On Spectral Triples in Quantum Gravity I
aa r X i v : . [ h e p - t h ] F e b On Spectral Triples in Quantum Gravity I
Johannes
Aastrup a , Jesper Møller Grimstrup b & Ryszard Nest c a SFB 478 ”Geometrische Strukturen in der Mathematik”Hittorfstr. 27, D-48149 M¨unster, Germany b The Niels Bohr InstituteBlegdamsvej 17, DK-2100 Copenhagen, Denmark c Matematisk InstitutUniversitetsparken 5, DK-2100 Copenhagen, Denmark
Abstract
This paper establishes a link between Noncommutative Geometryand canonical quantum gravity. A semi-finite spectral triple over aspace of connections is presented. The triple involves an algebra ofholonomy loops and a Dirac type operator which resembles a globalfunctional derivation operator. The interaction between the Dirac op-erator and the algebra reproduces the Poisson structure of GeneralRelativity. Moreover, the associated Hilbert space corresponds, up toa discrete symmetry group, to the Hilbert space of diffeomorphism in-variant states known from Loop Quantum Gravity. Correspondingly,the square of the Dirac operator has, in terms of canonical quantumgravity, the form of a global area-squared operator. Furthermore, thespectral action resembles a partition function of Quantum Gravity.The construction is background independent and is based on an in-ductive system of triangulations. This paper is the first of two paperson the subject. email: [email protected] email: [email protected] email: [email protected] ontents K i . . . . . . . . . . 25 p B n , D n , H n q A ˜ and generalised connections . . . . . . . . . . . 304.3 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . 324.4 The commutator between D ˜ and the algebra B ˜ . . . . . . . . 334.5 The role of the sequence t a n u . . . . . . . . . . . . . . . . . . 35 A a . . . . . . . . . . . . . . . . 446.1.2 Actions of the diffeomorphism group . . . . . . . . . . 456.2 Comparing the spaces A ˜ and A a . . . . . . . . . . . . . . . . 45 Including the sequences t a n u as dynamical variables 53 t a i u . . . . . . . . . . . . . 549.2 A spectral triple over R . . . . . . . . . . . . . . . . . . . . . 569.3 θ -summability . . . . . . . . . . . . . . . . . . . . . . . . . . . 599.4 The U p q -case . . . . . . . . . . . . . . . . . . . . . . . . . . . 599.5 The triple p B t , D t , H t q . . . . . . . . . . . . . . . . . . . . . . 61
10 Distances on A ˜ p △ q and the question of constraints . . . . . . . 6711.3 Background independence . . . . . . . . . . . . . . . . . . . . 6811.4 Additional degrees of freedom . . . . . . . . . . . . . . . . . . 69
12 Conclusion and outlook 71A A spectral triple without the basepoint 77B On diffeomorphism invariance 78C Symmetric states 80
Ever since the discovery of the Standard Model of Particle Physics physicistshave worked to understand the apparently arbitrary structure of this the-ory. Natures choice of gauge group, the Higgs sector, the 20-30 apparentlyunrelated parameters etc. almost begs for a deeper explanation.With the pioneering work of Alain Connes and co-workers on the StandardModel [1]-[8] such an explanation now appears to emerge. In Connes’ workthe Standard Model coupled to General Relativity is expressed as a singlegravitational theory. The language used for this unification is Noncommu-tative Geometry [1]. Within this framework, and under a few mathematicalassumptions, the Standard Model coupled to gravity can be shown to bealmost unique [7, 8].Noncommutative Geometry is based on the result [1, 9] that Riemannianspin geometry has an equivalent formulation in terms of commutative (cid:6) -3lgebras and Dirac operators. In this formulation it is the Dirac operator thatcarries metric information of the underlying manifold which is now writtenas the spectrum of the (cid:6) -algebra. In total, a Riemannian spin geometry canbe described in terms of a spectral triple which is the collection p B, H, D q ofthe algebra B , the Dirac operator D and the Hilbert space H which carriesa representation and action of B and D . To obtain equivalence the triple isrequired to satisfy a set of axioms of Noncommutative Geometry.The language of spectral triples has a natural generalisation which in-cludes also noncommutative (cid:6) -algebras and corresponding Dirac type opera-tors. It is this generalisation which leads to the aforementioned formulationof the Standard Model. It turns out that the classical action of the StandardModel coupled to the Einstein-Hilbert action emerges from a spectral actionprinciple applied to a specific spectral triple [3, 4, 5]. This triple involvesan almost commutative (cid:6) -algebra which is an algebra that factorises into acommutative part times a matrix factor. This means that the classical actiondescribing all fundamental physics emerges from an asymptotic expansionof the spectral action Tr φ p ˜ D { λ q (cid:16) ¸ λ n c n . Here ˜ D denotes the Dirac type operator D subjected to certain inner fluc-tuations stemming from the noncommutativity of the algebra. Without thematrix factor, that is, for a commutative (cid:6) -algebra, the same expression leadsto the Einstein-Hilbert action alone. Thus, it is the inclusion of noncommuta-tive (cid:6) -algebras in the language of spectral triples that permits the formulationof all fundamental forces and particles in terms of pure gravity.This success of Noncommutative Geometry as a framework to describefundamental physics in a unified manner raises, however, a fundamental ques-tion regarding quantization. The Standard Model by itself is a quantumfield theory. However, in its noncommutative formulation it arises as anintegrated part of a purely gravitational theory. This theory is essentiallyclassical. Quantum Field Theory enters the construction in a secondary stepwhere the spectral action has been expanded in a gravitational sector involv-ing only the metric field and a matter sector including all the fermionic andbosonic fields of the Standard Model. Quantization is applied to the lattersector only.So this is the question: how does the quantization procedure of QuantumField Theory fit into the language of Noncommutative Geometry? Since the Here we give the bosonic part only. SU p q . This means that the space of connections itselfis a pro-manifold, the projective limit of manifolds. This, in turn, permits aformulation of the quantization procedure on the level of finite graphs whosecomplexity are subsequently increased infinitely. This construction is due toAshtekar and Lewandowski [18, 19].In this paper we aim to construct a model which involves elements ofboth Noncommutative Geometry and Quantum Gravity. We use two centralelements of Loop Quantum Gravity to obtain such a model. First, the fact5hat gravity can be formulated in terms of Wilson loops leads us to considera spectral triple which involves an algebra of holonomy loops. These loopvariables serve as functions on an underlying space of connections. Second,we wish to exploit the pro-manifold structure of the space of connections todescribe the algebra of loops and to construct a Dirac type operator on thespace of connections. This means that we aim to construct a spectral tripleover each manifold associated to graphs in the projective system. Thesespectral triples are required to be compatible with all embeddings betweengraphs. This requirement will ensure that the limit where the complexity ofgraphs is increased infinitely gives rise to a limit spectral triple.This program was initiated first in [20] (see also [21]) where the authorsattempted to construct such a spectral triple. There the authors found thatthe inductive system of graphs used in Loop Quantum Gravity, which is thesystem of all piecewise analytic graphs, is too large to permit a Dirac typeoperator on the space of connections. Technically, the multitude of possibleembedding of different graphs was found to be too large for a Dirac typeoperator compatible with all embeddings to exist.In the present paper we return to this problem to consider now differentsystems of embedded graphs. In particular, we study the countable system ofembedded graphs given by a triangulation and its barycentric subdivisions.It turns out that this restricted system of graphs does permit a Dirac typeoperator on the associated projective system of manifolds. Furthermore, wefind that the limit of infinitely many barycentric subdivisions gives us anaccurate description of the full space of connections as well as the associatedalgebra of holonomy loops. The construction is general and only assumes thegauge group G to be compact.What we obtain is the following. Given a triangulation T and a compactLie group G , we construct a spectral triple p B ˜ , D ˜ , H ˜q , (1)where B ˜ is the (cid:6) -algebra of holonomy loops obtained via the inductive systemof triangulations. The algebra is represented on the separable Hilbert space H ˜ which carries an action of the Dirac type operator D ˜ . If we denote by A the space of smooth connections in a trivial principal bundle M (cid:2) G , where M is a manifold which corresponds to the triangulation T , then we findthat A is densely contained in the pro-manifold associated to the algebra B ˜ .This means that the spectral triple (1) is a geometrical construction over thespace of connections. 6he construction of the spectral triple (1) depends, as mentioned, cru-cially on the choice of graphs. This choice is closely related to the groupof diffeomorphisms acting on the Hilbert space H ˜ . We find that the choiceof a restricted system of graphs amounts to a type of gauge fixing of thediffeomorphism group. Thus, the Hilbert space H ˜ does not carry an actionof any smooth diffeomorphisms. Rather, it carries an action of a discretegroup of diffeomorphisms associated to the inductive system of graphs. Thismeans that the construction reduces the diffeomorphism group to a countablegroup.The spectral triple (1) has a clear interpretation in terms of a non-perturbative, background independent Quantum Field Theory related togravity. First of all, since the triple exists over a space of connections theDirac type operator D ˜ should be interpreted as a global functional deriva-tion operator. Also, the Hilbert space H ˜ has an inner product which involvesa functional integral. Next, we find that the interaction between the Diractype operator D ˜ and the loop algebra B ˜ reproduces the Poisson structureof General Relativity. Furthermore, the Hilbert space H ˜ is found to bedirectly related to the Hilbert space of (spatial) diffeomorphism invariantstates known from Loop Quantum Gravity [23]. The difference between thetwo is given by the group of discrete diffeomorphisms acting on H ˜ . Thus, weinterpret the Hilbert space H ˜ in terms of a partial solution to the (spatial)diffeomorphism constraint.The square of the Dirac type operator D ˜ has the form of an integral overthe underlying manifold M . The integrand is a quantity which, in terms ofcanonical quantum gravity, has an interpretation as an area-squared densityoperator. This operator resembles the area operators known in Loop Quan-tum Gravity [24]. Furthermore, we suggest that p D ˜q should be interpretedin terms of an action. This, in turn, gives the spectral action of D ˜ the formof a Feynman integral. Thus, at the core of the construction we find anobject which resembles a partition function related to Quantum Gravity.It is important to realize that the construction works in any dimensionand does not require a foliation of the underlying manifold.The construction of the Dirac type operator D ˜ involves an infinite di-mensional Clifford bundle. This structure entails the canonical anticommu-tation relations (CAR) algebra, which appears as a tensor factor acting onthe Hilbert space H ˜ .Technically, the triple (1) satisfies the requirements of a semi-finite spec-tral triple. This is due to the fact that the infinite dimensional Clifford bundle7ntails a large degeneracy of the spectrum of D ˜ which, naively, fails to havecompact resolvent. The solution to this problem is, in short, to integrateout the symmetry group related to this degeneracy. This process leads to asemi-finite spectral triple.The Dirac type operator D ˜ is gauge invariant. It is, however, not invari-ant under the group of discrete diffeomorphisms acting on H ˜ . This meansthat D ˜ is, by itself, not an observable in a physical sense. We suggest toapply a standard trick of Noncommutative Geometry: One obtains invari-ant quantities by multiplying the algebra of observables with the relevantsymmetry group. A prime example of this method is the identification oftwo points: By applying the noncommutative trick one obtains, besides in-variance under exchange of the points, additional degrees of freedom which,ultimately, leads to the Higgs mechanism. In the present case we expect thisgeneral mechanism to entail further degrees of freedom.Let us note in passing, that the algebra of holonomy loops that we workwith in this paper is smaller than the correponding algebra of Wilson loops.However, the missing information is recovered by the Hopf algebra structureof the algebra of holonomy loops. The missing information is coded in theextra structure coming from the fact that it is given as a (norm closure of a)group algebra C r f L , L loops on S ˜s , where S ˜ denotes the entire inductive system of triangulations. As such ithas a Hopf algebra structure and for example the connections on M can berecovered as Hopf algebra homomorphisms into the group algebra of G .The construction of the Dirac type operator D ˜ is not unique. In fact,we find a large class of Dirac type operators labelled by infinite sequences t a n u of real parameters. The operator D ˜ has compact resolvent wheneverthe sequence diverges sufficiently fast. These parameters are related to thescaling behaviour of the operator and are clearly of metric origin. Thus, inorder to obtain a Dirac type operator D ˜ we are forced to choose a certainscaling behaviour.It is clear that a correct interpretation of the sequence t a n u is imperativesince the existence of the operator D ˜ depends crucially hereon. In this paperwe present one possible solution as to how these free parameters should bedealt with. We propose an extension of the spectral triple (1) to includethe sequence t a n u as dynamical degrees of freedom. The result is a newtriple p B t , D t , H t q which is a fibration of spectral triples p B ˜ , D ˜ , H ˜q . We8 ǫ ǫ ǫ Figure 1: A graph Γ with edges t ǫ , ǫ , ǫ , ǫ u .emphasise, however, that the question as to how the sequence t a n u shouldbe understood and dealt with remains open.This paper is this first of two papers concerned with the spectral triple(1). This paper is primarily concerned with the general construction andphysical interpretation of the triple. The second paper [22] deals with theconcise mathematical construction. Before we go into details we first give a brief outline of the construction. Thefirst step is concerned with the formulation of a semi-finite spectral triple overa space of connections. The triple is, as mentioned, based on a (cid:6) -algebra ofloops which we denote by B ˜ . A smooth loop l gives a map from the spaceof smooth connections, denoted A , into the structure group Gl : ∇ Ñ Hol p ∇ , l q P G , where
Hol p ∇ , l q is the holonomy of the connection ∇ P A along l and G is acompact connected Lie group. In order to describe the algebra of holonomyloops we first restrict the loop algebra to a finite graph Γ with edges ǫ i (seefigure 1.1). Seen from Γ the connection ∇ can be seen as point ∇ in thespace G n ∇ (cid:16) p g , . . . , g n q P G n (cid:16) A Γ , n is the number of edges in Γ and where g i (cid:16) Hol p ∇ , ǫ i q is the holon-omy transform along the i ’th edge. That is, a connection is given by itsholonomy transforms along edges ǫ i . Clearly, A Γ is a highly inaccurate pic-ture of the full space A of connections. However, one can show [22] that fora suitable choice of embedded graphsΓ € Γ € Γ . . . € Γ n € . . . (2)the space A is densely contained in the limit space A (cid:16) lim n Ñ8 A Γ n . This provides us with the strategy to construct a spectral triple over A . Withthe system of graphs given by nested triangulations we construct, at the levelof each graph Γ n , a spectral triple p B n , D n , H n q Γ n , where the algebra B n is generated by loops in Γ n , the operator D n is someDirac operator on A Γ n (cid:16) G n and where H n (cid:16) L p G n , Cl p T (cid:6) G n qq . This tripleis almost canonical and the little choice one has is mostly eliminated by therequirement that the construction of the triple should be compatible withstructure maps P nm : A Γ n Ñ A Γ m , n ¡ m between the different coarse-grained spaces of field configurations.The Dirac type operator obtained in the limit is an operator on the space G and is of the general form a D (cid:0) a D (cid:0) . . . (cid:0) a k D k (cid:0) . . . where D k is an operator corresponding to a certain level in the projective sys-tem (2) and where the infinite sequence t a k u determines the weight assignedto each operator D k . We find that for sequences satisfyinglim k Ñ8 a k (cid:16) 8 sufficiently fast (3)the limit spectral triple p B ˜ , D ˜ , H ˜q : (cid:16) lim n Ñ8p B n , D n , H n q Γ n (4)10atisfies the requirements of a semi-finite spectral triple.The second part of the construction is more tentative. This part is con-cerned with the infinite sequence t a k u of free parameters which enters theconstruction of the semi-finite spectral triple p B ˜ , D ˜ , H ˜q . The sequence t a k u is readily seen to carry metric data since it determines the scaling behaviourof the operator D ˜ . Also, the sequence determines the measure that arisesin the square of D ˜ . Furthermore, the choice of parameters t a k u is relatedto the invariance properties of D ˜ . Based on these observations we choose toinclude the sequence t a k u as dynamical parameters in the construction. Thismeans that we first construct a spectral triple p A a , D a , H a q , (5)where elements in the algebra A a are functions f p x , x , . . . q on the modulispace of sequences t a k u satisfying condition (3). The construction of thetriple (5) is inspired by Higson and Kasparov [25]. The Hilbert space H a isa L space of functions over this moduli space and D a is a Dirac operatorhereon. Next, we merge the triple (5) with the triple (4) to obtain a totalspectral triple p B t , D t , H t q , (6)where the Dirac type operator D t combines the operators D ˜ and D a . Itis important to see that the first part of the Dirac operator, the operator D ˜ , depends on the parameters t a k u which the second part, the operator D a ,probes. The exact form of this interdependency, which makes the operator D t highly nontrivial, is governed by the requirement that H a is a Hilbertspace over an infinite dimensional space corresponding exactly to those se-quences t a k u which leaves the operator D ˜ θ -summable.The paper is organised as follows. Sections 2 - 4 give a self contained pre-sentation of the mathematical construction of the spectral triple p B ˜ , D ˜ , H ˜q .Here, section 2 introduces the basic machinery used in this paper: First theloop algebra associated to an abstract simplicial complex and the conceptof abstract connections associated to a simplicial complex. Thereafter pro-jective systems of simplicial complexes. In section 3 we construct a spectraltriple on the level of a simplicial complex and in section 4 we obtain thetriple p B ˜ , D ˜ , H ˜q via an inductive limit of simplicial complexes.Next, sections 5 - 8 are concerned with the physical significance of thespectral triple p B ˜ , D ˜ , H ˜q . In section 5 we show that the spectral triple11nvolves a representation of the Poisson algebra of General Relativity. Thenin section 6 we give a detailed comparison between the setup presented inthis paper and the setup used in Loop Quantum Gravity and find that thetwo constructions, on a technical level, differ primarily in the way the dif-feomorphism group is treated. Section 7 briefly reviews area operators inLoop Quantum Gravity and finds that the square of the Dirac operator D ˜ has a natural interpretation as a kind of global area operator. This leadsnaturally, in section 8, to an interpretation of the operator D ˜ in terms of anaction. This, in turn, gives the spectral action of D ˜ a strong resemblance toa partition function related to gravity.Then, in section 9, we describe the construction of the spectral triple p B t , D t , H t q which includes the sequence t a k u as dynamical variables. Finally,in section 10 we mention that the Dirac type operator D ˜ defines a distanceon the space A ˜ . Sections 11 and 12 contain a discussion and conclusion.Three appendices are added to outline first an extended setup which avoidsthe choice of a basepoint and next to discuss again a notion of diffeomorphisminvariance. The last appendix is concerned with certain symmetric states. As explained in the introduction, the aim is to study geometrical structuresover a space of connections. To do this we apply a dual picture of the space ofconnections. This means that, rather than studying the space of connectionsitself, we will work with an algebra of functions on the space. This algebrais an algebra of loops and has a natural interpretation in terms of holonomyloops.The purpose of this section is to introduce the machinery needed to de-scribe this algebra of loops. The strategy is to break up the algebra intofinite parts, introduce various geometrical structures on each finite part andfinally let the complexity of the construction increase infinitely to obtain thefull algebra of loops.To emphasise the purely combinatorial nature of the construction weadopt a formalism which, for the main part of the analysis, avoids any refer-ence to an underlying manifold. This means that we will work with abstractgraphs and their loop algebras. An alternative approach is to work directlywith graphs on a manifold and their loop algebras. This approach, whichmay be intuitively clearer, is applied in [22].12igure 2: An abstract simplicial complex with directed edges.
We first introduce the notion of an abstract graph and its associated loopalgebra. The abstract graphs we consider are given by simplicial complexes.Consider therefore first an abstract, finite, d -dimensional simplicial complex K with vertexes v i and directed edges ǫ j : t , u Ñ t v i u , ǫ j p q (cid:127)(cid:16) ǫ j p q , connecting the vertexes. We shall refer to the two elements of the set t , u as start and endpoint of the edge. The construction which we present worksfor a large class of simplicial complexes. We will, however, restrict ourselvesto simplicial complexes which corresponds to triangulations of d -dimensionalmanifolds.We will consider based simplicial complexes which means that the com-plex has a preferred vertex v .Given a simplicial complex K we wish to describe an algebra of basedloops living on K . First, a path is a finite sequence L (cid:16) t ǫ i , ǫ i , . . . , ǫ i n u ofedges in E K with the property that ǫ i k p q (cid:16) ǫ i k (cid:0) p q . Next, a loop is a path satisfying ǫ i p q (cid:16) ǫ i n p q , and a based loop is a loop satisfying the additional requirement ǫ i p q (cid:16) ǫ i n p q (cid:16) v . ǫ (cid:6) j we denote the edge ǫ j with reversed direction ǫ (cid:6) j p τ q (cid:16) ǫ j p (cid:1) τ q , τ P t , u . A path may contain both edges and their reverse. We discard trivial back-tracking by which we mean sequences that contain successions of edges ǫ i and their reverse ǫ (cid:6) i . Thus, we introduce the equivalence relation t . . . , ǫ j , ǫ k , ǫ (cid:6) k , ǫ l , . . . u (cid:18) t . . . , ǫ j , ǫ l , . . . u , and regard a path as an equivalence class with respect to this relation.We define a product between two based loops L i (cid:16) t ǫ j i u , i P t , u , simplyby gluing L (cid:5) L (cid:16) tt ǫ j u , t ǫ j uu . Notice that this product is noncommutative.The inversion of a based loop L (cid:16) t ǫ j , . . . , ǫ j i , . . . , ǫ j n u defined by L (cid:6) (cid:16) t ǫ (cid:6) j n , . . . , ǫ (cid:6) j i , . . . , ǫ (cid:6) j u , is again a based loop and satisfy the requirements of an involution p L (cid:6)q(cid:6) (cid:16) L , p L (cid:5) L q(cid:6) (cid:16) L (cid:6) (cid:5) L (cid:6) . Furthermore, we define the based identity loop L as the equivalence classthat includes the empty loop L (cid:16) t Ø u . The based identity loop clearly satisfies L (cid:5) L L based loops L . This, together with the observation that L (cid:6) (cid:5) L (cid:16) L (cid:5) L (cid:6) (cid:16) L , implies that, for based loops, the involution equals an inverse. This providesthe set of based loops with a group structure. We call the group of basedloops associated to K for the hoop group ( holonomy loops , see [18]), denoted HG K . 14or the remaining part of this paper we shall consider only based loopsand will therefore drop the prefix ’based’.We finally consider formal, finite series of loops living on a complex K a (cid:16) ¸ i a i L i , L i P HG K , a i P C . (7)The product between two elements a and b is defined by a (cid:5) b (cid:16) ¸ i,j p a i (cid:4) b j q L i (cid:5) L j , and the involution of a is defined by a (cid:6) (cid:16) ¸ i ¯ a i L (cid:6) i . The set of elements of the form (7) is a Æ -algebra. We denote this algebrawith B K . We now introduce the notion of an abstract connection. Let G be a compact,connected Lie-group and, for later reference, fix a matrix representation of G . Next we introduce a G -bundle over a smooth manifold M and a loop l in M . A smooth connection can be understood as a map ∇ : l Ñ ∇ p l q P G , (8)which satisfies the condition ∇ p l (cid:5) l q (cid:16) ∇ p l q (cid:4) ∇ p l q , (9)where l and l are loops in M . Here, the map (8) is the holonomy transformof ∇ along l , ∇ p l q (cid:16) Hol p ∇ , l q .This motivates the following definition of an abstract connection as a map ∇ : t ǫ j u Ñ G , (10)that associates to each edge ǫ j P E K a point g j P G . The map is required tosatisfy ∇ p ǫ j q (cid:16) p ∇ p ǫ (cid:6) j qq(cid:1) . A K the space of all abstract connections associated to K . Theaction of ∇ is extended to a path L (cid:16) t ǫ i , ǫ i , . . . , ǫ i n u simply by ∇ p L q (cid:16) ∇ p ǫ i q (cid:4) ∇ p ǫ i q (cid:4) . . . (cid:4) ∇ p ǫ i n q , (11)where the product on the rhs is matrix multiplication. This makes ∇ agroup homomorphism from the hoop group HG K into G which means that itsatisfies ∇ p L (cid:5) L q (cid:16) ∇ p L q (cid:4) ∇ p L q . This corresponds to condition (9) and justifies the terminology abstract con-nection.Via the space A K we can equip the Æ -algebra formed by elements of theform (7) with a natural norm given by } a } (cid:16) sup ∇ P A K } ¸ a i ∇ p L i q} G , (12)where the norm on the rhs of (12) is the matrix norm given by the repre-sentation of G . The closure of the Æ -algebra B K of loops with respect to thisnorm is a C Æ -algebra. We denote this loop algebra by B K .In fact, the algebra B K is a function algebra over the space A K with valuesin the matrix representation of the group G . A loop L gives rise to a function f L via f L p ∇ q : (cid:16) ∇ p L q , L P HG K , ∇ P A K . Notice that the algebra of functions f L with the natural product f L (cid:4) f L (cid:16) f L (cid:5) L , is noncommutative whenever G is non-Abelian.The space A K is identified as a manifold A K (cid:20) G n p K q , via the bijection A K Q ∇ Ñ p ∇ p ǫ q , . . . , ∇ p ǫ n p K qqq P G n p K q . (13)where n p K q denotes the number of edges in V K . This identification gives riseto various structures on A K . For example the topological structure is givenby the topological structure of G n p K q .16...Figure 3: The lhs shows a triangulation (dimension (cid:16)
2) and the rhs its twofirst barycentric subdivisions.A loop L (cid:16) t ǫ i , ǫ i , . . . , ǫ i k u gives, according to (13), rise a function on G n p K q given by f L : p g , g , . . . , g n p K qq Ñ g i (cid:4) g i (cid:4) . . . (cid:4) g i k P G .
We think of the space A K as a ”coarse grained version of a space of(smooth) connections”. To clarify this interpretation consider an embeddingof the simplicial complex into a triangulation φ : K Ñ T (14)and a trivial principal bundle P (cid:16) M (cid:2) G . Denote by A the space of smoothconnections in P . There is a natural map χ K : A Ñ A K , χ K p ∇ qp ǫ i q (cid:16) Hol p ∇ , φ p ǫ i qq , (15)where Hol p ∇ , φ p ǫ i qq denotes the holonomy of ∇ along the edge ǫ i which nowlives in M via the embedding φ . This means that points in A K can beunderstood in terms of connections. Clearly, the map (15) is not injectivewhich is exactly what is meant by ’coarse grained’. The idea is to graduallyincrease the complexity of the simplicial complex and thereby to turn the map(15) into an injection. Therefore, the next step is to introduce a refinementprocedure for the simplicial complex K . The key tool to refine the simplicial complex K is the barycentric subdivisionof simplexes. We will consider repeated barycentric subdivisions of K . Thus,the basic element in our analysis of geometrical structures over spaces ofconnections is an inductive systems of simplicial complexes. We start withthe following 17 efinition 2.3.1 An inductive system of simplicial complexes is a countableset pt K i u , t I jk uq of nested abstract simplicial complexes K i and embeddings I jk : K j Ñ K k , j k (16) so that I jk is either a barycentric subdivision or an inversion of edges. Theset is required to satisfy lim i n p K i q (cid:16) 8 . A barycentric subdivision of a simplicial complex K is here understood as thesimultaneous subdivision of all simplices in K . The simplest simplicial complex in the inductive system is called the initialcomplex and is denoted K o . The orientation of the initial complex is notimportant and one simply chooses one.The embedding (16) gives rise to a projection between spaces of abstractconnections P n ,n : A K n Ñ A K n , n ¥ n , which, via (13), is identified as a projection between manifolds P n ,n : G n Ñ G n , n ¥ n . (17)The projection P n ,n is given by composition of one or more of the followingoperations: • multiplying g i with g i . • inverting g i . • leaving out some g i in p g , . . . , g n q P G n .The embedding (16) commute by construction with the identification A K (cid:20) G n p K q , i.e. the diagram: A K i (cid:0) σ i (cid:0) ÝÑ G n p K i (cid:0) qÓ Ó A K i σ i ÝÑ G n p K i q where σ i is given by (13), commutes. This means that the limit space A ˜ : (cid:16) lim K Ý A K
18s a pro-manifold. That is, it is the projective limit of manifolds. This givesus immediate access to various structures on A ˜ . For instance, since theprojections P in (17) are continuous they give a topological structure on A ˜ .In general, the structure of a pro-manifold is a powerful tool that leads us toboth Hilbert space and metric structures on the limit space A ˜ .In section 4.2 we show that the space A of smooth connections is denselyembedded in A ˜ (once a suitable embedding of the simplicial complex hasbeen applied, see (14)). This fact is the very reason for studying the space A ˜ . It means that structures on A ˜ can equally be understood as structureson A . The aim is to construct a spectral triple at the level of each simplicial complexin an inductive system of simplicial complexes. The triples are required toinvolve the loop algebras B K i as function algebras over the associated spaces A K i of abstract connections. To ensure that the limit of increased complex-ity is well defined we require the spectral triples to be compatible with allprojections induced by the inductive system of complexes. This means thatgeometrical structures over A K i will converge to geometrical structures over A ˜ . Before we go into details let us outline the construction of the spectral tripleat the level of a simplicial complex K . The starting point is the manifold A K (cid:20) G n p K q . It is natural to consider first the Hilbert space H K (cid:16) L p G n p K qq , where L is with respect to the Haar measure on G n p K q . Since we wish toconstruct both a Dirac operator acting on H K and to have a representationof the algebra B K on H K we need to equip the Hilbert space with additionalstructure. Consider therefore instead the Hilbert space H K (cid:16) L p G n p K q , Cl p T (cid:6) G n p K qq b M l p C qq , (18)where l is the size of the representation of G and Cl p T (cid:6) G n p K qq is the Clif-ford bundle involving the cotangent bundle over G n p K q . If we recall that19oints in G n p K q represents homomorphisms ∇ from the hoop group into G we immediately have a representation of the loop algebra B K on H K f L (cid:4) ψ p ∇ q (cid:16) p b ∇ p L qq (cid:4) ψ p ∇ q , ψ P H K , (19)where the first factor acts on the Clifford part of the Hilbert space and thesecond factor acts by matrix multiplication on the matrix part of the Hilbertspace. Finally, we choose some Dirac operator D K on G n p K q and obtain thetriple p B K , D K , H K q , (20)on the level of the simplicial complex K .It is not difficult to construct the candidate (20) for a spectral triple onthe space A K . The crucial point, however, is to ensure that the constructionis compatible with the induced projections between the simplicial complexes(17). This requirement turns out to restrict the choice of Dirac operator on A K considerably.To ease the notation we shall from now on write A i for A K i , B i for B K i , H i for H K i , D i for D K i and n i for n p K i q . Before we proceed with the construction we need some preparations. A point p g , . . . , g n q P G n is denoted ¯ g . Let R g denote right translation on the group G , i.e. R g p h q (cid:16) hg . Accordingly, L g denotes left translation. We also denoteby R g the corresponding differential, R g : T h G Ñ T hg G . L g likewise. Givena cotangent vector φ at the identity we define the right translated cotangentvector field Rφ by: Rφ p g qp v q (cid:16) φ p R g (cid:1) p v qq , v P T g G .
The left translated cotangent vector fields are defined equivalently. Given aprojection P : G n Ñ G m we denote by P (cid:6) the corresponding differential P (cid:6) : T ¯ g G n Ñ T P p ¯ g q G m and by P (cid:6) the induced map on cotangent spaces P (cid:6) : T (cid:6) P p ¯ g q G m Ñ T (cid:6) ¯ g G n . P : G Ñ G ; p g , g q Ñ g (cid:4) g . An element in p v , v q P T p g ,g q G transforms according to P (cid:6)p v , v q Ñ p R g v (cid:0) L g v q , which is best seen by writing v (cid:16) 9 γ p τ q| τ (cid:16) where γ p τ q (cid:16) p γ p τ q , g q P G is the one parameter subgroup in G generated by v . P clearly maps γ p τ q into γ p τ q (cid:4) g (cid:16) R g γ p τ q P G . The same argument applies to v . Corre-spondingly, given an element of the cotangent bundle φ P T (cid:6) g (cid:4) g G the dualmaps yields P (cid:6) φ (cid:16) p R g (cid:1) φ, L g (cid:1) φ q P T (cid:6)p g ,g q G , (21)where R g (cid:1) φ p v q (cid:16) φ p R g v q , v P T g GL g (cid:1) φ p v q (cid:16) φ p L g v q , v P T g G .
To construct the Hilbert space related to a simplicial complex K i we firstconstruct an inner product on T (cid:6) G n i . We choose a left and right invariantmetric on G . The edges in K are numbered with 1 , . . . , n i . In the followingwe shall occasionally write n instead of n i (cid:16) n p K i q . Consider two elements φ , φ P T (cid:6) ¯ g G n i . We write φ k (cid:16) p φ k , . . . , φ kn q and define the inner productby x φ | φ y ¯ g (cid:16) N n ¸ j (cid:16) x φ j | φ j y Gg j , (22)where x(cid:4)|(cid:4)y Gg j is the inner product on T (cid:6) g j G . In (22) N is the number ofbarycentric subdivision between the simplicial complex K i and the initial There is some freedom in the choice of N . We could also introduce a factor N j asso-ciated to each individual edge ǫ j . Here, N j counts the number of barycentric subdivisionsthat lies between the edge ǫ j and the simplest complex K k ( k j ) that has the edge ǫ j asa segment of an edge in K k . This choice therefore associated different weights to edges ina simplicial complex according to their position in the complex. e i (cid:18) G (cid:18) G (cid:18) G (cid:18) G n Figure 4: an edge e i and its repeated barycentric subdivisions.complex K o . In [22] we prove that the inner product (22) is compatible withprojections induced by embeddings (16).With the inner product (22) on T (cid:6) G n we construct the Clifford bundle Cl p T (cid:6) G n q and define an inner product on the Hilbert space (18) x(cid:4)|(cid:4)y (cid:16) » dµ (cid:4) Tr (cid:4)x(cid:4)|(cid:4)y Cl , (23)where dµ is the Haar measure on G n , Tr is the trace on M l and x(cid:4)|(cid:4)y Cl is theinner product of the Clifford bundle Cl p T (cid:6) G n q . The inner product (23) iscompatible with projections (16). This gives us the Hilbert space (18). The Dirac-like operator on each space A i is required to be compatible withthe projections in the inductive system of complexes. Thus, the inducedmaps P (cid:6) : T (cid:6) G m Ñ T (cid:6) G n , m n , (24)give rise to the compatibility conditions P (cid:6)p D m v qp g , . . . , g n q (cid:16) D n p P (cid:6) v qp g , . . . , g n q , (25)where v P L p G m , Cl p T (cid:6) G m qq . These conditions largely restrict the Dirac-like operator D i on K i . On the initial complex K o with the corresponding22anifold G n o the Dirac operator is chosen to have the canonical form D o (cid:16) ¸ i,j e ij (cid:4) ∇ lc ˆ e ij , (26)where t e ij u is a global basis on T (cid:6) G corresponding to the i ’th copy of G . t ˆ e ij u is the corresponding basis on T G , obtained from t e ij u via the inner product(22). ∇ lc is the Levi-Civita connection on G .The problem is therefore to find a Dirac operator on G n i compatible withprojections (16). Here, it is sufficient to deal with an edge ǫ and its partitionsgenerated by barycentric subdivisions of the complex to which ǫ belongs, seefigure 4. Once the operator is constructed here we obtain an operator on theentire system of simplicial complexes by gluing the individual operators inan obvious manner.The generic problem is the projection P : G Ñ G ; p g , g q Ñ g (cid:4) g , (27)which we now consider together with the induced map between cotangentbundles (21). Let t e i u be an orthonormal basis of T (cid:6) id G and denote by e i p g q (cid:16) L g e i p id q (cid:17) e i the left-translated basis covectors at g P G . The push-forward of the basiscovector e i p g q in T (cid:6)p g ,g q G by P (cid:6) gives P (cid:6) e i p g (cid:4) g q (cid:16) p R g (cid:1) e i p g q , L g (cid:1) e i p g qq , where g (cid:16) g (cid:4) g . This suggests a natural orthonormal basis of T (cid:6)p g ,g q G given by E ,si (cid:16) p E i , (cid:8) E i q , (28)where E i p g q (cid:16) L g (cid:4) g R g (cid:1) e i p id q , E i p g q (cid:16) L g e i p id q , (29) There is some freedom of choice here since the Dirac operator on K o need not involvethe Levi-Civita connection. However, to obtain a self-adjoint operator certain invarianceproperties must be satisfied [22]. s in (28) represents the appropriate sign combinations s P tp(cid:0) , (cid:0)q , p(cid:0) , (cid:1)qu characterising the two orthonormal covectors. Denote by ˆ E ji and ˆ E ,si thecorresponding sections in T G and
T G respectively, defined via the innerproduct (22).We consider a Dirac operator corresponding to G of the form D (cid:16) ¸ s,i E ,si (cid:4) ∇ ˆ E ,si , (30)where ∇ is a connection on T (cid:6) G . It turns out that the operator (30)satisfies the compatibility condition (25) if the connection ∇ satisfies ∇ p ˆ E i , qp E i , q (cid:17) p ∇ lc ˆ E i E i , q , ∇ p , ˆ E i qp , E i q (cid:17) p , ∇ lc ˆ E i E i q . To obtain the general form of the Dirac like operator on G n we first definethe following twisted covectors on G E i p g q (cid:16) L g (cid:4) ... (cid:4) g n R p g (cid:4) ... (cid:4) g n q(cid:1) e i p id q ... E ji p g j q (cid:16) L g j (cid:4) ... (cid:4) g n R p g j (cid:0) (cid:4) ... (cid:4) g n q(cid:1) e i p id q ... E ni p g n q (cid:16) L g n e i p id q . (31)Next, we write E n,si (cid:16) p E i , (cid:8) E i , . . . , (cid:8) E ni q , where s (cid:16) p(cid:0) , (cid:8) , . . . , (cid:8)q is the sequence of signs which characterises thecovector. Again, we denote by ˆ E n,si the corresponding sections in T G n . Theglobal frames E n,si are found by repeated lifts of the covector e i p g q on G to G n . Therefore, they are constructed to satisfy E n,si (cid:16) P (cid:6)p E n { ,s i q , where the sequence s is obtained from the sequence s by replacing each signin s with the same sign twice, and where P : G n Ñ G n { ; p g , . . . , g n q Ñ p g (cid:4) g , . . . , g n (cid:1) (cid:4) g n q . (32)24he Dirac like operator on G n has the form D n (cid:16) n ¸ s,i E n,si (cid:4) ∇ n ˆ E n,si , (33)where the sum runs over i as well as all appropriate sign sequences s .We define the connections ∇ n recursively. That is, the action of ∇ n onbasis covectors E n,si are given recursively and thereafter extended via linearityand the requirements of a derivation to the entire Clifford bundle. Thus, werequire ∇ n P (cid:6) ˆ E p P (cid:6) E q (cid:16) P (cid:6)p ∇ n/2 ˆ E E q , ∇ n p P (cid:6) ˆ E qK p P (cid:6) E q (cid:16) , (34)where E and ˆ E are basis covectors and vectors, respectively, of the type p E i , (cid:8) E i , . . . , (cid:8) E n { i q at the level n . In equation (34) and in the following wedenote by p P (cid:6) ˆ E qK general elements in the orthogonal complement to vectors(and covectors) of the form P (cid:6) ˆ E . We fix the remaining freedom in ∇ n withthe additional condition ∇ n p P (cid:6) ˆ E qK p P (cid:6) ¯ E qK (cid:16) ∇ n p P (cid:6) ˆ E qp P (cid:6) ¯ E qK (cid:16) , which is required for the construction of the trace, see section 4.1 and [22].The properties given in (34) are again dictated by the requirement that theDirac operator (33) is compatible with the projections.The proof that the operator (33) satisfies the compatibility condition (25)is given in [22]. K i The Dirac type operator (33) is not the most general operator satisfying therequirements of compatibility with the structure maps (25). In fact, theserequirements renders substantial parts of the operator (33) free to modifi-cations. This observation is closely related to the fact that the Dirac typeoperator (33) will, at it stands, not descend to an operator with a compactresolvent in the inductive limit of repeated barycentric subdivisions. It turnsout that the additional degrees of freedom are exactly the necessary leverageneeded to obtain a well-behaved Dirac type operator in the limit.25et us start with the case n (cid:16)
2. We observe that a rescaling of parts ofthe operator (30) according to D (cid:16) ¸ i (cid:1)p E i , E i q (cid:4) ∇ p E i , E i q (cid:0) a p E i , (cid:1) E i q (cid:4) ∇ p E i , (cid:1) E i q(cid:9) . (35)where a P R , does not affect the compatibility with the embedding (21). Inthe general case, the modification of (33) is as follows: Let t a k u be an infinitesequence of real numbers and put a (cid:16)
1. Consider a single edge ǫ and itssubdivisions. Consider the n ’th subdivision. Let s be a finite sequence of n signs and denote by p s q the number of ’+’s in the beginning of the sequence.Define the number m p s q (cid:16) log (cid:1) N p s q (cid:9) . The modified Dirac type operator now has the form D n (cid:16) n ¸ s,i a m p s q E n,si (cid:4) ∇ n ˆ E n,si . (36)Let us consider the eigenvalues of this operator. For simplicity, considerthe Abelian case G (cid:16) U p q . If we define the product between a sequence s (cid:16) p(cid:0) , (cid:8) , (cid:8) , . . . q of signs with a sequence of real numbers p n , n , . . . q by p n , n , . . . q (cid:4) s (cid:16) n (cid:8) n (cid:8) . . . (37)where the signs on the rhs are read of the sequence s , then we can write thespectrum of the Dirac type operator (36) asspec p D q (cid:16) $&%(cid:8) N d¸ k a m p s k qpp n , n , . . . , n N q (cid:4) s k q ,.- , where s k is the sequence of signs corresponding to the k ’th orthonormalvector E n,s k . In this case, the number of eigenvalues of the limit operator D ˜ (cid:16) lim n D n in a range r , Λ s , Λ , is finite whenever a n (cid:16) n b n where lim n Ñ8 b n (cid:16) 8 . (38) or, to be general, any sequence of objects. G is a non-Abelian Lie-group is more complicated.The full analysis is given in [22] where we prove that for any compact Lie-group G there exist a sequence t a i u so that the number of eigenvalues withina finite range is finite up to a controllable degeneracy. This result will beclarified in the next section. p B n , D n , H n q Up till now we have considered a system of embedded, abstract simplicialcomplexes t K n u . To each simplicial complex K n we introduced a space A n which we interpreted as a coarse-grained version of a space of connections.The next step was to construct a spectral triple p B n , D n , H n q on each of thespaces A n . The spectral triple satisfies requirements of compatibility withthe operation of barycentric subdivision. This means that we can take thelimit of infinitely many barycentric subdivisions. The resulting triple, whichis denoted by p B ˜ , D ˜ , H ˜q , has the following elements:First, the Hilbert space H ˜ is constructed by adding all Hilbert spaces H n H K L p G n p K q , Cl p T (cid:6) G n p K qq b M l p C qq{ N , (39)where N is the subspace generated by elements of the form p . . . , v, . . . , (cid:1) P (cid:6) ij p v q , . . . q , where P (cid:6) ij are maps between Hilbert spaces H n induced by (24). The Hilbertspace H ˜ is the completion of H . The inner product on H ˜ is the inductivelimit inner product. This Hilbert space is manifestly separable. Next, thealgebra B ˜ : (cid:16) lim K ÝÑ B K . (40)contains loops defined on a simplicial complex K n in t K n u as well as theirclosure. Again, the algebra B ˜ is separable. Finally, the Dirac-like operator D n descends to a densely defined operator on the limit space H ˜ D ˜ (cid:16) lim K ÝÑ D n . (41)27 .1 A semi-finite spectral triple The question is whether the triple p B ˜ , D ˜ , H ˜q satisfies the requirements ofa spectral triple. Clearly, to have a true spectral triple is highly desirable,first of all to ensure that we are operating on mathematically safe ground.Also, the toolbox of Noncommutative Geometry is a rich one. It providesnot only metric structures to the underlying spaces involved, it also involvesaspects which does not exist in ordinary Riemannian geometry.To have a spectral triple p B, D, H q where B is a C Æ -algebra representedon a Hilbert space H on which an unbounded, selfadjoint operator D acts,means that the two requirements1. p λ (cid:1) D q(cid:1) , where λ R R , is a compact operator2. r b, D s , where b P B , is a bounded operator and where B is a dense Æ -subalgebra of B are satisfied. However, the triple p B ˜ , D ˜ , H ˜q is not spectral in this sense.The spectrum of D ˜ involves a large degeneracy due to the infinite dimen-sional Clifford bundle. This means that the resolvent of D will not be com-pact.There exist, however, another sense in which a triple p B, D, H q can becalled spectral. If there exist a trace on the algebra containing B and thespectral projections of D , then p B, D, H q is called a semi-finite spectral tripleif p λ (cid:1) D q(cid:1) is compact with respect to this trace.It turns out that the triple p B ˜ , D ˜ , H ˜q does form a semi-finite spectraltriple. The sufficient requirement is that the sequence t a k u satisfieslim k a k Ñ 8 , sufficiently fast . (42)A semi-finite spectral triple can be thought of as a spectral triple whichinvolves a redundant symmetry group. This symmetry group resembles agauge group and one must integrate out the extra degrees of freedom. In thepresent case the symmetry group is identified as the endomorphisms of theinfinite dimensional Clifford bundle. To deal with this redundancy we firstdefine a trace on the algebra containing B ˜ as well as the spectral projectionsof D ˜ .We will construct the trace at each level in the inductive system of sim-plicial complexes and subsequently take the limit of repeated barycentric28ubdivisions. For the construction we factorize the Hilbert space. Choose aglobal orthonormal frame in T (cid:6) e G . This choice gives rise to a decomposition L p G n , M l b Cl p T (cid:6) G n qq (cid:16) L p G n q b M l p C q b Cl p T (cid:6) e G n q . (43)We will construct the trace on the algebra C n (cid:16) K p L p G n qq b End p M l q b End p Cl p T (cid:6) e G n qq . where K here denotes the compact operators.Let T r op denote the operator trace on K p L p G n qq . For each n we havethe normalised trace tr on End p Cl p T (cid:6) e G n qq . We define the trace as
T r (cid:16)
T r op b T r l b tr. (44)Note that the trace is independent of the choice of global orthonormal framein T (cid:6) G , since a different choice of basis is given by a unitary transformation.In [22] we prove that the limit C ˜ : (cid:16) lim C n will be a C (cid:6) -algebra and that the trace T r gives a trace on C ˜ . Since C ˜ is contained in the weak closure of B ˜ the trace extends to a trace on B ˜ aswell.The important point in this construction is the normalisation of the trace tr on End p Cl p T (cid:6) e G n qq . To explain this consider the step going from n to n (cid:0) m . Let P λ be the spectral projection of D n for the eigenvalue λ . Goingfrom n to n (cid:0) m the spectral projection will roughly speaking be mappedto P λ b . Thus the size of the eigenspace λ grows in the same rate as thedimension of the Clifford bundle. To remedy this defect we must ensure that tr p q (cid:16) , which is what the normalised trace does.The proof that p B ˜ , D ˜ , H ˜q form a semi-finite spectral triple with respectto the trace (44) is given in [22]. It turns out that in general the Dirac-typeoperator (41) may require a perturbation at each level in the inductive sys-tem. This perturbation deals with the fact that the operators D n may, ingeneral, have nontrivial kernels which obstructs the control of the eigenvaluesof the limit operator D ˜ . These perturbations lift the entailed degeneracy,29hich would otherwise destroy the spectral properties of the triple. The re-quired perturbation is, at each level in the projective system, bounded anddoes not affect the commutator between D ˜ and the algebra B ˜ significantly.For the special case G (cid:16) SU p q we find that the operators D n have trivialkernels and therefore that no perturbations are needed.To recapitulate, we have successfully constructed a large class of semi-finite spectral triples p B ˜ , D ˜ , H ˜q . The triples are labelled by infinite se-quences t a k u of real numbers satisfying lim k a k Ñ 8 sufficiently fast.Let us end this section with a short discussion of the structure of theHilbert space H ˜ the role of the infinite dimensional Clifford bundle. Considerfirst the decomposition in (43) and rewrite it in the suggestive form L p G n q b M l p C q b Cl p T (cid:6) e G n q (cid:16) H n,b b H n,f , where H n,b (cid:16) L p G n q b M l p C q . In the inductive limit this leads to thedecomposition H ˜ (cid:16) H ˜ ,b b H ˜ ,f . Next we factorize the algebra H ˜ according to the above factorisation of theHilbert space. The result is C ˜ (cid:16) K p H ˜ ,b q b C, where C (cid:16) lim n End p Cl p T (cid:6) e G n qq , where the morphisms in this inductive system are the unital ones. In par-ticular C is a UHF-algebra and since Cl p T (cid:6) e G n q has dimension 2 n (cid:4) dim p G q , the C (cid:6) -algebra C is isomorphic to the CAR algebra.The CAR algebra is an integral element of fermionic Quantum Field The-ory. We find it interesting that this algebra naturally emerges from the triple p B ˜ , D ˜ , H ˜q which, a priori, is an entirely ’bosonic’ construction. In section11 we shall comment further on the appearance of the CAR algebra. A ˜ and generalised connections We have already indicated that the space A ˜ is a space of generalised connec-tions. This means that it is a closure of the space A of smooth G -connections.30.. M MT T Figure 5: The embedding of the system t K i u into M gives rise to an inductivesystem of triangulations t T i u .To see this consider the embedding φ of the simplicial complex K i T i : (cid:16) φ p K i q , where T i is a triangulation of the manifold M . There is a natural map χ ˜ : A Ñ A ˜ , χ ˜p ∇ qp φ p ǫ i qq (cid:16) Hol p ∇ , φ p ǫ i qq , (45)where Hol p ∇ , φ p ǫ i qq is the holonomy of ∇ along the edge φ p ǫ i q which now hasa location in M via the embedding φ . Further, if we are given two differentconnections ∇ , ∇ P A they will differ in a point, say m P M , and hencein a neighbourhood of m . We can therefore choose a small, directed edge φ p ǫ i q in a triangulation T i that is sufficiently refined, so that φ p ǫ i q lies in theneighbourhood of m where the connections ∇ and ∇ differ. Furthermore,because the system of triangulations contain edges in all directions in M wecan choose φ p ǫ i q so that Hol p ∇ , φ p ǫ i qq (cid:127)(cid:16) Hol p ∇ , φ p ǫ i qq . In other words, χ ˜ is an embedding, and hence the terminology generalisedconnection is justified.This means that we have successfully turned the map (15) into an injec-tion by repeating the barycentric subdivisions of the simplicial complexes.The identification of A ˜ as a space of connections provides us with a newunderstanding of the spectral triple p B ˜ , D ˜ , H ˜q . First, as already mentioned,31ccording to (45) the algebra of loops should be interpreted in terms ofholonomy loops f L p ∇ q (cid:18) Hol p L, ∇ q . It is a well known result (see [26] and [27] and references therein) that thecomplete set of holonomies, as well as their associated Wilson loops, containthe full information, up to gauge transformations, about the underlying spaceof smooth connections. The fact that the map (45) is an injection means thatthe algebra B ˜ , which involves a highly restricted set of loops, does in factcontain the same information about the underlying space of connections asdoes the full set of smooth or piece-wise analytic loops.Second, since the Dirac-type operator D ˜ is a derivation on the space A ˜ it should be understood in terms of a functional derivation operator D ˜ (cid:18) δδ ∇ in some integrated sense which shall become clearer soon. Finally, elementsin the Hilbert space H ˜ are functions over field configurations of connectionsand the inner product in H ˜ comes in the form of a functional integral x Ψ p ∇ q| . . . | Ψ p ∇ qy (cid:18) » A ˜ r d ∇ s . . . , Ψ P H ˜ . (46)That is, an integral over A ˜ .All together it is clear that the construction should be interpreted interms of Quantum Field Theory.Notice that the diffeomorphism group diff( M ) has no natural action on A ˜ or the algebra B ˜ , except for a few, discrete diffeomorphisms. We shallcomment on this fact in section 6.2. It remains to clarify whether the Dirac type operator D ˜ is gauge invariant.To this end let U be an element of the gauge group G of M (cid:2) P , i.e. U : This integral resembles functional integrals found elsewhere in physics. First, it issimilar to the inner product on the Hilbert space H diff of diffeomorphism invariant statesin Loop Quantum Gravity, see below. Second, it also resembles functional integrals inlattice gauge theories. Here, the main difference is the ’lattice spacing’ a in lattice gaugetheories which gives the continuum limit a Ñ Ñ G is a smooth function. Given a connection ∇ P A , U induces a gaugetransformed connection ˜ ∇ . Given a path L with startpoint x and endpoint x the holonomy along L transforms according to Hol p ∇ , L q Ñ U p x q Hol p ∇ , L q U (cid:6)p x q . To determine the properties of the Dirac type operator D ˜ when subjected toa gauge transformation we consider first the case A n (cid:16) G n that correspondsto n divisions of a single edge. Consider the general transformation U n : G n Ñ G n , p g , g , . . . , g n q Ñ p u g u (cid:1) , u g u (cid:1) , . . . , u n (cid:1) g n u (cid:1) n q , where u , u , . . . , u n are unitary group elements in G . This transformationgenerates a map U n : L p G n , Cl p T (cid:6) G n qq Ñ L p G n , Cl p T (cid:6) G n qq between Hilbert spaces. We need to check whether D n ξ (cid:16) U n D n U (cid:6) n ξ , ξ P L p G n , Cl p T (cid:6) G n qq . (47)In [22] we find that (47) holds whenever the connections used to construct D n satisfy a certain gauge compatibility condition. In particular, we findthat the special flat connections entering the construction of D n do satisfythis condition. This, in turn, implies that the full Dirac type operator D ˜ isgauge invariant. For the full analysis we refer the reader to [22]. D ˜ and the algebra B ˜ Section 5 is concerned with the relation between the Poisson algebra of Gen-eral Relativity and the algebra B ˜ . The point is that the interaction betweenthe Dirac type operator D ˜ and the algebra B ˜ equals the interaction betweenconjugate variables of gravity. Before we show this we need to calculate thecommutator between the operator (36) and a loop operator.First of all, the commutator is non vanishing r D ˜ , b s (cid:127)(cid:16) , b P B ˜ since, on the level of refinement corresponding to n edges, b is a non-trivialmatrix valued function on G n and D ˜ is the Dirac-type operator on G n .33onsider first the simple case where a loop L j corresponds to the functionon G n f L j : p g , . . . , g n q Ñ g j . Thus, the loop L j is really just the j ’th edge. We wish to calculate thecommutator r d ˆ E ji , f L j s ξ p g , . . . , g n q , where ξ P L p G n q . If we denote by e i the generators of the Lie algebra g ,then we introduce the twisted generators of gE ji (cid:16) g j (cid:0) g j (cid:0) . . . g n e i g (cid:1) n . . . g (cid:1) j (cid:0) corresponding to ˆ E ji . We now calculate r d ˆ E ji , f L j s ξ p g , . . . , g n q (cid:16) (cid:1) d ˆ E ji f L (cid:9) ξ p g , . . . , g n q(cid:16) ddt (cid:0) g j . . . g n exp p te i q g (cid:1) n . . . g (cid:1) j (cid:0) (cid:8) ξ p g , . . . , g n q(cid:16) g j E ji ξ p g , . . . , g n q . (48)It is important to notice that product in the last line of (48) is a matrixmultiplication.Equation (48) implies that r D n , f L j s (cid:16) n ¸ s,i (cid:8) a m p s q E n,si (cid:4) g j E ji . (49)The sign on the rhs of (49) correspond to the j ’th sign in the sequence s .Thus, for i (cid:16) (cid:0) ’sand (cid:1) ’s are equal.Next, the commutator between D ˜ and a general loop Lf L : p g , . . . , g n q Ñ g i g i . . . g i k . simply consist of repeated applications of p q according to r D ˜ , f L s (cid:16) r D ˜ , f L i s g i . . . g i k (cid:0) g i r D ˜ , f Li s . . . g i k (cid:0) . . . (50) we here consider the subdivision of an edge into n segments. Therefore, D n is theoperator given by (36). D n corresponding to the level of refinement given by the loop L .Thus, the action of D ˜ on a single loop is to insert the ’twisted’ generators E ji of the Lie-algebra into the loop at each vertex the loop passes throughand to multiply with an appropriate element in the Clifford bundle. t a n u In section 9 we will include the sequence t a n u as dynamical variables in theconstruction. To motivate this step we need a better understanding of thesequence t a n u and the role it plays in the spectral triple p B ˜ , D ˜ , H ˜q .Primarily, the role of the sequence t a n u is to shift an otherwise impossibledegeneracy in the spectrum of the operator D ˜ via the condition (42). Theparameters a i introduces a hierarchy between the eigenvalues of D ˜ on theinfinitely many copies of G in A ˜ . This hierarchy is closely related to thescaling behaviour of the construction.Consider first a line segment divided in two, corresponding to the projec-tion P : G Ñ G .
Functions on G naturally fall into two classes: push-forward of functions on G P (cid:6) : L p G q Ñ L p G q , p P (cid:6) f qp g (cid:4) g q , and the orthogonal complement hereof. We denote the former V and thelatter V K (cid:17) p P (cid:6) L p G qqK . Functions in V correspond to information which isalso contained in the less refined picture which involves only one copy of G .This is the simplest and most coarse-grained description of parallel transportsalong the line segment available. Function in V K , on the other hand, containadditional information which cannot be traced back to the simpler picture.Each additional division of the line segment refines the picture further.It is clear that any division of a line segment into two will involve newsegments which are shorter - independently of any choice of metric - thanthe original segment. Therefore, functions in V K correspond to informationabout A ˜ which is deeper , i.e. at a shorter distance, compared to informationcarried by functions in V .The role of the sequence t a i u is to take this scaling behaviour into account.The operator D ˜ weights functions on A ˜ according to where in the projectivesystem of spaces A i the information originates (see figure 6). Therefore, if35 i a i (cid:0) a i (cid:0) Figure 6: The role of the parameters t a i u is to weight the different segmentsof a given loop according to the refinement of the segments.a function ψ P L p A ˜q is the push-forward of functions on L p A i q then thecorresponding eigenvalues of ψ are weighted with the appropriate parameter a i . Thus, if we wish to probe A ˜ at a very short distance, this informationwill come with a very high weight factor a i corresponding to high energy.On the other hand, if A ˜ is probed at a more coarse-grained level, then thecorresponding eigenvalues are weighted with smaller weights a i .Let us end this subsection with a curious observation. If we start with ameter, then we find that it takes about 116 subdivisions of the meter to reachthe Planck length of 1 . (cid:2) (cid:1) meter. This corresponds then to 116 of theparameters t a i u . Thus, although the sequence t a i u is infinite, the number ofparameters involved when probing physical scales is certainly finite. In the next two sections we relate the construction of the spectral triple p B ˜ , D ˜ , H ˜q to canonical quantization of field theories and in particular tocanonical quantum gravity.The first section is concerned with the Poisson brackets of General Rel-ativity. We first introduce the formulation of General Relativity based onAshtekar variables. The fact that this formulation has a gauge connectionas a primary variable is the Raison D’´etre for the study of spaces of connec-tions in this paper. Next, we show that the interaction between the Dirac36ype operator D ˜ with the algebra B ˜ reproduces the structure of the Poissonbrackets of General Relativity when these are formulated in terms of loopvariables. This means that the triple p B ˜ , D ˜ , H ˜q includes a representationof the Poisson algebra of General Relativity.Section 6 is primarily concerned with a more detailed comparison betweenthe construction presented in this paper (the choice of graphs, the Hilbertspace) and the study of spaces of connections within Loop Quantum Gravity.It turns out that the key difference between the two lies in the treatment ofthe diffeomorphism group. We find that the Hilbert space H ˜ is directlyrelated to the Hilbert space of diffeomorphism invariant states known fromLoop Quantum Gravity. The difference between the two is a group of discretediffeomorphisms. This means that the representation of the Poisson bracketsgiven by the triple p B ˜ , D ˜ , H ˜q includes partial diffeomorphism invariance asa first principle.For the remaining part of this section we will, unless otherwise stated,restrict ourselves to G (cid:16) SU p q and dimension three. For an introduction to canonical gravity see for example [28, 29]. Let us firstfix some notation. We follow [29] and [30]. Consider the vierbein formulationof General Relativity where E Aµ is the vierbein and g µν (cid:16) E Aµ E Bν η AB is thecorresponding space-time metric where η AB (cid:16) t(cid:1) , , , u . Here µ, ν, . . . and A, B, ... denote curved and flat space-time indices respectively. We assumethat space-time can be foliated according to M (cid:16) Σ (cid:2) R where Σ is a spatialmanifold. Let m, n, . . . and a, b, . . . denote curved and flat spatial indices,respectively. Denote by e am the spatial dreibein. The spin connection ω µAB is given by ω µAB (cid:16) E νA ∇ µ E Bν , where ∇ µ is the covariant derivative which involves the Christoffel connec-tion.The canonical momenta Π ma corresponding to the dreibein are obtainedfrom the Einstein-Hilbert Lagrangian L [29]Π ma : (cid:16) δ L δ pB t e am q (cid:16) ee mb p K ab (cid:1) δ ab K q , where e (cid:16) det p e am q and K ab (cid:16) ω ab is the extrinsic curvature. We write37 (cid:16) K aa . The non-vanishing canonical Poisson brackets read t e ma p x q , Π nb p y qu (cid:16) δ ab δ nm δ p qp x, y q . (51)The Hamiltonian formulation of gravity involves a set of 3 constraints re-lated to the symmetries of the theory: the Gauss constraint correspondingto gauge invariance; the diffeomorphism constraint corresponding to spatialdiffeomorphisms within Σ; the Hamiltonian constraint encoding the full 4-dimensional diffeomorphism invariance and thus containing the dynamics ofGeneral Relativity. The diffeomorphism and Hamiltonian constraints corre-sponds to 4 of the 10 Einstein field equations. The Hamiltonian itself is alinear combination of the two constraints. In the quantum theory, this leadsto the famous Wheeler-DeWitt equation.A change of variables from the spatial spin connection and dreibein fieldto the connection A ma : (cid:16) (cid:1) ǫ abc ω mbc (cid:0) γK ma , (52)where ǫ abc is the totally antisymmetric symbol, and the inverse densitisedspatial dreibein ˜ E ma (cid:16) ee ma leads to the Poisson brackets (all other vanish) t A am p x q , ˜ E nb p y qu (cid:16) γδ ab δ nm δ p qp x, y q . (53)Often the variables t ˜ E, A u are contracted with Pauli matrices τ a accordingto ˜ E mαβ : (cid:16) ˜ E ma τ aαβ , A mαβ : (cid:16) A ma τ aαβ in order to replace Lorentz indices by spinorial SU p q indices.The variables t ˜ E, A u are the well-known Ashtekar variables [16, 17] andthe parameter γ (cid:127)(cid:16) A ma as a primary variable ofGeneral Relativity permits applications of techniques from Yang-Mills theory.In particular, one might shift focus from connections to their holonomies [11] Hol C p A q (cid:16) P exp » C A , where C is a curve in Σ, and express the Poisson brackets in terms ofholonomies and a set of conjugate variables. To find a suitable choice of The Gauss constraint corresponds to the connection-formalism, see below. S in Σ and define theflux vector F aS p ˜ E q : (cid:16) » S dF a . where the area element dF a is given by dF a (cid:16) ǫ mnp ˜ E ma dx n ^ dx p . (54)To obtain the Poisson brackets pick a curve C (cid:16) C (cid:4) C which intersects S at the single point C X C . Then the Poisson bracket between the newvariables reads [33] t F aS p ˜ E q , Hol C p A qu (cid:16) (cid:1) ι p C, S q γHol C p A q τ a Hol C p A q , (55)where ι p C, S q (cid:16) (cid:8) S and C ( ι p C, S q vanishes when C and S do not intersect).Notice that the structure of the bracket (55) is identical to structure of thecommutators (49) and (50) between the loop algebra B ˜ and the Dirac typeoperator D ˜ . Both set of commutators prescribe an insertion of an elementin the Lie-algebra into the loop or curve at an ”intersection point”: eitherat the intersection between the surface S and the curve C , or at the vertexbetween neighbouring edges in a given simplicial complex. To investigate thiscorrespondence we first need to consider the canonical quantization approachto Quantum Gravity. When the canonical quantization procedure is applied to gravity the philoso-phy is to rewrite Poisson brackets like (51), (53) or (55) as operator bracketsand represent the canonical variables on a suitable Hilbert space as multipli-cation and derivation operators. Clearly, there is a freedom in choice as towhich variables should be represented as multiplication operators and whichshould be represented as differential operators. Using Ashtekars variables itis possible to represent the holonomies
Hol C p A q as multiplication operators Hol C p A q Ñ C and the corresponding triad and flux variables ˜ E and F as differential oper-ators˜ E ma p x q Ñ E ma p x q (cid:16) ~ i δδA am p x q , F aS Ñ F aS (cid:16) » S ǫ mnp E ma dx n ^ dx p . (56)39 ǫ i Figure 7: Repeated division of an edge ǫ in the initial simplicial complex.on a suitable Hilbert space corresponding to a configuration space of connec-tions. In the following we refer to the setup used in Loop Quantum Gravity.We postpone to section 6 the construction of the Hilbert space on which theoperators C and F are represented within Loop Quantum Gravity. For nowit suffices to state that the Hilbert space is based on a projective system ofpiecewise analytic graphs. Therefore, we assume that the bracket r F aS , C s (cid:16) (cid:8) γ C τ a C , (57)where C (cid:16) C (cid:4) C is piecewise analytic, is defined as an operator bracket act-ing on a Hilbert space of functions over a configuration space of connections.We wish to show that the spectral triple p B ˜ , D ˜ , H ˜q reproduces theoperator bracket (57). To do this we will use the operators F aS to constructa new operator D . The commutator between this operator and an operator C is then shown to be identical to the commutator between the Dirac typeoperator D ˜ and an element of the algebra B ˜ .To proceed we restrict the algebra (57) to an inductive system of simplicialcomplexes t K i u and an embedding hereof φ p K i q (cid:16) T i P M . Thus, we consideronly curves C j which coincide with edges ǫ j in the some triangulation T i .To simplify matters further we first consider a single edge ǫ in the initialcomplex K . This edge corresponds to a sequence of edges t ǫ , ǫ , . . . , ǫ N u in the i ’th triangulation T i arising through N barycentric subdivisions, seefigure 7. For simplicity we set γ (cid:16)
1. For each edge ǫ j there exist a set ofsections ˆ E ji p g , . . . , g n q in the tangent bundle of the j ’th copy of G , see section3.4. Expand the corresponding generators E ji in terms of Pauli matrices E ji (cid:16) b ji,a τ a , To be exact, we here assume that the triangulation T i is piecewise analytic. i v i ǫ m ǫ n Figure 8: The surface S i is chosen to intersect only the vertex v i .and define the new operators F ji (cid:16) ¸ a b ji,a F aS j , where we introduce a set of surfaces t S k u chosen so that S k intersects thetriangulation T i at its vertices only. Choose the numbering of the surfaces sothat S k intersects the edge ǫ k at its endpoint, see figure 8. Then we obtain(no summation over repeated indices) r F ji , C j s (cid:16) C j E ji . (58)We chose S k in a way so that the sign in (58) is always positive when theorientation of the curve C k coincide with the orientation of the triangulation.The commutator (58) has the same structure as the commutator (48)between the vector field ˆ E ji and the group element corresponding to the j ’thcopy of G in G n . This suggest that the vector field ˆ E ji corresponds to theflux operator F ji F ji Ø ˆ E ji (59)when F j is restricted to the triangulation T i .Next, we define the operator D n (cid:16) n ¸ s,i a m p s q E n,si (cid:4) (cid:0)p F i , F i , F i , . . . q (cid:4) s (cid:8) , (60)where s is the sequence of signs p(cid:0) , (cid:8) , (cid:8) , . . . q corresponding to the vector E n,si and the ’ (cid:4) ’ in the bracket to the right is defined in (37). Then, the finalcommutator reads rD n , C j s (cid:16) n ¸ s,j a m p s q(cid:0) (cid:8) C j E ji (cid:8) (cid:4) E n,si . (61)41n (61) the sign on the rhs is again given by the i ’th entries in the sequences s . The commutator (61) has precisely the same form as the commutator (49).This shows that the operator D n corresponds exactly to the Dirac operator D n defined in equation (36): D n Ø D n . Notice also that the choice of surfaces S k has no importance for the defi-nition of (60). Only the intersection points, the vertices, counts. Therefore,the surfaces S k serve merely as labels of the vertices.The generalisation to the full picture where we consider the complete setof edges t ǫ k u in T i is straightforward. The corresponding operator, which weagain denote D n , is simply build from the operators associated to each edge ǫ k in the initial triangulation T . Further, we denote the limiting operatorwith D˜ . Again, the message is that the operator D˜ corresponds exactly tothe Dirac like operator D ˜ . The difference is that D˜ is written in terms ofvariables from canonical quantum gravity.This shows that the spectral triples p B ˜ , D ˜ , H ˜q constructed in section4.1 give a representation of the Poisson brackets (55). The representation isseparable and based on a system of graphs dense in M .In which sense does the representation of the Poisson algebra (55) givenby the triple p B ˜ , D ˜ , H ˜q differ by the representation used in Loop QuantumGravity? To answer this question we need a precise comparison of the gen-eral setup used in this paper with the setup used in Loop Quantum Gravity.This will be the topic of the next section.For later reference we need an expression for the square of the Dirac-likeoperator (60). We write D ˜ (cid:16) ¸ ij c j F ij F ij (cid:0) lower order , (62)where c j are constants depending on the parameters t a i u and where we bylower order refer to terms which are linear in F ij .42 Link to Canonical Quantum Gravity II. Dif-feomorphisms.
In this section we first introduce the inductive system of piecewise analyticgraphs applied in Loop Quantum Gravity. This system entail a space ofgeneralised connections much alike the space A ˜ constructed in this paper.We find that the essential difference between the two spaces is separability.This issue is closely related to the manner in which the diffeomorphism groupis treated in Loop Quantum Gravity and this paper respectively.These insights lead us to consider the role of diffeomorphism invariancein Quantum Gravity, in particular the diffeomorphism constraints found inLoop Quantum Gravity. We find that the Hilbert space H ˜ is, up to a discretesymmetry group, identical to the Hilbert space of diffeomorphism invariantstates in Loop Quantum Gravity. The following is a review of material known in Loop Quantum Gravity. Inparticular, we refer to the publications [18, 19, 34].Let M be a real analytic manifold and let P be the space of piecewiseanalytic directed paths on M . We will consider two paths in P to be the sameif they differs by trivial backtracking. P has a product simply by composingpaths. We define the space of generalised G -connections by A a (cid:16) Hom p P , G q , (63)where Hom means maps ∇ from P to G satisfying ∇ p P P q (cid:16) ∇ p P q ∇ p P q . The space A a was studied in [19].Again, we consider the space A of smooth G -connections. We repeat theargument of section 4.2: There is a natural map χ a : A Ñ A a given by χ a p ∇ qp P q (cid:16) Hol p ∇ , P q , where Hol p ∇ , P q denoted the holonomy of ∇ along P . If we are given twodifferent smooth connections ∇ , ∇ P A they are going to differ in a point,43et us say m , and hence also in a neighbourhood of m . We can thereforechoose a small directed analytic path P in a neighbourhood of m such that Hol p ∇ , P q (cid:127)(cid:16) Hol p ∇ , P q . In other words, χ a is an embedding, and hencethe terminology generalised connection is justified. A a . On M we will consider the system of connected piecewise analytic graphs.This system is directed under inclusions of graphs. Given a piecewise con-nected analytic graph Γ in M we denote by t ǫ , . . . , ǫ n p Γ qu the edges and by t v , . . . , v m p Γ qu the vertices. Let P Γ be the set of paths in Γ and define A Γ (cid:16) Hom p P Γ , G q . An inclusion of graphs Γ
Γ induces a projection P Γ Γ : A Γ Ñ A Γ . (64)Since every analytic path is a path in a piecewise analytic graph and viceversa we get A a (cid:16) lim Γ Ý A Γ , where the projective limit is taken over all piecewise analytic graphs.If we choose an orientation of the edges t ǫ , . . . , ǫ n p Γ qu we obtain again anidentification A Γ (cid:20) G n p Γ q via the map A Γ Q ∇ Ñ p ∇ p ǫ q , . . . , ∇ p ǫ n p Γ qqq P G n p Γ q . This identification gives rise to various structures on A Γ and therefore also on A a . For example the topological structure given by the topological structureof G n p Γ q . Also the projections P Γ Γ are continuous and hence give a topologicalstructure on the projective limit, i.e. A a . We note that A is dense in A a , see[22] and references therein.Another structure arising from this identification is the measure or morerelevant the Hilbert space structure. Since G is a compact group we canequip G n p Γ q uniquely with product Haar measure and define L p A Γ q (cid:16) L p G n p Γ qq . P Γ Γ induce embeddings of Hilbert spaces P (cid:6) Γ Γ : L p A Γ q Ñ L p A Γ q . With this we then define L p A a q (cid:16) lim Γ ÝÑ L p A Γ q . which is non-separable.The Hilbert space L p A a q is the Hilbert space that carries the representa-tion of the Poisson algebra (57) used in Loop Quantum Gravity. It is knownas the kinematical Hilbert space. Since an analytic diffeomorphism of M maps P to P we get an action of thegroup of analytic diffeomorphisms Dif f a p M q on A a . On the other hand thegroup of all diffeomorphisms Dif f p M q acts on A but does not extend to anaction on A a . There are several different way of completing A depending onthe choice of ”lattice”, for example piecewise analytic paths like in the casedescribed above, various kinds of smooth paths, where one gets an actionof the full diffeomorphism group. See [35, 36] for results and a thoroughdiscussion. The different completions all contain A but the crucial differenceseems to be the size of the symmetry group. A ˜ and A a So far we have introduced the three spaces A , A ˜ and A a . A is the space ofsmooth G -connections and we demonstrated that A ãÑ A ˜ , A ãÑ A a , which means that both A ˜ and A a are spaces of generalised connections.Furthermore, since it is known that A is dense in A a , and since there exist anatural surjection σ : A a Ñ A ˜ we know that A , too, is dense in A ˜ . To see how the surjection σ works recallthat both A ˜ and A a are spaces of homomorphisms A ˜ (cid:16) Hom p P ˜ , G q , A a (cid:16) Hom p P a , G q , (65)45nd since t P ˜u ãÑ t P a u then σ is just the corresponding surjection betweenthe spaces (65) of homomorphisms.To be precise we should state that we now think of the space A ˜ in termsof an embedding φ : K i Ñ T i of the projective system t K i u , and that we therefore identify A ˜ (cid:16) lim T A T . In the following we will demonstrate the relationship between and roleplayed by these three spaces of connections.First, it is important to see that the shift from A to a larger space ofgeneralised connections is necessary to equip A with topological and Hilbertspace structures. The identification of A ˜ and A a as pro-manifolds is the keystep to obtain this. The space A itself is not a pro-manifold.The key difference between the two spaces A ˜ and A a is that A a has anaction of the (analytic) diffeomorphism group; A ˜ does not. Further, theHilbert space structure associated to the two spaces are respectively non-separable for A a and separable for A ˜ . That is, the way A is completed isdecisive for how large a symmetry group remains. This shows that there is adirect link between separability of the Hilbert space structure and the actionof the diffeomorphism group.The following diagram illustrate the relationship between the three spacesof connections: A a diff a p M qÕ A σ Ó× A ˜ diff p △ q where we by diff p △ q refer to the discrete symmetries which remain on theinfinite triangulation underlying A ˜ .It is interesting that the construction of the Dirac-like operator (41) re-quires the shift from A a to A ˜ : In [20] the authors attempted to constructsuch an operator on A a and a corresponding Hilbert space. The attemptwas unsuccessful exactly because the number of possible projections (64) is we assume here that the edges in the triangulations T i are analytic. This depends onthe embedding φ : K i Ñ T i but is no real restriction. A a to A ˜ reduces the type of embeddingsbetween graphs sufficiently. We appear to find the following hierarchy: A : No Hilbert space structureNo Dirac-like operatorAction of diff( M ) A a : Hilbert space structure, non-separableNo Dirac-like operatorAction of analytic diffeomorphisms A ˜ : Hilbert space structure, separableDirac-like operatorAction of diff( △ )One way to think of the space A ˜ is to see it as the space of (smooth)connections subjected to a sort of gauge fixing of the diffeomorphism group .Here gauge fixing is meant in the sense that a symmetry group is (partly)removed while the integration at hand still involves the entire space, in thiscase the space of smooth connections A . In this sense the inner product (46)resembles a functional integral over A ”up to diffeomorphisms”.Another way to think of the space A ˜ is to relate it to the constructionof the Hilbert space H diff of diffeomorphism invariant states in Loop Quan-tum Gravity, see [23]. To do this we first notice that each triangulation inparticular is a graph, and we hence get an embeddinglim T ÝÑ L p A T q ι ãÑ lim Γ ÝÑ L p A Γ q . On the other hand consider the Hilbert space formally defined by H diff (cid:16) diffeomorphism invariant states in lim Γ ÝÑ L p A Γ q . The elaborate definition is written in [23]. There is a surjectionlim Γ ÝÑ L p A Γ q q Ñ H diff In fact, space A a is, in this line of thinking, subjected to another partial gauge fixingby reducing the symmetry group from the group of diffeomorphisms to the .group ofanalytical diffeomorphisms. T ÝÑ L p A T q Ξ ÝÑ H diff by composing. This map is also going to be a surjection since each graphcan, via a diffeomorphism, be mapped into a triangulated graph. The mapis not injective because of the symmetries of the triangulated graph.We have the following commutative diagramlim ÝÑ L p A Γ q ι Õ Ó q lim ÝÑ L p A T q Ξ ÝÑ H diff (66)The amount by which the map Ξ fails to be injective is exactly the symmetrygroup diff p △ q of discrete diffeomorphisms of triangulations. Therefore, wecan think of the Hilbert space L p A T q , and thereby also H ˜ , as Hilbert spacesof diffeomorphism invariant states up to the discrete group diff p △ q . In thispicture one should therefore think of a loop associated to a simplicial complexas a kind of equivalence class of smooth loops where the equivalence is withrespect to smooth diffeomorphisms.It is interesting that the construction of H ˜ relies on a partition of thediffeomorphism group in a ’countable’ and a ’over countable’ part. Here,the countable part is diff p △ q . This partition is a compromise between twoopposite considerations: • For the Hilbert space H ˜ to carry a representation of the loop algebrait must have an action of at least parts of the diffeomorphism groupsince the loop algebra itself is not diffeomorphism invariant. This re-quirement works to maximise the size of the remaining symmetry groupdiff p △ q . • The construction of a Dirac type operator acting on H ˜ requires a strictcontrol over permissible embeddings of the type (16). This requirementworks to minimise the size of diff p △ q .Therefore, the identification of the symmetry group diff p △ q is essential forthe construction of the spectral triple p B ˜ , D ˜ , H ˜q .Let us now return to the question regarding the different representa-tions of the Poisson algebras (53) and (55), the representation used in Loop48uantum Gravity and the representation contained in the triple p B ˜ , D ˜ , H ˜q .We find that the different representations correspond to different choices ofholonomy loops in (55). The representation used in Loop Quantum Gravityinvolves the non separable Hilbert space L p A a q whereas the representationpresented in this paper involves the separable Hilbert space L p A ˜q . Thecentral difference between the two representations lies therefore in the corre-sponding symmetry groups. The larger the class of loops is, the larger is thesymmetry group. The area operators are an important set of operators in Loop Quantum Grav-ity [24]. It turns out that the area operator also exist within the frameworkdescribed by the spectral triple p B ˜ , D ˜ , H ˜q . These operators are, however,not very natural objects to consider within this framework. However, we findthat the square of the Dirac-like operator D ˜ has a natural interpretation interms of a global area-squared operator. Classically, the area of a 2-dimensional surface S in Σ is given by A p S q (cid:16) » S ? dF a (cid:4) dF a . where the area element dF a was defined in (54). To convert this expressioninto a form suitable for quantization consider a partition of S into N smallersurfaces S n such that for any N we have ” n S n (cid:16) S . Then the area of S canbe written A p S q (cid:16) lim N Ñ8 N ¸ n (cid:16) b F iS n F jS n δ ij . (67)In Loop Quantum Gravity, the area operator is constructed by substitutingthe classical flux variable F iS n with the corresponding operators F iS n A p S q (cid:16) lim N Ñ8 N ¸ n (cid:16) b F iS n F jS n δ ij . (68)49he point here is that this operator, at the level of a given graph, is welldefined. This is due to the fact that the number of intersections of any graphwith the surface S is finite. Continued subdivisions of S are obsolete once theresolution is reached where each S n contains one intersection point. If thearea element S n does not involve an intersection with the graph Γ then thecorresponding operator F iS k vanishes. This means that the surface S has aminimal subdivision into elementary cells S n each containing one intersectionpoint with the graph. The area operator obtains the form A p S q (cid:16) ¸ i A p p i q , where the sum runs over intersection points p n and where A p p n q (cid:16) b F jS n F kS n δ jk . (69)The operator A p p n q basically assigns an area to the intersection point p n .The spectrum of the area operator A p S q , at the level of a given graph Γ, canbe computed and reads [15]Spec p A p S qq (cid:16) πγl P l ¸ p b j p p j p (cid:0) q+ , where p are intersection points between S and the graph Γ and j p are positivehalf integers (for details and subtleties we refer to [15]). l P l is the Plancklength.
The spectral triple p B ˜ , D ˜ , H ˜q involves another representation of the oper-ator algebra (57), now based on curves in the inductive system of triangu-lations. Therefore, it is possible to repeat the line of reasoning from LoopQuantum Gravity and define a second area operator, now based on the Dirac-like operator D ˜ and the vectors ˆ E ji .Recall the interpretation (59) which relates the vectors ˆ E ji to the fluxoperators F ji . This leads to an area operator A ˜p S q (cid:16) lim N Ñ8 ¸ n b F iS n F jS n δ ij . (70)50here F iS n are as explained in section 5.2.The difference between the operators (68) and (70) lies not in the expres-sion of the operators themselves but rather in the Hilbert spaces on whichthey act. The former acts on a non-separable Hilbert space whereas the latteracts on the separable Hilbert space H ˜ .The area operator (70), however, does not appear to be a natural objectin the construction presented here. It involves a surface S which has nonatural place in this setting.We can nevertheless apply the line of reasoning from Loop QuantumGravity to obtain a better understanding of the operator D ˜ . Consider againthe interpretation of D ˜ in terms of conjugate variables of canonical gravity.If we combine equation (62) with equation (69) we obtain D (cid:16) ¸ i c i A p v i q (cid:0) . . . (71)where, in the limit, the sum runs over all vertices in the inductive system oftriangulations t T i u . It is clear that the set of all vertices is a dense set M .Therefore, the sum over all vertices, weighted with the sequence t a i u and thea multiplicity factor given by the valency of the vertex, defines an integralover M ¸ v i ... Ñ » M d Vol . This provides us with a surprising interpretation of D and thereby of D ˜ .According to (71) it is an operator which is related to the area of the entiremanifold M D ˜ (cid:18) » M (cid:1) A p x q(cid:9) (cid:0) ... (72)Clearly, the spectrum of ∆ ˜ is discrete. The identification of the operator D ˜ as a global operator related to the areaof the underlying manifold leads us to a new interpretation of the spectraltriple p B ˜ , D ˜ , H ˜q . In this section we suggest that the operator D ˜ should bethought of as an action. Subsequently, we point out that the spectral action51f D ˜ resembles a partition function. The argumentation presented in thefollowing is tentative.The first indication that p D ˜q may be understood in terms of an action is,as already mentioned, that is has the form of an integral over an underlyingmanifold.The second indication is directly related to the classical Einstein-Hilbertaction. Within the language of Noncommutative Geometry the Einstein-Hilbert action has a natural interpretation as an area of the underlying man-ifold. With the previous section in mind this suggest that the operator p D ˜q is somehow related to the Einstein-Hilbert action.Let us go into some details. Consider a 4-dimensional, Riemannian spin-geometry described by the real, even spectral triple p A, D, H q where A is acommutative C (cid:6) -algebra. The Eucledian Einstein-Hilbert action on M canbe computed directly from D . It is proportional to the Wodzicki residue ofthe inverse square of the Dirac operator D [37]. If we denote by T r (cid:0) theDixmier trace and define the noncommutative integral (cid:31) by f : (cid:16) T r (cid:0) f | D |(cid:1) one has D (cid:16) (cid:1) π » M R ? gd x , where dv (cid:16) ? gd x is the volume form and R is the scalar curvature. Itis in this sense that one may interpret the Einstein-Hilbert action as the”two-dimensional measure of a four manifold”, the ”area” of M [38].This resembles the findings of the previous section where we saw that theoperator D ˜ has a natural interpretation in terms of a global area-squaredoperator over an underlying manifold. If we ignore the obvious issue of ’area’vs. ’area-squared’ one may speculate whether there is some deeper relationbetween the operator D ˜ and the Einstein-Hilbert action.These considerations entail an interesting interpretation of the spectralaction of D ˜ . Consider the quantity T r exp p(cid:1) sD ˜q , (73)where s is a real parameter, and let us perform a formal calculation to obtain52 better understanding of this quantity. We write T r exp p(cid:1) sD ˜q (cid:16) ¸ ψ n x ψ n | exp (cid:0)(cid:1) sD ˜(cid:8)| ψ n y(cid:16) » A ˜ r d ∇ sx δ ∇ | exp (cid:0)(cid:1) sD ˜(cid:8) | δ ∇ y(cid:16) » A ˜ r d ∇ s (cid:2)» A ˜ r d ∇ s δ ∇ p ∇ q exp (cid:0)(cid:1) sD ˜ δ ∇ (cid:8) p ∇ q(cid:10)(cid:16) » A ˜ r d ∇ s exp (cid:0)(cid:1) sD ˜ δ ∇ (cid:8) p ∇ q , (74)where t δ ∇ u is the orthogonal set of delta functions on the space A ˜ andwhere r d ∇ s denotes the measure on A ˜ introduced in section 4. The finalexpression in (74) may, strictly speaking, not make perfect mathematicalsense. However, we know that the initial quantity (73) is well defined. Thisformal calculations shows that the spectral action (73) has the form of aformal Feynman integral where the expression D ˜ δ ∇ plays the role of a classical action. t a n u as dynamicalvariables Up till this point we have constructed the spectral triple p B ˜ , D ˜ , H ˜q andrelated it to Quantum Gravity. We have shown that it contains informationof the Poisson brackets of canonical gravity and that it partly incorporatesdiffeomorphism invariance. Two questions remains to be addressed:1. The spectral triple relies on the divergent sequence t a i u . What struc-ture does this sequence represent and how do we deal with it?2. How do we incorporate the remaining diffeomorphisms contained in thegroup diff p △ q .One solution to the first question would be to fix the sequence t a i u in amanner which counts a partition of an edge with a factor . However, we53ill argue that there may be another solution which is more natural. Thissection is concerned with this issue.In section 5 we demonstrated that the triple p B ˜ , D ˜ , H ˜q has a naturalinterpretation in terms of Poisson brackets of canonical gravity. In this pic-ture the Dirac type operator D ˜ , including the sequence t a i u , is a linear sumof the flux operators involving the densitised vielbein operator, see equation(56). This suggest an interpretation of the parameters a i in terms of thedeterminant of the vielbein a i (cid:18) det p e am q . That the sequence t a i u carries metric information also comes out of thediscussion in subsection 4.5 where we relate the sequence t a i u to the scalingbehaviour of the construction.Thirdly, in section 7 we found that the square of D ˜ has the form of anintegral over the underlying manifold. Here, the measure involved in thisintegral is given by the sequence t a i u .This suggests, as a possible solution to the first question, that we shouldinclude the sequence t a i u in the construction as dynamical variables. If thesequence t a i u has a metric origin, then it seems natural to try to integrateover these degrees of freedom.Therefore we extend the spectral triple p B ˜ , D ˜ , H ˜q to include the se-quence t a k u as dynamical variables. By doing this we obtain a new triple,denoted p B t , D t , H t q , which is a fibration of triples p B ˜ , D ˜ , H ˜q over the spaceof permissible sequences t a k u . It turns out that there is a way to obtain thiswhich leaves the spectral triple p B t , D t , H t q with very few free parameters.This section gives a presentation of ideas and methods used to constructthe triple p B t , D t , H t q . A detailed account of the construction will be pre-sented elsewhere.To emphasise the dependency on the sequence t a i u we shall in the follow-ing write D ˜p a i q instead of D ˜ . t a i u We would like to think of the parameters a i as a coordinates in a space H ofpermissible sequences t a i u (that is, sequences that satisfy the condition givenby (42)) and to define a Dirac operator on this space. Clearly H € R . Thesequence t a i u is characterised by the condition that it diverges sufficiently54ast. This means that the inverse sequence t a (cid:1) i u converges towards zero:lim i a (cid:1) i (cid:16) . (75)This convergency condition is easier to work with than condition (42) sinceconvergent sequences can be understood in terms of a Hilbert space structure.Therefore, in a first step we consider L -sequences. That is, sequences t x k u (cid:16) p x , . . . , x n , . . . q P R which satisfy }t x k u} : (cid:16) ¸ k x k . (76)The space of sequences satisfying (76) is an infinite dimensional, real, sepa-rable Hilbert space which we also denote H .The exact relationship between vectors t x k u in H and the sequences t a k u satisfying condition (42) will be determined in the following.In order to construct a Dirac operator on H we first need a Hilbert spacestructure over H . The goal is to construct a spectral triple over H .The techniques used to construct L p H q and the corresponding spectraltriple are essentially the same techniques we used to construct the space A ˜ and the triple p B ˜ , D ˜ , H ˜q . The following is based on ideas by Higson andKasparov, see [25].The strategy is to consider first finite dimensional subspaces H n of H ,corresponding Hilbert spaces L p H n q and structure maps P (cid:6) m,n : L p H n q Ñ L p H m q , n ¤ m between Hilbert spaces. Next step is to construct a spectral triple over eachspace H n , ensure compatibility with the structure maps P (cid:6) m,n and finallyobtain a triple over H by taking the limit n Ñ 8 .There are two potential difficulties with this strategy: • The spaces H n are non-compact. The construction of the spectral triple p B ˜ , D ˜ , H ˜q relies strongly on the fact that G is compact. This is anecessary condition to construct the structure maps P (cid:6) (24) betweenHilbert spaces. Specifically, we used the fact that the constant functionis square integrable. This is no-longer the case when the underlyingspace is non-compact. 55 The Dirac type operator, which we aim to construct, will have a non-compact resolvent. This is the same problem we encountered whenwe constructed the operator D ˜ and a problem which will always arisewhen one constructs a Dirac like operator on an infinite dimensionalspace (see also [25]). This problem was solved in section 3 by intro-ducing the parameters t a i u , whose presence is the very reason why wenow wish to construct the extended triple p B t , D t , H t q . Therefore, itappears that we are in danger of a circular argument: Certainly, we donot wish to introduce yet another infinite sequence to ensure a well be-haved Dirac operator acting on L p H q entailing yet another extensionof p B t , D t , H t q and so forth.It turns out that there exist a way to construct the triple p B t , D t , H t q which avoids these two technical pitfalls.The first problem is related to the fact the canonical map from H n to H m is given by ξ Ñ ξ b n,m , where 1 n,m is the identity function on the m (cid:1) n dimensional orthogonal complement to H n in H m . However, the function1 n,m is not a L -function and therefore the canonical map is not a mapbetween Hilbert spaces. We can remedy this defect by finding a suitable L -function φ n,m to replace the constant function 1 n,m . The choice of function φ n,m is restricted by the requirement that it must lie in the kernel of theDirac type operator, which we wish to construct, when this is restricted tothe orthogonal complement to H n in H m .The second problem is related to the fact that we are dealing with aninfinite dimensional space. To solve this problem we modify the canonicalalgebra of functions on H n to include operators which project onto finitedimensional subspaces. R We first consider finite dimensional subspaces H n (cid:16) R n embedded in H by H n Q p x , . . . , x n q Ñ p x , . . . , x n , , . . . q P H .
Before we proceed with the construction we introduce some notation anddefinitions. The Gaussian functions on R φ α p x q (cid:16) p πα q { exp (cid:2) (cid:1) x α (cid:10) , α is a real positive number, is a square integrable normalised functionon R and the composition φ α p x q (cid:4) . . . (cid:4) φ α p x n q (cid:16) p πα q n { exp (cid:3)(cid:1) n ¸ i (cid:0) x i α (cid:11) is square integrable and normalised on R n . Define also φ n,m p x n (cid:0) , . . . , x m q (cid:16) φ α p x n (cid:0) q (cid:4) . . . (cid:4) φ α p x m q . This is a square integrable and normalised function on the orthogonal com-plement H n,m of H n in H m .Given a function ξ P L p H n q define a function P (cid:6) n,m p f q , n m , in L p H m q by P (cid:6) n,m p f qp x , . . . , x m q (cid:16) f p x , . . . , x n q (cid:4) φ n,m . (77)Note that P (cid:6) n,m defines an embedding of Hilbert spaces. We can thereforedefine the limit L p H q : (cid:16) lim Ñ L p H n q , which is a real, separable Hilbert space over H .Note that, as mentioned in the previous section, the Gauss distributionsplay the same role in the construction as the constant function did in theconstruction of H ˜ . This solves the first pitfall mentioned above.Since we wish to construct a Dirac type operator we need the Cliffordbundle. We define L p H, Cl p H qq : (cid:16) lim Ñ L p H n , Cl p H n qq , where we have used the unital embedding Cl p H n q Ñ Cl p H m q , n m .We identify Cl p H n q with the exterior product Λ (cid:6) H n and define the oper-ators ext p e q (cid:16) exterior multiplication by e on Λ (cid:6) H n , int p e q (cid:16) interior multiplication by e on Λ (cid:6) H n ,c (cid:8)p e q (cid:16) ext p e q (cid:8) int p e q . which means that c (cid:1)p e q is Clifford multiplication with e and c (cid:0)p e q is Cliffordmultiplication with e except that e (cid:16) e (cid:16) (cid:1)
1. Furthermore, oneeasily checks that t c (cid:0)p e q , c (cid:1)p e qu (cid:16) e i to be the vector which is 1 and the i ’th place and zero elsewhere.Define the operator D n on L p H n , Cl p H n qq by D n (cid:16) n ¸ i (cid:16) (cid:2) c (cid:1)p e i q BB x i (cid:0) c (cid:0)p e i q x i α (cid:10) . (78)This operator is known as the Bott-Dirac operator and its construction isdue to Higson and Kasparov [25]. It is proven in [25] that it is self adjointwith compact resolvent. It satisfies D n φ ,n p x , . . . , x n q (cid:16) . In fact, the kernel of D n is one dimensional and given by the function φ ,n p x , . . . , x n q . From this we get that P (cid:6) n,m p D n p ξ qq (cid:16) D m p ξ q and we obtain a self adjoint unbounded operator D on L p H, Cl p H qq .The square of D n is D n (cid:16) ¸ i (cid:2)(cid:1) BB x i BB x i (cid:0) x i x i α (cid:10) (cid:1) nα (cid:0) α N , (79)where N is the operator which assigns a differential form its degree. The firstpart of (79) is the harmonic oscillator Hamiltonian which has the spectrum0 , { α, { α, . . . .As anticipated, the operator D does not have compact resolvent sincethe group of all finite perturbations of N acts on its spectrum. A possiblesolution to this problem is to modify the canonical algebra of functions actingon L p H, Cl p H qq . In stead of functions on L p H, Cl p H qq we consider insteadfunctions on finite subspaces of L p H, Cl p H qq and extend these to the fullHilbert space by tensoring them with projection onto Gauss functions. Thus,effectively we include cut-off’s in the algebra. This will ensure that theresolvent of D , understood in terms of an interaction with the algebra, iscompact.Define Φ n,m : (cid:16) H n b P φ n,m P B p L p H m q , Cl p m qq , where P φ n,m is the orthogonal projection onto φ n,m in H n,m . Let A n bethe algebra of operators in L p H n , Cl p H n qq generated by tensor products ofelements in C p R q b Cl p q and Φ k,l . Note that A n embeds in A m , n m , by A n Q a Ñ a b H n,m (cid:4) Φ n,m . A (cid:16) lim Ñ A n . In particular, A consist of operators on L p H, Cl p H qq . Since the operatorsin A contain a cut-down to the Gauss distribution from a certain point on,we see that p A, D, L p H, Cl p H qqq is a spectral triple since a p λ (cid:1) D q(cid:1) , a P A , λ R R is a compact operator.Note that the semi-finite trick with the Clifford bundle, which workedfor the spaces of connections does not seem to work in this case, since thekernel of D is one dimensional. Therefore there is no symmetry which canbe discarded of. θ -summability We would like to interpret condition (76) in terms of θ -summability of theoperator D ˜p a i q . That is, we wish to establish a relation between sequences t x i u satisfying (76) and sequences t a i u which leave T r exp (cid:0)(cid:1) D ˜p a i q(cid:8) (80)finite. Thus, we seek a function f : R R which satisfies T r exp (cid:0)(cid:1) D ˜p a i q(cid:8) 8 f pt a i uq P l p N q . (81)This emphasis on θ -summability has a clear physical motivation. In sec-tion 8 we noted that the quantity (80) has the form of a path integral. Inthis light θ -summability simply means that this path integral is finite. U p q -case As a toy example let us consider the U p q -case and the simplified operator D a i q on U p q8 which is the limit of operators of the form D n : (cid:16) D (cid:0) a D (cid:0) . . . (cid:0) a n D n , D i is the Dirac operator on the i ’th copy of G . With this operator weconsider expression (80) and calculate T r exp (cid:0)(cid:1)p D ˜p a i q(cid:8) (cid:16) (cid:3) ¸ n P N exp (cid:1)(cid:1) ¸ a i n i (cid:9)(cid:11)(cid:16) Π n T r exp (cid:0)(cid:1) a n D (cid:8) , where D is the Dirac operator on U p q . Taking the logarithm we obtainln T r exp (cid:0)(cid:1)p D a i qq (cid:8) (cid:16) ¸ n ln T r exp (cid:0)(cid:1) a n D (cid:8)(cid:16) ¸ n ln ¸ k exp (cid:0)(cid:1) a n k (cid:8) . (82)Thus T r exp (cid:0)(cid:1)p D a i qq (cid:8) 8 if and only if ¸ n ¸ k ¡ exp (cid:0)(cid:1) a n k (cid:8) 8 . If we define the function f p a q (cid:16) sgn p a qd¸ k ¡ exp p(cid:1) a k q , we see that, for G (cid:16) U p q and for the modified operator, θ -summability of D a i q is directly related to the Hilbert space condition ¸ n f p a n q . This means that the sequence t a i u is related to the Hilbert state t x i u P H through the relation x i (cid:16) f p a i q . (83)60 .5 The triple p B t , D t , H t q We are now ready to combine the two spectral triples p B ˜ , D ˜ , H ˜q and p A, D, L p H, Cl p H qqq . A point t x n u in H gives a θ -summable spectral tripleon B ˜ via the Dirac operator on H ˜ defined by the sequence f (cid:1) pt x n uq . Letus denote this Dirac operator by D f (cid:1) pt x n uq . We define the operator D t actingon H t : (cid:16) L p H, Cl p H qq b H ˜ by D t p ξ b η qpt x n uq (cid:16) D p ξ q b η pt x n uq (cid:0) p(cid:1) q deg p ξ q ξ b D f (cid:1) pt x n uqp η qpt x n uq , (84)where deg p ξ q means the degree of ξ with respect to the degree in CL p H q .One of the requirement that this defines a semi-finite spectral triple over A b B ˜ : (cid:16) B t is that a (cid:0) D t , for all a P B t , is τ -compact, where τ is a trace defined in (44) tensored with the trace on B p L p H, Cl p H qqq where B denotes the bounded operators.We see that this requirement collide with the following symmetry prop-erty: All a P B t are from a certain step of the form b P φ n, and therefore the operator a (cid:0) D t is invariant under symmetries permuting all coordinates bigger than n .In order to remedy this we modify the construction of the spectral triple p A, D, L p H, Cl p H qqq slightly. This modification is based on the observationthat the parameter α in the Gauss distribution φ α p x q is in fact a free pa-rameter. Further, there are an infinite number of these free parameters inthe construction of the triple p A, D, L p H, Cl p H qqq , one for each dimensionin H . Let us denote these with t α i u . By choosing these parameters carefullywe can solve the problem mentioned above.For D t to have a T r -compact resolvent its eigenvalues must approachinfinity. However, for each copy of G we know that its eigenvalues scale with61 α i q(cid:1) . Thus, if we require the sequence t α i u to satisfylim i Ñ8 α i (cid:16) , then the triple p B t , D t , H t q may be spectral.Let us pause to comment on this idea. It appears that we have sim-ply exchanged one infinite sequence of free parameters t a i u with anotherinfinite sequence t α i u . However, we suggest that the sequence t α i u is not ar-bitrary but should be determined completely from symmetry considerations.The idea is that the α i ’s represent an “average background” over which theparameters a i are allowed to fluctuate. Therefore, we suggest to choose aspecific sequence t α i u satisfying the following requirements: • Copies of G which correspond to the same level in the projective systemof graphs should be assigned the same α i ’s. • Given two copies of G where one copy G corresponds to a subdivisionof the edge associated to the other copy G , then the correspondingparameters α , α should satisfy α (cid:16) α . The first point reflects a rotational symmetry within each graph. Thesecond point is related to the scaling properties of the construction: a divisionof a line segment corresponds to a factor (see section 4.5 and the discussionof the role of the parameters t a i u ).To implement this we enumerate the barycentric subdivisions with k andintroduce the modified Gauss distributions φ k p x q (cid:16) p πα k q { exp (cid:2) (cid:1) x α k (cid:10) , where α k (cid:16) (cid:1) k α . (85)This means that we associate to each edge in the simplicial complexes aparameter α k . In the initial complex each edge is associated the parameter α (cid:16) α . Edges arising through the first barycentric subdivision are thenassociated the parameter α (cid:16) α { p A, D, L p H, Cl p H qqq constructed with with thesenew Gauss distributions. This means that the Hilbert space embeddings (77)are modified together with the Bott-Dirac operator (78) which is now of theform D n (cid:16) N ¸ k (cid:16) n k ¸ i (cid:16) (cid:2) c (cid:1)p e i q BB x i (cid:0) c (cid:0)p e i q x i α k (cid:10) , where n (cid:16) ° k n k and where the first sum runs over the number of barycentricsubdivisions involved in the graph to which D n is associated, N being thetotal number of barycentric subdivisions. The second sum runs over thenumber of edges within a barycentric subdivision.One now repeats the above construction leading to the operator (84), nowwith the modified Bott-Dirac operator.The details concerning the spectral properties of the triple p B t , D t , H t q will appear elsewhere.
10 Distances on A ˜ On a Riemannian spin-geometry the Dirac operator D contains the geomet-rical information of the manifold M . In particular, distances can be formu-lated purely algebraically due to Connes [1]. Given two points x, y P M theirdistance is given by d p x, y q (cid:16) sup f P C M q | f p x q (cid:1) f p y q| }r D, f s} ¤ ( . (86)On a Noncommutative Geometry the state space replaces the notion ofpoints. It is possible to extend the notion of distance to the state spaceby generalising (86).In the present case it is natural to ask whether the Dirac-like operator D ˜ introduces a metric structure over the space A ˜ and if so, how distancesshould be interpreted in this setting.The possibility for a distance between smooth connections in A and someclosure hereof was first considered in [20]. There, however, the distancebetween two smooth connections was found to be infinite, since two smoothconnections will differ on infinitely many different loops. This entails a finitecontribution to the distance from infinitely many copies of G .In the present setting the situation is different due to the sequence t a i u .The role of the a i ’s is to assign a weight to each copy of G and therefore to63cale the corresponding distance on G . Therefore, given two points in A ˜ their distance d p ∇ , ∇ q (cid:16) sup a P B ˜ } ∇ p a q (cid:1) ∇ p a q} }r D ˜ , a s} ¤ ( , ∇ , ∇ P A ˜ , (87)where we choose }(cid:4)} as the operator norm on matrices in M l p C q , is well definedand finite, even for ∇ , ∇ P A . This is related to the fact that although ourgeometry on A ˜ is infinite dimensional it assigns A ˜ zero volume.The distance (87), when it is applied to smooth connections, is not inde-pendent of the embedding h of simplicial complexes into triangulations.With a metric structure on the space A ˜ it is natural to ask which con-nections are ’close’ to each other. Clearly, this depends on the choice of thesequence t a i u . However, it is possible to give some general remarks indepen-dently of this choice.Recall that the space A ˜ is the limit space lim n G n in an appropriate sense.The distance between ∇ , ∇ P A K i is simply the sum of geodesic distanceson each copy of G , weighted with the appropriate parameters a i . In theprevious section we found that the role of the parameters t a i u is to scalethe different copies of G according to their location in the projective systemof simplicial complexes. The larger a i ’s corresponds to short distances, nowwith respect to some manifold. This means that the the distance between ∇ and ∇ is larger if the connections differ mostly on those copies of G which are assigned small a i ’s. Correspondingly, the distance between ∇ and ∇ is smaller if ∇ and ∇ differ mostly on those copies of G which areassigned large a i ’s. Therefore, in general, the two generalised connectionswill be ’close’ to each other if they differ mostly at short scales.Via Levi-Civita connections we can interpret equation (87) as a distancebetween geometries . Again we can say that the distance between two ge-ometries depends on the scale on which they differ.Let us consider the Abelian case G (cid:16) U p q and two points ∇ , ∇ in A ˜ .In this case ∇ and ∇ are given by sequences of angles t θ i u and t θ i u whereeach angle θ ji corresponds to points exp p π i θ ji q on the i ’th copies of U p q . Letus for simplicity choose a coordinate system on G n so that θ ji corresponds tothe parameter a i . Thus, for example, if we consider G (cid:2) G parametrised byangles φ and φ we choose θ (cid:16) φ (cid:0) φ , θ (cid:16) φ (cid:1) φ In this case, however, the distance formula will be degenerate since different geometriesmay have identical Levi-Civita connections. ∇ and ∇ reads d p ∇ , ∇ q (cid:16) ¸ k k a k π | θ k (cid:1) θ k | , which should be read together with the condition (75).The notion of a distance on a space of connections is not a new one butwas discussed already by Feynman [39] and Singer [40]. See also [41] and ref-erences herein. However, these papers all deal with distances between gaugeequivalence classes of connections. The construction presented in this paperis quite different from these previous works since two connections which differby a gauge transformation will, in general, have a non vanishing distance.
11 Discussion
Before we end this paper we give a few remarks concerning the physicalinterpretation of the constructions presented.
As already mentioned, the shift to Ashtekar variables permits a formulationof General Relativity which is close to ordinary Yang-Mills theory. In theHamilton formulation the theory involves a configuration space of connec-tions corresponding to ordinary SU p q Yang-Mills theory. The essential dif-ference between General Relativity and Yang-Mills theory lies in the algebraof constraints which encode the symmetries of the theory. For General Rel-ativity the constraints encode diffeomorphism invariance. These constraintsare closely related to the foliation of the spacetime M according toΣ (cid:2) R Ñ M , (88)where Σ is a 3-dimensional hypersurface. Loosely stated, the constraintsencode information about diffeomorphisms within and perpendicular to Σ.Therefore, the key question in any quantization procedure of gravity is theimplementation of diffeomorphism invariance in the construction.To quantize a constrained theory there are in general two standard ap-proaches: 65 ) To eleminate first, at a classical level, the constraints and thereby findthe reduced phase space of the theory. The quantization procedure isapplied to the reduced phase space. b ) To construct first quantum kinematics for the full phase space by ig-noring the constraints, then construct operators corresponding to theclassical constraints and finally solve the quantum constraints to obtainthe physical Hilbert space.Loop Quantum Gravity follows the second approach which is due to Dirac(see [42] for a detailed exposition). In section 5.2 we demonstrated that thespectral triple p B ˜ , D ˜ , H ˜q involves a representation of the Poisson algebra(55) of General Relativity. In our interpretation, the construction shouldbe understood as a quantization scheme which lies somewhere between theapproaches a ) and b ).First of all, there is nothing in the geometrical construction that corre-sponds directly to the constraint algebra of General Relativity. However, insection 6 we related the Hilbert space H ˜ to the diffeomorphism invariantHilbert space H diff of Loop Quantum Gravity. The difference between H ˜ and H diff is the group diff( △ ) of discrete diffeomorphisms on the system ofsimplicial complexes. Therefore, we think of the spectral triple p B ˜ , D ˜ , H ˜q as a geometrical construction describing a quantization procedure of gravityon a Hilbert space which corresponds to a partial solution to the (spatial)diffeomorphism constraint. What remains then, in this picture, is the imple-mentation of the remaining diffeomorphisms encoded in the discrete groupdiff( △ ).The second point in which our construction differs from canonical quan-tum gravity is the issue of time. Clearly, a foliation of the manifold is essentialfor canonical gravity since the Hamilton formulation is based on a choice ofan explicit time direction. In contrast to this, the construction presented heredoes not a priori involve a foliation of space-time. Further, the dimension ofthe underlying space is completely free. The only restriction is the choice ofsymmetry group, which, for now, has to be compact . In particular, we canconsider triangulations of a 4-dimensional manifold.These observations suggest two possible interpretations of the construc-tion: First, the construction may be interpreted as a completely covariantconstruction in four dimensions, involving the full, four-dimensional group The case G (cid:16) SO p , q , for example, is not permitted.
66f diffeomorphisms. This interpretation leaves no room for a foliation andthus no Hamiltonian constraint. Therefore, a dynamical principle replacingthe Hamiltonian constraint must be sought elsewhere. This interpretationimplies that the group G corresponds to four-dimensional local Lorentz rota-tions. Next, if the loop algebra lives on a four-dimensional manifold, then theinterpretation of the Dirac type operator D ˜ in terms of canonical quantumgravity must involve a flux operator F aS where S is now a three-dimensionalhypersurface. Further, the interaction between D ˜ and the algebra of loopsshould be understood as a four-dimensional operator algebra containing arepresentation of the Poisson algebra of General Relativity as a kind of sub-algebra.Alternatively, we may interpret the construction as fundamentally three-dimensional, with G equal to SU p q . This interpretation does not necessarilymean that we consider a foliation of space-time according to (88). Rather,we would like time to emerge naturally from the construction.Indeed, let us end this discussion with the remark that the philosophy ofNoncommutative Geometry is to seek a time within the algebraic construc-tion rather than imposing it a priori. Here we think of the Tomita-Takesakitheory [43] which identifies a one-parameter automorphism group uniquelyup to inner automorphisms. It is an interesting question whether this groupof modular automorphisms is nontrivial in this or perhaps some similar con-struction. If the answer is in the affirmative, then this might provide us withan alternative to the foliation and thereby with a new dynamical principle. p △ q and the question of constraints Regardless of how one interprets the spectral triple p B ˜ , D ˜ , H ˜q it is clearthat one should seek to obtain invariance under the discrete group diff p △ q ofgraph preserving diffeomorphisms.Traditionally, given some space M , invariance under a symmetry group G is obtained by taking the the quotient M { G . However, Noncommutative Geometry gives an alternative approach to quo-tient spaces. A simple example is the space of two points identified. WithinNoncommutative Geometry this setup is described via two by two matriceswhere the off-diagonal entries represent the identification of the points. This67ntails a noncommutative algebra and a structure which, ultimately, leadsto the Higgs mechanism in the noncommutative formulation of the StandardModel.In the case of gravity the relevant symmetry group is the diffeomorphismgroup. This group is probably too large for an application of the noncom-mutative approach. However, in the present setup we have instead the muchsmaller group diff p △ q and one could speculate if the machinery of Noncom-mutative Geometry of quotient spaces could be successfully applied. Thisidea differs fundamentally from a Dirac-type quantization procedure.Thus, what we suggest is to obtain a formal diffeomorphism invarianceby considering the semi-direct product of the loop algebra with the groupdiff p △ q B ˜ (cid:11) diff p △ q , and thereafter building a spectral triple with an associated Hilbert space L p A q b L p diff p △ qq . Presumably, such a construction will give rise to addi-tional degrees of freedom through fluctuations around the Dirac type opera-tor. In section 6.2 we argued that the space A ˜ should be viewed as a space ofconnections subjected to a gauge fixing of the diffeomorphism group. Naively,it appears that this gauge fixing does not involve any choice of metric orbackground ’field’ on the manifold. Let us comment on this.The construction in section 2 depends primarily on the initial simplicialcomplex K o . The idea is that the basic data entering the construction is thetopology of the corresponding manifold. The initial complex gives rise to aninitial triangulation T o via a homeomorphism h : K Ñ M . This initial triangulation introduces a metric structure on M . The embed-ding A ãÑ A ˜ will, of course, depend crucially on the triangulation and inparticular on the homeomorphism h . However, the construction and spec-trum of the Dirac-type operator is independent of h and so is its interactionwith the algebra and the construction of the Hilbert space. In fact, the no-tion of a manifold may be left out altogether. It is only the identification of A ˜ as a space of generalised connections that requires a manifold.68nother question is the dependency on the sequence t a i u . The Diractype operator D ˜ depends crucially on this sequence as does its spectrum.Thus, one may argue that some degree of background dependency enters theconstruction with the sequence t a i u . In section 9 we attempted to free theconstruction of this dependency by making the a i ’s dynamical. The pricepaid, however, is the introduction of a new sequence t α i u determining theGauss distributions. Therefore, it seems that the construction, whether weconsider the triple p B ˜ , D ˜ , H ˜q or the triple p B t , D t , H t q , does posses somedegree of dependency on a parameter space which one may interpret in termsof a background. The exact nature and implication of this dependency is tobe clarified. The construction of the triples p B ˜ , D ˜ , H ˜q and p B t , D t , H t q takes canonicalquantum gravity as its point of departure. Thus, both spectral triples are apriori of purely gravitational origin. However, there are several indicationsthat the framework presented in this paper contains additional degrees offreedom, both bosonic and fermionic.First, recall that the framework of Noncommutative Geometry generallyinvolves fermionic degrees of freedom since it involves a Dirac type operatoracting on a Hilbert space . The ’fermions’ involved in the spectral triple p B ˜ , D ˜ , H ˜q are clearly very different from the fermions of the StandardModel since they live on a space of connections. However, this space of con-nections is of course linked to an underlying manifold. A classical limit willpresumable involve some kind of delta function on the space of connections(see next section for a discussion hereof) and therefore leave only space-timedegrees of freedom for the fermions. Therefore the interesting question iswhat structures may emerge from these ’fermions’ in a classical limit.Furthermore, during the construction of the the triple p B ˜ , D ˜ , H ˜q wefound that the CAR algebra emerged in an almost canonical fashion. Itseems plausible that almost any construction of a Dirac type operator onan infinite dimensional space will naturally entail an infinite dimensionalClifford bundle which, in turn, leads to the CAR algebra. Thus, it seemsthat Noncommutative Geometry provides us with a mechanism which equips For a commutative algebra the underlying space is a Riemannian geometry and theDirac operator acts on a Hilbert space of spinors. In the noncommutative formulation ofthe Standard Model the Hilbert space is labelled by the fermions in the Standard Model.
69 purely gravitational setting, the construction of a Dirac type operator onan infinite dimensional space of field configurations, with basic elements offermionic Quantum Field Theory. In section 4.1 we found that the Hilbertspace H ˜ factorizes into H ˜ (cid:16) H ˜ ,b b H ˜ ,f , where the CAR algebra acts on H ˜ ,f . This shows that H ˜ naturally involvesboth bosonic and fermionic sectors.Finally, let us consider the possibility that the noncommutativity of theloop algebra will generate an additional bosonic sector through the innerautomorphisms of the algebra. To explain this consider first a classical finite-dimensional, real spectral triple p A, D, H q . A noncommutative algebra A contains inner automorphisms of the form α u p x q (cid:16) uxu (cid:6) x P A , where u is an arbitrary element of the unitary group, uu (cid:6) (cid:16) u (cid:6) u (cid:16)
1. Achange of representation of the algebra A from π to π (cid:5) α (cid:1) u is equivalent tothe replacement of the Dirac operator D by˜ D (cid:16) D (cid:0) A J A J (cid:1) , (89)where A u r D, u (cid:6)s is a noncommutative one-form, and where J is a realstructure. In the noncommutative formulation of the Standard Model cou-pled to gravity [2] the entire bosonic sector of the Standard Model, includingthe Higgs boson, is generated by this type of fluctuations of the Dirac oper-ator by a general one-form A a i r D, b i s with a i , b i P A .This mechanism is general. The interesting question is what kind ofbosonic sector the inner automorphisms of the spectral triple p B ˜ , D ˜ , H ˜q will generate.The general idea behind these remarks is that pure Quantum Gravity may,as a ”free” spin-off, contain matter degrees of freedom. Thus, we believethat one should not attempt to couple matter degrees of freedom to theconstruction presented in this paper but rather hope to see matter emergedynamically. 70 In this paper we establish a link between the mathematics of Noncommu-tative Geometry and the field of canonical quantum gravity. We constructa spectral triple p B ˜ , D ˜ , H ˜q over a space of connections and show that thePoisson structure of General Relativity, formulated in terms of loop variables,is encoded in the interaction between the Dirac type operator D ˜ and theloop algebra B ˜ . The Hilbert space H ˜ corresponds to a partial solution ofthe diffeomorphism constraint of canonical gravity. The inner product of H ˜ involves a functional integral over a space of connections and the Dirac typeoperator D ˜ has the form of a global functional derivation. Consequently, weinterpret the triple in terms of a non-perturbative, background independent,quantum field theory.The construction is based on a projective system of simplicial complexes.The simplicial complexes are related through repeated barycentric subdivi-sions. The triple p B ˜ , D ˜ , H ˜q is the limit of spectral triples formulated atthe level of finite simplicial complexes. Since the operation of barycentricsubdivision is countable the limit triple is separable and spectral.The square of the Dirac type operator has, in terms of canonical quan-tum gravity, a natural interpretation as global operator related to the areaoperators known in Loop Quantum Gravity. We interpret the operator p D ˜q in terms of an action and show that the spectral action of D ˜ has the formof a Feynman path integral. Thus, at the core of the construction we find anobject which resembles a partition function related to Quantum Gravity.The construction of the spectral triple p B ˜ , D ˜ , H ˜q differs from a tradi-tional canonical quantization procedure of General Relativity in the way thegroup of diffeomorphisms is treated. Rather than encoding the symmetriesof the classical theory in a set of constraints the construction works directlyon a Hilbert space H ˜ which corresponds to a partial solution of the (spa-tial) diffeomorphism constraint. The existence of the spectral triple relieson a particular split of the diffeomorphism group into a countable and anover-countable part. The over-countable part is discarded of; the group ofcountable diffeomorphisms is the group diff p △ q of diffeomorphisms betweensimplicial complexes. This means that the Poisson algebra of General Rela-tivity is represented on the separable Hilbert space H ˜ .71he Dirac type operator D ˜ depends on an infinite sequence of parameters t a i u . These parameters determine the scaling behaviour of the construction.We believe that a correct understanding and treatment of these parametersis essential. In this paper we propose a possible way to treat the sequence t a i u . Since the sequence is seen to have metric origin we propose to includeit as a dynamical variable in the construction. This leads to a new triple,denoted p B t , D t , H t q , which includes the spectral triple p B ˜ , D ˜ , H ˜q as wellas a sector permitting the sequence t a i u to vary. The triple p B t , D t , H t q isconstructed to ensure that the operator D ˜ is θ -summable. Thus, we permitonly sequences t a i u which leaves the spectral action of D ˜ finite.Furthermore, the Dirac type operator D ˜ defines a distance on the un-derlying space of connections. Clearly, this distance depends strongly on thesequence t a i u . However, we find that a general feature of this distance func-tion is that two connections are ”close” if they differ mostly at short scales.It is possible to read this paper in a more conservative way, discarding therole of Noncommutative Geometry and reading it as a reformulation of LoopQuantum Gravity. If we ignore the construction of the Dirac type operator D ˜ and focus instead on the algebra B ˜ and the Hilbert space H ˜ , with-out the Clifford bundle, and consider the algebra of the vectors ˆ E ij , then, asalready mentioned, we obtain a representation of the Poisson algebra of Gen-eral Relativity on a separable Hilbert space. Therefore, this Hilbert space, letus denote it H , replaces the otherwise non-separable Hilbert space L p A a q known as the kinematical Hilbert space in Loop Quantum Gravity. In LoopQuantum Gravity the kinematical Hilbert space is the Hilbert space on whichthe constraints of General Relativity are defined. Thus, one could formulatethe complete set of constraints of General Relativity in terms of operatorsacting on the Hilbert space H . First, as explained in section 6, the Hilbertspace H has an action of the reduced set of diffeomorphisms diff p △ q . There-fore the spatial diffeomorphism constraint should be formulated in terms ofthe group diff p △ q . Next, one may likewise formulate the Gauss and Hamil-tonian constraints of Loop Quantum Gravity on H . The central messagehere is that it is possible to formulate Loop Quantum Gravity in terms ofa separable kinematical Hilbert space. It is an interesting question whetherthis observation will have an impact on any of the challenges facing LoopQuantum Gravity. 72et us finally remind the reader that the focus of this paper is the physi-cal significance of the spectral triple p B ˜ , D ˜ , H ˜q . The detailed mathematicalanalysis of the triple is given in [22]. Outlook
More analysis is needed to understand the physical and mathematicalsignificance of the spectral triple p B ˜ , D ˜ , H ˜q . First of all it is imperativeto understand the role and proper treatment of the sequence t a i u since thissequence is central to the existence of the triple. Let us here just say thatthe sequence t a i u should be understood in connection with the group ofdiffeomorphisms diff p △ q since elements hereof are given by rearrangement ofthe parameters a i . Having said this, let us list other issues which we thinkdeserves attention.- First, we have seen that the spectral action resembles a Feynman pathintegral. We believe that the computation and analysis of the spectralaction is the most interesting task to address at the present stage of theproject. Here one should consider the Dirac operator which involvesthe inner fluctuations described in section 11.- A prime issue for any theory or framework for non-perturbative Quan-tum Gravity is the formulation of a semi-classical limit. In the presentcase the aim is to obtain a classical limit which involves not only asmooth geometry - characterised by a commutative (cid:6) -algebra - but toobtain a limit which includes an additional matrix factor of the typethat characterises the almost commutative algebra in Connes’ formula-tion of the Standard Model. Indeed, since Connes’ geometrical realiza-tion of the Standard Model is so attractive and powerful as it stands,it remains to understand why the algebra which lies at the heart ofthis formulation should have this particular noncommutativity. Wesuggest that the source of this noncommutativity lies in pure Quan-tum Gravity. Recall that the loop algebra B ˜ is essentially an almostcommutative algebra over the space A of connections. That is, it isa product of smooth functions over A and a matrix factor M l p C q . Ifwe think of a classical limit as the emergence of a single geometry itseems reasonable to expect something close to a delta function peakedaround a connection ∇ . However, when we apply the loop algebra on a73elta function it reduces to the matrix factor M l p C q or some subalgebrahereof. Furthermore, if we keep in mind that the construction of thespectral triple p B ˜ , D ˜ , H ˜q involves a choice of a basepoint , then itseems possible that a similar construction which does not involve thisbasepoint will entail a smearing of the matrix factor over the manifold.Thus, we speculate that the matrix factor behind the noncommutativeformulation of the Standard Model emerges in the classical limit fromthe noncommutative algebra of holonomy loops.We suspect this issue to be related to the calculation of the spectralaction. However, one should also investigate whether ideas from LoopQuantum Gravity concerning coherent states [44] can be applied.- The question about time is fundamental to any general covariant theorysince such theories have no preferred time flow. This situation is evenmore complicated when one attempts to include quantum theory sincethis will presumable lead to a theory which involves some notion ofsuperpositions of geometries and thus does not permit any notion ofa predetermined time. A possible solution to this problem has beenproposed by Connes (see for example [38]. See also [45] for similarideas developed by Connes and Rovelli). The idea is that the conceptof time is intimately linked to the noncommutativity of the algebraof observables of Quantum Gravity. Specifically, it is a fundamentalproperty of von Neumann algebras that they possess a 1-parameterfamily of automorphisms which is unique up to inner automorphisms.Thus, the idea is that this group of automorphisms, the modular group,should be understood in terms of a time. It is therefore natural to askwhether the noncommutative (cid:6) -algebras introduced in this paper giverise to a nontrivial modular group and whether this can be interpretedas a time flow.Alternatively, one might try to exploit the fact that the constructionpresented in this paper is basically quantum mechanics on the group G taken infinitely many times. Here, each copy of G corresponds to adegree of freedom originating somewhere on the underlying manifold.Thus, one could consider the time evolution, with respect to one copyof G , given by the operator exp p i ∆ t q where ∆ is the Laplace operator. See appendix A for an extension of the triple p B ˜ , D ˜ , H ˜q which avoids this choice ofbasepoint. U p t q : (cid:16) exp p i p D ˜q t q may be thought of as a time evolution operator.- The construction of the spectral triple p B ˜ , D ˜ , H ˜q starts with a sim-plicial complex. It is an important question to determine the exactdependency of the triple on the choice of initial complex. For instance,consider two simplicial complexes K and K chosen so that their unionis the barycentric subdivision of yet another simplicial complex K .Does the construction depend on whether one chooses K or the unionof K and K as the initial complex? The answer is, a priori, yes, sincethe two choices will come with different sequences of parameters t a i u .The exact nature of this dependency needs to be clarified. Clearly, theidea is that the spectral triple should depend only on topological datacoming from the underlying manifold. A related issue is the mergingof different spectral triples based on different simplicial complexes. Bygluing complexes it should be possible to move from one topologicalsetting to another. These issues are all connected with the sequence t a i u and the group of diffeomorphisms in diff p △ q .- A related issue is the possibility to obtain a similar construction basedon a different projective system of graphs. The choice of simplicial com-plexes (or, triangulations) and barycentric subdivisions seems naturalbut is not compulsory. In [22] we provide certain necessary require-ments for a system of graphs to be suitable for the construction of aspectral triple. These requirements leave room for projective systemsof graphs which are not simplicial complexes. Again, more analysis isneeded to determine the dependency of the final construction on dif-ferent choices of graphs. It is clear that a projective system of graphsmust be countable in order to permit the construction of a spectraltriple.- It is desirable to be able to deal also with a non-compact gauge groupsuch as SO p , q . At present, the compactness of the gauge group isessential for the construction of the Hilbert space to work. Basically,we need the identity to be an L -function with respect to the Haarmeasure. One could speculate whether the techniques of Higson and75asparov [25] might be applied to resolve this issue. Here, the trick isto use a Dirac operator with a nontrivial square-integrable kernel andto define embeddings between Hilbert spaces via this kernel.- One should further clarify the relation between the construction pre-sented here and Loop Quantum Gravity. In particular, it would beinteresting to understand if there is a relation between the operator D ˜ and the Hamilton constraint. Here one should most likely consider thefluctuated version of D ˜ , with respect to inner automorphisms, sincethis operator involves also the loop algebra.- It is an interesting question whether the construction presented in thispaper has anything to say about nonperturbative gauge theory. Con-sider a single line segment and the sequence a n (cid:16) a n , where n cor-responds to the number of subdivisions of the segment. This is a nat-ural sequence to consider but we know that the asymptotic behaviour a n (cid:18) n is too weak for the Dirac type operator to have a compactresolvent. If we consider instead the sequence a n (cid:16) a p (cid:0) | ǫ |q n then D ˜ will have a compact resolvent as long as ǫ is non-zero. This setup isthen extended to all line segments in the projective system of graphs.As long as ǫ is non-zero the corresponding spectral action of D ˜ is welldefined and resembles a Feynman path integral over a space of connec-tions. The interesting question is what theory this object represents. Itseems clear that it should be understood as a non-perturbative Quan-tum Field Theory involving a gauge field. The setup resembles latticegauge theory with the crucial difference that a lattice spacing is ab-sent and that one does not have the freedom to choose an action. Itwould be interesting to calculate the spectral action and, subsequently,to take the limit ǫ Ñ
0. Presumably, this limit will lead to divergencessince the operator D ˜ ceases to have compact resolvent when ǫ (cid:16) p B t , D t , H t q should be carried out. A publication with these details is under prepa-ration. 76 cknowledgements We thank the following colleagues for fruitful discussions: Christian Fleis-chhack, Victor Gayral, Gerd Grubb, Troels Harmark, Thordur Jonsson,Mario Paschke, Adam Rennie, Carlo Rovelli, Thomas Schucker, ChristianVoigh and Raimar Wulkenhaar. Furthermore, we are grateful to the fol-lowing institutes for hospitality during visits: the Institute of Mathematicsin Reykjavik, Iceland; The Max Planck Institute for Mathematics in theSciences, Leipzig, Germany; the Isaac Newton Institute for MathematicalSciences, Cambridge, UK; the Institute of Theoretical Physics in Marseilles,France.Johannes Aastrup was funded by the German Research Foundation (DFG)within the research projects
Deformation Theory for Boundary Value Prob-lems and
Geometrische Strukturen in der Mathematik (SFB 478).
A A spectral triple without the basepoint
The diffeomorphism invariance obtained so far only includes diffeomorphismswhich preserve the basepoint introduced in section 2. The role of the base-point is to equip the algebra of loops with a product. Because the choice ofbasepoint partly breaks diffeomorphism invariance we would like to obtaina structure which avoids the basepoint. It turns out that such a construc-tion does exist. Instead of the group structure of loops this more generalconstruction is based on a groupoid structure of path.Once again we start with an abstract simplicial complex K with vertices t v i u and edges t ǫ j u . Consider the Hilbert space H K (cid:16) L pt v i u (cid:2) A K , M l p C qq , where we recall that A K (cid:16) G n p K q . For now we omit the Clifford bundle whichis not necessary for the construction and representation of the algebra.In fact, there are two natural algebras to consider. Denote by Ω the set ofloops in K with arbitrary basepoint and consider first the algebra generatedby these loops equipped with the product f L (cid:4) f L (cid:16) " f L (cid:4) L if basepoints coincide.77gain we can construct a norm via the matrix norm on G and we obtain a C (cid:6) -algebra which we denote B Ω .The second option, which is perhaps more natural, is to consider paths in K . Denote by P the set of paths in K and denote by B P the algebra generatedby such paths with a natural product. Clearly, B Ω ãÑ B P . Concretely, welet the algebra B P be given by its representation which is as follows. Givena path p that starts in v and ends in v we write p f p (cid:4) ξ qp v j , ∇ q (cid:16) " v j (cid:127)(cid:16) v ∇ p p q (cid:4) ξ p v , ∇ q if v j (cid:16) v , where ∇ p p q is defined as in (11), just with a path instead of a loop.In this setup a path can now be seen as an operator which combines theholonomy along the path with a matrix structure | v m yx v n | and we notice that loops with arbitrary basepoint are found on the diagonalof this matrix structure. Thus, a natural trace will pick out all the loops andthereby, in terms of holonomy loops, the gauge covariant elements.The inner product on the Hilbert space H K also involves a sum oververtices. The rest of the construction can be carried out in a similar mannerto the construction of the triple p B ˜ , D ˜ , H ˜q .This formulation seems better suitable for a semi-classical limit since itinvolves the points of the manifold, which, presumable, should emerge insuch a limit. B On diffeomorphism invariance
Although the Hilbert space L p A ˜q does not have an action of the full dif-feomorphism group it is possible to introduce a notion of analytic diffeomor-phisms on L p A ˜q via the space L p A a q . This, in turn, allows us to extendthe action of certain operators on L p A ˜q to the larger Hilbert space L p A a q and thereby introduce a notion of (analytic) diffeomorphism invariance forthese operators. This setup involves an embedding of general piecewise ana-lytic graphs Γ into triangulations T i coming from the projective system t K i u .Therefore, the action of these operators on L p A ˜q will depend on a choiceof embedding. It is important to realize that this construction does not work78or the operator D ˜ since we do not have a Clifford bundle over the space L p A a q .The following should be read as a rough idea or strategy as to how oneintroduces a notion of diffeomorphism invariance on L p A ˜q , rather than acomplete analysis.Let us go into details. Given an element ξ Γ P L p A a q associated to apiece-wise analytic graph Γ we choose an embedding of a suitable simplicialcomplex K i φ : K i Ñ T i , where T i is a triangulation in M , so that Γ lies in T i . Consider an operator O on L p A ˜q . This could for example be the Laplace operator. The actionof O on ξ Γ is defined as O p ξ Γ q : (cid:16) φ p O p φ (cid:1) ξ Γ qq . If we map d : ξ Γ Ñ ξ d p Γ q with a diffeomorphism d then the action of O changes accordingly O p ξ d p Γ qq : (cid:16) φ O pp φ ξ Γ qq , where φ is a new embedding. We obtain the diagram ξ Γ φ (cid:1) ÝÑ ξ ˜ ÝÑ O ξ ˜ φ ÝÑ p O ξ ˜q Γ d Ó d ˜ Ó d ˜ Ó d Ó ξ d p Γ q p φ ÝÑ ξ
1˜ ÝÑ O ξ φ O ξ d p Γ q (90)where d ˜ and d belong to diff p △ q . To have a diffeomorphism invariantstate means that the state is invariant under diff p △ q . This is exactly thespace H diff of diffeomorphism invariant states mentioned in the previoussubsection. A diffeomorphism invariant operator O is an operator for whichthe centre part of the diagram (90) commute ξ ˜ ÝÑ O p ξ ˜q d ˜ Ó ö d ˜ Ó ξ
1˜ ÝÑ O p ξ This reflects the simple observation that a loop is, by itself, not an observablesince it is not self-adjoint. Only the combination L (cid:0) L (cid:1) is self-adjoint,exactly because it is invariant with respect to the symmetry group of theloop. 79 Symmetric states
In Quantum Field Theory the vacuum state is defined as the unique trans-lational invariant state. In the present setting there are certain states whichdisplay a high degree of symmetry and which may be thought in terms of aground state.Consider first the spectral triple p B ˜ , D ˜ , H ˜q and the two states ψ p ∇ q (cid:16) ψ p g , g , . . . , g n , . . . q (cid:16) δ G p id q (cid:4) δ G p id q (cid:4) . . . (cid:4) δ G n p id q (cid:4) . . . , and φ p ∇ q (cid:16) . Here δ G n is the delta-function on the n ’th copy of G and c is a constant. Theaction of a loop L on the first state reads f L (cid:4) ψ p ∇ q (cid:16) ψ p ∇ q for any L in B ˜ . This means that the entire algebra B ˜ collapses into theidentity on the state ψ . This corresponds to flat space. This state, however,does not lie in the Hilbert space, nor in the domain of the operator D ˜ . Incontrast to this the state φ lies in the kernel of D ˜ . Both states ψ areinvariant under diff p △ q since they are invariant under any permutation ofthe argument p g , . . . , g n , . . . q .If we consider instead the spectral triple p B t , D t , H t q there are again twostates which are highly symmetric. Consider first H t Q Ψ p ∇ , ¯ x q (cid:16) ψ p ∇ q b ξ o p ¯ x q , where ξ p ¯ x q (cid:16) ξ p x , x , . . . , x n , . . . q (cid:16) δ x p α q (cid:4) δ x p α q (cid:4) . . . (cid:4) δ x n p α n q (cid:4) . . . This state fixes the sequence t a i u on the ’background’ sequence t α i u whichwas, in section 9.5, fixed according to relation (85). Again, this states doesnot lie in the domain of the Dirac operator D t . Notice however, that bychoosing the parameters a i ’s according to relation (85) one obtains a Diractype operator D ˜ which is, up to an overall factor, invariant under a shift inthe parameters a i . 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