Structural aspects of loop quantum gravity and loop quantum cosmology from an algebraic perspective
aa r X i v : . [ g r- q c ] A p r Structural aspects of loop quantum gravity and loop quantumcosmology from an algebraic perspective
Alexander Stottmeister ∗ and Thomas Thiemann † Institut für Quantengravitation, Lehrstuhl für Theoretische Physik III,Friedrich-Alexander-Universtität Erlangen-Nürnberg,Staudtstraße 7/B2, D-91058 Erlangen, Germany
May 30, 2018
Abstract
We comment on structural properties of the algebras A LQG/LQC underlying loop quantum gravity andloop quantum cosmology, especially the representation theory, relating the appearance of the (dynamicallyinduced) superselection structure ( θ -sectors) in loop quantum cosmology to recently proposed representationswith non-degenerate background geometries in loop quantum gravity with Abelian structure group. To thisend, we review and employ the concept of extending a given (observable) algebra with possibly non-trivialcentre to a (charged) field algebra with (global) gauge group. We also interpret the results in terms of thegeometry of the structure group G. Furthermore, we analyze the Koslowski-Sahlmann representations withnon-degenerate background in the case of a non-Abelian structure group. We find that these representationscan be interpreted from two different, though related, points view: Either, the standard algebras of loopquantum gravity need to be extended by a (possibly) central term, or the elementary flux vector fields needto acquire a shift related to the (classical) background to make these representations well-defined. Bothperspectives are linked by the fact that the background shift is not an automorphism of the algebras, butrather an affine transformation. A third perspective is offered by the recent construction of the holonomy-background flux-exponential algebra due to Campiglia and Varadarajan, which modifies the structure groupof the standard holonomy-flux algebra by an additional U (1) N -factor such that the Koslowski-Sahlmannrepresentations are applicable. Finally, we show how similar algebraic mechanisms that are used to explainthe breaking of chiral symmetry and the occurrence of θ -vacua in quantum field theory extend to loopquantum gravity. Thus, opening a path for the discussion of these questions in loop quantum gravity. Contents P LQG , A LQG & the AIL representation . . . . . . . . . . . 122.3 The algebra of loop quantum cosmology A LQC & the Bohr representation . . . . . . . . . . . . . 182.3.1 Dynamically induced “superselection” sectors in LQC . . . . . . . . . . . . . . . . . . . . 19 A λ LQC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Central operators in A LQG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.1 G ∼ = U (1) n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.2 G ∼ = K is compact, connected, simply connected and simple . . . . . . . . . . . . . . . . . 24 ∗ [email protected] † [email protected] θ -vacua in loop quantum gravity 33 A LQG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Chiral symmetry breaking and θ -vacua for Σ = R & P = R × G . . . . . . . . . . . . . . . . . . 35
Loop quantum gravity is based on a Hamiltonian formulation of general relativity in terms of a constrainedYang-Mills-type theory, i.e. in a field theoretic description the phase space of the classical theory is given bythe (densitiezed) cotangent bundle | Λ | T ∗ A P to the space of connections on a given (right) principal G-bundleP π → Σ, where Σ is the spatial manifold in a 3+1-splitting of a (globally hyperbolic) spacetime M ∼ = R × Σ. Ingeneral relativity, we have G = SU(2) , Spin , or central quotients of these groups.The basic variables, the theory is phrased in, are the Ashtekar-Barbero connection A ∈ A P and its conjugatemomentum E ∈ Γ (cid:0) T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ) (cid:1) . Strictly speaking, we further require E to be non-degenerate asa (densitiezed) section of the bundle of linear operators L(Ad(P) , T Σ). In general relativity, the existence of E is ensured by the triviality of the orthogonal frame bundle P SO (Σ). This mathematical setup also appearsto be valid in the context of the new variables proposed in [1, 2]. Here, Ad ∗ (P) = P × Ad ∗ g ∗ and | Λ | (Σ)denotes the bundle of 1-densities on Σ. Since A P is an affine space modeled on Ω (Ad(P)) = Γ( T ∗ Σ ⊗ Ad(P)),Ad(P) = P × Ad g , the following Poisson structure { E ai ( x ) , A jb ( y ) } = δ ab δ ji δ ( x, y ) (1.1)is meaningful in local coordinates φ : U ⊂ Σ → V ⊂ R subordinate to a local trivialization ψ : P | U → U × G,i.e. (( φ ◦ ψ ) − ) ∗ A | P | U = A jb dx b ⊗ τ j , ( φ ◦ ψ ) ∗ E | P | U = E ai ∂∂x a ⊗ τ ∗ i . (1.2)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 den-sitiezed dreibein for the spatial metric q ab E ai E bj = det( q ) δ ij , and A ia = Γ ia + K ia is built out of the Levi-Civitaconnection Γ of the spatial metric q and the extrinsic curvature K determined by the momentum P .What makes the variables ( A, E ) special, is that they allow to carry out a canonical quantization of generalrelativity, i.e. loop quantum gravity (cf. [3, 4] for general accounts on the topic). Especially, it is possible toconstruct mathematically well-defined operators for all constraints acting in a suitable Hilbert space within thisapproach, most prominently the Wheeler-DeWitt constraint (cf. [5–11]).The process of canonical quantization of constrained system in the sense of Dirac can roughly be divided intofour steps: First, a point-separating Poisson algebra of function(al)s on the classical phase space is identified.Second, an abstract quantum *-algebra based on the Poisson algebra is defined. Third, a representation of thequantum *-algebra is chosen. Fourth, operators corresponding to the constraint are constructed in the chosenrepresentation, and invariant (sub)spaces w.r.t. to these are selected as physical Hilbert spaces.In this article, we focus on the third step of this program. That is, we will analyze structures of loop quantumgravity related to the representation theory of a choice of quantum *-algebra. We will mainly work in the settingof the F/LOST theorem [12,13], which is an analog of the von Neumann uniqueness theorem for diffeomorphisminvariant theories. Therefore, the classical Poisson algebra will be given by the Ashtekar-Corichi-Zapata alge-bra [14–16], which is based on the Ashtekar-Isham configuration space [17] of generalized connections and itsassociated differential calculus [18] (see also [19]). As quantum *-algebra, we will use the holonomy-flux algebrain the semi-analytic category, which was defined in [12], or a certain Weyl form of this algebra [20]. Although,the F/LOST theorem states the uniqueness of a diffeomorphism invariant, pure state on the holonomy-fluxalgebra or a (concrete) Weyl form of it, which leads to a unitary implementation of the diffeomorphisms in theassociated Gel’fand-Naimark-Segal (GNS) representation, the Ashtekar-Isham-Lewandowski representation, itwas pointed out by several authors [21–24] that some of the underlying assumptions of the theorem have arather technical flavor and could be weakened from a physical perspective while others are not strictly necessaryto from a mathematical point of view to achieve a unitary implementation of the diffeomorphisms. Anotherissue, which was raised in [25] and followed upon in [23, 24], is the peculiar nature of the GNS vacuum ofthe Ashtekar-Isham-Lewandowski representation describing the extremely degenerate situation of an emptygeometry. While this appears to be a valid ground state for the deep quantum regime of a quantum theoryof gravity, where geometry is built from excitations of the gravitational field, such a state is not well suitedfor semi-classical considerations, where a classical background geometry needs to be approximated. Therefore,candidates for representations with ground states capturing information on a fixed background geometry wereproposed: The Koslowski-Sahlmann representations. Quite recently [26–31], these candidates were analyzedwith a focus on their applicability to asymptotically flat boundary conditions for the gravitational field, whichrequire a non-degenerate geometry at spatial infinity.Although, we will discuss certain aspects of the Koslowski-Sahlmann representations, and point out the needto extend the standard holonomy-flux algebra to make these representations well-defined, e.g. by admittingadditional “central terms” in the commutation relations of the fluxes or by the use of the holonomy-backgroundflux-exponential algebra, as recently pointed out by Campiglia and Varadarajan (cf. especially [28]), the mainfocus of the article lies on structural aspects of the quantum *-algebras, which are related to non-trivial geomet-rical and topological features of the structure group of the underlying Yang-Mills-type theory. More precisely,we observe that the use of a compact structure group G leads to a non-trivial center in the Weyl form ofthe holonomy-flux algebra, which clearly affects the representation theory, because central elements need tobe given by multiples of the identity in irreducible representations. Similar features are known in quantummechanical models [32–35]. Moreover, we point out distinctive features between the cases where G is Abelianor non-Abelian, and find that the representation theory is severely more constrained in the latter case. TheKoslowski-Sahlmann representation can be interpreted in this setting, as well. We also identify a purely topo-logical feature, which leaves its imprint in the representation theory. Namely, the existence of a sequence ofcoverings ˜G → ... → G → ... → ˜G /Z ( ˜G) , (1.3)where ˜G is the simply connected cover of G and Z ( ˜G) its center, accompanied by a sequence of non-trivialbundle coverings P ˜G → ... → P → ... → P ˜G /Z ( ˜G) (1.4)allows for the construction of a sequence of extensions of *-algebras A ˜G → ... → A → ... → A ˜G /Z ( ˜G) . (1.5)Such a sequence of extensions gives rise to another type of candidates for new representations of the quantum*-algebra, which are in some sense complementary to the Koslowski-Sahlmann representations.These structures resemble in many aspects a rigorous, fully quantum theoretical discussion of chiral symmetrybreaking and the related θ -vacua in quantum field theory [36,37]. That is, the existence of large gauge transfor-mations, π ( G ) = { } , is reflected in a non-trivial center of the (observable) algebra, and the anomalous chiralsymmetry does not leave the center point-wise invariant, thus leading to a spontaneous breakdown of the chiralsymmetry and the appearance of the θ -sectors. Interestingly, the main arguments of [37] can be transferred tothe framework of loop quantum gravity, if the existence of an anomalous chiral symmetry is assumed. Thisprovides a first step towards a discussion of anomalies in loop quantum gravity, which is a important issue inthe analysis of the semi-classical limit of the theory, especially in the presence of additional matter degreesof freedom. More precisely, since anomalies lead to non-trivial prediction concerning the matter content ofquantum field theory, it is necessary to establish a relation to such results in this limit. Thus, our observationwill allow to draw conclusions in loop quantum gravity similar those of quantum field theory, if the presence ofa chiral anomaly is achieved, either in full quantum theory or the semi-classical limit only. An arena for detailedinvestigations of these issue could be given by the so-called deparametrized models (see [38] for an overview).The article is organized as follows:In section 2, we provide a review of the mathematical background required to give precise definitions of thealgebraic structures employed in loop quantum gravity. Specifically, we use subsection 2.1 to recall some factsfrom the theory of (principal) fibre bundles, which are the basis for the (classical) phase space formulation ofloop quantum gravity. In subsection 2.2 and 2.3, we introduce the (quantum) algebras and states, which formthe standard setting of loop quantum gravity. Readers, which are familiar with these topics and/or are mainlyinterested in the results, can skip this section and use it as a reference.In section 3, we show that those algebras possess non-trivial centers, which are closely related to geometricand topological properties of the structure group, and affect their representation theory, e.g. by the appear-ance of the Koslowski-Sahlmann representations. Moreover, if the structure group is not simply connected, π (G) = { } , we provide a mechanism to construct extended (field) algebras, which admit automorphic actionsby the centers, and contain the original algebras in their fix-point algebras w.r.t. these actions (cp. [32, 35]).This, in turn, allows us to understand parts of the representation theory from a constructive point of view.In section 4, we analyze the Koslowski-Sahlmann representations in more detail, and point out the necessity toextend the algebras if G is non-Abelian, e.g. by admitting central terms in the basic commutation relations. Wecomment on the interpretation of the Koslowski-Sahlmann representations in terms of the holonomy-backgroundflux-exponential algebra in section 6. The case, when G is Abelian, can be understood in terms of the resultsof section 3.In section 5, we explain, how the algebraic explanation of chiral symmetry breaking and the occurrence of the θ -vacua in quantum field theory (cf. [36,37]) can be imported into the framework of loop quantum gravity. Again,the non-trivial structure in the representation theory, i.e. the θ -sectors, manifests itself as a consequence of anon-trivial center of the (quantum) algebra, which is closely related to topological properties of the structuregroup.Throughout the whole article, we choose units such that G = ~ = c = 1. Furthermore, we fix the Barbero-Immirzi parameter β = 1, although everything applies to the case β ∈ R =0 , as well. In this section, we review the definition of the (quantum) algebras P LQG and A LQG/LQC (cf. [4,13,39]) based onthe (classical) variables (
A, E ), and provide the necessary formalism for the analysis of the following sections.
Before we explain the construction of the algebras P LQG & A LQG/LQC , we need some formalism from thetheory of principal fibre bundles.As above, let P π → Σ be a principal G-bundle. Since A ∈ A P , it defines a parallel transport (or holonomy) h Ae : P | e (0) = π − ( e (0)) −→ P | e (1) = π − ( e (1)) (2.1)for every (broken, C ∞ ) path e : [0 , → Σ. Definition 2.1 (cf. [40, 41]):
Given a path e : [0 , → Σ , for every p ∈ P e (0) we consider the unique, horizontal (w.r.t. A ) lift ˜ e : [0 , → P defined by . ∀ t ∈ [0 ,
1] : A | ˜ e ( t ) (˜ e ′ ( t )) = 0 (2.2)2 . π ◦ ˜ e = e (2.3)3 . ˜ e (0) = p. (2.4) The parallel transport (or holonomy) of A along e is the map h Ae : P | e (0) / / P | e (1) p ✤ / / h Ae ( p ) = ˜ e (1) . (2.5)Clearly, the parallel transport is right equivariant, because the connection A is Ad-equivariant, i.e. ∀ g ∈ G : h Ae ◦ R g = R g ◦ h Ae , (2.6)and satisfies h Ae ◦ e = h Ae ◦ h Ae , h Ae − = ( h Ae ) − , (2.7)where e ◦ e is the composition of the paths e , e ( e (1) = e (0)), and e − is the reversion of the path e .To set up a correspondence between parallel transports, h Ae : P | e (0) → P | e (1) , and elements g ∈ G, we fix a setof reference points p x ∈ P | x , x ∈ Σ, and use the relation h Ae ( p e (0) ) = R g ( e,A, { p x } x ∈ Σ ) ( p e (1) ) (2.8)to define the element g ( e, A, { p x } x ∈ Σ ) ∈ G, which is well-defined by the free and fibre transitive action of G onP (cf. [12, 13])..1 Some fibre bundle theoretic digressions 5
Definition 2.2:
Given a path e : [0 , → Σ and set of reference points { p x } x ∈ Σ ⊂ P (2.8) defines the map g ( e, . , { p x } x ∈ Σ ) : A P / / G A ✤ / / g ( e, A, { p x } x ∈ Σ ) . (2.9)This map inherits the properties (2.7) in the following sense: g ( e ◦ e , A, { p x } x ∈ Σ ) = g ( e , A, { p x } x ∈ Σ ) g ( e , A, { p x } x ∈ Σ ) , (2.10) g ( e − , A, { p x } x ∈ Σ ) = g ( e, A, { p x } x ∈ Σ ) − . Due to the equivariance of the parallel transport, a change of reference points { p x } x ∈ Σ
7→ { p ′ x = p x g x } x ∈ Σ ,where the set { g x } x ∈ Σ ⊂ G is, again, well-defined by the free and fibre transitive action of G on P, effects themap (2.9) in a equivariant way: g ( e, A, { p ′ x } x ∈ Σ ) = g − e (1) g ( e, A, { p x } x ∈ Σ ) g e (0) . (2.11)For the purposes of loop quantum gravity, it is important that the map (2.9) separates points in A P , if we allowthe path e : [0 , → Σ to vary among a suitable class of paths P Σ (cf. [42]). Furthermore, (2.9) allows toidentify the space of generalized connections A with the groupoid homomorphisms Hom ( P Σ , G), A ∼ = Hom( P Σ , G ) (2.12)where G is considered as the action groupoid over a single object {∗} .Recall, that elements of A are defined as sets of parallel transports w.r.t. the class of paths P Σ . Definition 2.3:
A generalized connection A ∈ A is given by maps h Ae : P | e (0) −→ P | e (1) (2.13) for every e ∈ P Σ with the properties h Ae ◦ e = h Ae ◦ h Ae , h Ae − = ( h Ae ) − . (2.14)The space of connections A P is naturally identified with a subset of A via the holonomies.Next, let us consider the conjugate momentum E ∈ Γ (cid:0) T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ) (cid:1) , which similar to a connection A ∈ A P that is given as an Ad-equivariant 1-form on P with values in g , i.e. an element of Λ (P , g ) Ad , hasan interpretation as a geometric entity on P rather than on Σ. To this end, we need the following proposition(cf. [40, 43]). Proposition 2.4:
A section ω ∈ Γ(Λ k Σ ⊗ (P × ρ V )) =: Ω k (P × ρ V ) , where P × ρ V is the bundle associated with P via a (linear) rep-resentation ρ : G → Aut( V ) , corresponds in a one-to-one fashion to an element ˜ ω ∈ Λ k (P , V ) ρ , the horizontal, ρ -equivariant k -forms on P with values in V , or shortly ρ -tensorial k -forms on P . Proof:
Given an element ω ∈ Γ(Λ k Σ ⊗ (P × ρ V )), we define ˜ ω in the following way:˜ ω | p ( ˜ X , ..., ˜ X k ) = p − ω π ( p ) ( dπ | p ( ˜ X ) , ..., dπ | p ( ˜ X k )) , p ∈ P , ˜ X , ..., ˜ X k ∈ T p P , (2.15)where p − is the inverse of p : V → (P × ρ V ) | π ( p ) , p ( v ) = [( p, v )] ρ . Clearly, ˜ ω is well-defined and horizontal, as dπ | p : T p P → T π ( p ) Σ vanishes on vertical vectors, i.e. elements of T p (P) | π ( p ) . Furthermore, it is ρ -equivariant:(( R g ) ∗ ˜ ω ) | p ( ˜ X , ..., ˜ X k ) = ˜ ω | pg ( dR g | p ( ˜ X ) , ..., dR g | p ( ˜ X k )) (2.16)= ( pg ) − ω π ( pg ) ( dπ | pg ( dR g | p ( ˜ X )) , ..., dπ | pg ( dR g | p ( ˜ X k )))= ρ ( g − ) · p − ω π ( p ) ( dπ | p ( ˜ X ) , ..., dπ | p ( ˜ X k ))= ρ ( g − ) · ˜ ω | p ( ˜ X , ..., ˜ X k ) , since dπ | pg ◦ dR g | p = dπ | p and ( pg )( v ) = p ( ρ ( g ) v )..1 Some fibre bundle theoretic digressions 6Conversely, if ˜ ω ∈ Λ k (P , V ) ρ we construct ω by: ω | x ( X , ..., X k ) = p ˜ ω | p ( ˜ X , ..., ˜ X k ) , (2.17)for any p ∈ P | x and any ˜ X , ..., ˜ X k ∈ T p P, s.t. dπ | p ( ˜ X i ) = X i , i = 1 , ..., k , which is well-defined, because ˜ ω ishorizontal and ρ -equivariant.Similarly, we may set up a correspondence between sections X ∈ Γ( T Σ ⊗ (P × ρ V )) and horizontal, ρ -equivariantvector fields on P with values in V , ˜ X ∈ X ( P, V ) ρ , or ρ -tensorial vector fields on P for short. In contrast toΛ k (P , V ) ρ , X ( P, V ) ρ requires a connection A ∈ A P to be defined, as only horizontal k -forms and verticalvector fields on P are defined naturally. On the other hand, we expect this to be the case, as we expect theAshtekar-Barbero connection A and its conjugate momentum E to provide coordinates for T ∗ A P , and we have T A A P = Λ ( P, g ) Ad = ( X ( P, g ∗ ) Ad ∗ ) ∗ . (2.18) Proposition 2.5:
If we fix a connection A ∈ A P , there is a one-to-one correspondence between sections X ∈ Γ( T Σ ⊗ (P × ρ V )) and elements ˜ X of X ( P, V ) ρ . Proof:
Given X ∈ Γ( T Σ ⊗ (P × ρ V )), let X ∈ Γ( T P ⊗ (P × ρ V )) be its unique horizontal lift w.r.t. to A (cf. [40]), whichis right invariant, (( R g ) ∗ X ) | p = X | p , p ∈ P , g ∈ G , (2.19)by the Ad-equivariance of A . We define ˜ X | p = p − X | p , p ∈ P . (2.20)We only have to check ρ -equivariance.(( R g ) ∗ ˜ X ) | p = dR g | pg − ( ˜ X | pg − ) (2.21)= dR g | pg − (( pg − ) X | pg − )= ρ ( g ) · p − X | p = ρ ( g ) · ˜ X | p . Conversely, let ˜ X ∈ X ( P, V ) ρ , and set X | x = dπ | p ( p ˜ X | p ) (2.22)for an arbitrary p ∈ P | x . This is well-defined, because dπ | p ′ ( p ′ ˜ X | p ′ ) = dπ | pg ′ (( pg ′ ) ˜ X | pg ′ ) (2.23)= ( dπ | pg ′ ◦ dR g ′ | p )( p ( ρ ( g ′ ) · ρ ( g ′− ) · ˜ X | p ))= dπ | p ( p ˜ X | p ) , for any pair p, p ′ ∈ P | x . Clearly, (2.22) does not depend on the choice of connection A , which will be important in the follow-up. In analogy with the pairing between connections A ∈ A P and paths e : [0 , → Σ yielding group elements g ( e, A, { p x } x ∈ Σ ) ∈ G, A P × P Σ / / G( A, e ) ✤ / / g ( e, A, { p x } x ∈ Σ ) , (2.24)there is a pairing Γ( T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ)) × Γ(Ad(P)) × S Σ / / C ( E, n, S ) ✤ / / R S ∗ ( E ( n )) , (2.25).1 Some fibre bundle theoretic digressions 7where S Σ is a suitable class of hypersurfaces in Σ, E ( n ) ∈ Γ( T Σ ⊗| Λ | (Σ)) denotes the fibrewise pairing between E and n , and ∗ ( E ( n )) is the pseudo-2-form associated with E ( n ): ∗ ( E ( n )) = ε abc E ( n ) a dx b ∧ dx c (2.26)in local coordinates φ : U ⊂ Σ → V ⊂ R . Here, ε abc = δ a δ b δ c ] denotes the invariant pseudo tensor density ofweight −
1. Noteworthy, the dualityΓ( T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ)) × Γ(Ad(P)) / / Γ( T Σ ⊗ | Λ | (Σ))( E, n ) ✤ / / E ( n ) , (2.27)is compatible with the corresponding pairingΓ( T P ⊗| Λ | (P) , V ) Ad ∗ × Λ ( P, g ) Ad / / Γ( T P ⊗| Λ | (P)) G ( ˜ E, ˜ n ) ✤ / / ˜ E (˜ n ) , (2.28)in the sense, that dπ | p ( ˜ E (˜ n ) | p ) = E ( n ) | π ( p ) , p ∈ P | x . (2.29) In this subsection, we will analyze the behaviour of the variables( A, ˜ E ) ∈ | Λ | T ∗ A P = G A ∈ A P Γ( T P ⊗| Λ | (P) , g ∗ ) Ad ∗ (2.30)under the action of the gauge transformations G P of P , i.e. right equivariant diffeomorphisms of P,P (cid:8) λ / / R g (cid:15) (cid:15) P R g (cid:15) (cid:15) P λ / / P (2.31)that reduce to the identity π ◦ λ = π on Σ. First, let us derive some properties of gauge transformations. Lemma 2.6 (cf. [43]):
There is an isomorphism between the group of gauge transformations G P and the group of α -equivariant mapsfrom G to P , C (P , G) α , where α g : G / / G g ′ ✤ / / α g ( g ′ ) = gg ′ g − . (2.32) In analogy with proposition 2.4, we also have the isomorphism C (P , G) α ∼ = Γ(P × α G) . Proof:
Let λ ∈ G P and define f λ ∈ C (P , G) α by λ ( p ) = pf λ ( p ) , p ∈ P . (2.33) f λ is well-defined by right equivariance of λ and the free and fibre transitive action of G on P.Conversely, for f ∈ C (P , G) α we obtain λ f ∈ G P by λ f ( p ) = pf ( p ) , p ∈ P . (2.34) It is possible to consider the action of general bundle automorphism Aut(P) on | Λ | T ∗ A P (see e.g. [12, 44]). Furthermore, in localtrivialization ψ : P | U → U × G of P we have Aut(P | U ) ∼ = Diff( U ) ⋉ G P | U (cf. [4]). .1 Some fibre bundle theoretic digressions 8Similarly, we get s f ∈ Γ(P × α G) for the second isomorphism, s f ( x ) = pf ( p ) (2.35)for any p ∈ P | x . Here p : G → P × α G is the map we get from the associated bundle construction. (2.35) isindependent of the choice of p by α -equivariance, p ′ f ( p ′ ) = ( pg ′ ) f ( pg ′ ) = pα g ′ · α g ′− · f ( p ) (2.36)= pf ( p )for any pair p, p ′ ∈ P | x . The inverse of the second isomorphism is f s ( p ) = p − s ( π ( p )) , p ∈ P (2.37)for s ∈ Γ(P × α G).
Remark 2.7:
In general, the right action R g : P → P is not a gauge transformation, as this would require R g ◦ R g ′ = R g ′ g = R g ′ ◦ R g , ∀ g, g ′ ∈ G , (2.38)which holds if and only if G is Abelian.The Ad-tensorial 0-forms on P, Λ (P , g ) Ad , can be regarded as the Lie algebra of G P . Theorem 2.8 (cf. [43]): Λ (P , g ) Ad has a natural Lie algebra structure inherited from g , [˜ n, ˜ n ′ ] | p = [˜ n | p , ˜ n ′| p ] , ˜ n, ˜ n ′ ∈ Λ (P , g ) Ad , p ∈ P . (2.39) Proof:
Given ˜ n, ˜ n ′ ∈ Λ (P , g ) Ad , we need to verify that [˜ n, ˜ n ′ ] ∈ Λ (P , g ) Ad .( R g ) ∗ [˜ n, ˜ n ′ ] | p = [˜ n, ˜ n ′ ] | pg = [˜ n pg , ˜ n ′| pg ] = [Ad g − · ˜ n | p , Ad g − · ˜ n ′| p ] (2.40)= Ad g − · [˜ n, ˜ n ′ ] | p , p ∈ P . Definition 2.9 (cf. [43]):
The gauge algebra G P of P is the space Λ (P , g ) Ad of Ad -tensorial -forms on P with the Lie algebra structuregiven in theorem 2.8. Furthermore, there is an exponential map exp G P : G P −→ G P . (2.41) Theorem 2.10 (cf. [43]):
There is a map exp : Λ (P , g ) Ad → C (P , G) α defined by exp(˜ n )( p ) = exp G (˜ n | p ) , ˜ n ∈ Λ (P , g ) Ad , p ∈ P with theproperties . dd t | t =0 exp( t ˜ n ) = ˜ n (2.42)2 . d d t d s | t,s =0 α exp( t ˜ n ) (exp( s ˜ n ′ )) = [˜ n, ˜ n ′ ] . (2.43)exp : Λ (P , g ) Ad → C (P , G) α induces exp G P : G P −→ G P by exp G P (˜ n )( p ) = p exp(˜ n )( p ) . (2.44).1 Some fibre bundle theoretic digressions 9 Proof:
Clearly, α -equivariance of exp(˜ n ) follows from the properties of exp G : g → G.( R g ) ∗ exp(˜ n )( p ) = exp(˜ n )( pg ) = exp G (˜ n | pg ) = exp G (Ad g − · ˜ n | p ) = α g − (exp G (˜ n | p )) (2.45)= α g − · exp(˜ n )( p ) . The properties 1 . & 2 . are proved along the same lines, and are omitted at this point. exp G P : G P −→ G P iswell-defined by appealing to the isomorphism of lemma 2.6. Remark 2.11:
The notation ˜ n ∈ Λ (P , g ) Ad is intentional, when compared with (2.27) & (2.28), as it will be important toconsider ˜ n as generator of a gauge transformation in the regularization of the Poisson structure (1.1).Next, we define the (left) action of G P on | Λ | T ∗ A P . Definition 2.12:
The gauge transformations G P act on | Λ | T ∗ A P to the left by pullback and pushforward, i.e. L λ : | Λ | T ∗ A P / / | Λ | T ∗ A P ( A, ˜ E ) ✤ / / L λ ( A, ˜ E ) = (( λ − ) ∗ A, λ ∗ ˜ E ) . (2.46)This action is well-defined by the duality between pullback and pushforward(( λ − ) ∗ A )( λ ∗ ˜ E ) = A ( λ − ∗ ( λ ∗ ˜ E )) = A ( ˜ E ) = 0 , ( A, ˜ E ) ∈ | Λ | T ∗ A P . (2.47)By the definition 2.12, we only need the differential dλ : T P → T P to obtain an explicit expression for(( λ − ) ∗ A, λ ∗ ˜ E ). Lemma 2.13 (cf. [43]):
The differential dλ : T P → T P of λ ∈ G P is given by dλ | p ( ˜ X | p ) = dR f λ ( p ) | p ( ˜ X | p ) + ( dL f λ ( p ) − | f λ ( p ) ◦ df λ | p ( ˜ X | p )) ∗ λ ( p ) , p ∈ P , ˜ X | p ∈ T p P , (2.48) where ∗ : g → X ( P ) gives the fundamental vector fields: T ∗| p = dd t | t =0 p exp G ( tT ) , p ∈ P , T ∈ g , (2.49) which have the properties . ( R g ) ∗ T ∗| p = (Ad g − · T ) ∗| p (2.50)2 . λ ∗ T ∗| p = T ∗| p . (2.51) Proof:
Let λ ∈ G P and γ : [0 , → P , γ (0) = p, γ ′ (0) = ˜ X | p , then dλ | p ( ˜ X | p ) = dd t | t =0 λ ( γ ( t )) = dd t | t =0 γ ( t ) f λ ( γ ( t )) (2.52)= dR f λ ( p ) | p ( ˜ X | p ) + dd t | t =0 pf λ ( γ ( t )) = dR f λ ( p ) | p ( ˜ X | p ) + dd t | t =0 λ ( p ) L f λ ( p ) − ( f λ ( γ ( t )))= dR f λ ( p ) | p ( ˜ X | p ) + dλ ( p ) | e ◦ dL f λ ( p ) − | f λ ( p ) ◦ df λ | p ( ˜ X | p )= dR f λ ( p ) | p ( ˜ X | p ) + ( dL f λ ( p ) − | f λ ( p ) ◦ df λ | p ( ˜ X | p )) ∗| λ ( p ) . The properties of the fundamental vector fields are evident from their definition.
Corollary 2.14:
The action of the gauge transformations G P on | Λ | T ∗ A P is explicitly given as ( λ − ) ∗ A | p = A | λ − ( p ) ◦ dλ − | p (2.53)= Ad f λ ( p ) · A | p + dL f λ ( p ) | f λ ( p ) − ◦ df λ | p .1 Some fibre bundle theoretic digressions 10 λ ∗ ˜ E | p = dλ | λ − ( p ) ( ˜ E λ − ( p ) ) (2.54)= Ad ∗ f λ ( p ) · ( ˜ E | p + ( dR f λ ( p ) − | f λ ( p ) ◦ df λ | p ( ˜ E | p )) ∗| p ) . Proof:
Recall that A ( T ∗ ) = T, T ∈ g for A ∈ A P .The map exp G P : G P −→ G P allows us to derive the (infinitesimal) action of G P on | Λ | T ∗ A P . Lemma 2.15:
The explicit form of the action of G P on | Λ | T ∗ A P is dd t | t =0 (( λ t ˜ n ) − ) ∗ A = − ( d ˜ n + [ A, ˜ n ]) = − d A ˜ n (2.55)dd t | t =0 ( λ t ˜ n ) ∗ ˜ E = ( d ˜ n ( ˜ E )) ∗ + ad ∗ ˜ n · ˜ E, where λ t ˜ n = exp G P ( t ˜ n ) ∈ G P , ˜ n ∈ G P , and ad ∗ : g → g ∗ is the co-adjoint representation of g . Proof:
Note that for γ : [0 , → P , γ (0) = p, γ ′ (0) = ˜ X | p we havedd t | t =0 dR exp G ( − t ˜ n | p ) | exp G ( t ˜ n | p ) ◦ d exp G ( t ˜ n | ( . ) ) | p ( ˜ X | p ) = d d t d s | t,s =0 R exp G ( − t ˜ n | p ) (exp G ( t ˜ n | γ ( s ) )) (2.56)= d d s d t | t,s =0 R exp G ( − t ˜ n | p ) (exp G ( t ˜ n | γ ( s ) ))= d ˜ n | p ( ˜ X | p ) . Then apply corollary 2.14.For completeness, we also state the transformation behavior of ρ -tensorial k -forms on P, since T A A P ∼ =Λ (P , g ) Ad . Lemma 2.16 (cf. [43]):
The gauge transformations G P and the gauge algebra G P act on Λ k (P , V ) ρ (to the left) in the following way: ( λ − ) ∗ ˜ ω = ρ ( f λ ) · ˜ ω, dd t | t =0 (( λ t ˜ n ) − ) ∗ ˜ ω = dρ (˜ n ) · ˜ ω. (2.57) Here ω ∈ Λ k (P , V ) ρ , λ ∈ G P , ˜ n ∈ G P , λ t ˜ n = exp G ( t ˜ n ) , and dρ : g → End( V ) is the differential of ρ : G → Aut( V ) . Proof:
Use lemma 2.13 and ρ -equivariance of ˜ ω .Now, that we understand how the gauge transformations G P act on pairs ( A, ˜ E ) ∈ | Λ | T ∗ A P , we are able toderive their action on parallel transports h Ae , e ∈ P Σ , and projected, 1-density, vector fields E ∈ Γ( T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ)). Proposition 2.17 (cf. [12]):
A Gauge transformation λ ∈ G P affects the parallel transports h Ae , e ∈ P Σ , of a connection A ∈ A P viaconjugation, i.e. h ( λ − ) ∗ Ae = λ ◦ h Ae ◦ λ − . (2.58) The associated group elements g ( e, A, { p x } x ∈ Σ ) ∈ G (see definition 2.2) behave in an equivariant way, as well: g ( e, ( λ − ) ∗ A, { p x } x ∈ Σ ) = f λ ( p e (1) ) g ( e, A, { p x } x ∈ Σ ) f λ ( p e (0) ) − , (2.59) which is compatible with changes of reference points { p x } x ∈ Σ → { p ′ x } x ∈ Σ . The corresponding infinitesimalactions of ˜ n ∈ G P are: dd t | t =0 h (( λ t ˜ n ) − ) ∗ Ae ( p ) = (˜ n | h Ae ( p ) ) ∗| h Ae ( p ) − (˜ n | p ) ∗| h Ae ( p ) (2.60).1 Some fibre bundle theoretic digressions 11dd t | t =0 g ( e, (( λ t ˜ n ) − ) ∗ A, { p x } x ∈ Σ ) = ˜ n i | p e (1) R i | g ( e,A, { p x } x ∈ Σ ) − ˜ n j | p e (0) L j | g ( e,A, { p x } x ∈ Σ ) , (2.61) where { R i } , { L j } ⊂ X (G) are the right and left invariant vector fields on G associated with the generators { τ i } . Proof:
First, observe that λ ∈ G P acts on horizontal lifts in the appropriate way, i.e. if ˜ e : [0 , → P is a horizontallift of e : [0 , → P w.r.t A , then λ ◦ ˜ e : [0 , → P is a horizontal lift w.r.t ( λ − ) ∗ A by (2.48). Second, we have: h Ae (˜ e (0)) = ˜ e (1) = λ − ( λ (˜ e (1))) (2.62)= λ − ( h ( λ − ) ∗ Ae ( λ (˜ e (0)))) . (2.59) and compatibility follow from the right equivariance of λ ∈ G P resp. α -euqivariance of f λ ∈ C (P , G) α .To prove (2.60) & (2.61) we merely stick to the definition of fundamental, left invariant and right invariantvector fields. Remark 2.18:
The action (2.59) is opposite to the one employed in parts of the literature (cf. [4, 13]), where instead we find g ( e, ( λ − ) ∗ A, { p x } x ∈ Σ ) = f λ ( p e (0) ) g ( e, A, { p x } x ∈ Σ ) f λ ( p e (1) ) − . (2.63)This could be achieved if we worked with left principal bundles, or if we changed the defining identity (2.8) to h Ae ( p e (0) ) = R g ( e,A, { p x } x ∈ Σ ) − ( p e (1) ) . (2.64)The former would, in the case of trivial bundles, P ∼ = Σ × G, lead to a right action of the gauge transformations G P ∼ = C (Σ , G), which is not the typical choice in the majority of the literature. On the other hand, the latterwould make the homomorphism (2.9) an anti-homomorphism, i.e. reverse the order in the first line of (2.10).The actions of the gauge transformations G P and the gauge algebra G P on Γ( T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ)) remainto be discussed. Proposition 2.19:
The compatible actions of G P and G P on Γ( T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ)) are λ ⊲ E | x = p Ad ∗ f λ ( p ) · p − E | x , ˜ n ⊲ E | x = p ad ∗ ˜ n | p · p − E | x . (2.65) Here x ∈ Σ , p ∈ P | x and λ ∈ G P , ˜ n ∈ G P . As before, we regard p : G → Ad ∗ (P) as a map. Proof:
The actions are well-defined, i.e. independent of the choice of p ∈ P | x , because of α -equivariance of f λ resp.Ad-equivariance of ˜ n . To prove compatibility, we only need to combine proposition 2.5, corollary 2.14, lemma2.15 and the fact that dπ : T P → T Σ vanishes on vertical vectors. dπ | p ( p ( λ ∗ ˜ E ) | p ) = dπ | p ( p Ad ∗ f λ ( p ) · p − E | p ) = p Ad ∗ f λ ( p ) · p − dπ | p ( E | p ) (2.66)= p Ad ∗ f λ ( p ) · p − E | x , dd t | t =0 dπ | p ( p (( λ t ˜ n ) ∗ ˜ E ) | p ) = dπ | p ( p ad ∗ ˜ n | p · p − E | p ) = p ad ∗ ˜ n | p · p − dπ | p ( E | p ) (2.67)= p ad ∗ ˜ n | p · p − E | x , where ˜ E ∈ Γ( T P ⊗| Λ | (P) , g ∗ ) Ad ∗ corresponds to E via proposition 2.5.In view of lemma 2.16, identical formulas hold for k -forms in associated bundles. Proposition 2.20:
There are compatible actions of G P and G P on Ω k (P × ρ V ) : λ − ⊲ ω | x = p ρ ( f λ ( p )) · p − ω | x , − ˜ n ⊲ ω | x = p dρ (˜ n ) · p − ω | x , (2.68) where x ∈ Σ , p ∈ P | x and λ ∈ G P , ˜ n ∈ G P . As before, we regard p : V → P × ρ V as a map. Proof:
Just apply lemma 2.16 and proposition 2.4..2 The algebras of loop quantum gravity P LQG , A LQG & the AIL representation 12These induced actions on spaces of section in associated bundles have propertie that is essential in the followingsubsection 2.2.
Corollary 2.21:
The actions given in propositions 2.19 & 2.20 are transpose w.r.t. to the duality pairing (2.28) , i.e. ( λ ⊲ E )( n ) = E ( λ ⊲ n ) , (˜ n ′ ⊲ E )( n ) = E (˜ n ′ ⊲ n ) . (2.69)Let us make a closing remark for this subsection regarding the formalism in trivial bundles P ∼ = Σ × G. Remark 2.22:
If the bundle P is isomorphic to the trivial bundle Σ × G, the gauge transformations G P are isomorphic to theG-valued functions on Σ, C (Σ , G). The isomorphism is defined by the relation: f λ ( x, g ) = α g − ( g λ ( x )) , ( x, g ) ∈ Σ × G . (2.70) P LQG , A LQG & the AIL representation
In this subsection, we will stick to the semi-analytic category (cf. [12, 13] for the original utilization in thecontext of loop quantum gravity).Given a (right, semi-analytic) principal G-bundle P π → Σ (G compact Lie group), as before, we considerthe groupoid of (semi-analytic) paths P Σ in Σ. Fixing a system of reference points { p x } x ∈ Σ , we have theisomorphism A ∼ = Hom ( P Σ , G) (2.71)by definition 2.2.The construction of P LQG and A LQG is guided by the observation that A may be endowed with a compact,Hausdorff topology, which makes it accessible to measure theoretic consideration (cf. [17, 18, 45–48] for theoriginal literature). This topology is induced by giving an isomorphismHom( P Σ , G) ∼ = lim ←− l ∈ L Hom ( l, G) ⊂ Y l ∈ L Hom( l, G) , (2.72)where the projective limit is taken over subgroupoids L of P Σ generated by embedded, semi-analytic, compactlysupported graphs γ ∈ Γ sa0 in Σ. The projection p l : Hom( P Σ , G) → Hom( l, G) , l ∈ L , are simply therestrictions of the homomorphisms. It follows that the projective limit is a closed subset of the product space Q l ∈ L Hom( l, G), where the latter carries the Tikhonov topology. The spaces p l ( A ) =: A | l ∼ = Hom ( l, G) acquiretheir compact topology by the map (2.9) Hom( l, G) ∼ = G | E( γ l ) | , (2.73)which makes Q l ∈ L Hom( l, G) compact. Here, | E( γ l ) | denotes the number of edges in γ l . Furthermore, thisallows for the definition of a smooth and an analytic structure on A , since these structures are left and rightinvariant, and thus are invariant under a change of reference points { p x } x ∈ Σ [18, 19].Following this, let us introduce the basic building blocks of the algebra P LQG , which is constructed form certain(point-separating) functionals on | Λ | T ∗ A P . We loosely follow the notation of [4]. Definition 2.23 (Cylindrical functions):
The C ∗ -algebra Cyl is the closure of the cylindrical functions
Cyl = S l ∈ L C ( A l ) / ∼ in the sup -norm k . k ∞ .The equivalence is defined to be f l ∼ f l ′ : ⇔ ∃ l ′′ ⊇ l, l ′ : p ∗ l ′′ l f l = p ∗ l ′′ l ′ f l ′ , (2.74) where p l ′′ l : A l ′′ → A l , l, l ′′ ∈ L , is the restriction map. Every f ∈ Cyl is given as a projective family offunctions { f l } l ∈ L . Explicitly, we have f ( ¯ A ) = p ∗ l f l ( ¯ A ) = f l ( p l ( ¯ A )) = f l ( { h ¯ Ae } e ∈ E ( γ l ) ) (2.75)= F γ l ( { g ( e, ¯ A, { p x } x ∈ Σ ) } e ∈ E ( γ l ) ) . Here F γ l ∈ C (G | E ( γ l ) | ) is the function corresponding to f l ∈ C ( A l ) via (2.73) . .2 The algebras of loop quantum gravity P LQG , A LQG & the AIL representation 13It is well known that the spectrum of Cyl can be identified with the space of generalized connections A , thusleading to the isomorphism Cyl ∼ = C ( A ) . (2.76) Definition 2.24 (Flux vector fields, cf. [12]):
The flux vector fields X Flux on A considered as derivations on Cyl are the (regularized ) Hamiltonian vectorfields of the functions E n ( S ) = Z S ∗ ( E ( n )) (2.77) on T ∗ A defined by the pairing (2.25) , where S is a face, i.e. an embedded, semi-analytic, connected hyper-surface (without boundary) with oriented normal bundle N S , and n ∈ Γ sa0 (Ad(P | S )) , a compactly supported,semi-analytic section of adjoint pullback bundle P | S = ι ∗ S P . The action of the flux vector fields on f ∈ Cyl isobtained as follows:By proposition 2.4, we find a unique ˜ n ∈ Λ (P | S , g ) Ad , which gives rise to a -parameter group of gauge trans-formation λ t ˜ n ∈ G P | S by theorem 2.10. These gauge transformations define generalized gauge transformationson A in the following way: h λ ∗ t ˜ n ¯ Ae = h ¯ Ae ◦ ( λ t ˜ n ) ε ( e,S ) (2.78) where ε ( e, S ) = +1 e ∩ S = e (0) ∧ e is positively outgoing from S − e ∩ S = e (0) ∧ e is negatively outgoing from S e ∩ S = ∅ ∨ e ∩ ¯ S = e (2.79) is the indicator function of S w.r.t. to adapted edges . It is at this point, where semi-analyticity is crucial toensure that an arbitrary edge e ′ decomposes into a finite number of adapted edges e , which is necessary to get awell-defined action on Cyl . On the group elements g ( e, ¯ A, { p x } x ∈ Σ ) ∈ G this leads to g ( e, λ ∗ t ˜ n ¯ A, { p x } x ∈ Σ ) = g ( e, ¯ A, { p x } x ∈ Σ ) exp G ( 12 tε ( e, S )˜ n | p e (0) ) (2.80) A flux vector field E n ( S ) is the generator of a generalized gauge transformation on Cyl . ( E n ( S ) · F γ Sl )( { g ( e, ¯ A, { p x } x ∈ Σ ) } e ∈ E ( γ l ) ) = dd t | t =0 F γ Sl ( { g ( e, λ ∗ t ˜ n ¯ A, { p x } x ∈ Σ ) } e ∈ E ( γ l ) ) (2.81)= 12 X e ∈ E ( γ Sl ) ε ( e, S )˜ n i | p e (0) (L ei F γ Sl )( { g ( e, ¯ A, { p x } x ∈ Σ ) } e ∈ E ( γ Sl ) ) .F γ Sl denotes the representative of f ∈ Cyl w.r.t. an adapted decomposition γ Sl of an underlying graph γ l ∈ Γ sa0 and its associated groupoid l ∈ L . In the following, we will always assume to work with an adapted decompositionof a graph, when we consider the action of a flux vector field. Remark 2.25:
Note that we stick to the realization of the flux vector fields by left invariant vector fields on the structuregroup G. This is, again, due to the use of right principal bundles, and the requirement of an isomorphism A ∼ = Hom( P Σ , G) rather than an anti-isomorphism (cp. remark 2.18). Contrary, we could change the definitionof adapted edges in such a way that the non-vanishing contributions would be due to edges ending at a face,i.e. e ∩ S = e (1), if we wanted to arrive at a formulation in terms of right invariant vector fields on G.Our definition of the flux vector field appears to differ slightly from those existing in the literature (cf. especially[12]), but is nevertheless equivalent by the following lemma. See [4] for a detailed account of the regularization of (1.1). The factor in (2.78) is a remnant of the regularization procedure for the Hamiltonian vector field of E n ( S ) (see [4] for furtherexplanations). .2 The algebras of loop quantum gravity P LQG , A LQG & the AIL representation 14
Lemma 2.26:
Instead of defining the flux vector field in terms of S ⊂ Σ and n ∈ Γ sa0 (Ad(P | S )) , we may equivalently definethem by S ⊂ Σ and ˜ X ∈ X sa0 (P | S ) Gvert , a semi-analytic, compactly supported, right invariant, vertical vector fieldon P | S (cf. [12]).More precisely, we consider the flow φ ˜ Xt : P | S −→ P | S , t ∈ R (2.82) generated by ˜ X , which is a gauge transformation of P | S by the right invariance of ˜ X . Then, we may replace λ ± t ˜ n in (2.78) by φ ˜ X ± t and define flux vector fields E ˜ X ( S ) according to this relation. Proof:
Note that for ˜ n ∈ Λ (P | S , g ) Ad we have by theorem 2.10:dd t | t =0 exp G P | S ( t ˜ n )( p ) = dd t | t =0 p exp G ( t ˜ n | p ) (2.83)= (˜ n | p ) ∗| p , p ∈ P | S . Clearly, (˜ n ) ∗ ∈ X (P | S ) is semi-analytic and compactly supported if and only if ˜ n is. Moreover, due to thedefinition of ∗ : g → X (P | S ) G and the Ad-equivariance of ˜ n , (˜ n ) ∗ is right invariant and vertical.Conversely, since φ ˜ Xt , t ∈ R , is a 1-parameter group (connected to the identity, φ ˜ Xt =0 = id P | S ), we find acorresponding 1-parameter group f ˜ X,t ∈ C (P | S , G) α , t ∈ R , by lemma 2.6. Then, by theorem 2.10, we find ˜ n ˜ X ,s.t. φ ˜ Xt = exp G P ( t ˜ n ˜ X ) , ∀ t ∈ R . (2.84)In view of the calculations which will be performed in the following section of the article, we state a useful resultabout the flux vector fields. Lemma 2.27 (cf. [20]):
The action of the flux vector fields on
Cyl can be computed as follows: E n ( S ) · f = X x ∈ Σ X [ e ] x ∈ K ε ([ e ] x , S ) n i | p x L [ e ] x i | x f, (2.85) where ε denotes the indicator functions of S w.r.t. the edge germs [ e ] x , x ∈ Σ . The set of edge germs K x doesnot depend on x ∈ Σ in this setting. The action of the elementary vector fields L ix, [ e ] x is defined to be: L [ e ] x i | x p ∗ l f l = p ∗ l X ¯ e ∈ E ( γ l ) δ x, ¯ e (0) δ [ e ] x , [¯ e ] ¯ e (0) L ¯ ei f l , (2.86) where an adapted representative f l of f was chosen. The commutation relations between these vector fields are h L [ e ] x i | x , L [ e ′ ] x ′ j | x ′ i = 12 f ij k δ x,x ′ δ [ e ] x , [ e ′ ] x L kx, [ e ] x , (2.87) where [ τ i , τ j ] = f ij k τ k defines the structure constants of g . From the cylindrical functions Cyl and the flux vector fields (short: fluxes) we construct the *-algebra P LQG and a certain Weyl form A LQG of it. We denote by h X Flux i the Lie algebra span of X Flux . Definition 2.28 (The holonomy-flux algebra, cf. [12]):
The *-algebra P LQG is the *-algebra given by the quotient F / I of the tensor algebra F generated by Cyl ∞ and h X Flux i ⊂ X ( A ) by the two-sided -*-ideal I defined by the elements: V f − f V − V · f (2.88) V V ′ − V ′ V − [ V, V ′ ] X ( A ) f f ′ − f ′ f = 0 , ∀ f, f ′ ∈ Cyl ∞ , V, V ′ ∈ h X Flux i . The tensor product is taken relative to the algebra structure of
Cyl resp.
Cyl ∞ to make F a Cyl ∞ -module.The involution ∗ is defined by complex conjugation on Cyl ∞ , by V · f = V · f on h X Flux i , and extends to an .2 The algebras of loop quantum gravity P LQG , A LQG & the AIL representation 15 anti-automorphism of F .Note that the flux vector fields satisfy the reality condition E n ( S ) = − E n ( S ) ∗ , E n ( S ) ∈ X Flux . At this point, it is important to note, that there is a natural action by semi-analytic gauge transformations G saP and, more generally, semi-analytic automorphisms Aut sa (P) on the algebra P LQG . In general, the latter coverdiffeomorphisms, Diff sa (Σ), different from the identity:P χ / / π (cid:15) (cid:15) P π (cid:15) (cid:15) Σ φ χ / / Σ (2.89)with χ ∈ Aut sa (P) , φ χ ∈ Diff sa (Σ). For general bundles, it is not necessarily the case that every diffeomorphism φ ∈ Diff sa (Σ) is covered by an automorphism χ φ ∈ Aut sa (P), as this amounts to a non-trivial lifting problem(cf. [49]). P π (cid:15) (cid:15) P χ φ ? ? φ ◦ π / / Σ (2.90)In the smooth category, one finds a short exact sequence of NLF-manifolds [49]1 / / G ∞ P / / Aut ∞ (P) / / Diff ∞ ♮ (Σ) / / ∞ ♮ (Σ) of Diff ∞ (Σ) containing the connected component of the identity. This issuedoes not arise for the Ashtekar-Barbero variables, since then the bundle P comes from the natural bundleP SO (Σ) [50]. The actions of both groups of transformations on the basic elements, i.e. the cylindrical functionsand the fluxes, look as follows: Definition 2.29:
The transformations G saP and automorphisms Aut sa (P) have natural (right) actions on Cyl and X Flux inducedby those of corollary 2.14 and lemma 2.16. α λ ( f )( ¯ A ) = p ∗ l f l (( λ − ) ∗ ¯ A ) = F γ l ( { f λ ( p e (1) ) g ( e, ¯ A, { p x } x ∈ Σ )( f λ ( p e (0) )) − } e ∈ E ( γ ( l )) ) (2.92) α λ ( E n ( S )) = ( λ ⊲ E ) n ( S ) = E λ⊲n ( S ) α χ ( f )( ¯ A ) = p ∗ l f l (( χ − ) ∗ ¯ A ) = f l ( { χ ◦ h Aφ − χ ( e ) ◦ χ − } e ∈ E ( γ l ) ) (2.93)= F γ l ( { g ( e, ( χ − ) ∗ A, { p x } x ∈ Σ ) } e ∈ E ( γ l ) )= F φ − χ ( γ l ) ( { g χ (¯ e (1)) g (¯ e, A, { p x } x ∈ Σ ) g χ (¯ e (0)) − } ¯ e ∈ E ( φ − χ ( γ l )) ) ,α χ ( E n ( S )) = ( χ ∗ E ) n ( S ) = E χ ∗ n ( φ − χ ( S )) , f ∈ Cyl , E n ( S ) ∈ X Flux , λ ∈ G saP , χ ∈ Aut sa (P) , where g χ : Σ → G , s.t. χ ( p x ) = R g χ ( x ) ( p φ χ ( x ) ) , and χ ∗ n = χ − ◦ n ◦ φ χ . These actions extend to *-automorphicactions on P LQG . Since the algebra P LQG is supposed to be generated by the cylindrical functions and the fluxes, it is necessary toallow only semi-analytic gauge transformations or automorphisms, as otherwise the action of the transformationsgroups would not preserve the elementary operators of the algebra. Nevertheless, (distributional) extensionsof these transformations groups have been discussed in the literature [44, 51], and can be shown to have awell-defined action on Cyl ∞ , but which do not preserve X Flux . As an example, we show that, in case of a trivialbundle P ∼ = Σ × G, the extension of G saP ∼ = C sa (Σ , G) to G Σ = { g : Σ → G } leads to elements that are notgenerated from finite linear combinations of fluxes (unless Ad : G → Aut( g ) is trivial):Let us consider a flux E n ( S ) and, without loss of generality, a generalized gauge transformation { g x } x ∈ S , s.t. This implies: g χ − ◦ φ χ = g − χ . Furthermore, this definition has the necessary equivariance properties w.r.t. a change of referencesystem { p x } x ∈ Σ
7→ { p ′ x } x ∈ Σ . .2 The algebras of loop quantum gravity P LQG , A LQG & the AIL representation 16Ad g − ( n )( x ) = m ( x ) ∦ n ( x ) and ∀ y = x : Ad g − ( n )( y ) = n ( y ). Then the element α g ( E n ( S )) = E Ad g − ( n ) ( S ) (2.94)is not of the form required of a flux. Furthermore, this leads to[ E n ( S ) , α g ( E n ( S ))] = E [ n, Ad g − ( n )] ( | S | ) = X [ e ] x ∈ K ε ([ e ] x , S ) [ n, m ] j ( x ) L [ e ] x j | x , (2.95)which is a point-localized vector field on Cyl ∞ . Although it is not a point-localized flux, as it contains thesquared type indicator function of S . Clearly, such a point-localized object cannot be obtained from fluxes,as these are defined w.r.t. to open, semi-anlaytic surfaces S and compactly supported, semi-analytic functions n ∈ C sa0 ( S, g ) thereon.This subtlety indicates that the known proof of uniqueness of the Ashtekar-Isham-Lewandowski representation[12], which requires the algebra P LQG to be generated by finite linear combinations of products of the cylindricalfunctions and the fluxes, strictly speaking only holds without considering the action of G Σ . On the other hand,this subtlety poses no problem for the proof of uniqueness given in [13] involving a generalized Weyl form of P LQG .From a practical point of view the extension of G saP to G Σ appears to be unnecessary, because the action of thesemi-analytic gauge transformations is sufficiently localizable due to the existence of semi-analytic partitionsof unity (cf. [12] and references therein). Additionally, the use of G saP entails the occurrence of large gaugetransformation, i.e. gauge transformation not homotopic to the identity, which might be useful in the discussionof chiral symmetry breaking in loop quantum gravity (see below).The algebra A LQG is obtained by partially extending and exponentiating the generators of P LQG , and providingit with the formal commutation relations induced by the Lie bracket on X ( A ). The reason for not exponentiatingthe cylindrical functions is due to the fact, that they are essentially continuous functions of holonomies, thelatter being already a sort of exponential of the connection 1-form A . In contrast, the flux vector fields are notexponentiated up to this point, being essentially Hamiltonian vector fields of the (smeared) vector densities E . Definition 2.30 (The *-algebra in Weyl form, cf. [20]):
The *-algebra A LQG is generated by the elements of
Cyl and the Weyl elements W S ( tn ) = e tE n ( S ) = α λ ∗ t ˜ n subject to the following relations (cp. (2.78) & (2.92) ): f ∗ = f , f f ′ = f · Cyl f ′ , (2.96) W S ( tn ) ∗ = W S ( tn ) − = W S ( − tn ) , W S ( tn ) W S ( t ′ n ) = W S (( t + t ′ ) n ) ,W S ( tn ) f W S ( tn ) − = W S ( tn ) · f = α λ ∗ t ˜ n ( f ) , W S (0) = ,W S ( tn ) W S ′ ( t ′ n ′ ) W S ( tn ) − W S ′ ( t ′ n ′ ) − = α λ ∗ t ˜ n ◦ α λ ∗ t ′ ˜ n ′ ◦ α − λ ∗ t ˜ n ◦ α − λ ∗ t ′ ˜ n ′ , where f, f ′ ∈ Cyl and λ ∗ t ˜ n , λ ∗ t ′ ˜ n ′ are as in definition 2.24. The action of the Weyl elements on Cyl implementsthe formal identity W S ( tn ) · f = P ∞ k =0 t k k ! E n ( S ) k · f on Cyl ω . The set of Weyl elements will be denoted by W ,and the group generated by this set by h W i . Remark 2.31:
This definition of the algebra A LQG is not equivalent to the definition in [13], because we do not regard A LQG as a (closed) subalgebra of B ( L ( A , dµ )) (see below, (2.99)), and thus do not require all relations among thegenerating elements that would follow from such a definition. We will further explain the consequences of thisdifference in the next section.Additionally, we consider the extended algebra A extLQG of A LQG generated by elementary, point-localised fluxes(2.86) and the cylindrical functions. The extended algebra allows us to obtain an explicit expression for thecommutator between the fluxes[ E n ( S ) , E n ′ ( S ′ )] = X x ∈ S ∩ S ′ X [ e ] x ∈ K ε ([ e ] x , S ) ε ([ e ] x , S ′ )[ n, n ′ ] k ( x ) L [ e ] x k | x , (2.97)which will be important in the following sections. It is interesting to note that the commutator does notclose among the fluxes in the non-Abelian case precisely because of the indicator function ε , i.e. the product In the Abelian case the relation (2.97) is trivially closed, i.e. [ E n ( S ) , E n ′ ( S ′ )] = 0. .2 The algebras of loop quantum gravity P LQG , A LQG & the AIL representation 17 ε ([ e ] x , S ) ε ([ e ] x , S ′ ) is in general not of the form ε ([ e ] x , S ′′ ) for a suitable surface S ′′ . Although, there are certainspecial cases where iterated commutators lead to fluxes again, e.g.[ E n ( S ) , [ E n ′ ( S ) , E n ′′ ( S )]] = X x ∈ S ∩ S ′ X [ e ] x ∈ K ε ([ e ] x , S )[ n, [ n ′ , n ′′ ] k ( x ) L [ e ] x k | x = E [ n, [ n ′ ,n ′′ ]] ( S ) , (2.98)since ε ([ e ] x , S ) = ε ([ e ] x , S ). This is a feature that is missed by a restriction to Abelian groups G , e.g. G = U (1) (“Abelian artifact”, cf. [52]).Typically, the Ashtekar-Isham-Lewandowski representation is invoked as a Hilbert space representation of P LQG resp. A LQG , defined by the irregular (algebraic) state ω ( f E n ( S ) ...E n j ( S j )) = ( µ ( f ) if { , .., j } = ∅ , ∀ f ∈ Cyl ∞ , E n ( S ) ...E n j ( S j ) ∈ X Flux , (2.99) ω ( f W S ( n ) ...W S j ( n j )) = µ ( f ) , ∀ f ∈ Cyl , W S ( n ) ...W S j ( n j ) ∈ A LQG , where µ denotes the Ashtekar-Isham-Lewandowski measure induced by the Haar measure on G . In terms of(gauge-variant) spin network functions T s , s ∈ S , which form a special orthonormal basis in H ω ∼ = L ( A , dµ ),(2.99) reads ω ( T s E n ( S ) ...E n j ( S j )) = ( δ s, if { , .., j } = ∅ , ∀ f ∈ Cyl , E n ( S ) ...E n j ( S j ) ∈ X Flux , (2.100) ω ( T s W S ( n ) ...W S j ( n j )) = δ s, , ∀ s ∈ S, W S ( n ) ...W S j ( n j ) ∈ A LQG , where s = 0 denotes the spin network label corresponding to the empty graph γ = ∅ . This representation enjoysa uniqueness property under certain natural assumptions [12].At this point, we want to state a short lemma regarding the regularity and gauge invariance of states on P LQG and A LQG for compact, connected G.
Lemma 2.32:
Let ω be a gauge invariant state, i.e. ω ◦ α λ = ω, λ ∈ G saP , on P LQG or A LQG . Then, ω is irregular w.r.t. thegauge variant spin network functions (cf. [4]), T γ,~π,~m,~n ( ¯ A ) = Y e ∈ E ( γ ) p dim( π e ) π e ( g ( e, ¯ A, { p x } x ∈ Σ )) m e ,n e , (2.101) with γ ∈ Γ sa0 , { [ π e ] } e ∈ E ( γ ) ∈ ( ˆ G \ { [ π triv ] } ) | E ( γ ) | , m e , n e = 1 , ..., dim( π e ) . Here, irregularity is understood in thesense that for any π e = π triv there exist m e , n e = 1 , ..., dim( π e ) , s.t. [0 , ∋ s ω ( T e s ,π e ,m e ,n e ) , e ∈ P Σ , e s ( t ) = e ( st ) , t ∈ [0 ,
1] (2.102) is not continuous from the right in [0 , at s = 0 . Proof:
The action of the gauge transformations λ ∈ G saP on the gauge variant spin network functions looks as follows: α λ ( T γ,~π,~m,~n )( ¯ A ) = Y e ∈ E ( γ ) p dim( π e ) π e ( f λ ( p e (1) ) g ( e, ¯ A, { p x } x ∈ Σ ) f λ ( p e (0) ) − ) m e ,n e (2.103)= Y e ∈ E ( γ ) p dim( π e ) dim( π e ) X k e ,l e =1 π e ( f λ ( p e (1) )) m e ,k e π e ( g ( e, ¯ A, { p x } x ∈ Σ )) k e ,l e π e ( f λ ( p e (0) ) − ) l e ,n e . Now, let us choose maximal torus T ⊂ G and consider spin network functions T e s ,π e ,m e ,n e defined on singleedges { e s } s ∈ [0 , ⊂ P Σ , and gauge transformations λ e s (1) localized at the vertex e s (1) of e s , s.t. ∀ ≥ s > f λ ( p e s (1) ) = t = 1 G ∈ T ⊂ G and f λ ( p e s (0) ) = 1 G . Such gauge transformations exist because of the existence ofsemi-analytic partitions of unity [12, 13]. Next, we notice that [ π e ] = [ π triv ] implies the non-triviality of π e | T : T → Aut( V π e ) , ∃ t ∈ T : π e | T ( t ) = V πe , (2.104)since every g ∈ G is conjugate to some t g ∈ T [53]. Thus, we obtain, by diagonalizing the representation of T,.3 The algebra of loop quantum cosmology A LQC & the Bohr representation 18a non-trivial decomposition V π e ∼ = M ρ e V ρ e , π e | T ∼ = M ρ e ρ e , (2.105)where ρ e : T → T , dim( V ρ e ) = 1 , are irreducible representations of T, i.e. characters of the maximal torus, ρ e ∈ ˆT. From (2.103), (2.104) and (2.105), we conclude that we find an element t ∈ T, s.t. ω ( T e s ,π e ,m e ,n e ) = ω ( α λ s ( T e s ,π e ,m e ,n e )) (2.106)= ρ e ( t ) | {z } =1 ω ( T e s ,π e ,m e ,n e ) , s ∈ [0 , , for some m e , n e , and we have ∀ ≥ s > ω ( T e s ,π e ,m e ,n e ) = 0. We may even choose m e = n e . But, ω ( T e ,π e ,m e ,n e ) = p dim( π e ) δ m e ,n e , because g ( e , A, { p x } x ∈ Σ ) = 1. Thus, discontinuity follows for diagonalexpectation value functions [0 , ∈ s ω ( T e s ,π e ,m e ,m e ) , m e = 1 , ..., dim( π e ).This result is inspired by a similar statement in the algebraic formulation of quantum gauge field theories [33].Interestingly, in quantum field theory the only way to avoid irregular representations of the gauge field variable A ∈ A P , seems to be the use of indefinite inner product (Krein) spaces (cf. [33, 34]). A LQC & the Bohr representation
The algebra A LQC of (homogeneous, isotropic) loop quantum cosmology is given by the Weyl algebra associatedwith the space R = { ( λ, θ ) | λ, θ ∈ R } with the (canonical) symplectic structure (cf. [32]): σ (( λ , θ ) , ( λ , θ )) = λ θ − λ θ . (2.107) Definition 2.33:
The algebra A LQC is the *-algebra generated by the elements U ( λ ) = e iλb , λ ∈ R , and V ( θ ) = e iθν , θ ∈ R ,subject to the relations U ( λ ) ∗ = U ( − λ ) = U ( λ ) − , U (0) = V ( θ ) ∗ = V ( − θ ) = V ( θ ) − , V (0) = (2.108) U ( λ ) U ( λ ) = U ( λ + λ ) , V ( θ ) V ( θ ) = V ( θ + θ ) , U ( λ ) V ( θ ) = e − iλθ V ( θ ) U ( λ ) . A LQC can be made a C ∗ -algebra by completing it w.r.t. the maximal C ∗ -norm (cf. [54]) || W || max = sup {|| W || | || . || is a C ∗ − norm on A LQC } , W ∈ A LQC . (2.109)The generators b, ν defined w.r.t. a regular representations are related to the Hubble parameter and theoriented volume respectively. These elementary variables are related to those in standard treatments of LQC,where { b, v } = 2 γ , by rescaling the volume ν = γ v . Alternatively, this algebra is written in terms of thecombined operators W ( λ, θ ) = e i ( λb + νθ ) = e i λθ U ( λ ) V ( θ ) , ( λ, θ ) ∈ R . W ( λ, θ ) ∗ = W ( − λ, − θ ) = W ( λ, θ ) − , W (0 ,
0) = (2.110) W ( λ , θ ) W ( λ , θ ) = e − i σ (( λ ,θ ) , ( λ ,θ )) W ( λ + λ , θ + θ ) = e − iσ (( λ ,θ ) , ( λ ,θ )) W ( λ , θ ) W ( λ , θ ) . This algebra is obtained by restricting the holonomies to a cubic graph and the fluxes to surfaces dual to thisgraph, and exploiting isotropy to reduce from SU (2) to U (1), followed by a “decompactification” to R Bohr (cf. [39, 52]).In analogy with the Hilbert space representation typically chosen for A LQG , one selects a preferred (irregular)representation induced by the (algebraic) state ω ( W ( λ, θ )) = δ λ, , ∀ λ, θ ∈ R . (2.111)The representation of this state can be understood in terms of Besicovitch’s almost-periodic functions, i.e. H ω ∼ = L ( R Bohr , dµ
Bohr ) (cf. [32, 55]). The uniqueness of this state was recently justified [56] along the samelines as the uniqueness of the Ashtekar-Isham-Lewandowski representation for A LQG [12]. In contrast, the usualFock or Schrödinger representation is obtained from the (regular) state ω F ( W ( λ, θ )) = e − λ θ , ∀ λ, θ ∈ R . (2.112)9 The quantization of the (gravitational) Hamiltonian constraint H in the spatially flat case ( k = 0) that isderived w.r.t. the GNS representation ( H ω , π ω , Ω ω ) of the state (2.111) takes the form (up to numericalconstants) [39]: H ∼ ν sin( λ b ) λ ν sin( λ b ) λ ∼ ν ℑ ( U ( λ )) ν ℑ ( U ( λ )) , λ ∈ R , (2.113)where ℑ ( U ( λ )) denotes the imaginary part, λ is a minimal length scale connected to the (kinematical) minimalarea eigenvalue of loop quantum gravity , and ν is the (densely defined) generator of the 1-parameter group { π ω ( V ( θ )) } θ ∈ R , which exists by the continuity of the state w.r.t. θ (cf. [57] for details regarding the domain D ( H )). It is easy to see that H commutes with π ω ( V ( θ = πλ )), and the representation admits a direct sumdecomposition w.r.t. the spectrum of the latter ( σ ( π ω ( V ( θ = πλ ))) = S = { e iϑ | ϑ ∈ [0 , π ) } ): H ω ∼ = M ϑ ∈ [0 , π ) H ϑ , π ω ∼ = M ϑ ∈ [0 , π ) π ϑ , (2.114)where the summands ( H ϑ , π ϑ ) , ϑ ∈ [0 , π ) , are preserved by the subalgebra A U (1)LQC ⊂ A LQC A λ LQC = h{ W (2 λ n, θ ) | ( n, θ ) ∈ Z × R ⊂ R }i , (2.115)i.e. the Weyl algebra associated with the cotangent bundle T ∗ U (1) ∼ = S × R . Thus, the parameter λ is onehalf of the inverse radius of the S -factor, and plays the role of a “compactification scale”. In the literature, it isargued that this gives rise to a “superselection” structure induced by H , and Dirac observables are computedw.r.t. one of the ϑ -sectors (cf. [39, 58]).In the following sections we will explain further similarities between this structure in loop quantum cosmologyand the Koslowki-Sahlmann representations [24]. The algebras A LQG and A λ LQC have a common feature that will be in focus of this section. Namely, bothalgebras, as defined in 2.33 and 2.30, have non-trivial centers, which implies that they have no irreducible,faithful representations. Furthermore, there is a common cause for the appearance of non-trivial central elementsin these algebras, as both can be related to quantizations of the cotangent bundle of a compact group, i.e. thestructure group G and the dual U (1) of the invariance group Z of H respectively. This feature affects therepresentations theory of both algebras, since irreducible representation require that elements of the center arerepresented by multiples of the identity (superselection structure). In the subsequent discussion of these aspects,we will repeatedly encounter a unifying algebraic structure consisting of the following data (cf. [32]):1. An algebra A of “observables” with a non-trivial center Z .2. An extended algebra F ⊃ A , which is called the “field algebra”.3. A group of automorphisms G representing the adjoint action of Z on F , s.t. A ⊆ F G is contained in thefix-point algebra of F w.r.t. this action. G is called “global gauge group”.4. A group of automorphisms of A , which does not leave the center Z pointwise invariant, i.e. ρ ( Z ) = Z for ρ ∈ C , Z ∈ Z . Elements of C are called “charge automorphisms”.Let us briefly explain the nomenclature (see also remark 3.6 below): The algebra A is the algebra of which weintend to understand the representation theory. In many cases this will be the algebra of observables of a givenquantum system. The center Z of A reflects the superselection structure of the quantum system. Moreover, ifwe are dealing with a gauge theory, we will have to deal with the question of gauge invariance, and the issue ofcharged fields (charged with respect to the global gauge group G ) from which we construct observables. As theobservables should be invariant under the global gauge group G , and the charged fields cannot be part of A , butshould belong to an (extended) field algebra F , we need to require that A ⊂ F G . On the other hand, the center Z embeds into the gauge group G via the adjoint action, and is, therefore, part of the global gauge group, whilethe adjoint action of a (unitary) charged field can lead to an automorphism of A (charge and conjugate chargecombine to zero charge) that moves the elements of center among each other, which amounts to a shift in the The presence of the minimal length scale λ serves as an argument for the use of the irregular representation, as the limit λ → .1 Central operators in A λ LQC F from a given algebra A with center Z provides candidatesof charged automorphisms, ρ ∈ C , which can be combined with known representations π of A to give new,inequivalent representations π ◦ ρ .Furthermore, the inequivalence of the representations π and π ◦ ρ can be understood as an instance of spontaneoussymmetry breaking (in the algebraic sense, cf. [59]). Namely, in spite of the fact that ρ is an automorphisms (asymmetry) of A , a pure (or primary) algebraic state ω on A cannot be invariant w.r.t. ρ , i.e. ω ◦ ρ = ω . To seethis, we recall that a state ω is pure (or primary) if and only if the associated GNS representation ( π ω , H ω , Ω ω )has a 1-dimensional commutant π ω ( A ) ′ = C · H ω (or center Z ω = π ω ( A ) ′ ∩ π ω ( A ) ′′ = C · H ω ). But, this impliesnon-invariance of ω , because π ω ( Z ) ⊂ π ω ( A ) ′ = C · H ω , and ρ acts non-trivially on Z , which are incompatiblerequirements.If we intended to consider an invariant state ω on A , we will have an extremal (or central) decomposition intopure (or primary) states ω x , x ∈ X, : ω = Z X ω x dµ ( x ) . (3.1)The central decomposition of ω is related to the central decomposition of the von Neumann algebra π ω ( A ) ′′ with respect to its center Z ω = π ω ( A ) ′ ∩ π ω ( A ) ′′ , which is especially important in physics, because elements ofthe center describe invariants of the quantum system, which assume specific values in the states ω x , x ∈ X [59].This point of view will be important in the discussion of chiral symmetry breaking in section 5.The algebraic formulation of spontaneous symmetry breaking connects with the standard formulation that asymmetry of the Hamiltonian of a quantum system does not entail a symmetric (ground) state of the latter inthe following way: Symmetries ρ of the algebra A of observables, to which the Hamiltonian is affiliated, are notnecessarily symmetries of a state ω on A . A λ LQC
Our analysis of the algebra A λ LQC , and how the structure of its center Z λ LQC reflects the decomposition (2.114),follows closely the analysis of the Weyl algebra for a quantum particle on a circle as given in [32].The center Z λ LQC is generated by the element W (0 , πλ ), and we may regard the algebra A λ LQC as the fix-pointalgebra A Z LQC w.r.t the Z -action α m ( W ( λ, θ )) = e i πλ mλ W ( λ, θ ) , m ∈ Z . (3.2)In this setting, Z is called the global gauge group, and A LQC the field algebra. The extension of A λ LQC to A LQC is minimal in a precise sense (cf. [60]). Clearly, the action of the global gauge group is implemented by theadjoint action generator W (0 , πλ ) of the center Z λ LQC on A LQC : α m ( W ( λ, θ )) = W (0 , πλ ) m W ( λ, θ ) W (0 , − πλ ) m . (3.3)We observe that the irregular state ω (2.111) is gauge invariant, in contrast to the regular Fock state ω F (2.112).On the other hand, requiring a gauge invariant state ω ◦ α m = ω immediately leads to ω ( W ( λ, θ )) = 0 if λ / ∈ λ Z .Additionally, we have the so-called charged automorphisms of A λ LQC : ρ ϑ ( W (2 λ n, θ )) = e − iϑ λ π θ W (2 λ n, θ ) , ϑ ∈ [0 , π ) , (3.4)which are necessarily outer automorphisms, as they do not the leave the center Z λ LQC pointwise invariant. Theseautomorphisms are inner in the larger algebra A LQC ρ ϑ ( W (2 λ n, θ )) = W ( ϑ λ π , W (2 λ n, θ ) W ( − ϑ λ π , , (3.5)and they intertwine inequivalent irreducible representations of A λ LQC . The latter follows, because every irre-ducible representation π requires that we have for the generator of the center π ( W (0 , πλ )) = e iϑ , ϑ ∈ [0 , π ).Thus, an irreducible representation π ϑ is labeled by an “angle” ϑ ∈ [0 , π ), and we find that π = π ϑ ◦ ρ ϑ (3.6)is a representation satisfying π ( W (0 , πλ )) = 1 . (3.7).2 Central operators in A LQG A λ LQC supplemented by (3.7), which are regular w.r.t.the 1-parameter group { W (0 , θ ) } θ ∈ R , are unitarily equivalent. The representations π ϑ can be realized by theGNS representation of the state ω ϑ ( W (2 λ n, θ )) = e iϑ λ π θ δ n, , ∀ n ∈ Z , θ ∈ R . (3.8)The difference between the representations with distinct values of ϑ are also seen on the level of the generator ν ϑ of { π ϑ ( V ( θ )) } θ ∈ R , i.e. we have H ϑ ∼ = L ([0 , πλ ) , db ) and ν θ is the self-adjoint extension of − i ∂∂b subject tothe boundary condition ψ ( πλ ) = e iϑ ψ (0).The occurrence of these structures can be related to the topology of group U (1), which is the dual of theinvariance group Z acting according to (3.2) (cf. [35]). The Z -action corresponds geometrically to translationsof the variable b , i.e. b b + πλ m, m ∈ Z . In the restricted setting of the algebra A λ LQC , it can be interpretedas the action of the large gauge transformations with winding number m , which are the rotations by 2 πm ofthe underlying circle group U (1) ⊂ T ∗ U (1). As argued in [33, 34, 61], there is a strong analogy between thesealgebraic structures and those present in the context of chiral symmetry breaking and the vacuum structure ofQCD (see section 5). To this end the following remark is in order:Although the charge automorphisms ρ ϑ , which play the role of the chiral automorphisms of QCD, and gaugeautomorphisms α m commute, i.e. ρ ϑ ◦ α m = α m ◦ ρ ϑ , ϑ ∈ [0 , π ) , m ∈ Z , (3.9)the implementers of the gauge transformations W (0 , m πλ ) ∈ Z λ LQC are not invariant under the charge auto-morphisms ρ ϑ by (3.4). Thus, the charge symmetry is necessarily spontaneously broken in any irreduciblerepresentation π ϑ of A λ LQC .Interestingly, there is also way to relate the ϑ -sectors to a purely imaginary topological term contributing tothe action of a free particle on the circle via the functional integral point of view (cf. [32]): S = m Z ˙ x ( τ ) dτ + iϑ λ π Z ˙ x ( τ ) dτ. (3.10)We conclude this subsection by pointing out how the appearance of these structures differs in the GNS repre-sentation of the Fock state (2.112) from that in the Bohr state (2.111). The Fock state leads to a representation( H F , π F , Ω F ) of A LQC that is unitarily equivalent to the Schrödinger representation by von Neumann’s unique-ness theorem, but this representation is reducible for A λ LQC . In fact, we have a central decomposition of therepresentation over the spectrum of W (0 , πλ ): H F ∼ = Z ϑ ∈ [0 , π ) H ϑ dϑ, π F ∼ = Z ϑ ∈ [0 , π ) π ϑ dϑ. (3.11)In comparison with (2.114), which reflects that π ω ( W (0 , πλ )) has only pure point spectrum, the spectrum of π F ( W (0 , πλ )) is purely absolutely continuous, and the GNS vectors Ω ϑ are its improper eigenvectors. A LQG
In the construction of A LQG , it is assumed that the structure group of the principal bundle P is a compact Liegroup G. By compactness, G is the finite extension of its (connected) identity component G by G / G ∼ = π (G).On the other hand, it is well-known [53] that G is isomorphic to the quotient of the product of a n -torus U (1) n and a compact, connected, simply connected Lie group K by a central, finite, Abelian subgroup A. Furthermore,K is isomorphic to a finite product of compact, connected, simply connected, simple Lie groups.G ∼ = (K × U (1) n ) / A (3.12)Therefore, we will give separate discussions of the structure of A LQG in the two cases:1. G ∼ = U (1) n for some n ∈ N .2. G ∼ = K is compact, connected, simply connected and simple.In the second case, we will also comment on the case G ∼ = K / A , A ⊂ Z (K) & finite, i.e. π ( G ) = { } ..2 Central operators in A LQG ∼ = U (1) n If G ∼ = U (1) n , we notice that the only non-trivial relation among the generators of A LQG is (cp. (2.96)) W S ( tn ) f = α λ ∗ t ˜ n ( f ) W S ( tn ) . (3.13)Furthermore, the generators { τ i } i =1 ,...,n ⊂ u (1) ⊕ n ( u (1) = Lie( U (1)) = i R ) can be treated independently,because U (1) n is Abelian. Thus, it is sufficient to discuss the relation (3.13) for n = 1.Since the C ∗ -algebra C ( U (1)) is generated by the characters ( . ) n : U (1) → C , g g n , let us consider (3.13)for the spin (or charge) network functions T γ,~m ( ¯ A ) = Y e ∈ E ( γ ) g ( e, ¯ A, { p x } x ∈ Σ ) m e , γ ∈ Γ sa0 , ¯ A ∈ A , (3.14)where ~m = ( m e ) e ∈ E ( γ ) ∈ Z | E ( γ ) |6 =0 : W S ( tn ) T γ,~m = Y e ∈ E ( γ S ) e m Se tε ( e,S )˜ n | pe (0) T γ,~m W S ( tn ) . (3.15)The labels { m Se } e ∈ E ( γ S ) are those defined by T γ,~m for the adapted graph γ S . This relation basically resemblesthe commutation relations of A λ LQC (cp. (2.108)), apart from the complication due to the intersection propertiesof γ and S . Therefore, the center Z LQG of A LQG is generated by elements W S ( tn ) with t ˜ n | p x = 4 πi ∀ x ∈ S . Butby definition 2.24, this is only possible if S is closed and compact, as otherwise n ∈ Γ(Ad(P)) is not allowed tobe constant on S . Examples of such closed and compact S are given by embedded compact Riemann surfaces,e.g. S = S or T . Thus, we have: Z LQG = h W S (4 πi ) i , S closed and compact . (3.16)As in the previous subsection, we conclude that in any irreducible representation π of A LQG the generators of Z LQG are represented by multiples of the identity, i.e. π ( W S (4 πi )) = e iϑ S , ϑ S ∈ [0 , π ) ∀ S closed and compact , (3.17)which implies that any irreducible representation π = π ϑ is labeled by family of “angles” ϑ = { ϑ S } S , where weset ϑ S = 0 if S is not closed and compact.If we define a type of (charged) automorphisms ρ ϑ,E (0) = { ρ ϑ S ,E } S by ρ ϑ S ,E (0) ( W S ( tn )) = ( e i ϑS π vol ∗ E (0)( i )( S ) t R S ∗ E (0) ( n ) W S ( tn ) S closed and compact W S ( tn ) otherwise , (3.18)for some E (0) ∈ Γ( T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ)) , we find a relation analogous to (3.6): π = π ϑ ◦ ρ ϑ,E (0) . (3.19)Clearly, the Ashtekar-Isham-Lewandowski representation (2.99) is a representation with ϑ = { } S , and is singledout by automorphism invariance or diffeomorphism and gauge invariance (cf. [12, 13]). The question, if this isthe only representation with ϑ = { } S , is more subtle, and will be discussed elsewhere. Inspecting (3.18) moreclosely, we may even choose ϑ S = 0 for arbitrary faces S , and set ϑ S = ϑ ∀ Sρ ϑ ,E (0) ( W S ( tn )) = e i ϑ π vol ∗ E (0)( i )( S ) t R S ∗ E (0) ( n ) W S ( tn ) ∀ S, (3.20)which would lead us to the Koslowksi-Sahlmann representations π ϑ ,E (0) = π ω ◦ ρ − ϑ ,E (0) [24] (see below).Following the discussion of the previous subsection, we can also ask, whether we can regard A LQG as the fix-point algebra of a larger field algebra F LQG under the adjoint action of the generators of the center Z LQG . Tothis end, we exploit the similarity of (3.15) and (2.108):First, we use the covering homomorphism π : R → U (1) , ϕ e iϕ , which coincides with the exponential map vol ∗ E (0) ( i ) ( S ) = R S ∗ E (0) ( i ) is the volume of S relative to the pairing of E (0) and the generator i of u (1), and serves as anormalization factor. .2 Central operators in A LQG u (1) → U (1), to lift the functions F γ l on U (1) | E ( γ l ) | ∼ = Hom( l, U (1)) to functions ˜ F γ l = F γ l ◦ π ×| E ( γ l ) | to R | E ( γ ) | ∼ = Hom( l, R ). Clearly, the lifting is isometric w.r.t the sup-norm and compatible with the projectivestructure of Hom( P Σ , R ). Thus, we are allowed to consider ˜ F γ l as defining a cylindrical function on the lattervia p l : Hom( P Σ , R ) → Hom( l, R ). This is possible, because the construction of A does not require thecompactness of G. Only the construction of the Ashtekar-Isham-Lewandowski measure requires a compactstructure group. Especially, we may lift the spin network functions ˜ T γ,~m , which form a subset of the Fouriernetwork functions on Hom( P Σ , R ):˜ T γ,~β ( { ϕ e } e ∈ E ( γ ) ) = Y e ∈ E ( γ ) e iα e ϕ e , γ ∈ Γ sa0 , ~β = ( β e ) e ∈ E ( γ ) ∈ R | E ( γ ) |6 =0 . (3.21)Second, we note that the action of the Weyl elements W S ( tn ) on the cylindrical functions, which defines thecommutation relation (3.13), is compatible with lift through ξ : R → U (1), as well.( W S ( tn ) · ˜ F γ Sl )( { ϕ e } e ∈ E ( γ Sl ) ) = ˜ F γ Sl ( { ϕ e − i tε ( e, S )˜ n | p e (0) } e ∈ E ( γ Sl ) ) = F γ Sl ( { e iϕ e e tε ( e,S )˜ n | pe (0) } e ∈ E ( γ Sl ) ) . (3.22)Third, we define the field algebra F LQG to be generated by the Fourier network functions ˜ T γ,~β and the Weylelements W S ( tn ) subject to the equivalent set of relations as in (2.96), but involving the lifted action (3.22). Remark 3.1:
The lifting of the structure group U (1) to R by the covering homomorphism π , requires on the classical level, i.e.for the construction to be related to structures in principal G-bundles, the existence of a non-trivial covering ofthe principal U (1)-bundle P by a principal R -bundle P R P R (cid:8) ξ / / π R (cid:15) (cid:15) P π (cid:15) (cid:15) Σ id Σ / / Σ P R (cid:8) ξ / / R ϕ (cid:15) (cid:15) P R ξ ϕ ) (cid:15) (cid:15) P R ξ / / P (3.23)with a diagram of fibrations: R (cid:15) (cid:15) ξ / / U (1) (cid:15) (cid:15) Z ♣♣♣♣♣♣♣ ' ' ◆◆◆◆◆◆ P R ξ / / π R ' ' ◆◆◆◆◆◆ P π w w ♥♥♥♥♥♥♥ Σ (3.24)By construction, the adjoint action of the generators of Z LQG on F LQG fixes the algebra A LQG8 , which we inferfrom: α Sm ( T γ,~β ) = W S (4 πi ) m ˜ T γ,~β W S ( − πi ) m = Y e ∈ E ( γ S ) e πiβ Se ε ( e,S ) m ˜ T γ,~β , (3.25) α Sm ( W S ′ ( tn )) = W S (4 πi ) m W S ′ ( tn ) W S ( − πi ) m = W S ′ ( tn ) , where we defined Z -actions α S : Z → Aut( F LQG ) (the “global” gauge group) for every closed, compact face S . On the other hand, we get (charged) automorphisms by the adjoint action of the Fourier network functions˜ T γ,~β , ~β = ( ϑ e π ) e ∈ E ( γ ) ∈ [0 , | E ( γ ) | on A LQG : ρ γ~β ( T γ ′ ,~m ) = ˜ T γ,~β T γ ′ ,~m ˜ T γ, − ~β = T γ ′ ,~m , (3.26) ρ γ~β ( W S ( tn )) = ˜ T γ,~β W S ( tn ) ˜ T γ, − ~β = Y e ∈ E ( γ S ) e − β Se tε ( e,S )˜ n | pe (0) W S ( tn ) . Strictly speaking, it fixes an algebra containing the lift of A LQG in F LQG . .2 Central operators in A LQG Z LQG these automorphisms lead to ρ γ~β ( W S (4 πi )) = e − i P e ∈ E ( γS ) ε ( e,S ) ϑ Se W S (4 πi ) = e − iϑ S W S (4 πi ) , (3.27)where we defined ϑ S = P e ∈ E ( γ ) ε ( e, S ) ϑ Se . Thus, we arrive at a second type of (charged) automorphisms (cp.(3.18) & (3.20)) labeled by a graph γ ∈ Γ sa0 and an associated set of angles { ϑ e } e ∈ E ( γ ) . As in the previoussection, we have (cp. (3.9)) ρ ϑ S ,E (0) ◦ α Sm = α Sm ◦ ρ ϑ S ,E (0) , (3.28) ρ γ~β ◦ α Sm = α Sm ◦ ρ γ~β . Similar to the discussion of ϑ -representations of A λ LQC , the difference between representations of A LQG withdistinct labels { ϑ S } S , which are regular w.r.t. the Weyl elements W S ( tn ), can be seen on the level of the fluxes,e.g. in the GNS representation of ω (2.99): E ϑ,E (0) S ( n ) = E S ( n ) + i ϑ π vol ∗ E (0) ( i ) ( S ) Z S ∗ E (0) ( n ) (3.29) E γ,ϑS ( n ) = E S ( n ) + π X e ∈ E ( γ S ) ε ( e, S ) ϑ Se ˜ n | p e (0) . Actually, this is the starting point for the construction of Koslowski-Sahlmann representations (see below). ∼ = K is compact, connected, simply connected and simple
In this subsection, we assume that G ∼ = K is a compact, connected, simply connected and simple Lie group,which is the most important case for loop quantum gravity, because in a version of the theory in the Ashtekar-Barbero variables K = SU (2). A variant of loop quantum gravity w.r.t. the new variables has G = Spin [1],which is compact, connected, simply connected and semi-simple, because Spin ∼ = SU (2) × SU (2) [62], and thuscan be reduced to the simple case.In view of the previous subsection, we have additional non-trivial relations among the generators of A LQG (cp.(2.96)): W S ( tn ) f W S ( tn ) − = W S ( tn ) · f = α λ ∗ t ˜ n ( f ) (3.30) W S ( tn ) W S ′ ( t ′ n ′ ) W S ( tn ) − W S ′ ( t ′ n ′ ) − = α λ ∗ t ˜ n ◦ α λ ∗ t ′ ˜ n ′ ◦ α − λ ∗ t ˜ n ◦ α − λ ∗ t ′ ˜ n ′ . As in the case of the Weyl algebra associated with a linear symplectic space (cp. (2.110)), the second relationsets up a strong relation between the product of Weyl elements W S ( tn ) and the composition of the maps α λ ∗ t ˜ n ,and thus the group product of K. But, the relation leaves room for the existence of non-trivial central elements.To be more precise, the existence of the central elements is due to the relations (3.30) and the fact that fora compact Lie group we can find 0 = X ∈ k s.t. exp K ( X ) = 1 K , because there exist maximal tori in K.Moreover, since we assume K to be simple, it has a non-degenerate, negative definite (by compactness) Killingform ( X, Y ) k = tr gl ( k ) (ad X ◦ ad Y ), which is Ad-invariant, i.e. Ad K ⊂ SO ( k ). This implies, that all elements inthe adjoint orbit of X ∈ k , s.t. exp K = 1 K , are mapped to 1 K :exp K (Ad g ( X )) = α g (exp K ( X )) = α g (1 K ) = 1 K , ∀ g ∈ K . (3.31)In general, a similar observation can be made for all elements g ∈ Z (K), since then Stab(g) = K. But by thefirst line of (3.30), only the cut locus exp − ( { K } ) of 1 K in k will define central elements of A LQG : Z LQG = h W S ( tn ) i , t ˜ n | p x = X ∈ exp − ( { K } ) for all reference points p x , S closed and compact . (3.32)The restriction to closed and compact faces S is again necessary, because of the support properties of n . By(3.31), the Ad-equivariance of ˜ n implies that t ˜ n | p ∈ exp − ( { K } ) ∀ p ∈ P. In the case K = SU (2), we have:exp − ( { K } ) = (cid:8) π ( ~e · ~τ ) | ~e ∈ S ⊂ R , τ i = − i σ i (cid:9) , (3.33).2 Central operators in A LQG { σ i } i =1 , , are the Pauli matrices. As above, we conclude that Z LQG is non-trivial, and that in anyirreducible representation π the identities π ( W S (2 X )) = e iϑ S (2 X ) , ϑ S (2 X ) ∈ [0 , π ) ∀ X ∈ exp − ( { K } ) , S closed and compact , (3.34)hold, with ϑ S (2 X + 2 X ′ ) = ϑ S (2 X ) + ϑ S (2 X ′ ) mod 2 π if X = µX ′ for some µ ∈ R . Unfortunately, wecannot define (charged) isomorphisms by the analog of (3.18), because we have non-trivial relations amongWeyl elements.For example, since we have that K is simply connected, we know that Ad(P) is spin [62]. Therefore, we findthat Ad(P | S ) ∼ = S × k [62], which gives the identification Γ sa0 (Ad(P | S )) ∼ = C sa0 ( S, k ). Now, we specialized toK = SU (2) choose two constant, ortho-normalized functions f in ∈ C sa0 ( S, k ) , i = 1 ,
2, i.e. ( f in , f jn ) k = − δ ij and f in ( x ) = X i ∈ k ∀ x ∈ S , and consider the associated Weyl element W S ( n i ) , i = 1 ,
2. By the second line of (3.30)and the Ad-equivariance of ˜ n i , i = 1 ,
2, we obtain W S (4 πn ) W S ( − n ) W S (4 πn ) − W S ( − n ) − = W S (2 n ) . (3.35)On the level of the holonomy-flux algebra, the presence of non-trivial relations is exemplified by (2.98). Clearly,an analogue of this construction works for G = Spin by the isomorphism Spin = SU (2) × SU (2). Wesummarize this observation in the following proposition. Proposition 3.2:
For G compact, connected and simply connected, assume that ∀ X ∈ g ∃ g X : Ad g X ( X ) = − X . Then, the subgroup h W i ⊂ h W i , generated by the Weyl elements W S ( tn ) , S closed and compact , n ∈ Γ sa0 (Ad(P | S )) constant w.r.t. some triv. of Ad(P | S ) , (3.36) is perfect, i.e. h [ h W i , h W i ] i = h W i . Proof:
From the simply connectedness of G, we deduce the triviality of Ad(P | S ) ∼ = S × g , as above. Thus, W S ( tn ) ∈ h W i is determined by a constant function f n : S → g . By assumption, we are allowed to choose an element g n ∈ G,s.t. Ad g n ( f n ) = − f n , and by compactness and connectedness of G, we find 0 = X n ∈ g , s.t. exp G ( X n ) = g n . Ifwe define a constant section s n ∈ Γ sa0 (Ad(P | S )) by S ∋ x X n ∈ g , we have W S ( s n ) W S ( − tn ) W S ( s n ) − W S ( − tn ) − = W S ( tn ) . (3.37)This implies the proposition. Corollary 3.3:
By proposition 3.2, representations π of A LQG with ϑ S = 0 , for some closed and compact S , cannot be inducedby character automorphisms ρ χ ( W S ( tn )) = χ ( W S ( tn )) W S ( tn ) (3.38) of the Weyl group h W i , where χ : h W i → U (1) is a character. Proof:
The generators of the center Z LQG are contained in the subgroup h W i , which is perfect. Therefore, the restriction χ |h W i is trivial. Remark 3.4:
The proof of the proposition 3.2 clearly fails in this form for G = SU (2), if we remove the condition that n isconstant. To see this, choose S ∼ = S subordinate to a coordinate chart of Σ, and define n : S → g ∼ = R tobe the (outward) unit normal vector field on S . This implies that n ⊥| x ∼ = T x S . But, T S admits no nowherevanishing, continuous section m : S → T S , since the Euler characteristic is positive, χ ( S ) = 2 [63].A similar result can formulated for the algebra of flux vector fields h X Flux i (cp. (2.98)). Proposition 3.5:
Assume that g is perfect, which will be the case, if g is simple or semi-simple, i.e. [ g , g ] = g . Then, the subalgebra h X Flux , i ⊂ h X Flux i generated by the elements E n ( S ) , S closed and compact , n ∈ Γ sa0 (Ad(P | S )) constant w.r.t. some triv. of Ad(P | S ) , (3.39) is perfect, i.e. h [ h X Flux , i , h X Flux , i ] i = h X Flux , i . .2 Central operators in A LQG Proof:
This follows immediately from equation (2.98) and the perfectness of g . Remark 3.6:
The subalgebra h X Flux , i and subgroup h W i also exist in the Abelian case, where they admit a natural, heuristicinterpretation in terms of Gauß’ law. Formally, we have W S ( tn ) = e tE n ( S ) , and for S closed and compact , n ∈ Γ sa0 (Ad(P | S )) constant, we find from the (classical) formula (2.77) and the Gauß’ theorem: Z V (div T Σ ( E ))( n ) = Z S ∗ E ( n ) , ∂V = S, (3.40)where n is extended constantly to the region V bounded by S , and the adjoint bundle is assumed to trivializeover V , i.e. Ad(P | V ) ∼ = V × g . Thus, h X Flux , i and subgroup h W i are quantizations of the smeared Gauß’constraints (div T Σ ( E )) n ( V ) = R V (div T Σ ( E ))( n ), and can serve as implementers of the gauge transformationsgenerated by n ∈ Γ(Ad(P | V )) , n = constant. This justifies the terminology “global” gauge group for the Z -automorphisms α S defined by the adjoint action of the center Z LQG on the field algebra F LQG : Z LQG ⊂ G LQG = h W i = α G sa , . (3.41)The relations (3.28) generalize accordingly ρ ϑ S ,E (0) ◦ α W S ( n ) = α W S ( n ) ◦ ρ ϑ S ,E (0) , (3.42) ρ γ~β ◦ α W S ( n ) = α W S ( n ) ◦ ρ γ~β . We conclude that the charged automorphisms are spontaneously broken w.r.t. gauge invariant, pure states ω .An interpretation along these lines is not available in the non-Abelian setting, because the Gauß’ law is givenby the vanishing of the smeared horizontal (or covariant) divergences: Z V (div AT Σ ( E ))( n ) = − Z V ˜ E ( d A ˜ n ) = 0 , ( A, ˜ E ) ∈ | Λ | T ∗ A P , (3.43)which spoils the applicability of the Gauß’ theorem. Here, ˜ E ( d A ˜ n ) denotes the projection of the right invariantdensity ˜ E ( d A ˜ n ) on P to Σ (cp. (2.29)). Corollary 3.7:
There are no representations π of P LQG satisfying π ( E n ( S )) = π ω ( E n ( S )) + c n ( S ) · H ω , c n ( S ) ∈ C , (3.44) with c n ( S ) = 0 for closed and compact S, n ∈ Γ sa0 (Ad(P | S )) constant w.r.t. some triv. of Ad(P | S ) . ω is theAshtekar-Isham-Lewandowski state (2.99) Proof:
Let E n ( S ) ∈ h X Flux , i . Then, we have by proposition 3.5 and (2.98): π ( E n ( S )) = π ([ E n ′ , [ E n ′′ ( S ) , E n ′′′ ( S )]]) = [ π ( E n ′ ( S )) , [ π ( E n ′′ ( S )) , π ( E n ′′′ ( S ))]] (3.45)= [ π ω ( E n ′ ( S )) , [ π ω ( E n ′′ ( S )) , π ω ( E n ′′′ ( S ))]] = π ω ([ E n ′ ( S ) , [ E n ′′ ( S ) , E n ′′′ ( S )]])= π ω ( E n ( S )) , where we chose n ′ , n ′′ , n ′′′ , s.t. [ n ′ , [ n ′′ , n ′′′ ]] = n by the perfectness of g . Thus, c n ( S ) = 0.Applying the same reasoning to general E n ( S ) ∈ h X i , we conclude, that for arbitrary faces S , we are forcedto set c n ( S ) = 0 ∀ n ∈ Γ sa0 (Ad(P | S )), s.t. ˜ n ∈ [ G sa , | S , [ G sa , | S , G sa , | S ]], where G sa , | S denotes the semi-analytic,compactly supported gauge algebra of P | S (see definition 2.9).Finally, we want to consider the case G ∼ = K / A, for some Abelian, finite group A ⊂ Z (K). With minormodifications, similar results holds for semi-simple K, e.g. Spin . In the same way, as in the discussion ofG = U (1), we use the covering homomorphism ξ A : K → G to construct a lift of the algebra A LQG to anextended algebra F LQG . Because G is compact, the spin network functions of G, T γ,~π,~m,~n ( ¯ A ) = Y e ∈ E ( γ ) p dim( π e ) π e ( g ( e, ¯ A, { p x } x ∈ Σ )) m e ,n e , (3.46).2 Central operators in A LQG G by the Peter-Weyl theorem, where we introduced the notation Cyl G to indicate theLie group the cylindrical functions are based on. Here, we denote by π e ( . ) m e ,n e , e ∈ E ( γ ) , a matrix entryof a non-trivial, unitary, irreducible representation of G. Therefore, we only need to define the lifts of thesefunctions and the Weyl elements. The lift of a spin network function is defined via pullback:˜ T γ,~π,~m,~n ( { g e } e ∈ E ( γ ) ) = Y e ∈ E ( γ ) p dim( π e ) π e ( ξ A ( k e )) m e ,n e , { k e } e ∈ E ( γ ) ∈ K | E ( γ ) | , (3.47)which embeds these function, isometrically w.r.t. sup-norm, into the spin network functions of K, and iscompatible with the projective structure of Hom( P Σ , K):˜ T γ,~η,~i,~j ( { k e } e ∈ E ( γ ) ) = Y e ∈ E ( γ ) p dim( η e ) η e ( k e ) i e ,j e , { [ η e ] } e ∈ E ( γ ) ∈ ( ˆK \ { [ η triv ] } ) | E ( γ ) | , { k e } e ∈ E ( γ ) ∈ K | E ( γ ) | . (3.48)The naturalness of the exponential maps, exp G ◦ dξ A | K = ξ A ◦ exp K , and the fact that dξ A | K : k → g is anisomorphism, gives rise to a compatible action of the Weyl elements:( W S ( tn ) · ˜ T γ S ,~π,~m,~n )( { k e } e ∈ E ( γ S ) ) = ˜ T γ S ,~π,~m,~n ( { k e exp K ( tε ( e, S )( dξ A | K ) − (˜ n | p e (0) )) } e ∈ E ( γ S ) ) (3.49)= T γ S ,~π,~m,~n ( { ξ A ( k e ) exp G ( tε ( e, S )˜ n | p e (0) ) } e ∈ E ( γ S ) ) . This action respects the Ad-equivariance of ˜ n , because Ad k = ( dξ A | K ) − ◦ Ad ξ A ( g ) ◦ dξ A | K , k ∈ K. The fieldalgebra F LQG is defined as the algebra generated by the cylindrical functions on Hom( P Σ , K) and the Weylelements of A LQG subject to the relations (2.96) and the compatible action (3.49).In view of remark 3.1, the construction requires on the level of the principal G-bundle P the existence of anon-trivial covering P K (cid:8) ξ / / π K (cid:15) (cid:15) P π (cid:15) (cid:15) Σ id Σ / / Σ P K (cid:8) ξ / / R k (cid:15) (cid:15) P R ξ A( k ) (cid:15) (cid:15) P K ξ / / P (3.50)with a diagram of fibrations: K (cid:15) (cid:15) ξ A / / G (cid:15) (cid:15) A ♦♦♦♦♦♦ ' ' ❖❖❖❖❖❖ P K ξ / / π K ' ' ❖❖❖❖❖❖ P π x x qqqqqq Σ (3.51)The fact that Ad : K → Aut(K) descends to the central quotient G ∼ = K / A, since ker(Ad) = Z (K), implies theequivalence of the adjoint bundles of P and P K :Ad(P) ∼ = Ad(P K ) . (3.52)Thus, we can regard the field algebra F LQG as the a Weyl algebra of P K . The algebra A LQG embeds into F LQG via the lifting procedure, and its image is contained in fix-point algebra under the adjoint action of thegenerators of Z LQG , i.e. W S ( tn ) , t ˜ n = X ∈ exp − ( { G } ) , S closed and compact: α Sm ( T γ,~η,~i,~j ) = W S ( tn ) m ˜ T γ,~η,~i,~j W S ( − tn ) m = W S ( tn ) m · ˜ T γ,~η,~i,~j , (3.53) α Sm ( W S ′ ( t ′ n ′ )) = W S ( tn ) m W S ′ ( t ′ n ′ ) W S ( − tn ) m = W S ′ ( t ′ n ′ ) , W S ( tn ) ∈ Z LQG , m ∈ Z . In fact, the first line of (3.53) is trivial for those irreducible representations η e , e ∈ E ( γ ) , of K that are trivialon A, which are precisely the irreducible representation of G. Moreover, the algebra F LQG is not equal tothe fix-point algebra, which can be seen by considering the action (3.53) on spin network functions ˜ T e,η e ,i e ,j e defined on single edges e ∈ P Σ . The actions of the gauge transformations G saP K and G saP are compatible withthe covering, as we will show next..2 Central operators in A LQG Lemma 3.8:
Given a bundle covering ξ : P K → P as in (3.50) & (3.51) , every λ K ∈ G saP K induces a ( λ K ) G ∈ G saP by ( λ K ) G ( p ) = ξ ( λ K ( q )) (3.54) for some q ∈ P K , s.t. ξ ( q ) = p . The map ( . ) G : G saP K → G saP is a homomorphism. Proof:
Clearly, (3.54) is well-defined: If q ′ ∈ P K is another element, s.t. ξ ( q ′ ) = p , we know by (3.51) that q ′ = qa, a ∈ A. This implies: ξ ( λ K ( q ′ )) = ξ ( λ K ( qa )) = ξ ( λ K ( q ) a ) = ξ ( λ K ( q )) ξ A ( a ) (3.55)= ξ ( λ K ( q )) . Semi-analyticity follows from the semi-analyticity of the involved maps, and we have ( λ K ) − = ( λ − ) G , ( λ K ◦ λ ′ K ) G = ( λ K ) G ◦ ( λ ′ K ) G . Now, we only need to verify that ( λ K ) G is a right equivariant bundle map covering theidentity. π (( λ K ) G ( p )) = π ( ξ ( λ K ( q ))) = π K ( λ K ( q )) = π K ( q ) = π ( ξ ( q )) (3.56)= π ( p ) , q ∈ P K : ξ ( q ) = p ( λ K ) G ( pg ) = ξ ( λ K ( qk )) = ξ ( λ K ( q )) ξ A ( k ) (3.57)= ( λ K ) G ( p ) g, q ∈ P K : ξ ( q ) = p, k ∈ K : ξ A ( k ) = g. The lemma tells us that the action of G saP K on A LQG ⊂ F LQG descends to the action of induced gauge transfor-mations in G saP . If we assume that P K and P are path-connected, we may conclude that ( . ) G : G saP K → G saP isonto. Proposition 3.9 (Lifting of gauge transformations):
Assume that P K and P are path-connected. Given λ G ∈ G saP and two points q, q ′ ∈ P K , s.t. λ G ( ξ ( q )) = ξ ( q ′ ) ,there exist a unique lift ˜ λ G ∈ G saP K , s.t. λ G ◦ ξ = ξ ◦ ˜ λ G and ˜ λ G ( q ) = q ′ . The diagram of pointed spaces is (P K , q ′ ) ξ (cid:15) (cid:15) (P K , q ) ˜ λ G λ G ◦ ξ / / (P , p ) (3.58) Proof:
Form the assumptions, we deduce the existence of a lift ˜ λ G ∈ Diff sa ((P K , q ) , (P K , q ′ )) by the lifting theorem forcovering spaces [64], which applies, because [ λ G ◦ ξ ]( π (P K , q )) = [ ξ ]( π (P K , q ′ )) since λ G is a diffeomorphisms.That ˜ λ G covers the identity, is evident from the definition of the lift: π K ◦ ˜ λ G = π ◦ ξ ◦ ˜ λ G = π ◦ λ G ◦ ξ = π ◦ ξ (3.59)= π K . Finally, we need to check that ˜ λ G is right equivariant. We know that ∀ k ∈ K : R k ◦ ˜ λ G and ˜ λ G ◦ R k are liftsof λ G ◦ R ξ A ( k ) by the equivariance of λ G . Furthermore, we find R k (˜ λ G ( q )) = q ′ k and ˜ λ G ( R k ( q )) = q ′ ka ( q ) forsome continuous a : P K → A. But, A is discrete by assumption, which implies ∀ q ∈ P K : a ( q ) = a ∈ A. Thus,we have ∀ k ∈ K : ˜ λ G ◦ R k = R a ◦ R k ◦ ˜ λ G , which leads to a = 1 K for k = 1 K and the free action of K on P K .A similar reasoning applies to the actions of the automorphisms Aut sa (P K ) and Aut sa (P).Candidates for (charged) automorphisms of A LQG can be defined with the help of unitary cylindrical functions f on Hom( P Σ , K) that do not descend through A, i.e. f = p ∗ l f l ∈ Cyl K s.t. f l ¯ f l = 1 and f l does not define afunction on G | E ( γ l ) | : ρ f ( T γ,~π,~m,~n ) = f T γ,~π,~m,~n f ∗ = T γ,~π,~m,~n , (3.60) ρ f ( W S ( tn )) = f W S ( tn ) f ∗ = f ( W S ( tn ) · f ∗ ) W S ( tn ) . .2 Central operators in A LQG f ℜ γ,~η,~i,~j = e i ℜ ( T γ,~η,~i,~j ) , f ℑ γ,~η,~i,~j = e i ℑ ( T γ,~η,~i,~j ) , (3.61)for irreducible representations η e , e ∈ E ( γ ) , of K that do not reduce to G. To arrive at a true automorphism of A LQG , we need to ensure that ∀ W S ( tn ) : f ( W S ( tn ) · f ∗ ) gives a cylindrical function on Hom( P Σ , G). We callfunctions f ∈ Cyl K satisfying these requirements (K , A)-admissible, and denote them by U (Cyl K ) A . Examplesof ( K, A )-admissible functions could be generated from 1-dimensional, unitary representations χ : K → T , s.t. χ | A = 1. But, unfortunately compact, connected, semi-simple Lie groups are (topologically) perfect, and thusdo not posses non-trivial 1-dimensional, unitary representations [65]. Therefore it seems possible that there areno ( K, A )-admissible functions, i.e. U (Cyl K ) A = ∅ . Nevertheless, we observe that U (Cyl K ) A is preserved bygauge transformations and automorphisms (see lemma 3.8).Thus, we conclude the section with the observation that for structure groups G admitting non-trivial coverings ξ : K → G, together with a bundle covering (3.50) & (3.51), we can construct an embedding of algebras A LQG ⊂ F LQG , which allows to construct candidates for (charged) automorphisms ρ f , f ∈ Cyl K as in (3.60).If we find among the latter a true automorphism of A LQG that acts non-trivially on the center Z LQG , we willobtain a new irreducible representations of A LQG from the state (cp. (2.99)): ω f = ω ◦ ρ f . (3.62)Let us also shortly comment on the issue of gauge and automorphism invariance of the state ω f . From theinvariance of ω , we find: ω f ◦ α λ = ω α λ − ( f ) , ω f ◦ α φ = ω α φ − ( f ) (3.63)for λ ∈ G sa P K , φ ∈ Diff sa (Σ). Thus, gauge invariance could be achieved by the additional requirement α λ ( f ) = f ∀ λ ∈ G sa P K , although it is not obvious that this condition can be satisfied non-trivially in combination with theadditional constraints on f . Requiring automorphism invariance poses a much more severe constraint, becausethe analogous requirement α χ ( f ) = f ∀ χ ∈ Aut sa (P) can probably not be satisfied non-trivially [66], i.e. itleads to f ≡ H ω , π ω , Ω ω ) of the field algebra F LQG , w.r.t.which the (charged) automorphisms ρ f , f ∈ U (Cyl K ) A , of A LQG are unitarily implemented, because they areinner automorphisms of F LQG . Therefore, we find A LQG -invariant subspaces H f = π ω ( f ∗ )( π ω ( A LQG )Ω ω ) , π f = π ω | H f . (3.64)The implementers of the gauge transformations and automorphisms U ( λ ), λ ∈ G sa P K , and U ( χ ), χ ∈ Aut sa (P K )map these subspaces into each other according to (3.63): U ( λ ) H f = U ( λ ) π ω ( f ∗ )( π ω ( A LQG )Ω ω ) (3.65)= π ω ( α λ ( f ∗ ))( π ω ( α λ ( A LQG ))Ω ω )= π ω ( α λ ( f ) ∗ )( π ω ( A LQG )Ω ω )= H α λ ( f ) U ( χ ) H f = U ( χ ) π ω ( f ∗ )( π ω ( A LQG )Ω ω )= π ω ( α χ ( f ∗ ))( π ω ( α χ ( A LQG ))Ω ω )= π ω ( α χ ( f ) ∗ )( π ω ( A LQG )Ω ω )= H α χ ( f ) . If we denote by [ f ] the equivalence class of f ∈ U (Cyl K ) A under the actions of G saP K and Aut sa (P K ), we canform the direct sum H [ f ] = M f ′ ∈ [ f ] H f ′ , π [ f ] = M f ′ ∈ [ f ] π f ′ . (3.66)This gives us a (reducible) representation of A LQG with a unitary implementation of G saP K and Aut sa (P K ).On it, we can apply the usual group averaging procedure to obtain gauge or automorphism invariant spaces0( H G ) , ( H ) Aut (cf. [4, 14]).
Now, we turn to the discussion of the Koslowski-Sahlmann representations, mainly for non-Abelian structuregroup G, [24] in view of the results of the previous section. The discussion will be split into two parts relatedto a similar division in [24]:1. “Central extensions” of holonomy-flux algebras an non-degenerate backgrounds,2. Weyl forms of the holonomy-flux algebra and non-degenerate backgrounds.
The holonomy-flux algebras considered in [24] are essentially of the form, we defined in subsection 2.2 (see 2.28).That is, the algebras are generated by elements f ∈ Cyl ∞ , Y n ( S ), S a face, n ∈ Γ sa0 (Ad(P | S )) together with thecommutation relation [ Y n ( S ) , f ] = E n ( S ) · f, [ f, f ′ ] = 0 (4.1)and the reality and linearity conditions f ∗ = ¯ f , Y n ( S ) ∗ = − Y n ( S ) , (4.2) Y n + n ′ ( S ) = Y n ( S ) + Y n ′ ( S ) . (4.3)But, in contrast to definition 2.28, the higher commutation relations for the elements Y n ( S ) are not specified,but only required to satisfy the Jacobi identity, i.e.[[ Y n ( S ) , Y n ′ ( S ′ )] , f ] = [ Y n ( S ) , [ Y n ′ ( S ′ ) , f ]] − [ Y n ′ ( S ′ ) , [ Y n ( S ) , f ]] (4.4)= [ E n ( S ) , E n ′ ( S )] X ( A ) · f, and similar higher order relations. While, in the case of an Abelian structure group, e.g. G = U (1), this posesno specific constraints on the algebraic relations for the elements Y n ( S ) to make the Koslowski-Sahlmannrepresentations well-defined, this is not the case for a non-Abelian structure group, e.g. G = SU (2). In thelatter case, we find (cp. (2.98)):[[ Y n ( S ) , [ Y n ′ ( S ) , Y n ′′ ( S )]] , f ] = [ E n ( S ) , [ E n ′ ( S ) , E n ′′ ( S )] X ( A ) ] X ( A ) · f (4.5)= E [ n, [ n ′ ,n ′′ ]] ( S ) · f = [ Y [ n, [ n ′ ,n ′′ ]] ( S ) , f ] . Thus, we are forced to require the additional relation[ Y n ( S ) , [ Y n ′ ( S ) , Y n ′′ ( S )]] = Y [ n, [ n ′ ,n ′′ ]] ( S ) + c [ n, [ n ′ ,n ′′ ]] ( S ) , (4.6)where c [ n, [ n ′ ,n ′′ ]] ( S ) is an element of the algebra that commutes with the subalgebra Cyl ∞ . If we, additionally,assume that c [ n, [ n ′ ,n ′′ ]] ( S ) commutes with the generators Y m ( S ), the Jacobi identity will give us a “co-cyclecondition”: c [ n, [ n ′ ,n ′′ ]] ( S ) + c [ n ′ , [ n ′′ ,n ]] ( S ) + c [ n ′′ , [ n,n ′ ]] ( S ) = 0 . (4.7)Clearly, c [ n, [ n ′ ,n ′′ ]] ( S ) needs to satisfy linearity conditions related to (4.3) as well.This said, we return to the Koslowski-Sahlmann representations, which are proposed to be defined by an E (0) ∈ Γ( T Σ ⊗ Ad ∗ (P) ⊗ | Λ | (Σ)): π E (0) ( Y n ( S )) = π ω ( E n ( S )) + i Z S ∗ E (0) ( n ) · H ω , π E (0) ( f ) = π ω ( f ) , (4.8) Although, we are allowed to consider modifications, e.g. a central extension [ Y n ( S ) , Y n ′ ( S ′ )] = c [ n,n ′ ] ( S, S ′ ). .1 “Central extensions” of holonomy-flux algebras and non-degenerate backgrounds 31w.r.t. to the Ashtekar-Isham-Lewandowski representation ( π ω , H ω , Ω ω ). A similar construction applies inthe temporal gauge to algebraic formulation of quantum electrodynamics [33]. The interpretation of theserepresentation is obtained from the consideration of the limitlim R →∞ ( ω ◦ ρ E (0) )( W S R ( n )) = e iθ ( n ) (4.9)in the Abelian case (cp. (3.18)) for Σ = R , where we chose S R = S R (2-sphere), n = n ( ϑ, ϕ ), (˜ n, ˜ n ) g = 1, E (0) ( r, ϑ, ϕ ) ∼ θ ( ϑ, ϕ ) r − . Thus, the choice of E (0) affects the asymptotic flux configuration (cp. [67]).We will analyze the Koslowski-Sahlmann representations of the holonomy-flux algebras, in the above sense, fromtwo different, though related, points of view. First, we will argue that the Koslowski-Sahlmann representationsrequire a modification of the commutation relations by a non-trivial “central term”, if we want the Y n ( S )to correspond to the fluxes E n ( S ). Second, we will show, that we can interpret the Y n ( S ) as shifted fluxes E n ( S ) + i R S ∗ E (0) ( n ), which leads to the conclusion that the Koslowski-Sahlmann representations are theAshtekar-Isham-Lewandowski representations w.r.t. the shifted fluxes. The two points are related by theobservation that the shift transformation ρ E (0) : E n ( S ) E n ( S ) + i Z S ∗ E (0) ( n ) (4.10)is not a *-automorphism of P LQG but only an affine transformation. Thus, in contrast to section 3 the chargetransformations ρ E (0) are already broken on the level of the algebra P LQG , and not on the level of a state orrepresentation.As we discussed in subsection 3.2.1, the Koslowski-Sahlmann representations can be understood in terms ofcharged automorphisms of the Weyl algebra A LQG ( c ≡
0) in the Abelian case (cp. (3.20) & (3.29)). In thenon-Abelian setting, the question, whether (4.8) defines representations of a holonomy-flux algebra is moresubtle, because of (4.6): π E (0) ([ Y n ( S ) , [ Y n ′ ( S ) , Y n ′′ ( S )]]) = [ π E (0) ( Y n ( S ) , )[ π E (0) ( Y n ′ ( S )) , π E (0) ( Y n ′′ ( S ))]] (4.11)= π ω ([ E n ( S ) , [ E n ′ ( S ) , E n ′′ ( S )] X ( A ) ] X ( A ) )= π ω ( E [ n, [ n ′ ,n ′′ ]] ( S )) ,π E (0) ([ Y n ( S ) , [ Y n ′ ( S ) , Y n ′′ ( S )]]) = π E (0) ( Y [ n, [ n ′ ,n ′′ ]] ( S )) + π E (0) ( c [ n, [ n ′ ,n ′′ ]] ( S )) (4.12)= π ω ( E [ n, [ n ′ ,n ′′ ]] ( S )) + i Z S ∗ E (0) ([ n, [ n ′ , n ′′ ]]) · H ω + π E (0) ( c [ n, [ n ′ ,n ′′ ]] ( S )) . Therefore, we find, that the Koslowski-Sahlmann representations require the presence of a non-trivial “centralterm” in the higher commutation relations (cp. corollary 3.7): π E (0) ( c [ n, [ n ′ ,n ′′ ]] ( S )) = − i Z S ∗ E (0) ([ n, [ n ′ , n ′′ ]]) · H ω . (4.13)This relation could be easily satisfied by c E (0) [ n, [ n ′ ,n ′′ ]] ( S ) = − i Z S ∗ E (0) ([ n, [ n ′ , n ′′ ]]) · , π E (0) ( c [ n, [ n ′ ,n ′′ ]] ( S )) = π ω ( c E (0) [ n, [ n ′ ,n ′′ ]] ( S )) , (4.14)which satisfies the “co-cycle condition” due to the linearity of the integral and the Jacobi identity of [ . , . ] : g × g → g . But, we would still have to check that there is compatible definition for [ Y n ( S ) , Y n ′ ( S )], and thatthere are no other higher order relations in conflict with it. Moreover, we have to extend the actions of thegauge transformation G saP and automorphisms Aut sa (P) to account for the “central term”. α λ ( c [ n, [ n ′ ,n ′′ ]] ( S )) = c λ⊲ [ n, [ n ′ ,n ′′ ]] ( S ) (4.15) α χ ( c [ n, [ n ′ ,n ′′ ]] ( S )) = c χ ∗ [ n, [ n ′ ,n ′′ ]] ( φ − χ ( S )) , λ ∈ G saP , χ ∈ Aut sa (P) , where we assumed that the actions are natural w.r.t. to the generators Y n ( S ), i.e. identical to those onthe flux vector fields E n ( S ) (see definition 2.29). Unfortunately, this leads to the conclusion that (4.14) and(4.15) are incompatible, i.e. the “central term” cannot be proportional to the unit element. Moreover, theunitary implementers of the gauge transformations and automorphisms in the Ashtekar-Isham-Lewandowski.1 “Central extensions” of holonomy-flux algebras and non-degenerate backgrounds 32representation are not compatible with the choice (4.13), and an extension like (4.15) for generic E (0) : π E (0) ( α λ ( c [ n, [ n ′ ,n ′′ ]] ( S ))) = − i Z S ∗ E (0) ( λ ⊲ [ n, [ n ′ , n ′′ ]]) · H ω (4.16)= − i Z S ∗ ( λ ⊲ E (0) )([ n, [ n ′ , n ′′ ]]) · H ω = U ω ( λ ) (cid:18) − i Z S ∗ E (0) ([ n, [ n ′ , n ′′ ]]) · H ω (cid:19) U ω ( λ ) ∗ = U ω ( λ ) π E (0) ( c [ n, [ n ′ ,n ′′ ]] ( S )) U ω ( λ ) ∗ .π E (0) ( α χ ( c [ n, [ n ′ ,n ′′ ]] ( S ))) = − i Z φ − χ ( S ) ∗ E (0) ( χ ∗ [ n, [ n ′ , n ′′ ]]) · H ω (4.17)= − i Z S ∗ ( χ ∗ E (0) )([ n, [ n ′ , n ′′ ]]) · H ω = U ω ( χ ) (cid:18) − i Z S ∗ E (0) ([ n, [ n ′ , n ′′ ]]) · H ω (cid:19) U ω ( χ ) ∗ = U ω ( χ ) π E (0) ( c [ n, [ n ′ ,n ′′ ]] ( S )) U ω ( χ ) ∗ . The latter issue can be fixed in the same way a proposed in [24], i.e. the unitary implementers intertwinebetween representations with different background. π E (0) ( α λ ( Y n ( S ))) = U ω ( λ ) π λ ∗ E (0) ( Y n ( S )) U ω ( λ ) ∗ , (4.18) π E (0) ( α λ ( c [ n, [ n ′ ,n ′′ ]] ( S ))) = U ω ( λ ) π λ ∗ E (0) ( c [ n, [ n ′ ,n ′′ ]] ( S )) U ω ( λ ) ∗ ,π E (0) ( α χ ( Y n ( S ))) = U ω ( χ ) π χ ∗ E (0) ( Y n ( S )) U ω ( χ ) ∗ , (4.19) π E (0) ( α χ ( c [ n, [ n ′ ,n ′′ ]] ( S ))) = U ω ( χ ) π χ ∗ E (0) ( c [ n, [ n ′ ,n ′′ ]] ( S )) U ω ( χ ) ∗ . The second way of thinking about the Koslowski-Sahlmann representations, which is more along the lines ofsection 3, is offered by the following observation:There is a shift transformation of the holonomy-flux algebra P LQG defined by: ρ E (0) ( E n ( S )) = E n ( S ) + i Z S E (0) ( n ) · , ρ E (0) ( f ) = f. (4.20)It resembles the (classical) moment map problem, i.e. the association of a phase space function with a Hamil-tonian vector field is only unique up to constant terms. By the same argument as before, this transformationis not a *-automorphism of the holonomy-flux algebra P LQG , but only an affine transformation: ρ E (0) ([ E n ( S ) , [ E n ′ ( S ) , E n ′′ ( S )]]) = [ ρ E (0) ( E n ( S )) , [ ρ E (0) ( E n ′ ( S )) , ρ E (0) ( E n ′′ ( S ))]] , (4.21)and we will be forced to introduce “central” elements, if we want it to be a *-isomorphism. Thus, the shifttransformations ρ E (0) can be considered as charge transformations that are already broken on the level of thealgebra P LQG . The Koslowski-Sahlmann representations arise by the identification Y n ( S ) = ρ E (0) ( E n ( S )) andthe use of the Ashtekar-Isham-Lewandowski representation ( π ω , H ω , Ω ω ). In terms of an algebraic state ω ,we have: ω ( f Y n ( S ) ...Y n j ( S j )) = ( µ ( f ) (cid:16) i R S E (0) ( n ) (cid:17) ... (cid:16) i R S j E (0) ( n j ) (cid:17) if { , .., j } = ∅ , (4.22)for every f ∈ Cyl ∞ , Y n ( S ) ...Y n j ( S j ) ∈ X Flux .Let us summarize our findings in this subsection:1. The Koslowski-Sahlmann representations require, at least, the modification of the commutation relationsof the standard holonomy-flux algebra (see definition 2.28) by a “central term” (4.6) to be well-defined(cp. corollary 3.7) w.r.t. the identification Y n ( S ) = E n ( S ). But, it is so far unclear, whether the additionof a “central term” suffices to satisfy all relation imposed by higher order commutators.2. If the extension exists, and the “central term” commutes with the generators Y n ( S ), it has to satisfythe “co-cycle condition” (4.7). The actions of the gauge transformations and automorphisms have to bemodified to account for the presence of the “central term”. If the gauge transformations and automor-.2 Weyl form of the holonomy-flux algebras and non-degenerate backgrounds 33phisms are supposed to act naturally on the generators Y n ( S ), i.e. the actions are identical to thoseon the flux vector fields E n ( S ), the standard unitary implementers of both groups of transformations inthe Koslowski-Sahlmann representations do not implement the modified actions, but intertwine betweendifferent backgrounds.3. The Koslowski-Sahlmann representations can be interpreted as the Ashtekar-Isham-Lewandowski rep-resentation after shifting the generators of P LQG by ρ E (0) . But, the shift transformation is not a *-automorphism, and thus broken on the level of the algebra.4. An important difference between the first and the second point of view is the relation to gauge andautomorphism invariance, because the second perspective does not allow to treat different choices of E (0) as representations of the same generators, i.e. ρ E (0) ( E n ( S )) = ρ E ′ (0) ( E n ( S )) for generic E (0) = E ′ (0) .Although, the various generators are realized in the Ashtekar-Isham-Lewandowski representation, thereis only the standard vacuum Ω ω . Thus, a treatment along the lines of [24] requires the first attitudetowards the Koslowski-Sahlmann representations. Regarding the Weyl form of the holonomy-flux algebra in relation to non-degenerate backgrounds, we will onlycomment on the version defined by Fleischhack in [13]. Koslowski and Sahlmann also consider a different versiongenerated by “exponentials of area operators”, which we will not discuss in this article (cf. [24]).The C ∗ -Weyl algebra defined by Fleischhack is similar to the concrete realization of the algebra A LQG forG = SU (2) via the Ashtekar-Isham-Lewandowski representation, i.e. π ω ( A LQG ) || . || B ( H ω ⊂ B ( H ω ) . (4.23)Especially, both algebras contain the perfect subgroup of Weyl elements π ω ( h W i ) (see proposition 3.2). Thus,by corollary 3.3, there is a severe constraint on the definition of new representations via character automorphisms(3.38) as suggested in [24]. This observation is in accordance with the result of the previous subsection thatthe Koslowski-Sahlmann representations for holonomy-flux algebras cannot be defined for the algebra P LQG .Therefore, it appears to be necessary to look for Weyl forms of the (possibly) modified holonomy-flux algebrasproposed above. On the other hand, there is the second possibility, in analogy with the preceding discussion,to consider the representation of Weyl form A LQG defined by the state ω ( f V S ( n ) ...V S j ( n j )) = µ ( f ) e i R S ∗ E (0) ( n ) · ... · e i R Sj ∗ E (0) ( n j ) , ∀ f ∈ Cyl , V S ( n ) ...V S j ( n j ) ∈ A LQG (4.24)w.r.t. the shifted generators V n ( S ) = e i R S ∗ E (0) ( n ) W n ( S ). The GNS representation realizes the Koslowski-Sahlmann representation with E (0) for A LQG : π ω ( V n ( S )) = e i R S ∗ E (0) ( n ) π ω ( W n ( S )) , π ω ( f ) = π ω ( f ) , (4.25)but again the shift transformation W n ( S ) V n ( S ) is not a *-automorphism of A LQG . θ -vacua in loop quantum gravity In the last section of this article, we would like to present another application of the relation between centraloperators and representation theory (see section 3), and outline a setup for the discussion of chiral symmetrybreaking and occurrence of θ -vacua in the framework of loop quantum gravity. This setup is inspired by andstrongly resembles a discussion of these topics in the setting of algebraic quantum field theory, which was givenby Morchio and Strocchi in [37] (see also [36] for the original account on the ideas involving the topology of thegauge group without the use of semi-classical approximations).Let us briefly, recall the problem of chiral symmetry breaking and the θ -vacuum structure in quantum fieldtheory. If we consider a field theory on Minkowski space M in the temporal gauge given in terms of gauge fieldvariables ( A, E ) chirally coupled to fermion field variables (Ψ , ¯Ψ) (notably the standard model), we will have achiral symmetry associated with the transformation ρ ζ (Ψ) = e ζγ Ψ , ρ ζ ( A ) = A, ρ ζ ( ¯Ψ) = ¯Ψ e ζγ , ρ ζ ( E ) = E, (5.1)where γ ∗ = − γ . If this symmetry were preserved in the quantization of the field theory, we would expect thepresence of associated parity doublets. In the case of the standard model, such parity doublets are missing,and the chiral symmetry is said to be broken. Since the standard model is also missing Goldstone bosons.1 An extension of the algebra A LQG U (1)-problem [68], the solution ofwhich is argued to be the chiral anomaly and its relation to the large gauge transformations [36,69] in standardtreatments. The arguments goes, loosely speaking, as follows [36]:The regularized expression for the symmetry generating axial current j µ = i ¯Ψ γ γ µ Ψ acquires the famous gaugedependent axial anomaly, which is crucial for the theoretical explanation of the π → γγ decay: ∂ µ j µ = − P = − ∂ µ C µ , (5.2)where P denotes the Pontryagin density, which equals the divergence of the Chern-Simons form C µ = − π ε µνρσ tr( F Aνρ A σ − A ν A ρ A σ ) (5.3)Thus, the conserved current J µ = j µ + 2 C µ gives rise to a gauge dependent symmetry generator Q = Z Σ J d x λ Z Σ J + 2 n [ λ ] , λ ∈ G (5.4)and is therefore rejected. Here, n [ λ ] is the winding number of the extension of λ to the 1-point compactification˙ R = R ∪ {∞} = S , i.e. λ : S = ˙ R → G, which is defined by λ ( ∞ ) = 1 G , since λ differs from 1 G only on acompact set.On the other hand, it is argued in [37] that this line of thought is incomplete in view of the results of Bardeen [72],who showed that J µ gives rise to a well-defined symmetry on the observable algebra in perturbation theory inlocal gauges. Furthermore, in [37] Morchio and Strocchi put forward a way to close this gap, which we will arguecould apply in the framework of loop quantum gravity, as well. This is of particular interest in the setting ofdeparametrizing models, which provide an arena for the discussion of the standard model and related theoriesin the context of loop quantum gravity (see [38] for a review), and for which the Ashtekar-Isham-Lewandowskirepresentation can provide the physical Hilbert space.The main ingredients necessary for a discussion of chiral symmetry breaking along the lines of [37], are analgebra of (localized) observables A , containing unitary elements U ( λ ) implementing the (localized) gaugetransformations G , and a 1-parameter group of chiral automorphisms ρ ζ : A → A , interacting non-triviallywith elements associated with large gauge transformation λ, = [ λ ] ∈ π (G)( ∼ = Z in many relevant cases, e.g. SU ( n ) , n ≥ ρ ζ ( U ( λ )) = e − i ζn [ λ ] U ( λ ) . (5.5)In the following, we will argue that, if we assume the existence of a 1-parameter group of automorphisms ofthe form (5.5) for the algebra A LQG (or a slightly extended version of it), we will have all ingredients at ourdisposal. A discussion of the possibility to obtain a chiral symmetry (5.5) in loop quantum gravity will be givenelsewhere. A LQG
We start our discussion with the observation that the algebra A LQG admits an extension by operators U ( λ ) , λ ∈ G sa , , representing the semi-analytic, compactly supported gauge transformations, in the following way: Definition 5.1:
The extension G sa , ⋊ A LQG of A LQG is given along the lines of definition 2.30, but with the additional elements U ( λ ) , λ ∈ G sa , and relations U ( λ ) ∗ = U ( λ − ) , U ( λ ◦ λ ′ ) = U ( λ ′ ) U ( λ ) , (5.6) U ( λ ) f = α λ ( f ) U ( λ ) , U ( λ ) W S ( tn ) = α λ ( W S ( tn )) U ( λ ) , for any f, W S ( tn ) ∈ A LQG . The action of the automorphims
Aut sa (P) extends to this algebra by conjugation onthe gauge transformations G sa , , i.e. α χ ( U ( λ )) = U ( χ − ◦ λ ◦ χ ) . (5.7)Evidently, there is an analogous construction on the basis of the holonomy-flux algebra P LQG , and it is possibleto extend by the automorphisms Aut sa (P) in a similar way. As a simple corollary we have: The expression for Q is heuristic, but there are known strategies to regularize such expressions [70, 71]. .2 Chiral symmetry breaking and θ -vacua for Σ = R & P = R × G 35
Corollary 5.2:
The Ashtekar-Isham-Lewandowski representation ( π ω , H ω , Ω ω ) extends to a representation of G sa , ⋊ A LQG of A LQG , which can be defined by the (algebraic) state: ω ext0 ( f W S ( n ) ...W S ′ ( n ′ ) U ( λ )) = µ ( f ) , ∀ f ∈ Cyl , W S ( n ) ...W S ′ ( n ′ ) , U ( λ ) ∈ G sa , ⋊ A LQG . (5.8) Proof:
This is immediate from the invariance properties of ω .Interestingly, if we extend the algebra A LQG only by the subgroup of gauge transformations close to the identityGauss P = exp G P ( G P ), this will correspond to the inclusion of (smeared) generators of the gauge transformationinto P LQG , and fits with their separate quantization (cf. [4], cp. also (2.61)): G V (Λ) = Z V div AT Σ ( E )(Λ) , Λ ∈ Γ sa0 (Ad(P | V )) , V ⊂ Σ open and semi-analytic . (5.9)Thus, the extension Gauss P ⋊ A LQG appears to be natural from the point of view that the algebra A LQG contains(smeared) functions of the (classical) variables ( A, ˜ E ) ∈ | Λ | T ∗ A P . θ -vacua for Σ = R & P = R × G For the discussion of chiral symmetry breaking in the context of loop quantum gravity it is important to note,that the formalism, recalled here, is capable of treating gravitational and Yang-Mills degrees of freedom at thesame time. At the given structural level, this is reflected in the choice of structure group G. Further differenceswould arise at the level of dynamics and the associated Hamiltonian constraints. In the following, we will notdistinguish between the different types of degrees of freedom, and therefore in principle allow for chiral symmetrybreaking w.r.t. the gravitational degrees of freedom. Furthermore, it is possible to include fermions into thetreatment (cf. [73]), which points out a potential direction to investigate the existence of a gauge dependentchiral symmetry (5.5). Interestingly, the anomaly (5.2) can be generated by a gauge invariant regularizationprocedure by point-split objects like j µ ( e (1) , e (0)) = i ¯Ψ( e (1)) γ γ µ hol Ae Ψ( e (0)) , (5.10)which have natural analogs in the loop quantum gravity framework (cf. [74–76]). But, these objects behave com-plicated w.r.t. general automorphisms Aut sa (P) [73], which might restrict their applicability to deparametrizedmodels.Let us now turn to the mechanism for chiral symmetry breaking in the loop quantum gravity framework. Tosimplify the discussion, we will restrict to a spatial manifold Σ = R and a trivial bundle P = R × G. Thequantum field algebra will be Gauss P ⋊ A LQG or G sa , ⋊ A LQG , and the existence of a chiral symmetry { ρ ζ } ζ ∈ R with the property (5.5) will be assumed in the latter case.This has the important implication, that G sa , ∼ = C sa0 ( R , G). Thus every λ ∈ G sa , determines uniquely a map(see above) g λ : ˙ R = S → G , (5.11)and a homotopy class [ λ ] = [ g λ ] ∈ π (G). From this point on, let us assume that π (G) ∼ = Z , which holds forG = SU ( n ) , n ≥ SO ( n ) , n ≥ , n = 4 . Then, [ λ ] is uniquely determined by the winding numberor instanton number [77] n [ λ ] = 124 π Z R tr( g − λ dg λ ∧ g − λ dg λ ∧ g − λ dg λ ) . (5.12)Gauge transformations λ with n [ λ ] = 0 are called large gauge transformations.Next, we analyze the difference between Gauss P -invariance and gauge invariance for Gauss P ⋊ A LQG . Again, theargument follows Morchio and Strocchi [37], who exploit the localization properties of operators in the quantumalgebra, which is also possible for the algebra Gauss P ⋊ A LQG . Lemma 5.3:
Any
Gauss P -invariant state ω on Gauss P ⋊ A LQG is also gauge invariant, and the large gauge transformations Note that this excludes the case G = Spin , π (Spin ) ∼ = Z × Z , which is important in the treatment of the new variables [1]. .2 Chiral symmetry breaking and θ -vacua for Σ = R & P = R × G 36 are unitarily implemented in the GNS representation ( π ω , H ω , Ω ω ) . Furthermore, any Gauss P -invariant operatorin Gauss P ⋊ A LQG is also gauge invariant.
Proof:
Let λ ∈ C sa0 ( R , G) be a large gauge transformation, and define λ a ( x ) = λ ( x − a ) , x, a ∈ R . Then, λ · λ − a and λ − a · λ are Gauss P transformations. This implies( ω ◦ α λ )( f ) = ( ω ◦ α λ )( α λ − af ( f )) = ( ω ◦ α λ − af · λ )( f ) (5.13)= ω ( f )( ω ◦ α λ )( W S ( tn )) = ( ω ◦ α λ )( α λ − aS,n ( W S ( tn ))) = ( ω ◦ α λ − aS,n · λ )( W S ( tn )) (5.14)= ω ( W S ( tn )) . ( ω ◦ α λ )( U ( λ ′ )) = ( ω ◦ α λ )( α λ − aλ ′ ( U ( λ ′ ))) = ( ω ◦ α λ − aλ ′ · λ )( U ( λ ′ )) (5.15)= ω ( U ( λ ′ )) , where f, W S ( tn ) , U ( λ ′ ) are generators of Gauss P ⋊ A LQG , and we chose a f , a S,n , a λ ′ ∈ R in accordance withthe respective localization regions. The unitary implementability follows from a standard argument. The otherstatement follows from the same argument.The implementers of the (large) gauge transformations are unique up to phases in irreducible (or factorial)representations, i.e. w.r.t. to pure (or primary), Gauss P -invariant states ω , of Gauss P ⋊ A LQG .In view of this result, and corollary 5.2, we will use the algebra G sa , ⋊ A LQG to discuss the spontaneousbreakdown of the chiral symmetry and its relation to the topology of G sa , . To this end, we need a furtherresult concerning the implementers of the (large) gauge transformations. Proposition 5.4 (cp. [37]):
In a GNS representation of a
Gauss P -invariant state with Gauss P -invariant GNS-vacuum π ω ( U ( λ ))Ω ω = Ω ω , λ ∈ Gauss P , (5.16) the implementers π ω ( U ( λ )) of the gauge transformations λ ∈ G sa , are of the form: π ω ( U ( λ ))Ω ω = C ωn [ λ ] Ω ω . (5.17) The non-trivial elements C ωn [ λ ] are central, and belong to the strong closure of π ω ( G sa , ⋊ A LQG ) . Furthermore,we have C ωn [ λ ] C ωn [ λ ′ ] = C ωn [ λ ] + n [ λ ′ ] . (5.18) Proof:
For λ ∈ G sa , , we (densely) define S ω ( λ ) π ω ( O )Ω ω = π ω ( α λ ( O ))Ω ω , O ∈ G sa , ⋊ A LQG . (5.19)Clearly, S ω ( λ ) is isometric on dense subspace of H ω , and extends to an unitary element of B ( H ω ), which wedenote by S ω ( λ ), as well. Then, by the same argument as in lemma 5.3, we have S ω ( λ ) = s-lim | a |→∞ π ω ( U ( λ − a · λ )) . (5.20)This operator has the properties S ω ( λ ) π ω ( O ) S ω ( λ ) ∗ = π ω ( α λ ( O )) , O ∈ G sa , ⋊ A LQG , (5.21) S ω ( λ )Ω ω = Ω ω . These allow us to define C ωn [ λ ] = π ω ( U ( λ )) S ω ( λ ) ∗ , which are central and belong to the strong closure of π ω ( G sa , ⋊ A LQG ). Clearly, the C ωn [ λ ] ’s depend only on the topological quantities n [ λ ] and satisfy (5.18), sincefor any λ, λ ′ ∈ G sa , with n [ λ ] = n [ λ ′ ] the operator π ω ( U ( λ ′ ) ∗ ) π ω ( U ( λ )) represents a Gauss P transformation,which leaves Ω ω invariant. The (general) non-triviality of the elements C ωn [ λ ] follow from (5.5).7The proposition implies that the central elements { C ωn } n ∈ Z represent the quotient G sa , / Gauss P . Similar tothe preceding sections, we find non-trivial, central elements associated with the algebra G sa , ⋊ A LQG , reflectingthe topology of the group of gauge transformations G sa , . The property (5.5) of the chiral automorphisms leadsto their spontaneous breakdown w.r.t. pure (or primary), Gauss P -invariant states, and the appearance of the θ -sectors. Corollary 5.5 (cp. [37]):
Given a pure (or primary),
Gauss P -invariant state ω on G sa , ⋊ A LQG , the chiral automorphisms { ρ ζ } ζ ∈ R arenecessarily spontaneouly broken. Moreover, every such state is labeled by an angle θ ∈ [0 , π ) , C ωn = e i nθ · H ω .The GNS representation of a chirally invariant, Gauss P -invariant state ω ′ admits a central decomposition, w.r.t. C ω ′ H ω ′ = Z [0 ,θ ) H θ dµ ( θ ) , C ω ′ n H θ = e i nθ H θ , (5.22) with translation invariant measure µ . Proof:
Assume that the chiral symmetry is unbroken. Then, we find a 1-parameter group of unitaries { U ω ( ζ ) } ζ ∈ R thatimplements the symmetry by conjugation π ω ( ρ ζ ( O )) = U ω ( ζ ) π ω ( O ) U ω ( ζ ) ∗ , O ∈ G sa , ⋊ A LQG , ζ ∈ R . (5.23)This leads to a unique extension of the ρ ζ ’s to the strong closure of π ω ( G sa , ⋊ A LQG ), and we find by (5.5) and(5.20) (since n [ λ ] = n [ λ a ] ): ρ ζ ( S ω ( λ )) = S ω ( λ ) , ζ ∈ R . (5.24)This implies, again by (5.5) and the definition of C ωn : ρ ζ ( C ωn [ λ ] ) = e − i ζn [ λ ] C ωn [ λ ] , ζ ∈ R , (5.25)which is incompatible with the purity (or primarity) of ω , as this implies irreducibility (or factoriality) of( π ω , H ω , Ω ω ), and thus C ωn [ λ ] = e i n [ λ ] θ · H ω , θ ∈ [0 , π ).The central decomposition (5.22) follows from the observation that (5.24) implies σ ( C ω ′ ) = { e i θ | θ ∈ [0 , π ) } .The unitaries U ω ′ ( ζ ) act as intertwiners between the θ -sectors: U ω ′ ( ζ ) H θ = H θ − ζ mod π . (5.26) To conclude the article, we comment on our findings in the various sections, and offer some future perspectives.Section 2 mainly provided a review of the mathematical structures behind the (canonical) formulation of loopquantum gravity with two exceptions: Equation (2.98), which states an algebraic relation among the fluxvector fields that affects the representation theory of the holonomy-flux algebra P LQG in a non-trivial way (seesection 4), and lemma 2.32, which shows that Hilbert space representations of P LQG and its Weyl form A LQG induced by gauge invariant states ω are necessarily discontinuous w.r.t. to the spin network functions, i.e. thetwo-point function “ ω ( A ( x ) A ( y ))” of the “quantum connection” A cannot exist in such representations. Thelatter result is in accordance with results in of quantum field theory in the temporal gauge [33], where the onlyalternative appears to be the use of (non-positive) Krein space representations, e.g. the Feynman-Gupta-Bleulerquantization of QED. Thus, it would be interesting, whether such an alternative is possible in loop quantumgravity, as well, and how it connects to the standard approach.In section 3, we focused on aspects of the representation theory of A LQG with an emphasis on the presenceof non-trivial central operators, and their relation to topological and geometrical structures of the structuregroup G. We found, that a non-trivial first homotopy group π (G), supplemented by an associated bundlecovering, can be related to the existence of a field algebra extension F LQG , that can be used to generate new,inequivalent representations from existing ones with the help of charge automorphisms defined by the adjointaction of unitary, charged fields. While this construction works well for Abelian structure groups, where it offersa new perspective on the Koslowski-Sahlmann representations and the ε -sectors of loop quantum cosmology,it is accompanied by further difficulties in the non-Abelian case, which are due to restrictive topological and8geometrical properties of G (see proposition 3.2 & 3.5). Especially, there might exist no suitable unitary, chargedfields in the extension F LQG to define charge automorphisms. In the future, it could be gratifying to investigatethe algebraic structure of A LQG resp. P LQG on a deeper level, e.g. its structure of ideals, its universal envelopingvon Neumann algebra etc., to improve control on the representation theory and the possible dynamics supportedby the algebra. Especially, in view of the deparametrizing models (see [38] for an overview), where A LQG and P LQG become algebras of elementary observables, instead of purely kinematical objects, such an analysis willoffer immediate insight into physical questions.We continued our analysis of the Koslowski-Sahlmann representations, started in section 3 for Abelian structuregroups, in section 4, where we concentrated on the non-Abelian case. We showed, that the general line of thought,which places these representations into the framework of section 3, bifurcates for non-Abelian structure groups,and one is left with two possible interpretations:1. The Koslowski-Sahlmann representations are defined for an (centrally) extended algebra, and the elemen-tary operators Y n ( S ) are identified with the fluxes E n ( S ).2. The Koslowski-Sahlmann representations are defined for the holonomy-flux algebra, but the elementaryoperators Y n ( S ) are identified with shifted fluxes E n ( S ) + i R S ∗ E (0) ( n ).This bifurcation is explained by the fact that the shift transformation ρ E (0) : E n ( S ) E n ( S ) + i R S ∗ E (0) ( n ) isnot a *-automorphism of P LQG , but only an affine transformation, in the non-Abelian setting. Thus, the firstpoint of view represents the idea to define a modified holonomy-flux algebra P E (0) LQG , s.t. ρ E (0) : P E (0) LQG → P LQG becomes a *-isomorphism, while the second point of view changes the interpretation of the elementary operators Y n ( S ) of the Koslowski-Sahlmann framework. Clearly, the second option avoids the obstruction posed by corol-lary 3.7, and shows that the Koslowski-Sahlmann representations reduce to the Ashtekar-Isham-Lewandowskirepresentation for the shifted fluxes, but it forbids the treatment of gauge and automorphism invariance alongthe lines of [24], as well (see the summary at the end of section 4). The first option, which offers a richermathematical structure, suffers from the fact that the “central extension” of the holonomy-flux algebra is onlya necessary ingredient, but probably not sufficient due to further higher order commutation relations imposedby the basic commutation rule [ Y n ( S ) , f ] = E n ( S ) · f. (6.1)Thus, this approach is weakened, because control on all higher order relation appears to be out of reach at thepresent stage.Nevertheless, it is interesting to analyse the recent work on the Koslowski-Sahlmann representation, which isfocused on the implementation of diffeomorphisms and possible applications to asymptotically flat scenarios[26–31], in view of our findings. Especially, in [28] it has been pointed out that it is possible to introducea slightly modified algebra, A BLQG , called holonomy-background exponential-flux algebra, which admits theKoslowski-Sahlmann representation as a true representation. The latter is possible because of a modification ofthe generators of A BLQG in comparison to A LQG , not only involving the fluxes, E n ( S ), but also the cylindricalfunctions, which are made background dependent, E (0) , by means of so-called background exponentials β E (0) ( A ) = e i R Σ E ( A ) . (6.2)These background exponentials lead to an additional U (1) N -factor accompanying the structure group G , onwhich the cylindrical functions are based ( N is the number of background fields). Thus, the modification ofthe fluxes can be realised by additional derivations that act on the U (1) N -factors, avoiding our corollary 3.7 oncentral extensions of A LQG . Put differently, the additional U (1) N -structure resolves the obstruction posed by(4.5), rendering it invalid in A BLQG .Finally, in section 5, we applied the general formalism of section 3 to adapt the discussion of chiral symmetrybreaking and θ -vacua by Morchio and Strocchi [37] to the framework of loop quantum gravity. We showed thatunder the assumption of an anomalous, chiral symmetry (5.5) this adaption is possible, and has some of theexpected properties (a discussion of the Goldstone spectrum of the generator of the chiral symmetry is missing).Our analysis is intended to stimulate the discussion of gauge anomalies in loop quantum gravity, especiallyin the matter sector, because anomalies have important physical consequences for the matter content of thestandard model. Thinking of the semi-classical limit of loop quantum gravity, it is necessary to make contactwith the predictions of quantum field theory, and to offer an explanation of the consequences of anomalies inthe latter, e.g. the solution of the U (1)-problem and the restriction of matter to so-called safe representations.Thus, in spite of the fact that an anomaly like (5.5) appears to be a rather strong requirement, we wouldexpect that a structure of this type arises in loop quantum gravity, at least in a limiting sense connected tothe aforesaid semi-classical limit. A natural starting point for an investigation, of how anomalies could occurin loop quantum gravity, is suggested by symmetry generating currents of the form (5.10), which, on the one9hand, are natural objects in the framework of loop quantum gravity and, on the other hand, are the centralobjects in the study of anomalies in quantum field theory. More precisely, an understanding of the coincidencelimit of these point-split currents, possibly in combination with a semi-classical limit, could offer first insights.At a preliminary stage, it might be easiest to consider these objects in the context of deparametrizing models(see [38] for an overview), which avoid complications due to the diffeomorphism and Hamiltonian constraints.A further simplification might be achieved, if the discussion was restricted to cosmological or other symmetryreduced models. We thank Norbert Bodendorfer, Detlev Buchholz and Hanno Sahlmann for helpful comments and suggestions.Furthermore, we thank Stefan Hollands for pointing out the importance of anomalies in the relation to quantumfield theory to one of us. AS gratefully acknowledges financial support by the Ev. Studienwerk e.V.. This workwas supported in parts by funds from the Friedrich-Alexander-University, in the context of its Emerging FieldInitiative, to the Emerging Field Project “Quantum Geometry”. [1] N. Bodendorfer, T. Thiemann, A. Thurn, New variables for classical and quantum gravity in all dimensions: I. Hamiltonian analysis,Classical and Quantum Gravity 30 (4) (2013) 045001. doi:10.1088/0264-9381/30/4/045001 .URL http://iopscience.iop.org/0264-9381/30/4/045001 [2] N. Bodendorfer, T. Thiemann, A. Thurn, New variables for classical and quantum gravity in all dimensions: II. Lagrangian analysis,Classical and Quantum Gravity 30 (4) (2013) 045002. doi:10.1088/0264-9381/30/4/045002 .URL http://iopscience.iop.org/0264-9381/30/4/045002 [3] C. Rovelli, Quantum Gravity, Cambridge Monographs on Mathematical Physics, Cambridge UniversityPress, 2007.URL [4] T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge Monographs on MathematicalPhysics, Cambridge University Press, 2008.URL [5] T. Thiemann, Quantum spin dynamics (QSD), Classical and Quantum Gravity 15 (4) (1998) 839. doi:10.1088/0264-9381/15/4/011 .URL http://iopscience.iop.org/0264-9381/15/4/011 [6] T. Thiemann, Quantum spin dynamics (QSD): II. The kernel of the Wheeler - DeWitt constraint operator,Classical and Quantum Gravity 15 (4) (1998) 875. doi:10.1088/0264-9381/15/4/012 .URL http://iopscience.iop.org/0264-9381/15/4/012 [7] T. Thiemann, Quantum spin dynamics (QSD): III. Quantum constraint algebra and physical scalar product in quantum general relativity,Classical and Quantum Gravity 15 (5) (1998) 1207. doi:10.1088/0264-9381/15/5/010 .URL http://iopscience.iop.org/0264-9381/15/5/010 [8] T. Thiemann, Quantum spin dynamics (QSD): IV. Euclidean quantum gravity as a model to test Lorentzian quantum gravity,Classical and Quantum Gravity 15 (5) (1998) 1249. doi:10.1088/0264-9381/15/5/011 .URL http://iopscience.iop.org/0264-9381/15/5/011 [9] T. Thiemann, Quantum spin dynamics (QSD): V. Quantum gravity as the natural regulator of the Hamiltonian constraint of matter quantum field theories,Classical and Quantum Gravity 15 (5) (1998) 1281. doi:10.1088/0264-9381/15/5/012 .URL http://iopscience.iop.org/0264-9381/15/5/012 [10] T. Thiemann, Quantum spin dynamics (QSD): VI. Quantum Poincaré algebra and a quantum positivity of energy theorem for canonical quantum gravity,Classical and Quantum Gravity 15 (6) (1998) 1463. doi:10.1088/0264-9381/15/6/005 .URL http://iopscience.iop.org/0264-9381/15/6/005 [11] T. Thiemann, Quantum spin dynamics (QSD): VII. Symplectic structures and continuum lattice formulations of gauge field theories,Classical and Quantum Gravity 18 (17) (2001) 3293. doi:10.1088/0264-9381/18/17/301 .URL http://iopscience.iop.org/0264-9381/18/17/301 doi:10.1007/s00220-006-0100-7 .URL http://link.springer.com/article/10.1007%2Fs00220-006-0100-7 [13] C. Fleischhack, Representations of the Weyl algebra in Quantum Geometry, Communications in Mathe-matical Physics 285 (1) (2009) 67–140. doi:10.1007/s00220-008-0593-3 .URL http://link.springer.com/article/10.1007/s00220-008-0593-3 [14] A. Ashtekar, J. Lewandowski, Quantum theory of geometry: I. Area operators, Classical and QuantumGravity 14 (1A) (1997) A55. doi:10.1088/0264-9381/14/1A/006 .URL http://iopscience.iop.org/0264-9381/14/1A/006 [15] A. Ashtekar, J. Lewandowski, Quantum theory of geometry. II. Volume operators, Advances in Theoreti-cal and Mathematical Physics 2 (1) (1998) 388–429.URL http://intlpress.com/site/pub/pages/journals/items/atmp/content/vols/0002/index.html [16] A. Ashtekar, A. Corichi, J. A. Zapata, Quantum theory of geometry: III. Non-commutativity of Riemannian structures,Classical and Quantum Gravity 15 (10) (1998) 2955. doi:10.1088/0264-9381/15/10/006 .URL http://iopscience.iop.org/0264-9381/15/10/006 [17] A. Ashtekar, C. J. Isham, Representations of the holonomy algebras of gravity and nonAbelian gauge theories,Classical and Quantum Gravity 9 (6) (1992) 1433. doi:10.1088/0264-9381/9/6/004 .URL http://iopscience.iop.org/0264-9381/9/6/004 [18] A. Ashtekar, J. Lewandowski, Differential geometry on the space of connections via graphs and projective limits,Journal of Geometry and Physics 17 (3) (1995) 191–230. doi:10.1016/0393-0440(95)00028-G .URL [19] M. C. Abbati, A. Manià, On differential structure for projective limits of manifolds, Journal of Geometryand Physics 29 (1–2) (1999) 35–63. doi:10.1016/S0393-0440(98)00030-8 .URL [20] H. Sahlmann, T. Thiemann, On the superselection theory of the Weyl algebra for diffeomorphism invariant quantum gauge theories,arXiv preprint gr-qc/0302090.URL http://arxiv.org/abs/gr-qc/0302090 [21] M. Varadarajan, Towards new background independent representations for loop quantum gravity, Classi-cal and Quantum Gravity 25 (10) (2008) 105011. doi:10.1088/0264-9381/25/10/105011 .URL http://iopscience.iop.org/0264-9381/25/10/105011 [22] M. Dziendzikowski, A. Okołów, New diffeomorphism invariant states on a holonomy-flux algebra, Classicaland Quantum Gravity 27 (22) (2010) 225005. doi:10.1088/0264-9381/27/22/225005 .URL http://iopscience.iop.org/0264-9381/27/22/225005 [23] H. Sahlmann, On loop quantum gravity kinematics with a non-degenerate spatial background, Classicaland Quantum Gravity 27 (22) (2010) 225007. doi:10.1088/0264-9381/27/22/225007 .URL http://iopscience.iop.org/0264-9381/27/22/225007 [24] T. A. Koslowski, H. Sahlmann, Loop Quantum Gravity Vacuum with Nondegenerate Geometry, Sym-metry, Integrability and Geometry: Methods and Applications 8 (026). arXiv:1109.4688 , doi:10.3842/SIGMA.2012.026 .URL [25] T. A. Koslowski, Dynamical Quantum Geometry (DQG Programme), arXiv:0709.3465.URL http://arxiv.org/abs/0709.3465 [26] M. Varadarajan, The generator of spatial diffeomorphisms in the Koslowski–Sahlmann representation,Classical and Quantum Gravity 30 (17) (2013) 175017. doi:10.1088/0264-9381/30/17/175017 .URL http://iopscience.iop.org/0264-9381/30/17/175017 [27] M. Campiglia, M. Varadarajan, The Koslowski–Sahlmann representation: gauge and diffeomorphism invariance,Classical and Quantum Gravity 31 (7) (2014) 075002. doi:10.1088/0264-9381/31/7/075002 .URL http://iopscience.iop.org/0264-9381/31/7/075002 [28] M. Campiglia, M. Varadarajan, The Koslowski–Sahlmann representation: quantum configuration space,Classical and Quantum Gravity 31 (17) (2014) 175009. doi:10.1088/0264-9381/31/17/175009 .URL http://iopscience.iop.org/0264-9381/31/17/175009 http://arxiv.org/abs/1412.5527 [30] S. Sengupta, Quantum geometry with a nondegenerate vacuum: a toy model, Physical Review D 88 (6)(2013) 064016. doi:10.1103/PhysRevD.88.064016 .URL http://prd.aps.org/abstract/PRD/v88/i6/e064016 [31] S. Sengupta, Asymptotic flatness and quantum geometry, Classical and Quantum Gravity 31 (8) (2014)085005. doi:10.1088/0264-9381/31/8/085005 .URL http://iopscience.iop.org/0264-9381/31/8/085005/ [32] F. Strocchi, An introduction to the mathematical structure of quantum mechanics: a short course for mathematicians,2nd Edition, Vol. 28 of Advanced Series in Mathematical Physics, World Scientific, 2008.URL [33] J. Löffelholz, G. Morchio, F. Strocchi, Mathematical structure of the temporal gauge in quantum electrodynamics,Letters in Mathematical Physics 44 (11) (2003) 5095–5107. doi:10.1063/1.1603957 .URL http://scitation.aip.org/content/aip/journal/jmp/44/11/10.1063/1.1603957 [34] F. Strocchi, An Introduction to Non-perturbative Foundations of Quantum Field Theory, Vol. 158 of In-ternational Series of Monographs on Physics, Oxford University Press, 2013.URL http://ukcatalogue.oup.com/product/9780199671571.do [35] G. Morchio, F. Strocchi, Quantum mechanics on manifolds and topological effects, Letters in Mathemati-cal Physics 82 (2-3) (2007) 219–236. doi:10.1007/s11005-007-0188-5 .URL http://link.springer.com/article/10.1007/s11005-007-0188-5 [36] R. W. Jackiw, Topological investigations of quantized gauge theories, in: S. B. Treiman, R. W. Jackiw,B. Zumino, E. Witten (Eds.), Current Algebra and Anomalies, Princeton Series in Physics, World Scientific,1985, pp. 211–359.URL [37] G. Morchio, F. Strocchi, Chiral symmetry breaking and θ vacuum structure in QCD, Annals of Physics324 (10) (2009) 2236–2254. doi:10.1016/j.aop.2009.07.005 .URL [38] K. Giesel, T. Thiemann, Scalar Material Reference Systems and Loop Quantum Gravity, arXiv:1206.3807.URL http://arxiv.org/abs/1206.3807 [39] A. Ashtekar, P. Singh, Loop quantum cosmology: a status report, Classical and Quantum Gravity 28 (21)(2011) 213001. arXiv:1108.0893 , doi:10.1088/0264-9381/28/21/213001 .URL http://iopscience.iop.org/0264-9381/28/21/213001/ [40] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Volume I, Wiley Classics Library, JohnWiley & Sons, 1963.URL http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471157333.html [41] R. L. Bishop, R. J. Crittenden, Geometry of Manifolds, Vol. 15 of Pure and Applied Mathematics, AcademicPress, 1964.[42] R. Giles, Reconstruction of gauge potentials from Wilson loops, Physical Review D 24 (8) (1981) 2160. doi:10.1103/PhysRevD.24.2160 .URL http://journals.aps.org/prd/abstract/10.1103/PhysRevD.24.2160 [43] D. Bleecker, Gauge Theory and Variational Principles, no. 1 in Advanced Graduate-level Text, Addison-Wesley, 1981.[44] B. Bahr, T. Thiemann, Automorphisms in loop quantum gravity, Classical and Quantum Gravity 26 (23)(2009) 235022. doi:10.1088/0264-9381/26/23/235022 .URL http://iopscience.iop.org/0264-9381/26/23/235022 [45] A. Ashtekar, J. Lewandowski, Representation Theory of Analytic Holonomy C*-algebras, in: J. C. Baez(Ed.), Knots and Quantum Gravity, Oxford University Press, 1994, Ch. Representation Theory of AnalyticHolonomy C*-algebras, pp. 21–62.URL http://math.ucr.edu/home/baez/kqg.html doi:10.1007/BF00761713 .URL http://link.springer.com/article/10.1007%2FBF00761713 [47] D. Marolf, J. M. C. Mourão, On the support of the Ashtekar-Lewandowski measure, Communications inMathematical Physics 170 (3) (1995) 583–605. doi:10.1007/BF02099150 .URL http://link.springer.com/article/10.1007%2FBF02099150 [48] A. Ashtekar, J. Lewandowski, Projective techniques and functional integration, Journal of MathematicalPhysics 36 (5) (1995) 2170–2191. doi:10.1063/1.531037 .URL http://scitation.aip.org/content/aip/journal/jmp/36/5/10.1063/1.531037 [49] M. C. Abbati, R. Cirelli, A. Mania, P. W. Michor, The Lie group of automorphisms of a principle bundle,Journal of Geometry and Physics 6 (2) (1989) 215–235. doi:10.1016/0393-0440(89)90015-6 .URL [50] I. Kolár, J. Slovák, P. W. Michor, Natural Operations in Differential Geometry, Springer, 1993. doi:10.1007/978-3-662-02950-3 .URL [51] J. M. Velhinho, Functorial aspects of the space of generalized connections, Modern Physics Letters A20 (17n18) (2005) 1299–1303. doi:10.1142/S0217732305017767 .URL [52] M. Bojowald, Mathematical structure of loop quantum cosmology: Homogeneous models, Symmetry, In-tegrability and Geometry: Methods and Applications 9 (082). arXiv:1206.6088 .URL http://dx.doi.org/10.3842/SIGMA.2013.082 [53] T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups, Vol. 98 of Graduate Texts in Mathe-matics, Springer, 1985.URL [54] C. Bär, N. Ginoux, F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lecturesin Mathematics and Physics, European Mathematical Society Publishing House, 2007. doi:10.4171/037 .URL [55] J. Löffelholz, G. Morchio, F. Strocchi, Spectral stochastic processes arising in quantum mechanical models with a non- L ground state,Letters in Mathematical Physics 35 (3) (1995) 251–262. doi:10.1007/BF00761297 .URL http://link.springer.com/article/10.1007/BF00761297 [56] A. Ashtekar, M. Campiglia, On the uniqueness of kinematics of loop quantum cosmology, Vol. 29, 2012,p. 242001. doi:10.1088/0264-9381/29/24/242001 .URL http://iopscience.iop.org/0264-9381/29/24/242001 [57] W. Kamiński, J. Lewandowski, The flat FRW model in LQC: self-adjointness, Classical and QuantumGravity 25 (3) (2008) 035001. doi:10.1088/0264-9381/25/3/035001 .URL http://iopscience.iop.org/0264-9381/25/3/035001 [58] A. Ashtekar, T. Pawlowski, P. Singh, Quantum Nature of the Big Bang, Physical Review Letters 96 (14)(2006) 141301. doi:10.1103/PhysRevLett.96.141301 .URL http://prl.aps.org/abstract/PRL/v96/i14/e141301 [59] O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1: C ∗ -and W ∗ -Algebras, Symmetry Groups, Decomposition of States, 2nd Edition, Texts and Monographs in Physics,Springer, 1987.[60] F. Acerbi, G. Morchio, F. Strocchi, Infrared singular fields and nonregular representations of canonical commutation relation algebras,Journal of Mathematical Physics 34 (3) (1993) 899. doi:10.1063/1.530200 .URL http://scitation.aip.org/content/aip/journal/jmp/34/3/10.1063/1.530200 [61] F. Acerbi, G. Morchio, F. Strocchi, Theta vacua, charge confinement and charged sectors from nonregular representations of CCR algebras,Letters in Mathematical Physics 27 (1) (1993) 1–11. doi:10.1007/BF00739583 .URL http://link.springer.com/article/10.1007/BF00739583 [62] M.-L. Michelson, H. B. Lawson Jr., Spin geometry, Vol. 38 of Princeton Mathematical Series, PrincetonUniversity Press, 1989.URL http://press.princeton.edu/titles/4573.html http://press.princeton.edu/titles/1973.html [64] G. E. Bredon, Topology and Geometry, Vol. 139 of Graduate Texts in Mathematics, Springer, 1993.URL [65] M. Gotô, A Theorem on compact semi-simple groups, Journal of the Mathematical Society of Japan 1 (3)(1949) 270–272. doi:10.2969/jmsj/00130270 .URL [66] J. M. C. Mourão, T. Thiemann, J. M. Velhinho, Physical properties of quantum field theory measures,Journal of Mathematical Physics 40 (5) (1999) 2337–2353. doi:10.1063/1.532868 .URL http://link.aip.org/link/?JMAPAQ/40/2337/1 [67] D. Buchholz, The physical state space of quantum electrodynamics, Communications in MathematicalPhysics 85 (1) (1982) 49–71. doi:10.1007/BF02029133 .URL http://link.springer.com/article/10.1007/BF02029133 [68] S. Weinberg, The U (1) problem, Physical Review D 11 (12) (1975) 3583. doi:10.1103/PhysRevD.11.3583 .URL http://prd.aps.org/abstract/PRD/v11/i12/p3583_1 [69] G. t’Hooft, Symmetry Breaking through Bell-Jackiw Anomalies, Physical Review Letters 37 (1) (1976) 8–11. doi:10.1103/PhysRevLett.37.8 .URL http://prl.aps.org/abstract/PRL/v37/i1/p8_1 [70] B. Schroer, P. Stichel, Current commutation relations in the framework of general quantum field theory,Communications in Mathematical Physics 3 (4) (1966) 258–281. doi:10.1007/BF01649524 .URL http://link.springer.com/article/10.1007/BF01649524 [71] G. Morchio, F. Strocchi, Charge density and electric charge in quantum electrodynamics, Journal of Math-ematical Physics 44 (12) (2003) 5569–5587. doi:10.1063/1.1623928 .URL http://link.aip.org/link/?jmp/44/5569/1 [72] W. A. Bardeen, Anomalous currents in gauge field theories, Nuclear Physics B 75 (2) (1974) 246–258. doi:10.1016/0550-3213(74)90546-X .URL [73] T. Thiemann, Kinematical Hilbert spaces for fermionic and Higgs quantum field theories, Classical andQuantum Gravity 15 (6) (1998) 1487. doi:10.1088/0264-9381/15/6/006 .URL http://iopscience.iop.org/0264-9381/15/6/006 [74] H. A. Morales Técotl, C. Rovelli, Loop space representation of quantum fermions and gravity, NuclearPhysics B 451 (1) (1995) 325–361. doi:10.1016/0550-3213(95)00343-Q .URL [75] H. A. Morales Técotl, C. Rovelli, Fermions in quantum gravity, Physical Review Letters 72 (23) (1994)3642. doi:10.1103/PhysRevLett.72.3642 .URL http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.72.3642 [76] J. C. Baez, K. V. Krasnov, Quantization of diffeomorphism-invariant theories with fermions, Journal ofMathematical Physics 39 (3) (1998) 1251–1271. doi:10.1063/1.532400 .URL http://scitation.aip.org/content/aip/journal/jmp/39/3/10.1063/1.532400 [77] D. Chruściński, A. Jamiołkowski, Geometric phases in classical and quantum mechanics, Vol. 36 ofProgress in Mathematical Physics, Birkhauser, 2004. doi:10.1007/978-0-8176-8176-0 .URL.URL