σ , and weassociate with it the following functionals of ( A, E ) ∈ Γ : g e ( A ; σ )( s, t ) ∈ G, s ≤ t ∈ [0 ,
1] s.t. hol Ae ( s ( e ( s ))) = R g e ( A ; σ )( s,t ) ( s ( e ( t ))) , (2.3) P eX ( A, E ; σ ) = Z S e ∗ (cid:16)(cid:16) Ad ∗ g e ( A ; σ )( s e , g ρe ( A ; σ )(0 , ( E σ ) (cid:17) ( X ) (cid:17) , X ∈ g , where σ : Σ → P is a (measurable) section, which defines a (measurable) equivalences Σ × G ∼ σ P , ψ σ ( x, g ) = R g ( σ ( x )) , and Σ × g ∗ ∼ σ Ad ∗ (P) , Ad ∗ ( ψ σ )( x, θ ) = [( σ ( x ) , θ )] . In this sense, E σ isthe vector density valued section of Σ × g corresponding to E , and ∗ ( E σ ( X )) is the pseudo-2-formdual the vector density E ( X ) . The parameter s e ∈ [0 , is determined from e ( s e ) = x e .The set of all finite, oriented, semi-analytic structured graphs is called L , and we abbreviate g e ( A ; σ )(0 ,
1) = g e ( A, σ ) in the following. The images of Γ under the functionals (2.3) (s=0,t=1),when varied w.r.t. l ∈ L , are the truncated phase spaces Γ l := × e ∈ E ( γ ) ( T ∗ G ) e ∼ = T ∗ G | E ( γ ) | (w.r.t.the right trivialisation). The images of Γ under the holonomy functionals alone are the truncatedconfigurations spaces C l := × e ∈ E ( γ ) G e ∼ = G | E ( γ ) | , which are naturally covered by the Γ l via thecotangent bundle projection T ∗ G → G . A detailed discussion of these functionals, and the issue of imposing boundary conditions on (
A, E ),can be found in , where a regularised Poisson (even symplectic) structure on the truncated phasespaces Γ l = T ∗ G ×| E ( γ ) | , l = ( γ, P γ , Π γ ) ∈ L , compatible with (2.1) is derived, as well: Proposition II.2 (cf. , Theorem 3.2): Let l = ( γ, P γ , Π γ ) ∈ L and f, f ′ ∈ C ∞ ( G ) , then the regularised Poisson structure on Γ l w.r.t. tothe functionals (2.3) agrees with the Poisson structure on (smooth) polynomial symbols coming fromthe canonical symplectic form on T ∗ G ×| E ( γ ) | ∼ = ( G × g ) ×| E ( γ ) | (cp. theorem III.14 and equations(3.61), (3.62) & (3.65) of our second article ) on , i.e.: { f ( g e ( . ; σ )) , f ′ ( g e ′ ( . , σ )) } Γ l ( A, E ) = 0 , (2.4) { P eX ( . , . ; σ ) , f ′ ( g e ′ ( . , σ )) } Γ l ( A, E ) = δ e,e ′ ( R X f ′ )( g e ′ ( A ; σ )) , { P eX ( . , . ; σ ) , P e ′ Y ( . , . ; σ ) } Γ l ( A, E ) = − δ e,e ′ P e [ X,Y ] ( A, E ; σ ) . Since the functionals (2.3) provide coordinates for the truncated phase spaces Γ l , l ∈ L , we mayextend the Poisson structure (2.4) to C ∞ ( T ∗ G | E ( γ ) | ) ∼ = C ∞ ( T ∗ G ) ˆ ⊗| E ( γ ) | , γ ∈ l (the isomorphismis to be understood in the nuclear Fréchet space topology, cf. ).The functionals (2.3) behave naturally w.r.t. to composition and inversion of edges: Lemma II.3:
Let e = e ◦ e , i.e. e ( t ) = e (2 t ) , t ∈ [0 , ] , e ( t ) = e (2 t − , t ∈ [ , , and e − ( t ) = e (1 − t ) , t ∈ [0 , , then we have: g e ( A ; σ )( s, t ) = g e ( A ; σ )(2 s, t ) : s ≤ t ≤ g e ( A ; σ )(0 , t − g e ( A ; σ )(2 s,
1) : s ≤ ≤ tg e ( A ; σ )(2 s − , t −
1) : ≤ s ≤ t , (2.5) g e − ( A ; σ )( s, t ) = g e ( A ; σ )(1 − t, − s ) − and P e − X ( A, E ; σ ) = − P e Ad ge ( A ; σ ) ( X ) ( A, E ; σ ) . (2.6) Proof: (2.5) is a simple consequence of the properties of the holonomy map of a connection A ∈ A P . (2.6)follows from (2.5), if we assume that the polyhedronal decomposition of an edge-inverted graph γ − (corresponding to γ ) is chosen s.t. S e − = S e carries the orientation opposite to S e , and thesystems of paths satisfy ρ e − ( x ) = ρ e ( x ) , x ∈ S e , s e − = 1 − s e .The behaviour under the group of gauge transformations G P is natural as well and even vertexlocal . Lemma II.4:
Let λ ∈ G P , and denote by f λ ∈ C (P , G ) α the corresponding α -equivariant G -valued function on P ( α is the conjugation in G ). Then, the transformations, A ( λ − ) ∗ A, (2.7) E λ ∗ E, where E is identified with its Ad ∗ -equivariant extension to P , induce the following transformationson the functionals (2.3) : g e (( λ − ) ∗ A ; σ )( s, t ) = f λ ( σ ( e ( t ))) g e ( A ; σ ) f λ ( σ ( e ( s ))) − , (2.8) P eX (( λ − ) ∗ A, λ ∗ E ; σ ) = P e Ad fλ ( σ ( e (1))) − ( X ) ( A, E ; σ ) . Proof:
This is a simple application of the transformation behaviour of (
A, E ) ∈ | Λ | T ∗ A P .Let us add an extend remark concerning the structure of the functionals (2.3), and their dependenceon the choice of an auxiliary (measurable) section σ : Σ → P. Remark II.5:
The space of connections A P can be modelled as an affine space on the space of Ad-equivariant,horizontal, g -valued 1-forms on P, Λ (P , g ) Ad , which serves as configuration space in the Ashtekar-Barbero formulation of general relativity (and also in higher dimensional generalisations, cf. ).The momentum variables, on the other hand, are elements of Γ (cid:0) T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ) (cid:1) . But, tomake the (densitiesed) cotangent bundle structure explicit, i.e. ( A, E ) ∈ | Λ | T ∗ A P , we have to iden-tify Γ ( T Σ ⊗ Ad ∗ (P)) with T ∗ A A P = (Λ (P , g ) Ad ) ∗ , which, indeed, is possible because (Λ (P , g ) Ad ) ∗ is isomorphic with the space of Ad ∗ -equivariant, horizontal (w.r.t. A ), g -valued vector fields on P, X (P , g ) Ad ∗ . Thus, E can be realised as an element of | Λ | T ∗ A A P , and does depend on the base point A .Therefore, the impression that the use of the position functionals g e ( A ; σ ) is the sole source ofdependence of the momentum functional P eX ( A, E ; σ ) on A is only apparent, because the usualdefinition of ( A, E ) employs the trivialisation | Λ | T ∗ A P ∼ = A P × Γ (cid:0) T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ) (cid:1) , whichis structurally similar to the trivialisation T ∗ G ∼ = G × g ∗ , shrouding the (densitiesed) cotangentbundle structure.This said, we may appreciate the special form of the Poisson algebra (2.4) generated by the func-tionals (2.3), which intertwines the (right) trivialisations of | Λ | T ∗ A P and T ∗ G ×| E ( γ ) | .Now, the dependence of the functionals (2.3) on the (measurable) section σ : Σ → P remains to beclarified:Clearly, if we are given two sections σ, σ ′ : Σ → P, the fibre-transitive (right) action of G on Pwill provide us with a measurable function g : Σ → G , s.t. σ ′ ( x ) = R g ( x ) ( σ ( x )). Inspecting thedefinitions (2.3) closely, and making use of the equivariance of the constructions, we find: g e ( A ; σ ′ )( s, t ) = g ( e ( t )) − g e ( A ; σ )( s, t ) g ( e ( s )) , (2.9) P eX ( A, E ; σ ′ ) = P e Ad g ( e (1)) ( X ) ( A, E ; σ ) . Thus, comparing (2.8) and (2.9), we see that a change of section from σ to σ ′ is similar in effectto a gauge transformation, which could also be inferred from the observation that λ ◦ σ defines asection of P for λ ∈ G P . But, although these operations of changing the section σ and acting withgauge transformation λ effect the functionals (2.3) in a similar fashion, they are strictly speakingnot equivalent, because the transformations induced by changes of sections are only measurable,while the gauge transformations come with additional regularity properties (semi-analytic in ourcase), which are influenced by the possible non-triviality of the bundle P. Nevertheless, as long aswe are concerned with a finite collection of structured graphs, or at least locally finite collections ,and the regularity properties allow for a suitable localisation of gauge transformations, e.g. semi-analyticity, the overall effect of the gauge group G P accounts for all possible changes of sections σ ,as well, due to the vertex local character of (2.9) and (2.8).But, if the action of the gauge group G P is essentially equivalent to the action of all (measurable)maps g : Σ → G , we may wonder, whether the topological (and differential geometric) properties ofthe bundle P are in any way reflected in the quantum theory, and thus if (principal) fibre bundles areimportant to question in the quantum theory at all. The answer to this question is quite subtle, butit can be shown that non-trivial topological properties of the gauge group G P (e.g. existence of largegauge transformation) leave an imprint on the structure of the algebra of observables (e.g. θ -sectors)under certain conditions (e.g. chirally coupled fermions with chiral anomaly ). An observationalong similar lines was made by Landsman , i.e. domains of definition for (unbounded) observablesof (quantum) particles coupled to external gauge fields can be affected by topological properties ofthe (classical) bundle P.Coming back to Poisson relation (2.4), proposition II.2 tells us that it makes sense to identifythe functional P e ( . ) ( . , . ; σ ) with the momentum map of the strongly Hamiltonian left G -action L ∗ ( . ) − on the e -th component of Γ l (cp. equations (3.61) & (3.62) of our companion article ).Furthermore, the behaviour under edge inversion (2.6), e e − ( l l ′ in L ), is precisely suchthat it turns P e − ( . ) ( . , . ; σ ) into the momentum map of the compatible right action R ∗ ( . ) − : { P e − X ( . , . ; σ ) , f ′ ( g e ′− ( . , σ )) } Γ l ′ ( A, E ) (2.10)= δ e,e ′ ( R X f ′ )( g e ′− ( A ; σ ))= δ e,e ′ ddt | t =0 f ′ (exp( tX ) g e ′ ( A ; σ ) − )= δ e,e ′ ddt | t =0 ( f ′ ◦ ( . ) − )( g e ′ ( A ; σ ) exp( − tX ))= δ e,e ′ ddt | t =0 ( f ′ ◦ ( . ) − )(exp( − t Ad g e ′ ( A ; σ ) ( X )) g e ′ ( A ; σ ))= − δ e,e ′ ( R Ad ge ′ ( A ; σ ) ( X ) ( f ′ ◦ ( . ) − ))( g e ′ ( A ; σ ))= {− P e Ad ge ( . ; σ ) ( X ) ( . , . ; σ ) , ( f ′ ◦ ( . ) − )( g e ′ ( . , σ )) } Γ l ( A, E ) . In the (right) trivialisation T ∗ G | E ( γ ) | ∼ = ( G × g ∗ ) | E ( γ ) | these actions are given explicitly as: L ∗ h − ( θ, g ) = (Ad ∗ h ( θ ) , hg ) , R ∗ h − ( θ, g ) = ( θ, gh − ) , (2.11)where g, h ∈ G, θ ∈ g ∗ . This, in turn, allows us to associate with each edge e of γ ∈ l the C ∗ -dynamical system ( C ( G ) , G, α L ), which is determined by the integrated form of (2.4) (see equations(3.8) of our second article ), and thus the transformation group C ∗ -algebra C ( G ) ⋊ L G . The edgeinversion fits into this (global) picture in the following sense: Proposition II.6:
Given a structured graph l ∈ L , we consider the collection of C ∗ -dynamical systems ( C ( G ) , G, α L ) e , e ∈ E ( γ ) , associated with Γ l via the functionals (2.3) . The edge inversion, e e − , induces theisomorphism ( C ( G ) , G, α L ) e − ∼ = ( C ( G ) , G, α R ) e , e ∈ E ( γ ) . (2.12) Additionally, we have a natural isomorphism ( C ( G ) , G, α L ) ∼ = ( C ( G ) , G, α R ) , which induces anisomorphism of the assignment e ( C ( G ) , G, α L ) e form the edge inversion.From the collection of C ∗ -dynamical systems ( C ( G ) , G, α L ) e , e ∈ E ( γ ) , we may form the tensorproduct of the associated transformation group C ∗ -algebras, A l := ( C ( G ) ⋊ L G ) ⊗| E ( γ ) | (which isunambiguous, because C ( G ) ⋊ L G is nuclear (cf. , G is amenable), and the order of the tensorfactors is irrelevant, because associativity and commutativity for the tensor product, e.g. the spatialtensor product, are implemented by natural isomorphisms (cf. )). The latter satisfies: ( C ( G ) ⋊ L G ) ⊗| E ( γ ) | ∼ = C ( C l ) ⋊ L C l . (2.13) Proof:
The isomorphism (2.12) is immediate from the behaviour of the functionals (2.3) under edge in-version and the comment preceding the proposition. Thus, we only need to prove the isomorphism( C ( G ) , G, α L ) ∼ = ( C ( G ) , G, α R ). We do this via the natural left and right regular integrated repre-sentations on L ( G ),( ρ L ( f )Ψ) ( g ) = Z G dh f ( h, g )Ψ( h − g ) , ( ρ R ( f )Ψ) ( g ) = Z G dh f ( h, g )Ψ( gh ) (2.14)for f ∈ C ( G, C ( G )) , Ψ ∈ L ( G ), which provide isomorphisms of the transformation group C ∗ -algebras C ( G ) ⋊ L G and C ( G ) ⋊ R G with K ( L ( G )) (see definition II & theorem II.7 of ourcompanion article ).The algebras C ∗ -algebras C ( G ) ⋊ L G and C ( G ) ⋊ R G are defined as completions of C ( G, C ( G )) inthe universal C ∗ -norm together with an involution and a convolution product (see definition II.4of our companion article ), which involve the left respectively right action of G on itself. We provethe isomorphism by providing an isomorphism of C ( G, C ( G )) that intertwines these structures via ρ L and ρ R . ∀ f ∈ C ( G, C ( G )) : I ( f )( h, g ) := f ( α g ( h − ) , g ) = f ( gh − g − , g ) , (2.15) I − ( f )( h, g ) = f ( α g − ( h − ) , g ) , h, g ∈ G. Clearly, I : C ( G, C ( G )) → C ( G, C ( G )) is an isomorphism (with inverse I − ), because group mul-tiplication and inversion are continuous. Next, let us see how ρ L and ρ R are related via I :( ρ R ( I ( f ))Ψ) ( g ) = Z G dh I ( f )( h, g )Ψ( gh ) = Z G dh f ( α g ( h − ) , g )Ψ( gh ) (2.16)= Z G dh f ( h − , g )Ψ( hg ) = Z G dh f ( h, g )Ψ( h − g )= ( ρ L ( f )Ψ) ( g ) . The involutions and multiplications are intertwined w.r.t. I as well: ∀ f ∈ C ( G, C ( G )) : I ( f ∗ L ) ( h, g ) = f ∗ L ( α g ( h − ) , g ) (2.17)= f ( α g ( h ) , α g ( h ) g )= f ( α gh ( h ) , gh )= f (( α gh ( h − )) − , gh )= I ( f )( h − , gh )= I ( f ) ∗ R ( h, g ) , ∀ f.f ′ ∈ C ( G, C ( G )) : I ( f ∗ L f ′ ) ( h, g ) = ( f ∗ L f ′ )( α g ( h − ) , g ) (2.18)= Z G dk f ( k, g ) f ′ ( k − α g ( h − ) , k − g )= Z G dk f ( α g ( k ) , g ) f ′ ( α g ( k − h − ) , gk − )= Z G dk f ( α g ( k − ) , g ) f ′ ( α gk (( k − h ) − ) , gk )= Z G dk I ( f )( k, g ) I ( f ′ )( k − h, gk )= ( I ( f ) ∗ R I ( f ′ )) ( h, g ) . (2.13) follows from the isomorphisms C ( G ) ⋊ L G ∼ = K ( L ( G )) and K ( L ( G )) ⊗ ∼ = K ( L ( G × ))(cf. ). Remark II.7:
Noteworthy, the isomorphism (2.12) (or its inverse) reflects the cotangent bundle structure T ∗ G ∼ = G × g ∗ on the (global) level of G × G , because it is related to the momentum maps (in the righttrivialisation, see equations (3.61) & (3.44) of our second article ): f ( h, g ) = f (exp( X h ) , g ) , (2.19)= Z g ∗ dθ (2 π ) n e iθ ( X h ) ˆ f ( θ, g ) , = Z g ∗ dθ (2 π ) n e iθ ( X h ) ˆ f (cid:16) J L ∗ ( . ) − ( θ, g ) , g (cid:17) ,I ( f )( h, g − ) = f ( α g − ( h − ) , g − )= f (exp( − Ad g − ( X h )) , g − )= Z g ∗ dθ (2 π ) n e iθ ( X h ) ˆ f ( − Ad ∗ g − ( θ ) , g − )= Z g ∗ dθ (2 π ) n e iθ ( X h ) ˆ f (cid:16) J R ∗ ( . ) − ( θ, g ) , g − (cid:17) , where f exp ( X, g ) = f (exp( X ) , g ) and exp( X h ) = h .So far, we have only analysed the relations between the functionals (2.3) associated with structuredgraphs l, l ′ ∈ L that are related via edge inversion, but it is possible to introduce partial orders, ≤ and . , on L that leads to a projective structure on the collection of truncated phase spacesΓ l , l ∈ L (see below). ≤ turns out to be compatible with the Poisson algebra (2.4) and certaingeneralisations of the C ∗ -dynamical systems introduced in proposition II.6. This will also explain,why we have not made the dependence of the functionals (2.3) on l ∈ L explicit, but only indicateda dependence on γ ∈ l . Definition II.8 (cp. & ): Given two structured graphs l = ( γ, P γ , Π γ ) , l ′ = ( γ ′ , P ′ γ ′ , Π ′ γ ′ ) ∈ L , we say that l ≤ l ′ if γ ⊆ γ ′ ,i.e. the oriented graph γ ∈ l is an oriented subgraph of the oriented graph γ ′ ∈ l ′ .If l ≤ l ′ and l ′ ≤ l , we say that l and l ′ are equivalent, l ∼ l ′ .Alternatively, we say that l . l ′ , if | γ | ⊆ | γ ′ | (the non-oriented graphs underlying γ and γ ′ agree).If l . l ′ and l ′ . l , we say that l and l ′ are equivalent up to orientation, l ≃ l ′ .Two other, but somewhat different, partial orders are the following (cf. , p. 52-53):We say l ⋖ L l ′ , if | γ | ⊆ | γ ′ | , and ∀ e ∈ E ( γ ) : ∃ e ′ ∈ E ( γ ′ ) : ∃ s ∈ [0 ,
1) : e | [ s , = e ′ . (2.20) We write, l . = L l ′ , if l ⋖ L l ′ and l ′ ⋖ L l .We say l ⋖ R l ′ , if | γ | ⊆ | γ ′ | , and ∀ e ∈ E ( γ ) : ∃ e ′ ∈ E ( γ ′ ) : ∃ s ∈ (0 ,
1] : e | [0 ,s ] = e ′ . (2.21) We write, l . = R l ′ , if l ⋖ R l ′ and l ′ ⋖ R l .Clearly, l . = L l ′ or l . = R l ′ , if and only if γ = γ ′ ( s = 0 is necessary). Loosely speaking, ⋖ L and ⋖ R encode the condition that a graph γ ′ , which is finer (and possibly larger) than another graph γ , contains an (oriented) edge e ′ ∈ E ( γ ′ ) corresponding to the last respectively first part of an(oriented) edge e ∈ E ( γ ) . It follows from the discussion in that ( L , ≤ ) and ( L , . ) are partially ordered sets . Moreover,( L , . ) is directed, in contrast with ( L , ≤ ), which follows, because any two non-oriented, finite,semi-analytic graphs γ, γ ′ have a common refined graph γ ′′ , that admits an orientation and a dual0polyhedronal decomposition (cf. , see also , Section 6.2.2).It is also easy to see that ( L , ⋖ L ) and ( L , ⋖ R ) are partially ordered and directed (cf. ), because γ ′′ can be oriented s.t. (2.20) respectively (2.21) are satisfied w.r.t. the edge sets E ( γ ), E ( γ ′′ ) and E ( γ ′ ), E ( γ ′′ ).In case we are not in danger of ambiguities, we will use L as a short hand for ( L , ≤ ), ( L , . ),( L , ⋖ L ) and ( L , ⋖ R ).In the next theorem (II.9), we show that the partial orders ≤ , ⋖ L and ⋖ R on L are compatiblewith the Poisson structures defined on Γ l , l ∈ L . We also show, why we have, at this point, todeal, with oriented graphs, Γ sa , ↑ , although edge inversion e → e − induces and isomorphism of Γ l and Γ l ′ (see lemma II.3 & (2.10)), when γ and γ ′ agree up to some edge orientations.The reason for this lies in a compatibility condition of edge inversion and composition, that is notnecessarily satisfied for the corresponding maps between the truncated phase spaces Γ l , l ∈ L .In contrast, the Ashtekar-Isham-Lewandowski Hilbert space, L ( A ), which is a fundamental build-ing block of loop quantum gravity, arises from a projective structure constructed w.r.t. finite,non-oriented, semi-analytic graphs Γ sa0 instead of L (cf. for a general exposition, and originalreferences). Thus, there seems to be a certain tension between the phase space quantisation forloop quantum gravity presented here (cf. ), and the framework based on the holonomy-fluxalgebra and its Hilbert space representation on L ( A ).The link between the two can be roughly understood as follows:If we consider only the images of the holonomy functionals (2.3), we will obtain the truncatedconfiguration spaces C l ∼ = G | E ( γ ) | , l ∈ L , which admit the coarsening . of the partial order ≤ introduced above. Then, we will have projections p ll ′ : C l ′ → C l , if l . l ′ , which are compatiblewith . , i.e. p ll ′′ = p ll ′ ◦ p l ′ l ′′ for l . l ′ . l ′′ . Furthermore, these maps can be lifted to symplecticprojections ˜ p ll ′ : Γ l ′ → Γ l , but these lifts are not unique, and therefore turn out to be only compat-ible with ≤ , ⋖ L and ⋖ R instead of . .We will further comment on the implications of this issue on the relation between phase spacequantisation and holonomy-flux algebras in theorem II.10 and the outlook III. Theorem II.9:
Given l, l ′ ∈ L , s.t. l . l ′ , we have smooth projections ˜ p cll ′ : Γ l ′ → Γ l , c = { c e ′ ( l, l ′ ) } e ′ ∈ E ( γ ′ ) ⊂ R ,defined by g e ( A ; σ ) = g e ′ m ( A ; σ ) s m ...g e ′ ( A ; σ ) s , (2.22) P eX ( A, E ; σ ) = c e ′ m ( l, l ′ ) P e ′ smm X ( A, E ; σ ) + ... + c e ′ ( l, l ′ ) P e ′ s Ad ( ge ′ m ( A ; σ ) sm ...ge ′ A ; σ ) s − ( X ) ( A, E ; σ ) , m X n =1 c e ′ n ( l, l ′ ) = 1 , where e ∈ E ( γ ) , e ′ , ..., e ′ m ∈ E ( γ ′ ) s.t. e = e ′ s m m ◦ ... ◦ e ′ s with s n ∈ {± } ∀ n = 1 , ..., m , because | γ | ⊆ | γ ′ | .If c e ′ n = 1 for n = 1 or n = m in the decomposition of an edge e = e ′ s m m ◦ ... ◦ e ′ s , we denote thecorresponding maps by p R ll ′ and p L ll ′ , which have the properties:1. The dual maps ˜ p R , L ∗ ll ′ : C ∞ (Γ l ) → C ∞ (Γ l ′ ) are injective, continuous Poisson maps w.r.t. (2.4) : ∀ f, f ′ ∈ C ∞ (Γ l ) : ˜ p R , L ∗ ll ′ { f, f ′ } Γ l = { ˜ p R , L ∗ ll ′ f, ˜ p R , L ∗ ll ′ f ′ } Γ l ′ . (2.23)1
2. In case, l is equivalent to l ′ ( l ≃ l ′ ), ˜ p R ∗ ll ′ = ˜ p L ∗ ll ′ and ˜ p R ∗ l ′ l = ˜ p L ∗ l ′ l , are Poisson isomorphismsthat are inverse to one another.3. If l ≤ l ′ ≤ l ′′ , the maps ˜ p R , L ll ′ , ˜ p R , L l ′ l ′′ and ˜ p R , L ll ′′ are compatible with transitivity of ≤ , i.e. ˜ p R , L ll ′ ◦ ˜ p R , L l ′ l ′′ = ˜ p R , L ll ′′ (edge orientations coincide, ∀ n = 1 , ..., m : s n = 1 ).If l ⋖ R , L l ′ ⋖ R , L l ′′ , the maps ˜ p R , L ll ′ , ˜ p R , L l ′ l ′′ and ˜ p R , L ll ′′ are compatible with transitivity of ≤ , i.e. ˜ p R , L ll ′ ◦ ˜ p R , L l ′ l ′′ = ˜ p R , L ll ′′ (edge orientations coincide for n = m respectively n = 1 , i.e. s m = 1 or s = 1 ).4. If γ ⊂ γ ′ and l − , l ′− denote the structured graphs with all edge orientations reversed, and p l − l : Γ l → Γ l − , l ∈ L , are the edge inversion maps ( ˜ p R l − l = ˜ p L l − l ), we have: Γ l ′ ˜ p l ′− l ′ / / ˜ p L ll ′ (cid:15) (cid:15) Γ l ′− ˜ p R l − l ′− (cid:15) (cid:15) Γ l ˜ p l − l / / Γ l − (2.24) In general, p c ∗ ll ′ , l ≤ l ′ will only be Poisson for those choices of c , s.t. for every composition e = e ′ s m m ◦ ... ◦ e ′ s , we have c e ′ n ( l, l ′ ) = 1 for some n = 1 , ..., m (all other c e ′ n ( l, l ′ ) ’s vanish). Proof:
Let us first explain, why the maps ˜ p cll ′ are natural lifts of the maps p ll ′ (the latter arise from theholonomy part of (2.22)), i.e. Γ l ′ ˜ p cll ′ / / ̟ l ′ (cid:15) (cid:15) Γ l̟ l (cid:15) (cid:15) C l ′ p ll ′ / / C l (2.25)If we consider a function f on G e , where e ∈ E ( γ ) decomposes in γ ′ as e = e ′ s m m ◦ ... ◦ e ′ s for some e ′ , .., e ′ m ∈ E ( γ ′ ), we can pull it back to × mn =1 G e ′ n via p ll ′ (to this end, we extend f by 1 on theother copies of G in Γ l ). Especially, we may pull back R eX f for some X ∈ g e , where R e is the rightinvariant derivation on the e -th copy of G :( p ∗ ll ′ ( R X f )) ( g e ′ m , ..., g e ′ ) = ddt | t =0 f (cid:16) e tX g s m e ′ m ...g s e ′ (cid:17) (2.26)= ddt | t =0 f (cid:16) e tc e ′ m X g s m e ′ m ...e tc e ′ Ad ( gsme ′ m ...gs e ′ − ( X ) g s e ′ (cid:17) = (cid:16)(cid:16) c e ′ m R e ′ smm X + ... + c e ′ R e ′ s Ad ( gsme ′ m ...gs e ′ − ( X ) (cid:17) ( p ∗ ll ′ f ) (cid:17) ( g e ′ m , ..., g e ′ )for P mn =1 c e ′ m = 1. But, since the right invariant derivation R e is generated by the Poisson bracketwith P e , if we interpret f as a function on Γ l via the cotangent bundle projection, we see that P e should arise from the P e ′ n , n = 1 , ..., m, in precisely the way given in (2.22).2That ˜ p cll ′ : Γ l ′ → Γ l , c = { c e ′ } e ′ ∈ E ( γ ′ ) are projections is obvious from the definition (we suppress thepossible dependence of c on l, l ′ at this point), and the fact, that a subgraph | γ | ⊆ | γ ′ | is obtainedby removing and composing edges. Smoothness of ˜ p cll ′ is implied by the smoothness of the groupoperations.Next, we show that ˜ p c ∗ ll ′ is Poisson if and only if c e ′ n = 1 for some n = 1 , ..., m (all other c ’s vanish).To deduce this, we realise that any ˜ p cll ′ , l . l ′ , is obtained from the successive application of threefundamental operations:1. Removal of an edge e ′ from γ ′ , i.e. ˜ r : T ∗ G × → T ∗ G, ˜ r (( θ , g ) , ( θ , g )) = ( θ , g ).2. Composition of two edges, e = e ◦ e , i.e. ˜ p c : T ∗ G × → T ∗ G ,˜ p c (( θ , g ) , ( θ , g )) = ( c Ad ∗ g ( θ ) + (1 − c ) θ , g g ).3. Inversion of an edge e e − , i.e. ˜ ι : T ∗ G → T ∗ G, ι ( θ, g ) = ( − Ad ∗ g − ( θ ) , g − ).It is obvious, that ˜ r ∗ : C ∞ ( T ∗ G ) → C ∞ ( T ∗ G × ) is Poisson, i.e. ∀ f, f ′ ∈ C ∞ ( T ∗ G ) : { ˜ r ∗ f, ˜ r ∗ f ′ } T ∗ G × (( θ , g ) , ( θ , g )) = ˜ r ∗ ( { f, f ′ } T ∗ G ) (( θ , g ) , ( θ , g )) , (2.27)because ˜ r ∗ f and ˜ r ∗ f ′ depend only on ( θ , g ). A short calculation shows, what conditions on c areimplied, if ˜ p c ∗ : C ∞ ( T ∗ G ) → C ∞ ( T ∗ G × ) is assumed to be Poisson. { ˜ p c ∗ f, ˜ p c ∗ f ′ } T ∗ G × (( θ , g ) , ( θ , g )) (2.28)= h ∂ θ ˜ p c ∗ f, R ˜ p c ∗ f ′ i (( θ , g ) , ( θ , g )) −h ∂ θ ˜ p c ∗ f ′ , R ˜ p c ∗ f i (( θ , g ) , ( θ , g )) − θ ([ ∂ θ ˜ p c ∗ f, ∂ θ ˜ p c ∗ f ′ ](( θ , g ) , ( θ , g )))+ h ∂ θ ˜ p c ∗ f, R ˜ p c ∗ f ′ i (( θ , g ) , ( θ , g )) −h ∂ θ ˜ p c ∗ f ′ , R ˜ p c ∗ f i (( θ , g ) , ( θ , g )) − θ ([ ∂ θ ˜ p c ∗ f, ∂ θ ˜ p c ∗ f ′ ](( θ , g ) , ( θ , g )))= ˜ p c ∗ ( h ∂ θ f, Rf ′ i − h ∂ θ f ′ , Rf i ) (( θ , g ) , ( θ , g )) − (2 − c ) c Ad ∗ g ( θ ) ((˜ p c ∗ [ ∂ θ f, ∂ θ f ′ ])(( θ , g ) , ( θ , g ))) − (1 − c ) θ ((˜ p c ∗ [ ∂ θ f, ∂ θ f ′ ])(( θ , g ) , ( θ , g )))= ˜ p c ∗ ( h ∂ θ f, Rf ′ i − h ∂ θ f ′ , Rf i ) (( θ , g ) , ( θ , g )) − (2 − c ) θ ˜ p c (( θ ,g ) , ( θ ,g )) ((˜ p c ∗ [ ∂ θ f, ∂ θ f ′ ])(( θ , g ) , ( θ , g )))+ (1 − c ) θ ((˜ p c ∗ [ ∂ θ f, ∂ θ f ′ ])(( θ , g ) , ( θ , g )))= ˜ p c ∗ ( { f, f ′ } T ∗ G ) (( θ , g ) , ( θ , g )) − (1 − c ) θ ˜ p c (( θ ,g ) , ( θ ,g )) ((˜ p c ∗ [ ∂ θ f, ∂ θ f ′ ])(( θ , g ) , ( θ , g )))+ (1 − c ) θ ((˜ p c ∗ [ ∂ θ f, ∂ θ f ′ ])(( θ , g ) , ( θ , g ))) , for all f, f ′ ∈ C ∞ ( T ∗ G × ). Here, we used the formula for the canonical Poisson structure on T ∗ G for { , } T ∗ G and { , } T ∗ G × (see theorem III.14 of our second article ). The last line shows, thatthe only possible choices for c , to make ˜ p c ∗ a Poisson map, are c = 1 or c = 0. Clearly, a similarphenomenon occurs for compositions involving more than 2 edges. Even, if we were to relax thecondition P mn =1 c e ′ n = 1, this phenomenon would persist.Another short calculation shows that ˜ ι ∗ : C ∞ ( T ∗ G ) → C ∞ ( T ∗ G ) is Poisson. { ˜ ι ∗ f, ˜ ι ∗ f ′ } T ∗ G ( θ, g ) = h ∂ θ ˜ ι ∗ f, R ˜ ι ∗ f ′ i ( θ, g ) − h ∂ θ ˜ ι ∗ f ′ , R ˜ ι ∗ f i ( θ, g ) − θ ([ ∂ θ ˜ ι ∗ f, ∂ θ ˜ ι ∗ f ′ ]( θ, g )) (2.29)= ˜ ι ∗ ( h ∂ θ f, Rf ′ i )( θ, g ) − ˜ ι ∗ ( h ∂ θ f ′ , Rf i )( θ, g ) + Ad ∗ g − ( θ )(˜ ι ∗ [ ∂ θ f, ∂ θ f ′ ]( θ, g ))3= ˜ ι ∗ { f, f ′ } T ∗ G ( θ, g ) . If l and l ′ are equivalent, l ≃ l ′ , ˜ p cll ′ = ˜ p ll ′ (no c -dependence) is induced from the map ˜ ι on singleedges. Since ˜ ι is an involution, ˜ ι ◦ ˜ ι = id T ∗ G , we conclude that ∀ c : ∀ l ≃ l : (˜ p ll ′ ) − = ˜ p l ′ l . (2.30)Moreover, because ˜ ι ∗ is Poisson, ˜ p ∗ ll ′ is Poisson for all l ≃ l ′ .To understand what conditions are imposed on the set c = { c e ′ } e ′ ∈ E ( γ ′ ) by demanding compatibilitywith transitivity w.r.t. ≤ , ⋖ R or ⋖ L , we take a look at the implications coming from the associativityof edge composition. This certainly encompasses the case of composing three edges in the forms( e ◦ e ) ◦ e = e ′ ◦ e = e and e ◦ ( e ◦ e ) = e ◦ e ′ = e :˜ p c ′ : T ∗ G ′ × T ∗ G → T ∗ G, (2.31)˜ p c ′ (( θ ′ , g ′ ) , ( θ , g )) = ( c ′ Ad ∗ g ′ ( θ ) + (1 − c ′ ) θ ′ , g ′ g ) , ˜ p c (32)1 : T ∗ G × T ∗ G × T ∗ G → T ∗ G ′ × T ∗ G , ˜ p c (32)1 (( θ , g ) , ( θ , g ) , ( θ , g )) = (( c (32)12 Ad ∗ g ( θ ) + (1 − c (32)12 ) θ , g g ) , ( θ , g )) , ˜ p c ′ : T ∗ G ′ × T ∗ G → T ∗ G, ˜ p c ′ (( θ , g ) , ( θ , g ) , ( θ , g )) = ((1 − c ′ )Ad ∗ g ′ ( θ ) + c ′ θ ′ , g ′ g ) , ˜ p c : T ∗ G × T ∗ G × T ∗ G → T ∗ G × T ∗ G ′ , ˜ p c (( θ , g ) , ( θ , g ) , ( θ , g )) = (( θ , g )( c Ad ∗ g ( θ ) + (1 − c ) θ , g g )) , ˜ p c : T ∗ G × T ∗ G × T ∗ G → T ∗ G, ˜ p c (( θ , g ) , ( θ , g ) , ( θ , g )) = ( c Ad ∗ g g ( θ ) + c Ad ∗ g ( θ ) + c θ , g g g ) , ⇒ (˜ p c ′ ◦ p c (32)1 )(( θ , g ) , ( θ , g ) , ( θ , g )) = (˜ p c ′ ◦ ˜ p c )(( θ , g ) , ( θ , g ) , ( θ , g )) (2.32)= ˜ p c (( θ , g ) , ( θ , g ) , ( θ , g )) ⇔ c = c ′ = (1 − c ′ ) c ∧ c = (1 − c ′ ) c (32)12 = (1 − c ′ )(1 − c ) ∧ c = (1 − c ′ )(1 − c (32)12 ) = c ′ , where we used the constraints c + c + c = 1 , c ′ + c ′ ′ = 1 , c ′ ′ + c ′ = 1 , c (32)12 + c (32)13 = 1and c + c = 1.Since we are only interested in Poisson maps, the only interesting cases to check are c n = δ n , c n = δ n and c n = δ n , n = 1 , ,
3. The first imposes ˜ p c ′ = ˜ p R2 ′ , ˜ p c (32)1 = ˜ p R(32)1 or ˜ p L(32)1 ,˜ p c ′ = ˜ p R31 ′ , ˜ p c = ˜ p R3(21) . The second case forces us to set ˜ p c ′ = ˜ p L2 ′ , ˜ p c (32)1 = ˜ p R(32)1 , ˜ p c ′ = ˜ p L31 ′ ,˜ p c = ˜ p L3(21) . The third case gives ˜ p c ′ = ˜ p L2 ′ , ˜ p c (32)1 = ˜ p L(32)1 , ˜ p c ′ = ˜ p L31 ′ , ˜ p c = ˜ p L3(21) or ˜ p R3(21) .Thus, we infer that choosing ˜ p L ll ′ or ˜ p R ll ′ for all l ≤ l ′ generates a system of maps compatible withtransitivity of ≤ , because outer left or right (w.r.t. edge orientation, i.e. n = 1 or n = m in acomposition chain) θ -labels are preserved in composition sequences. This property is not affectedby edge removal, because the latter only generates new (left or right) edge boundaries, which must4be present in a subgraph independent of the specific sequence of composing and removing edges.An analogous argument works for ⋖ R and ⋖ L in combination with ˜ p R ll ′ and ˜ p L ll ′ respectively, becausethese partial orders preserve the notion of first respectively last part of an edge between an orientedgraph and its oriented subgraphs (cf. , p. 52-53).The continuity of ˜ p L ∗ ll ′ can be reduced to the continuity of ˜ p L ∗ : C ∞ ( T ∗ G ) → C ∞ ( T ∗ G × ),(˜ p L ∗ f )(( θ , θ ) , ( g , g )) := f ( θ , g g ) , which corresponds to the fundamental operation of compos-ing two edges e = e ◦ e . || ˜ p L ∗ f || m, ( K ,K ) = sup α ,α ,β ,β ∈ N n | α | + | α | + | β | + | β |≤ m sup ( θ ,θ ∈ K × K g ,g ∈ G (cid:12)(cid:12)(cid:12)(cid:16) R α R α ∂ β θ ∂ β θ (˜ p L ∗ f ) (cid:17) (( θ , θ ) , ( g , g )) (cid:12)(cid:12)(cid:12) (2.33)= sup α ,α ,β ∈ N n | α | + | α | + | β |≤ m sup θ ∈ K g ,g ∈ G (cid:12)(cid:12)(cid:12)(cid:16) R α R α ∂ β θ f (cid:17) ( θ , g g ) (cid:12)(cid:12)(cid:12) ≤ C m sup α ,α ,β ∈ N n | α | + | α | + | β |≤ m sup θ ∈ K g ∈ G (cid:12)(cid:12)(cid:12)(cid:16) R α + α ∂ β θ f (cid:17) ( θ, g ) (cid:12)(cid:12)(cid:12) = C m || f || m,K , for some C m > , where m ∈ N and K , K < g ∗ are compact. To arrive at the inequality in the next to lastline, we used the fact that the commutator [ R i , R j ] = − f kij R k reduces the order of derivatives.The proof of the continuity of ˜ p R ∗ ll ′ is analogous: First, we reduce it to showing that the map˜ p R ∗ : C ∞ ( G ) → C ∞ ( G × ) , (˜ p R ∗ f )(( θ , θ ) , ( g , g )) := f (Ad ∗ g ( θ ) , g g ) , is continuous. Second,we use that Ad ∗ : G → GL ( g ∗ ) gives orthogonal transformations w.r.t. to some G -invariant metricon g ∗ , and ( θ, g )
7→ || Ad ∗ g ( θ ) || g ∗ is bounded on G × K for some compact subset K < g ∗ .Finally, commutativity of the diagram (2.24) follows from:(˜ p R l − l ′− ◦ ˜ p l ′− l ′ )( { ( θ e ′ , g e ′ ) } e ′ ∈ E ( γ ′ ) ) = ˜ p R l − l ′− ( { ( − Ad ∗ g − e ′ ( θ e ′ ) , g − e ′ ) } e ′ ∈ E ( γ ′ ) ) (2.34)= { (Ad ∗ g − e ′ e ...g − e ′ ( m − e ( − Ad ∗ g e ′ me ( θ e ′ me )) , g − e ′ e ...g − e ′ me ) } e ∈ E ( γ ) = { ( − Ad ∗ ( g e ′ me ...g e ′ e ) − ( θ e ′ me ) , ( g e ′ me ...g e ′ e ) − ) } e ∈ E ( γ ) = ˜ p l − l ( { ( θ e ′ me , g e ′ me ...g e ′ e ) } e ∈ E ( γ ) )= (˜ p l − l ◦ ˜ p L ll ′ )( { ( θ e ′ , g e ′ ) } e ′ ∈ E ( γ ′ ) ) . On the (quantum) level of continuous, linear operators L ( C ∞ ( C l )) (which is associated with C ( G ) ⋊ L G ⊗| E ( γ ) | in a natural way) acting on C ∞ ( C l ) ⊂ L ( C l ), we have *-morphisms α R , L l ′ l corresponding to the Poisson maps ˜ p R , L ∗ ll ′ : C ∞ (Γ l ) → C ∞ (Γ l ′ ) via quantisation (Kohn-Nirenbergand Weyl, see paragraph III.A.2 of our second article ). Theorem II.10:
Given l, l ′ ∈ L , s.t. l . l ′ , we represent the continuous, linear operators in L ( C ∞ ( C l )) and L ( C ∞ ( C l ′ )) by their (left) convolution kernels (obtained from Schwartz’ kernel theorem). Then, we have injective *-morphisms (*-isomorphisms for l ≃ l ′ ) α R , L l ′ l : D ′ ( C l ) ˆ ⊗ C ∞ ( C l ) → D ′ ( C l ′ ) ˆ ⊗ C ∞ ( C l ′ ) , (2.35) induced from the four fundamental injective *-morphisms: η : L ( C ∞ ( G )) → L ( C ∞ ( G )) , η ( F )(( h , g ) , ( h , g )) := δ e ( h ) F ( h , g ) , (2.36) γ : L ( C ∞ ( G )) → L ( C ∞ ( G )) , γ ( F )( h, g ) := I ( F )( h, g − ) = F ( α g − ( h − ) , g − ) ,α R : L ( C ∞ ( G )) → L ( C ∞ ( G )) , α R ( F )(( h , g ) , ( h , g )) := δ e ( h )(( α ∗ g ⊗ L ∗ g ) F )( h , g )= δ e ( h ) F ( α g ( h ) , g g ) ,α L : L ( C ∞ ( G )) → L ( C ∞ ( G )) , α L ( F )(( h , g ) , ( h , g )) := δ e ( h )( R ∗ g F )( h , g )= δ e ( h ) F ( h , g g ) , for F ∈ D ′ ( G ) ˆ ⊗ C ∞ ( G ) . Furthermore, we have commutative diagrams for l . l ′ : ˆ E ′ U l ( c ∗ l ) ˆ ⊗ C ∞ ( C l ) ˜ p R , L ∗ ll ′ / / F ( W ) ,ε (cid:15) (cid:15) ˆ E ′ U l ′ ( c ∗ l ′ ) ˆ ⊗ C ∞ ( C l ′ ) F ( W ) ,ε (cid:15) (cid:15) D ′ ( C l ) ˆ ⊗ C ∞ ( C l ) α R , L l ′ l / / D ′ ( C l ′ ) ˆ ⊗ C ∞ ( C l ′ ) , (2.37) where we used the notation of paragraph III.A.2 of our second article , and c ∗ l , l ∈ L , denotes thedual of the Lie algebra of C l .The maps α R , L l ′ l respect transitivity of ≤ , i.e. α R , L l ′′ l ′ ◦ α R , L l ′ l = α R , L l ′′ l for l ≤ l ′ ≤ l ′′ .Also for ⋖ R , L , we have transitivity of the corresponding collections of maps α R , L l ′ l , i.e. α R , L l ′′ l ′ ◦ α R , L l ′ l = α R , L l ′′ l for l ⋖ R , L l ′ ⋖ R , L l ′′ .If γ ⊂ γ ′ and l − , l ′− denote the structured graphs with all edge orientations reversed, and α ll − : D ′ ( C l − ) ˆ ⊗ C ∞ ( C l − ) → D ′ ( C l ) ˆ ⊗ C ∞ ( C l ) , l ∈ L , are the edge inversion *-isomorphisms( α R ll − = α L ll − ), we have: D ′ ( C l − ) ˆ ⊗ C ∞ ( C l − ) α ll − / / α L l ′− l − (cid:15) (cid:15) D ′ ( C l ) ˆ ⊗ C ∞ ( C l ) α R l ′ l (cid:15) (cid:15) D ′ ( C l ′− ) ˆ ⊗ C ∞ ( C l ′− ) α l ′ l ′− / / D ′ ( C l ′ ) ˆ ⊗ C ∞ ( C l ′ ) (2.38) Proof:
Since l . l ′ , we know that | γ | ⊆ | γ ′ | , i.e. γ is obtained from γ ′ by removing, inverting and composingedges. These operations are modelled by the four fundamental maps (2.36). Therefore, we only needto understand how an operator in L ( C ∞ ( G )) behaves w.r.t these, and whether the prescriptionsreally define *-morphisms. Thus, we may reduce the proof to showing that the maps (2.36) defineinjective *-morphisms.Let us first show injectivity: The injectivity of γ follows from the injectivity of I (see proposition II.66and (2.15)). Injectivity of α L , α R & η can be deduced in the following way: Assume we are given F, F ′ ∈ D ′ ( G ) ˆ ⊗ C ∞ ( G ) s.t. α ( F ) = α ( F ′ ). Then, we define Ψ ∈ C ∞ ( G × ) : Ψ ( g , g ) := Ψ( g )and Ψ ∈ C ∞ ( G × ) : Ψ ( g , g ) := Ψ( g g ) for Ψ ∈ C ∞ ( G ). Applying ρ L ( η ( F )) and ρ L ( η ( F ′ )) toΨ , we find:( ρ L ( η ( F ))Ψ )( g , g ) = Z G dh Z G dh η ( F )(( h , g ) , ( h , g ))Ψ ( h − g , h − g ) (2.39)= Z G dh F ( h , g )Ψ( h − g )= ( ρ L ( F )Ψ)( g )( ρ L ( η ( F ′ ))Ψ )( g , g ) = Z G dh Z G dh η ( F ′ )(( h , g ) , ( h , g ))Ψ ( h − g , h − g )= Z G dh F ′ ( h , g )Ψ( h − g )= ( ρ L ( F ′ ))Ψ)( g ) , which shows that ρ L ( F ) = ρ L ( F ′ ), and therefore F = F ′ .Applying ρ L ( α L ( F )) and ρ L ( α L ( F ′ )) to Ψ , we find:( ρ L ( α L ( F ))Ψ )( g , g ) = Z G dh Z G dh α L ( F )(( h , g ) , ( h , g ))Ψ ( h − g , h − g ) (2.40)= Z G dh F ( h , g g )Ψ( h − g g )= ( ρ L ( F )Ψ)( g g )( ρ L ( α L ( F ′ ))Ψ )( g , g ) = Z G dh Z G dh α L ( F ′ )(( h , g ) , ( h , g ))Ψ ( h − g , h − g )= Z G dh F ′ ( h , g g )Ψ( h − g g )= ( ρ L ( F ′ )Ψ)( g g ) , which shows that ρ L ( F ) = ρ L ( F ′ ), and therefore F = F ′ . An analogous calculation works for α R .The *-morphism property needs to be proved w.r.t. to the involution and convolution product of C ( G ) ⋊ R / L G (see definition II.4 of our companion article ), because we work with (left) convolutionkernels (linearity of η, γ & α R , L is evident). ∀ F, F ′ ∈ D ′ ( G ) ˆ ⊗ C ∞ ( G ): α L ( F ∗ L F ′ )(( h , g ) , ( h , g )) = δ e ( h )( F ∗ L F ′ )( h , g g ) (2.41)= δ e ( h ) Z G dh F ( h, g g ) F ′ ( h − h , h − g g )= Z G dh ′ Z G dh δ e ( h ′ ) F ( h, g g ) δ e ( h − h ) F ′ ( h − h , h − g h ′− g )= Z G dh ′ Z G dh α L ( F )(( h, g ) , ( h ′ , g )) × α L ( F ′ )(( h − h , h − g ) , ( h ′− h , h ′− g ))= ( α L ( F ) ∗ L α L ( F ′ ))(( h , g ) , ( h , g )) . α L ( F ∗ L )(( h , g ) , ( h , g )) = δ e ( h ) F ∗ L ( h , g g )= δ e ( h ) F ( h − , h − g g )= δ e ( h − ) F ( h − , h − g h − g )= α L ( F )(( h − , h − g ) , ( h − , h − g ))= α L ( F ) ∗ L (( h , g ) , ( h , g )) ,α R ( F ∗ L F ′ )(( h , g ) , ( h , g )) = δ e ( h )( F ∗ L F ′ )( α g ( h ) , g g ) (2.42)= δ e ( h ) Z G dh F ( h, g g ) F ′ ( h − α g ( h ) , h − g g )= Z G dh ′ Z G dh δ e ( h ′ ) F ( h, g g ) δ e ( h ′− h ) F ′ ( h − α g ( h ) , h ′− h − g g )= Z G dh ′ Z G dh δ e ( h ′ ) F ( α h ′− g ( h ) , g g ) δ e ( h ′− h ) × F ′ ( α h ′− g ( h − h ) , h ′− g h − g )= Z G dh ′ Z G dh α R ( F )(( h ′ , g ) , ( h, g )) × α R ( F ′ )(( h ′− h , h ′− g ) , ( h − h , h − g ))= ( α R ( F ) ∗ L α R ( F ′ ))(( h , g ) , ( h , g )) .α R ( F ∗ L )(( h , g ) , ( h , g )) = δ e ( h ) F ∗ L ( α g ( h ) , g g )= δ e ( h ) F ( α g ( h ) − , α g ( h ) − g g )= δ e ( h − ) F ( α h − g ( h ) − , h − g h − g )= α R ( F )(( h − , h − g ) , ( h − , h − g ))= α R ( F ) ∗ L (( h , g ) , ( h , g )) ,η ( F ∗ L F ′ )(( h , g ) , ( h , g )) = δ e ( h )( F ∗ L F ′ )( h , g ) (2.43)= δ e ( h ) Z G dh F ( h, g ) F ′ ( h − h , h − g )= Z G dh ′ Z G dh δ e ( h ′ ) F ( h, g ) δ e ( h ′− h ) F ′ ( h − h , h − g )= Z G dh ′ Z G dh η ( F )(( h, g ) , ( h ′ , g )) × η ( F ′ )(( h − h , h − g ) , ( h ′− h , h ′− g ))= ( η ( F ) ∗ L η ( F ′ ))(( h , g ) , ( h , g )) .η ( F ∗ L )(( h , g ) , ( h , g )) = δ e ( h ) F ∗ L ( h , g )= δ e ( h ) F ( h − , h − g )= δ e ( h − ) F ( h − , h − g )= η ( F )(( h − , h − g ) , ( h − , h − g ))= η ( F ) ∗ L (( h , g ) , ( h , g )) , γ ( F ∗ L F ′ )( h, g ) = I ( F ∗ L F ′ )( h, g − ) (2.44)= Z G dk F ( k, g − ) F ′ ( k − α g − ( h − ) , k − g − )= Z G dk F ( α g − ( k ) , g − ) F ′ ( α g − ( hk ) − , ( kg ) − )= Z G dk F ( α g − ( k − ) , g − ) F ′ ( α g − k ( k − h ) − , ( k − g ) − )= Z G dk γ ( F )( k, g ) γ ( F ′ )( k − h, k − g )= ( γ ( F ) ∗ L γ ( F ′ ))( h, g ) .γ ( F ∗ L )( h, g ) = F ∗ L ( α g − ( h − ) , g − )= F ( α g − ( h ) , g − h )= γ ( F )( h − , h − g )= γ ( F ) ∗ L ( h, g ) . Now, let σ ∈ ˆ E ′ U ( g ∗ ) ˆ ⊗ C ∞ ( G ).Then, ˜ p L ∗ σ ∈ ˆ E ′ U × U (( g ∗ ) × ) ˆ ⊗ C ∞ ( G × ), because ( θ , θ ) σ ( θ , g g ) is analytic for any g , g ∈ G , with constant growth bound in θ , and ( g , g ) σ ( θ , g g ) is smooth for any θ ∈ g ∗ . Similarly, ˜ p R ∗ σ, ˜ r ∗ σ ∈ ˆ E ′ U × U (( g ∗ ) × ) ˆ ⊗ C ∞ ( G × ) and˜ ι ∗ σ ∈ ˆ E ′ U ( g ∗ ) ˆ ⊗ C ∞ ( G ) (the coadjoint action, Ad ∗ , is analytic). Finally, we observe: F W,ε ˜ p L ∗ σ (( h , g ) , ( h , g )) = Z g ∗ dθ (2 πε ) n Z g ∗ dθ (2 πε ) n e iε ( θ ( X h )+ θ ( X h )) (2.45) × (˜ p L ∗ σ )(( θ , q h − g ) , ( θ , q h − g ))= Z g ∗ dθ (2 πε ) n Z g ∗ dθ (2 πε ) n e iε ( θ ( X h )+ θ ( X h )) σ ( θ , q h − g q h − g )= δ ( n )0 ( X h ) Z g ∗ dθ (2 πε ) n e iε θ ( X h ) σ ( θ , q h − g q h − g )= j (0)=1 δ e ( h ) Z g ∗ dθ (2 πε ) n e iε θ ( X h ) σ ( θ , q h − g g )= δ e ( h ) F W,εσ ( h , g g )= α L ( F W,εσ )(( h , g ) , ( h , g )) ,F W,ε ˜ p R ∗ σ (( h , g ) , ( h , g )) = Z g ∗ dθ (2 πε ) n Z g ∗ dθ (2 πε ) n e iε ( θ ( X h )+ θ ( X h )) (2.46) × (˜ p R ∗ σ )(( θ , q h − g ) , ( θ , q h − g ))= Z g ∗ dθ (2 πε ) n Z g ∗ dθ (2 πε ) n e iε ( θ ( X h )+ θ ( X h )) σ (Ad ∗ g ( θ ) , q h − g q h − g )= δ ( n )0 ( X h ) Z g ∗ dθ (2 πε ) n e iε Ad ∗ g − ( θ )( X h ) σ ( θ , q h − g q h − g )9= j (0)=1 δ e ( h ) Z g ∗ dθ (2 πε ) n e iε θ ( X αg h ) σ ( θ , g q h − g − g g )= g √ h − g − = √ α g ( h ) − δ e ( h ) F W,εσ ( α g ( h ) , g g )= α R ( F W,εσ )(( h , g ) , ( h , g )) ,F W,ε ˜ r ∗ σ (( h , g ) , ( h , g )) = Z g ∗ dθ (2 πε ) n Z g ∗ dθ (2 πε ) n e iε ( θ ( X h )+ θ ( X h )) (2.47) × (˜ r ∗ σ )(( θ , q h − g ) , ( θ , q h − g ))= Z g ∗ dθ (2 πε ) n Z g ∗ dθ (2 πε ) n e iε ( θ ( X h )+ θ ( X h )) σ ( θ , q h − g )= δ ( n )0 ( X h ) Z g ∗ dθ (2 πε ) n e iε θ ( X h ) σ ( θ , q h − g )= j (0)=1 δ e ( h ) Z g ∗ dθ (2 πε ) n e iε θ ( X h ) σ ( θ , q h − g )= η ( F W,εσ )(( h , g ) , ( h , g )) ,F W,ε ˜ ι ∗ σ ( h, g ) = Z g ∗ dθ (2 πε ) n e iε θ ( X h ) (˜ ι ∗ σ )( θ, √ h − g ) (2.48)= Z g ∗ dθ (2 πε ) n e iε θ ( X h ) σ ( − Ad ∗ g − √ h ( θ ) , g − √ h )= Z g ∗ dθ (2 πε ) n e − iε Ad ∗ √ h − g ( θ )( X h ) σ ( θ, g − √ h )= Z g ∗ dθ (2 πε ) n e iε θ ( X αg − √ h ( h − ) σ ( θ, g − √ h )= Z g ∗ dθ (2 πε ) n e iε θ ( X αg − h − ) σ ( θ, g − √ hgg − )= F W,εσ ( α g − ( h − ) , g − )= γ ( F W,εσ )( h, g ) , which proves (2.37) for Weyl quantisation. The proof for the Kohn-Nirenberg quantisation isanalogous.To show the transitivity property, we argue in the same fashion as in the proof of theorem II.9.First, we analyse the maps (cp. (2.31)): α L2 ′ : D ′ ( G ) ˆ ⊗ C ∞ ( G )) → D ′ ( G ′ × G ) ˆ ⊗ C ∞ ( G ′ × G )) , (2.49) α L2 ′ ( F )(( h ′ , g ′ ) , ( h , g )) = δ e ( h ) F ( h ′ , g ′ g ) ,α L(32)1 : D ′ ( G ′ × G ) ˆ ⊗ C ∞ ( G ′ × G )) → D ′ ( G × G × G ) ˆ ⊗ C ∞ ( G × G × G )) ,α L(32)1 ( F )(( h , g ) , ( h , g ) , ( h , g )) = δ e ( h ) F (( h , g g ) , ( h , g )) , α L31 ′ : D ′ ( G ) ˆ ⊗ C ∞ ( G )) → D ′ ( G × G ′ ) ˆ ⊗ C ∞ ( G × G ′ )) ,α L31 ′ ( F )(( h , g ) , ( h ′ , g ′ )) = δ e ( h ′ ) F ( h , g g ′ ) ,α L3(21) : D ′ ( G × G ′ ) ˆ ⊗ C ∞ ( G × G ′ )) → D ′ ( G × G × G ) ˆ ⊗ C ∞ ( G × G × G )) ,α L3(21) ( F )(( h , g ) , ( h , g ) , ( h , g )) = δ e ( h ) F (( h , g ) , ( h , g g )) ,α L321 : D ′ ( G ) ˆ ⊗ C ∞ ( G )) → D ′ ( G × G × G ) ˆ ⊗ C ∞ ( G × G × G )) ,α L321 ( F )(( h , g ) , ( h , g ) , ( h , g )) = δ e ( h ) δ e ( h ) F ( h , g g g ) ⇒ ( α L(32)1 ◦ α L2 ′ )( F ) = ( α L3(21) ◦ α L31 ′ )( F ) = α L321 ( F ) . From this we understand, that the α L l ′ l embed the h -dependence of an edge splitting into the outmosttensor factor corresponding to the final edge in a composition chain. Since this property is preservedunder successive splittings and adding of new edges to a composition chain (this is capture by η ),we obtain transitivity of the α L l ′ l w.r.t. ≤ . A similar argument works for the maps α R l ′ l , l ≤ l ′ , as in this case the embedding, which arises from edge splitting, is into the outmost tensor factorcorresponding to the initial edge in a composition chain. As before, the argument also works for ⋖ R and ⋖ L in combination with α R l ′ l and α L l ′ l respectively, because these partial orders preserve thenotion of first respectively last part of an edge between an oriented graph and its oriented subgraphs(cf. , p. 52-53). If we take edge inversion into account, the situation will change, because the notionof initial and finial edge in a composition chain gets permuted. This is essentially captured in thediagram (2.38), which follows from: ∀ F ∈ D ′ ( C l ) ˆ ⊗ C ∞ ( C l ):( α R l ′ l ◦ α ll − )( F ) (cid:0) { ( h e ′ , g e ′ ) } e ′ ∈ E ( γ ′ ) (cid:1) = (cid:16) Y e ∈ E ( γ ) δ e ( h e ′ e ) ...δ e ( h e ′ me ) (cid:17) (2.50) × α ll − ( F ) (cid:0) { ( α g e ′ me ...g e ′ e ( h e ′ e ) , g e ′ me ...g e ′ e ) } e ∈ E ( γ ) (cid:1) = (cid:16) Y e ∈ E ( γ ) δ e ( h e ′ e ) ...δ e ( h e ′ me ) (cid:17) × F (cid:0) { ( α g − e ′ e ( h − e ′ e ) , g − e ′ e ...g − e ′ me ) } e ∈ E ( γ ) (cid:1) = (cid:16) Y e ∈ E ( γ ) δ e ( α g − e ′ e ( h − e ′ e )) ...δ e ( α g − e ′ me ( h − e ′ me )) (cid:17) × F (cid:0) { ( α g − e ′ e ( h − e ′ e ) , g − e ′ e ...g − e ′ me ) } e ∈ E ( γ ) (cid:1) = α L l ′− l − ( F ) (cid:0) { ( α g − e ′ ( h − e ′ ) , g − e ′ ) } e ′ ∈ E ( γ ′ ) (cid:1) = ( α l ′ l ′− ◦ α L l ′− l − )( F )( { ( h e ′ , g e ′ ) } e ′ ∈ E ( γ ′ ) ) . The next corollary explains how the system of Hilbert spaces { L ( C l ) } l ∈ L fits into the picture.On the one hand, the inductive limit of { L ( C l ) } l ∈ L w.r.t. . is the Ashtekar-Isham-LewandowskiHilbert space, i.e. lim −→ ( L , . ) L ( C l ) = L ( A ).The same holds for the inductive limits w.r.t. ⋖ L and ⋖ R lim −→ ( L , ⋖ L , R ) L ( C l ) = L ( A ) , (2.51)1although the edge inversion is not explicitly implemented. This can be inferred from the followingargument:If we consider the structured graphs l, l − obtained from a single edge e : [0 , → Σ, i.e. γ = e and γ − = e − , we can find a decomposition of e into edges e , e , s.t. e = e ◦ e − . Therefore, if weconstruct a structured graph l ′ from the edges e , e , we will have l ⋖ L l ′ and l − ⋖ L l ′ , as well as l ⋖ R ( l ′ ) − and l − ⋖ R ( l ′ ) − . But then, we have for Ψ ∈ L ( G ):( p ∗ ll ′ Ψ)( g , g ) = Ψ( g g − ) (2.52)= ( ι ∗ Ψ)( g g − )= ( p ∗ l − l ′ ( ι ∗ Ψ))( g , g ) , ( p ∗ l − ( l ′ ) − Ψ)( g , g ) = Ψ( g g − ) (2.53)= ( ι ∗ Ψ)( g g − )= ( p ∗ l ( l ′ ) − ( ι ∗ Ψ))( g , g ) . Thus, the edge inversion ι ∗ : L ( G ) → L ( G ) , ( ι ∗ Ψ)( g ) := Ψ( g − ) , is automatically enforced asa symmetry in the limit (2.51), i.e. p ∗ ll ′ ( L ( C l )) = p l − l ′ ( ι ∗ ( L ( C l − ))) and p ∗ l − ( l ′ ) − ( L ( C l − )) = p l ( l ′ ) − ( ι ∗ ( L ( C l ))). This justifies, why we do not differentiate between ⋖ L and ⋖ R on the Hilbertspace level.Moreover, the action of Weyl (and Kohn-Nirenberg) quantisation F W,ε via ρ L on the scale of Hilbertspaces { L ( C l ) } l ∈ L is compatible with this additional symmetry (cp. diagram (2.59)). Explicitly,we have for σ ∈ ˆ E ′ U ( g ∗ ) ˆ ⊗ C ∞ ( G ) and Ψ ∈ C ∞ ( G ) w.r.t. ⋖ L : (cid:0) Q Wε (˜ p L ∗ ll ′ σ )( p ∗ ll ′ Ψ) (cid:1) ( g , g ) = Z G dh Z G dh F W,ε ˜ p L ∗ ll ′ σ (( h , g ) , ( h , g ))( p ∗ ll ′ Ψ)( h − g , h − g ) (2.54)= Z G dh Z G dh α L l ′ l ( F W,εσ )(( h , g ) , ( h , g ))Ψ( h − g g − h )= Z G dh Z G dh δ e ( h ) F W,εσ ( h , g g − )Ψ( h − g g − h )= Z G dh F W,εσ ( h , g g − )Ψ( h − g g − )= ( p ∗ ll ′ ( Q Wε ( σ )Ψ))( g , g )= Z G dh F W,εσ ( h , g g − )Ψ( h − g g − )= Z G dh F W,εσ ( α g g − ( h − ) , g g − )Ψ( α g g − ( h − ) g g − )= Z G dh Z G dh δ e ( h ) F W,εσ ( α g g − ( h − ) , g g − )Ψ( h − g g − h )= Z G dh Z G dh δ e ( h ) γ ( F W,εσ )( h , g g − )Ψ( h − g g − h )= Z G dh Z G dh δ e ( h ) F W,ε ˜ ι ∗ σ ( h , g g − )Ψ( h − g g − h )2= Z G dh Z G dh α L l ′ l − ( F W,ε ˜ ι ∗ σ )(( h , g ) , ( h , g ))( ι ∗ Ψ)( h − g g − h )= Z G dh Z G dh F W,ε ˜ p L ∗ l − l ′ ˜ ι ∗ σ (( h , g ) , ( h , g )) × ( p ∗ l − l ′ ι ∗ Ψ)(( h , g ) , ( h , g ))= (cid:0) Q Wε (˜ p L ∗ l − l ′ ˜ ι ∗ σ )( p ∗ l − l ′ ι ∗ Ψ) (cid:1) ( g , g ) , where we used the invariance properties of the Haar measure on G . Similarly, we have w.r.t. ⋖ R : (cid:16) Q Wε (˜ p R ∗ l − ( l ′ ) − ˜ ι ∗ σ )( p ∗ l − ( l ′ ) − Ψ) (cid:17) ( g , g ) = (cid:16) p ∗ l − ( l ′ ) − ( Q Wε ( σ )Ψ) (cid:17) ( g , g ) (2.55)= (cid:16) Q Wε (˜ p R ∗ l ( l ′ ) − ˜ ι ∗ σ )( p ∗ l ( l ′ ) − ι ∗ Ψ) (cid:17) ( g , g ) . On the other hand, the would-be inductive limitlim −→ ( L , ≤ ) L ( C l ) = L ( A ↑ ) (2.56)of { L ( C l ) } l ∈ L w.r.t. ≤ might give rise to a Hilbert space on “oriented” generalised connections A ↑ (edge inversion is not necessarily a symmetry). We refer to (2.56) as a would-be inductive limit,because ( L , ≤ ) is not directed, and thus the limit is does not necessarily exist in the category ofHilbert spaces. In analogy with the C ∗ -algebraic construction of A , it could be possible to realise A ↑ as the spectrum of the C ∗ -closure of (2.56) (in the inductive sup-norms), if the limit existed. Corollary II.11:
For l ∈ L , the action of L ( C ∞ ( C l )) on C ∞ ( C l ) ⊂ L ( C l ) via D ′ ( C l ) ˆ ⊗ C ∞ ( C l )) is induced from theintegrated left regular representation ρ L : D ′ ( G ) ˆ ⊗ C ∞ ( G )) → L ( C ∞ ( G )) : (cid:0) ρ l L ( F )Ψ (cid:1) (cid:0) { g e } e ∈ E ( γ ) (cid:1) = Z G ×| E ( γ ) | (cid:18) Y e ∈ E ( γ ) dh e (cid:19) F (cid:0) { ( h e , g e ) } e ∈ E ( γ ) (cid:1) Ψ (cid:0) { h − e g e } e ∈ E ( γ ) (cid:1) (2.57) for all F ∈ D ′ ( C l ) ˆ ⊗ C ∞ ( C l ) , Ψ ∈ C ∞ ( C l ) . Moreover, the action is covariant w.r.t. to the families of maps { p R , L ∗ ll ′ , α R , L l ′ l } l ≤ l ′ , { p L ∗ ll ′ , α L l ′ l } l ⋖ L l ′ and { p R ∗ ll ′ , α R l ′ l } l ⋖ R l ′ (all (sub)diagrams commute): C ∞ (Γ l ) ⊃ ˆ E ′ U l ( c ∗ l ) ˆ ⊗ C ∞ ( C l ) ˜ p R , L ∗ ll ′ (cid:15) (cid:15) ˜ p R , L ∗ ll ′′ * * F ( W ) ,ε / / Q ( W ) ε + + D ′ ( C l ) ˆ ⊗ C ∞ ( C l ) α R , L l ′ l (cid:15) (cid:15) α R , L l ′′ l v v ρ l L / / L ( C ∞ ( C l )) (cid:8) C ∞ ( C l ) ⊂ L ( C l ) p ∗ ll ′ (cid:15) (cid:15) p ∗ ll ′′ t t C ∞ (Γ l ′ ) ⊃ ˆ E ′ U l ′ ( c ∗ l ′ ) ˆ ⊗ C ∞ ( C l ′ ) ˜ p R , L ∗ l ′ l ′′ (cid:15) (cid:15) F ( W ) ,ε / / Q ( W ) ε D ′ ( C l ′ ) ˆ ⊗ C ∞ ( C l ′ ) α R , L l ′′ l ′ (cid:15) (cid:15) ρ l ′ L / / L ( C ∞ ( C l ′ )) (cid:8) C ∞ ( C l ′ ) ⊂ L ( C l ′ ) p ∗ l ′ l ′′ (cid:15) (cid:15) C ∞ (Γ l ′′ ) ⊃ ˆ E ′ U l ′′ ( c ∗ l ′′ ) ˆ ⊗ C ∞ ( C l ′′ ) F ( W ) ,ε / / Q ( W ) ε D ′ ( C l ′′ ) ˆ ⊗ C ∞ ( C l ′′ ) ρ l ′′ L / / L ( C ∞ ( C l ′′ )) (cid:8) C ∞ ( C l ′′ ) ⊂ L ( C l ′′ )(2.58) Consistency w.r.t. to edge inversion and the partial order . is so far only achieved in the fol-lowing sense (all (sub)diagrams commute, cp. (2.38) , we abuse notation and use the same nota-tion for the *-morphisms between operators spaces L ( C ∞ ( C l )) as for those between kernel spaces D ′ ( C l ) ˆ ⊗ C ∞ ( C l ) , l ∈ L ): C ∞ ( C l ) (cid:9) L ( C ∞ ( C l )) p ∗ ll ′ (cid:15) (cid:15) α R , L l ′ l (cid:15) (cid:15) p ∗ ll − + + α l − l + + ˆ E ′ U l ( c ∗ l ) ˆ ⊗ C ∞ ( C l ) Q ( W ) ε o o ˜ p ∗ ll − / / ˜ p R , L ∗ ll ′ (cid:15) (cid:15) ˆ E ′ U l − ( c ∗ l − ) ˆ ⊗ C ∞ ( C l − ) Q ( W ) ε / / ˜ p L , R ∗ l − l ′− (cid:15) (cid:15) L ( C ∞ ( C l − )) (cid:8) C ∞ ( C l − ) α L , R l ′− l − (cid:15) (cid:15) p ∗ l − l ′− (cid:15) (cid:15) C ∞ ( C l ′ ) (cid:9) L ( C ∞ ( C l ′ )) p ∗ l ′ l ′− α l ′− l ′ ˆ E ′ U l ′ ( c ∗ l ′ ) ˆ ⊗ C ∞ ( C l ′ ) Q ( W ) ε o o ˜ p ∗ l ′ l ′− / / ˆ E ′ U l ′− ( c ∗ l ′− ) ˆ ⊗ C ∞ ( C l ′− ) Q ( W ) ε / / L ( C ∞ ( C l ′− )) (cid:8) C ∞ ( C l ′− )(2.59) Proof:
The statements follow from theorems II.9 and II.10, and because the ˜ p (R , L) ll ′ are lifts of the p ll ′ . B. Inductive limit and non-commutative phase spaces
This subsection is devoted to the question whether it is possible to construct inductive limitsof C ∗ -algebras from the A l , l ∈ L , which serve as “non-commutative topological phase spaces”underlying loop quantum gravity.At the level of operators on C ∞ ( G ) via ρ L , the *-morphisms α R , L , γ, η : L ( C ∞ ( G )) → L ( C ∞ ( G × ))4are explicitly given by: ρ L ( α R ( F )) = U α R ( ⊗ ρ L ( F )) U ∗ α R , ( U α R Ψ)( g , g ) := Ψ( g , g g ) , Ψ ∈ L ( G × ) , (2.60) ρ L ( α L ( F )) = U α L ( ρ L ( F ) ⊗ ) U ∗ α L , ( U α L Ψ)( g , g ) := Ψ( g g , g ) , Ψ ∈ L ( G × ) ,ρ L ( γ ( F )) = U ι ρ L ( F ) U ∗ ι , ( U ι Ψ)( g ) := Ψ( g − ) , Ψ ∈ L ( G ) ,ρ L ( η ( F )) = ρ L ( F ) ⊗ , U α R , L ∈ U B ( L ( G × )) , U ι ∈ U B ( L ( G )) . In this sense α R and α L are “twisted” versions (by the left respectively right action) of the embed-ding on the second respectively first tensor factor.It is interesting to note that these maps, apart from γ , cannot be defined at the level of trans-formation group C ∗ -algebras C ( G ) ⋊ L G and C ( G ) ⋊ L ( G ) ⊗ , which feature in proposition II.6( A l , l ∈ L ), because these algebras are not unital. More precisely, for A ∈ K ( L ( G )), operatorsof the form ⊗ A or A ⊗ are not compact, and therefore not in K ( L ( G × )). Thus, if weintend to define a directed system of C ∗ -algebras ( { C l } l ∈ L , { α R , L l ′ l } ) as non-commutative analogueof | Λ | T ∗ A , we need to extend the algebras A l to make sense out of (2.60).One way to achieve this, which is inspired by the compactification of A to A , is to choose uni-tisations i l : A l ֒ → C l , l ∈ L (corresponding to compactifications of the state spaces S l of the A l ), i.e. embeddings of A l into unital C ∗ -algebras C l s.t. i l ( A l ) is an essential ideal in C l (cf. ).At this point it is not clear which unitisations should be chosen, although it is easy to see thatthe minimal unitisations A l ∼ = C + K ( L ( C l )) via adjoining identites are not sufficient, because ⊗ A and A ⊗ are not in C + K ( L ( G × )) for A ∈ K ( L ( G )). Therefore, we stick to theunique maximal unitisations M ( A l ) ∼ = B ( L ( C l )), the multiplier algebras of A l . The latter can bedefined as the C ∗ -algebras of adjointable operators B ad ( A l ) on A l as a (left) Hilbert module overitself. The unitisations are then the embeddings i L l : A l ֒ → B ad ( A l ) , i L l ( a ) b = ab, a, b ∈ A l , via(left) multiplication. M ( A l ) enjoys several useful properties:1. (unique extension of morphisms): For every C ∗ -algebra B , X Hilbert B -module and non-degenerate *-morphism π l : A l → B ad ( X ), there exists a unique extension ¯ π l : M ( A l ) → B ad ( X ) s.t. ¯ π l ◦ i L l = π l . Moreover, if π l is faithful (surjective), so is ¯ π l , because i L l ( A l ) ⊂ M ( A l ) is essential ( A l is σ -unital, cf. ).2. (embedding of C l and C ( C l )): The building blocks C l ∼ = G ×| E ( γ ) | and C ( C l ) are embedded in M ( A l ) in the following sense (a similar statement for A l is not true, cf. ): There exist a non-degenerate, faithful *-morphism i C ( C l ) : C ( C l ) ֒ → M ( A l and a strictly continuous, injectivemorphism i C l : C l ֒ → U M ( A l ) (taking values in unitaries), s.t.(a) ∀ F ∈ C ( C l , C ( C l )) , g, h ∈ C l and f ∈ C ( C l ):( i C ( C l ) ( f ) F )( h ) = f F ( h ) , , ( i C l ( g ) F )( h ) = α L ( g )( F ( g − h )) . (2.61)(b) The maps are covariant, i.e. i C ( C l ) ( α L ( g )( f )) = i C l ( g ) i C ( C l ) ( f ) i C l ( g ) ∗ . (2.62)(c) For every non-degenerate, covariant representation ( π, U ) of ( C ( C l ) , C l , α L ), the uniqueextension ¯ ρ of its integrated form ρ satisfies:¯ ρ ( i C ( C l ) ( f )) = π ( f ) , ¯ ρ ( i C l ( g )) = U g . (2.63)53. (recovery of S l ): The state space S l ⊂ A ∗ l can be recovered from M ( A l ) in the followingway. Since A l ∼ = K ( L ( C l )), we have have A ∗ l ∼ = S ( L ( C l )) (the trace class operators on L ( C l )). On the other hand, M ( A l ) ∼ = B ( L ( C l )), and therefore M ( A l ) inherits the structureof a von Neumann algebra with predual M ( A l ) ∗ ∼ = S ( L ( C l )). Now, S l corresponds to theset of positive, normalised elements of S ( L ( C l )) (density matrices), which are equivalentlycharacterised as the normal or σ -weakly continuous states on B ( L ( C l )). But, σ -weakly con-tinuous functionals are the same as σ -strongly* continuous functionals on B ( L ( C l )) (cf. ),and the σ -strong* topology on B ( L ( C l )) coincides with the strict topology coming from M ( A l ) ∼ = B ( L ( C l )). Thus, S l can be characterised as the set of strictly continuous states on M ( A l ). In general, the norm dual of a C ∗ -algebra B is isometrically isomorphic to the strictdual of M ( B ) with the strong topology (cf. ).Using property 1, we see that the maps (2.60) can be defined for M ( C ( G ) ⋊ L G ) and M ( C ( G × ) ⋊ L ( G × )), if we replace ρ L by its unique extension ¯ ρ L . Actually, ¯ ρ L provides the natural isomorphism M ( C ( G ) ⋊ L G ) ∼ = B ( L ( G )). Unfortunately, there is also a drawback in passing from A l to M ( A l ).Namely, while all the A l ’s are nuclear, and thus are very well-behaved as C ∗ -algebras (an inductivelimit of such algebras would preserve this porperty), the M ( A l )’s are generically not (unless G isfinite), as these are type I | G | factors ( | G | is the number of elements in G , with | G | = ∞ for non-finitegroups). Therefore, in the present case, it might be advantageous to consider the M ( A l )’s as vonNeumann algebras, and thus as “non-commutative measure spaces” in the usual philosophy. Thisis further supported by the observation M ( A l ⊗ A l ′ ) ∼ = M ( K ( L ( C l )) ⊗ K ( L ( C l ′ ))) ∼ = M ( K ( L ( C l ) ⊗ L ( C l ′ ))) (2.64) ∼ = B ( L ( C l ) ⊗ L ( C l ′ )) ∼ = B ( L ( C l )) ¯ ⊗ B ( L ( C l ′ )) ∼ = M ( K ( L ( C l ))) ¯ ⊗ M ( K ( L ( C l ′ ))) ∼ = M ( A l ) ¯ ⊗ M ( A l ′ ) , where ¯ ⊗ is the (spatial) tensor product of von Neumann algebras. Commentary II.12:
Before we proceed with the definition of the non-commutative analogue of | Λ | T ∗ A , we add afurther comment concerning the partial orders ≤ and . . We have noted after equation (2.56), thatthe partial order ≤ is not directed as opposed to . , ⋖ L and ⋖ R .Therefore, the existence of inductive limits w.r.t. ( L , ≤ ) is not ensured, and we would like topass to limits w.r.t. ( L , . ). Unfortunately, in view of the non-trivial compatibility conditionsbetween edge composition and inversion, as depicted by the diagrams (2.24), (2.38) and (2.59), itnot obvious that this is possible. Nevertheless, we will assume in remainder of this remark, thatcompatible collections of maps { ˜ p ∗ ll ′ } l . l ′ and { α l ′ l } l . l ′ exist. We will further comment on this issuein the outlook III.Alternatively, we may define inductive limits w.r.t. the partial orders ⋖ L and ⋖ R (cf. ). Clearly, ⋖ L and ⋖ R do not require the implementation of edge inversions as isomorphisms on the level ofphase spaces Γ l as well as algebras A l ( or M ( A l )), because two structured graphs l, l ′ are onlyconsidered to be equivalent in case the underlying oriented graphs are equal γ = γ ′ .In summary, we obtain directed systems of (unital) C ∗ -algebras ( { M ( A l ) } l ∈ L , { α l ′ l } l . l ′ ),( { M ( A l ) } l ∈ L , { α R l ′ l } l ⋖ R l ′ ) and ( { M ( A l ) } l ∈ L , { α L l ′ l } l ⋖ L l ′ ), which define unique (unital) inductive6limit C ∗ -algebras (cf. , Section 11.4.) C ∗ − lim −→ l ∈ ( L , . ) M ( A l ) = A α , C ∗ − lim −→ l ∈ ( L , ⋖ R , L ) M ( A l ) = A α R , L . (2.65)Since we have compatible directed systems of faithful representations ( { (¯ ρ l L , L ( C l )) } l ∈ L , { U l ′ l } l ≤ l ′ ),( { (¯ ρ l L , L ( C l )) } l ∈ L , { U l ′ l } l ⋖ R l ′ ) and ( { (¯ ρ l L , L ( C l )) } l ∈ L , { U l ′ l } l ⋖ L l ′ ), we have unique inductive limitrepresentationslim −→ l ∈ ( L , . ) (¯ ρ l L , L ( C l )) = (¯ ρ L , L ( A )) , lim −→ l ∈ ( L , ⋖ R , L ) (¯ ρ l L , L ( C l )) = (¯ ρ L , L ( A )) . (2.66)The isometries U l ′ l : L ( C l ) → L ( C l ′ ) are those induced from the maps p ∗ ll ′ : C ∞ ( C l ) → C ∞ ( C l ′ ). A α (R , L) could be interpreted as the “non-commutative topological phase space” of loop quantumgravity (or a “non-commutative measure space” associated with it, if we take the von Neumannalgebra point of view).It can be characterised as the C ∗ -algebra such that for each l ∈ L there exists an injective*-morphism α (R , L) l : A l → A α (R , L) satisfying α (R , L) l = α (R , L) l ′ ◦ α (R , L) l ′ l for l . l ′ ( l ⋖ R , L l ′ ), and S l ∈ L α (R , L) l ( A l ) ⊂ A α (R , L) is everywhere dense.We also have directed systems of states spaces ( { S l } ) l ∈ L , { α ∗ l ′ l } l . l ′ ), ( { S l } ) l ∈ L , { α R , L ∗ l ′ l } l ⋖ R , L l ′ )and ( { S ( M ( A l )) } ) l ∈ L , { α ∗ l ′ l } l . l ′ ), ( { S ( M ( A l )) } ) l ∈ L , { α R , L ∗ l ′ l } l ⋖ R , L l ′ ) by duality, which give rise toprojective limit state spaceslim ←− l ∈ ( L , . ) S l = S Nα , lim ←− l ∈ ( L , . ) S ( M ( A l )) = S α , (2.67)lim ←− l ∈ ( L , ⋖ R , L ) S l = S Nα R , L , lim ←− l ∈ ( L , ⋖ R , L ) S ( M ( A l )) = S α R , L . (2.68) S α (R , L) is isomorphic (continuously in weak* topology) with the state space of A α (R , L) , and S Nα (R , L) is basic in the latter (cf. . This implies that we can find a representation π (R , L) of A α (R , L) s.t.the distinguished states (density matrices) of π (R , L) ( A α (R , L) ) coincide with S Nα (R , L) . Then, thenormal states of the weak closure, π (R , L) ( A α (R , L) ) ′′ , of π (R , L) ( A α (R , L) ) are the σ -weakly continuousextensions of elements in S Nα (R , L) , and W ∗ − lim −→ l ∈ ( L , . ) M ( A l ) = π ( A α ) ′′ , W ∗ − lim −→ l ∈ ( L , ⋖ R , L ) M ( A l ) = π R , L ( A α R , L ) ′′ , (2.69)which are unique up to (normal) isomorphisms .In view of (2.67), a state ω on A α (R , L) can be determined from a collection of states { ω l } l ∈ L , ω l ∈ S l or S ( M ( A l )) , subject to the consistency conditions coming from the collection of *-morphisms { α l ′ l } l . l ′ (or { α R , L l ′ l } l ⋖ R , L l ′ ), i.e. ω l = ω ◦ α (R , L) l , ω l = ω l ′ ◦ α (R , L) l ′ l .That the consistency conditions are non-trivial, is due to the composition maps, α R , L , (2.36) usedto built the collection { α l ′ l } l . l ′ (or { α R , L l ′ l } l ⋖ R , L l ′ ), as those make A α (R , L) different from an infinitetensor product N l ∈ L M ( A l ), which would be the result of (2.65), if we were to use only the map η as a building block.7On the infinite tensor product any collection of states { ω l } l ∈ L , ω l ∈ S l or S ( M ( A l )) , would giverise to a infinite product state N l ∈ L ω l (cf. ).It is easy to see that the representations ρ l L : A l → K ( L ( C l )) arise from the consistent collectionof states ω l ( F ) = Z G × | E ( γ ) | (cid:18) Y e ∈ E ( γ ) dh e dg e (cid:19) F (cid:0) { ( h e , g e ) } e ∈ E ( γ ) (cid:1) , F ∈ C ( C l , C ( C l )) . (2.70)The corresponding consistent collection of complex regular Borel measures { µ l } l ∈ L , µ l ∈ C ( C l ) ∗ ob-tained from the embeddings i C ( C l ) : C ( C l ) ֒ → M ( A l ) determines the Ashtekar-Isham-Lewandowskimeasure. C. Gauge transformations
Finally, we analyse the behaviour of gauge transformation w.r.t. the Weyl quantisation and theprojective limit structure.In lemma 2.8, we have seen how the functionals (2.3) transform w.r.t. gauge transformations λ ∈ G P . On the truncated phase spaces Γ l , l ∈ L , this action corresponds to an action of G l := G | V ( γ ) | via the strongly Hamiltonian G-actions (2.10):˜ λ l : G l × Γ l → Γ l , (2.71)( { g v } v ∈ V ( γ ) , { ( θ e , g e ) } e ∈ E ( γ ) ) ˜ λ l (cid:0) { g v } v ∈ V ( γ ) , { ( θ e , g e ) } e ∈ E ( γ ) (cid:1) = { L ∗ g − e (1) R ∗ g − e (0) ( θ e , g e ) } e ∈ E ( γ ) = { ( Ad ∗ g e (1) ( θ e ) , g e (1) g e g − e (0) ) } e ∈ E ( γ ) . For l . l ′ , we define (smooth) projections π ll ′ : G l ′ → G l , π ll ′ ( { g v ′ } v ′ ∈ V ( γ ′ ) ) = { g v ′ } v ′ = v ∈ V ( γ ) ,which are compatible with the projections ˜ p R , L ll ′ : Γ l ′ → Γ l , because G l ′ × Γ l ′ ˜ λ l ′ / / π ll ′ × ˜ p R , L ll ′ (cid:15) (cid:15) Γ l ′ ˜ p R , L ll ′ (cid:15) (cid:15) G l × Γ l ˜ λ l / / Γ l (2.72)are commutative diagrams for l . l ′ . Furthermore, we have the transitivity property π ll ′′ = π ll ′ ◦ π l ′ l ′′ for l . l ′ . l ′′ , and ˜ λ l is compatible with the cotangent bundle projection, i.e. it is a lift of theaction λ l : G l × C l → C l .Again, the action (2.71) can be transferred to the “quantum” level, where G l acts in an automorphicfashion: α λl : G l → Aut( A l ) , α λl ( { g v } v ∈ V ( γ ) )( F )( { (( h e , g e ) } e ∈ E ( γ ) ) (2.73) α λl : G l → D ′ ( C l ) ˆ ⊗ C ∞ ( C l ) , = F ( { ( α g − e (1) ( h e ) , g − e (1) g e g e (0) ) } e ∈ E ( γ ) )8for F ∈ C ( C l , C ( C l )) or D ′ ( C l ) ˆ ⊗ C ∞ ( C l ).The automorphism group α λl ( G l ), as well as its extension ¯ α λl ( G l ) to M ( A l ), is unitarily implementedin ( ρ l L , L ( C l )) respectively (¯ ρ l L , L ( C l )): U λl : G l → U B ( L ( C l )) , (cid:0) U λl ( { g v } v ∈ V ( γ ) )Ψ (cid:1) ( { g e } e ∈ E ( γ ) ) = Ψ( { g − e (1) g e g e (0) } e ∈ E ( γ ) ) , (2.74)for Ψ ∈ L ( C l ). Since ¯ ρ l L is an isomorphism, the implementation is even inner in M ( A l ).Furthermore, the various actions (2.71), (2.73) & (2.74) are consistent with ( L , ≤ ), ( L , ⋖ L ) and( L , ⋖ R ) and quantisation via F ( W ) ,ε (or Q ( W ) ε ). We refrain from displaying the projection betweenthe G l ’s. ˆ E ′ Ul ( c ∗ l ) ˆ ⊗ C ∞ ( Cl )˜ p R , L ∗ ll ′ (cid:15) (cid:15) ˜ p R , L ∗ ll ′′ ' ' F ( W ) ,ε / / D ′ ( Cl ) ˆ ⊗ C ∞ ( Cl ) (cid:8) C ∞ ( Cl ) α R , L l ′ l (cid:15) (cid:15) α R , L l ′′ l & & p ∗ ll ′ (cid:15) (cid:15) p ∗ ll ′′ z z Gl × ˆ E ′ Ul ( c ∗ l ) ˆ ⊗ C ∞ ( Cl )˜ λ ∗ l ❢❢❢❢❢❢❢ ˜ p R , L ∗ ll ′ (cid:15) (cid:15) ˜ p R , L ∗ ll ′′ ' ' F ( W ) ,ε / / Gl × ( D ′ ( Cl ) ˆ ⊗ C ∞ ( Cl ) (cid:8) C ∞ ( Cl )) α R , L l ′ l (cid:15) (cid:15) α R , L l ′′ l ' ' p ∗ ll ′ (cid:15) (cid:15) p ∗ ll ′′ | | ( αλl ,λ ∗ l ) ❞❞❞❞❞❞❞❞❞ ˆ E ′ Ul ′ ( c ∗ l ′ ) ˆ ⊗ C ∞ ( Cl ′ )˜ p R , L ∗ l ′ l ′′ (cid:15) (cid:15) F ( W ) ,ε / / D ′ ( Cl ′ ) ˆ ⊗ C ∞ ( Cl ′ ) (cid:8) C ∞ ( Cl ′ ) α R , L l ′′ l ′ (cid:15) (cid:15) p ∗ l ′ l ′′ (cid:15) (cid:15) Gl ′ × ˆ E ′ Ul ′ ( c ∗ l ′ )ˆ ⊗ C ∞ ( Cl ′ )˜ λ ∗ l ′ ❢❢❢❢❢❢❢ ˜ p R , L ∗ l ′ l ′′ (cid:15) (cid:15) F ( W ) ,ε / / Gl ′ × ( D ′ ( Cl ′ ) ˆ ⊗ C ∞ ( Cl ′ ) (cid:8) C ∞ ( Cl ′ )) α R , L l ′′ l ′ (cid:15) (cid:15) p ∗ l ′ l ′′ (cid:15) (cid:15) ( αλl ′ ,λ ∗ l ′ ) ❞❞❞❞❞❞❞❞❞ ˆ E ′ Ul ′′ ( c ∗ l ′′ ) ˆ ⊗ C ∞ ( Cl ′′ ) F ( W ) ,ε / / D ′ ( Cl ′′ )ˆ ⊗ C ∞ ( Cl ′′ ) (cid:8) C ∞ ( Cl ′′ ) Gl ′′ × ˆ E ′ Ul ′′ ( c ∗ l ′′ ) ˆ ⊗ C ∞ ( Cl ′′ )˜ λ ∗ l ′′ ❢❢❢❢❢❢ F ( W ) ,ε / / Gl ′′ × ( D ′ ( Cl ′′ ) ˆ ⊗ C ∞ ( Cl ′′ ) (cid:8) C ∞ ( Cl ′′ ))( αλl ′′ ,λ ∗ l ′′ ) ❞❞❞❞❞❞❞❞❞ (2.75) III. CONCLUSIONS AND PERSPECTIVES
The Weyl quantisation for loop quantum gravity-type models, which we have constructed in thiswork, is in some aspects similar to the coherent state techniques, which are usually employed inthe discussion of semi-classical limits of loop quantum gravity, because both are based on the trun-cated phase space formalism for . Therefore, both methods are best suited to handle semi-classicallimits of graph-preserving operators. Clearly, the computation of coherent state expectation valuesw.r.t. to coherent states, that are built on a single (structured) graph, as was, for example, donein , is only meaningful for graph-preserving operators. This was also nicely pointed out in areview on deparametrising models , which require the use of graph-preserving operators. Namely,in the Ashtekar-Isham-Lewandowski representation, diffeomorphism-invariant operators have to begraph-preserving. Thus, if the (active) diffeomorphism invariance of the physical Hamiltonian indeparametrising models is to be retained upon quantisation, it is necessary to implement it in agraph-preserving manner. Although, if the framework of algebraic quantum gravity and its infi-nite tensor product representation is used, it is sufficient for an operator to preserve the underlying,infinite abstract graph, but not necessarily all possible subgraphs associated with different sectorsof the infinite tensor product. On the other hand, if a quantisation by means of graph-changingoperators is to be constructed, as e.g. in , where the (physical) Hamiltonian is implemented viagraph-changing operators on certain classes of partially diffeomorphism-invariant states, new toolswill be necessary.Nevertheless, some preliminary results on the semi-classical limit of graph-changing operators mightbe obtained through the use of the proposed Weyl quantisation in the following way:Let us consider a graph-changing operator O and a finite scale of Hilbert spaces, L ( A . l ) =9 S l ′ . l L ( C l ′ ), which is a subspace of L ( A ). Since we have L ( C l ′ ) ⊂ L ( C l ) for all l ′ . l , andthus L ( A . l ) = L ( C l ), we will obtain a non-trivial restriction O | L ( A . l ) for a large and refinedenough “cut-off graph” l , which is amenable to Weyl quantisation w.r.t. Γ l . Therefore, the Weylquantisation can be used to study the family { O | L ( A . l ) } l ∈ L of “graph cut-off” operators associ-ated with O .A similar scheme can be applied to the coherent state quantisation based on the Segal-Bargmann-Hall transform (see definition III.38 of our second article ).A somewhat different line of thought that could be pursued further concerns the compactnessproblem, which affects the flexibility of the potential implementation of space-adiabatic perturba-tion theory in loop quantum gravity-type models. Namely, it would be interesting to find out,whether it is possible to choose modified Ashtekar-Barbero variables for loop quantum gravity thatare connected to a nilpotent or, more generally, an exponential Lie group. This would make theexponential map a diffeomorphism and lift the problems related to the discrete nature the spaceof coadjoint orbits. Clearly, such variables would render the Ashtekar-Isham-Lewandowski repre-sentation ill-defined, but the construction of the quantum algebras A l ∼ = C ( C l ) ⋊ L C l ( l ∈ L , astructured graph) would still be possible, and the construction of a suitable new representationcould be discussed in terms of the projective limit of state spaces, S = lim ←− l ∈ L S l (cf. for a similarpoint of view). In view of full loop quantum gravity, we have not said much about the problem ofrecovering the phase space Γ = | Λ | T ∗ A P of the continuum theory. We have mainly pointed outthat the compatibility of the Weyl quantisation is a minimal requirement to discuss the continuumlimit by the techniques presented here. In principle, it should be possible to obtain Γ along thelines of , but the correct interplay of the procedure proposed therein with the methods of space-adiabatic perturbation should be verified. Additionally, it might be necessary to adapt the Weylquantisation to infinite graphs and the associated infinite tensor product construction to allowfor a discussion of infinite volume limits.At this point, we also want to address the technical issue concerning the construction of the projec-tive limit, Γ = lim ←− l ∈ L Γ l , over structured graphs l ∈ L in the (truncated) phase space quantisation,as it is of utmost importance to the validity of this approach. We have observed in section II (seee.g. commentary II.12), that the projective structure on the family of (truncated) phase spaces { Γ l } l ∈ L is only compatible with the (finer) partial orders ≤ , ⋖ L and ⋖ R and not necessarily withthe partial order . . But, the relations . , ⋖ L and ⋖ R are those, that identify the Ashtekar-Isham-Lewandowski Hilbert space, L ( A ) = lim −→ l ∈ L L ( C l ), as a representation space for the quantumalgebra A = lim −→ l ∈ L A l . This, is in compliance with the fact that a generalised connection ¯ A ∈ A is completely determined by its values on an oriented representative of a non-oriented graph class.In the commentary II.12, we only assumed that its is possible to choose collections of maps, { ˜ p ll ′ : Γ l ′ → Γ l } l . l ′ and { α l ′ l : L ( C ∞ ( C l )) → L ( C ∞ ( C l ′ )) } l . l ′ , that satisfy all transitivity conditioninduced by the directed partial order . . Since the main complication in providing such a choicecomes from the non-trivial interaction of composition and inversion of edges (diagrams (2.24), (2.38)and (2.59)), it is obvious that any oriented representative γ , together with its oriented subgraphs γ ′ ⊂ γ , of a given non-oriented graph | γ | can be given a consistent (w.r.t. ≤ ) choice of maps, { ˜ p ll ′ } l ≤ l ′ and { α l ′ l } l ≤ l ′ . But, then its is conceivable that, due to the mutual exchange of left andright composition under edge inversion, it is possible to generate relatively consistent choices ofmaps w.r.t. . , because any other oriented representative ˜ γ of | γ | can be accessed from γ via a finitenumber of single edge inversions.Clearly, this argument only makes the existence of a . -compatible choice of maps for a singlenon-oriented graph, and its non-oriented subgraphs, plausible. A statement regarding the set of all0graphs appears to be difficult. For example, transfinite induction, which would be available, because( L , . ) is well-founded ( l ∈ L with γ l = ∅ is a minimal element), is not applicable in this case, asit is not necessarily possible to obtain a consistent choice of maps for a given non-oriented graphfrom its already consistently labelled non-oriented subgraphs without allowing for a relabelling ofthe latter.on the one hand, it must be admitted that a reconciliation of the (truncated) phase space approachw.r.t. the partial order . with the usual treatment in terms of the holonomy-flux algebra is still anopen problem, which might require further attention. Clearly, this problem also affects the coherentstate formalism, which is also based on the (truncated) phase space quantisation considered here.But, on the other hand, in case we use the relation ⋖ L and ⋖ R , everything works fine, and weobtain quantum algebras A L and A R , that act in a well-defined fashion on L ( A ). The dichotomybetween ⋖ L and ⋖ R reflects the fact, that we have to choose between the left and the right actionof the structure group G on itself. These actions are equal for Abelian groups, but they are for onlyisomorphic via group inversion for non-Abelian groups, which explains why the projective structuresw.r.t. ⋖ L and ⋖ R are related via edge inversion (cp. 2.24 & 2.38).In respect of the aforesaid, it would be interesting to check, whether the Weyl quantisation proposedin this work is also compatible with the projective family of phase spaces constructed in . At firstsight this seems possible, because the (truncated) phase spaces used therein are of the cotangentbundle form, Γ η ∼ = T ∗ G n η ( η is some label).With a Weyl quantisation, which is compatible with the (truncated) phase space approach to loopquantum gravity-type models, at our disposal, it appears to be possible to investigate symmetricobservables at the classical and quantum level simultaneously. More precisely, if we realise a sym-metry by a subgroup of the spatial diffeomorphisms, which act in a natural way on the truncatedphase spaces Γ l , l ∈ L , by permutations on the label set L (structured graphs), we will be ina position to talk about symmetric functions in C ∞ (Γ l ), and thus in Cyl ∞ (Γ) := lim −→ l ∈ L C ∞ (Γ l ).Moreover, we expect the Weyl quantisation to be covariant w.r.t. to the action of the spatial dif-feomorphisms due to the formula (2.57), which would entail the invariance of any operator arisingas the quantisation of a symmetric function.But, in view of the solution of the spatial diffeomorphism constraint in loop quantum gravity, whichmakes use of the distributional dual of the linear span of spin network functions, it might be nec-essary to adapt the Weyl quantisation to handle a suitable distributional extension of Cyl ∞ (Γ).Insights into this aspects of the Weyl quantisation could shed a light onto the question of how toimplement semi-classical techniques in a diffeomorphism invariant setting, as well (see above).In view of the extraction of quantum field theory on curved spacetimes from loop quantum gravity,we have to face yet another type of difficulty, which originates in the well-know methods used toconstruct linear quantum field theories. On the one hand, space adiabatic perturbation theoryrelies on the construction of a bundle of Hilbert spaces, H γ ∼ = H f , of the fast sector over the phasespace, Γ, of the slow variables via the (principal symbol of the) projection onto the adiabaticallydecoupled subspace, 0 → H f → π H → Γ →
0. Moreover, it is necessary to require the existence ofunitary maps between the fibres, H γ , which is typically obstructed by a version of Haag’s theorem,unless we allow for some sort of regularisation (in the toy models without regularisation, every fibrecarries a quantum field with a different “mass”). On the other hand, the usual constructions in loopquantum gravity, which are invoked to quantise gravity-matter systems (see also ), are notanticipated to have this problem due to a natural regularisation of the matter fields by means of thequantisation scheme employed in the gravitational sector, and the use of irregular representations.Thus, further work needs to be invested to gain a better understanding of how the typically regularrepresentations of quantum field theory on curved spacetimes arise in a semi-classical limit of loop1quantum gravity with matter content .Another related problem is the actual construction of quantum field theories on specific curvedspacetimes via the space-adiabatic approach to loop quantum gravity with matter. Namely, thederivation of spacetime metrics will only be possible, if we extract effective Hamiltonian equationsfor the gravitational degrees of freedom that give rise to a correspondence between the slow sec-tor’s phase space points and said spacetime metrics. But, effective equations, that are obtained inspace-adiabatic perturbation theory via a semi-classical limit (Egorov’s hierachy), are tied to almostinvariant subspaces, which are constructed from spectral bands of the (principal) Hamiltonian sym-bol that defines the quantum field theories in the fibres of the adiabatic bundle. The upshot of thisis, that the resulting spacetime metrics might have a spectral dependence on the quantum matterfields, so-called rainbow metrics . Clearly, a further investigation into this aspect is desirable.But, it should be said, that a direct attempt to tackle this problem is only conceivable in symmetryreduced loop quantum cosmology-type models, at the moment. Nevertheless, it would already be amajor achievement to rigorously derive the recently constructed loop quantum cosmology-extensionof cosmological perturbation theory , which invoke a test field approximation (no back reaction),from a model of quantum field theory on a quantum cosmological spacetime (including back reac-tion) by the methods of space-adiabatic perturbation theory. Regarding full loop quantum gravity,we expect further progress on the extraction of a continuum phase space to be necessary before-hand (see above). To this end, it is legitimate to say, that the program of adiabatic perturbationtheory will require some (substantial) modifications, if a fully satisfactory derivation of quantumfield theory on curved spacetimes inside loop quantum gravity is to be obtained along its lines. ACKNOWLEDGMENTS
We thank Suzanne Lanéry for sharing her expertise on directed partial orders on families ofdecorated graphs. AS gratefully acknowledges financial support by the Ev. Studienwerk e.V.. Thiswork was supported in parts by funds from the Friedrich-Alexander-University, in the context ofits Emerging Field Initiative, to the Emerging Field Project “Quantum Geometry”.
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