Nonperturbative Quantum Field Theory and Noncommutative Geometry
aa r X i v : . [ m a t h - ph ] D ec Nonperturbative Quantum Field Theoryand Noncommutative Geometry
Johannes
Aastrup a & Jesper Møller Grimstrup b a Mathematisches Institut, Universit¨at Hannover,Welfengarten 1, D-30167 Hannover, Germany. b QHT Gruppen, Copenhagen, Denmark.
This work is financially supported by Ilyas Khan,St Edmunds College, Cambridge, United Kingdom.
Abstract
A general framework of non-perturbative quantum field theory on acurved background is presented. A quantum field theory is in this set-ting characterised by an embedding of a space of field configurationsinto a Hilbert space over R ∞ . This embedding, which is only localup to a scale that we interpret as the Planck scale, coincides in thelocal and flat limit with the plane wave expansion known from canon-ical quantisation. We identify a universal Bott-Dirac operator actingin the Hilbert space over R ∞ and show that it gives rise to the freeHamiltonian both in the case of a scalar field theory and in the case of aYang-Mills theory. These theories come with a canonical fermionic sec-tor for which the Bott-Dirac operator also provides the Hamiltonian.We prove that these quantum field theories exist non-perturbativelyfor an interacting real scalar theory and for a general Yang-Mills the-ory, both with or without the fermionic sectors, and show that thefree theories are given by semi-finite spectral triples over the respec-tive configuration spaces. Finally, we propose a class of quantum fieldtheories whose interactions are generated by inner fluctuations of theBott-Dirac operator. email: [email protected] email: [email protected] ontents S . . . . . . . . . . . . . . . . 19 A . . . . . . . . . . . . . . . . 24 M In quantum mechanics on for example the real line R one usually consid-ers L ( R ) together with the operators x and i ddx satisfying the Heisenbergrelation [ i ddx , x ] = i. (1)Alternatively one can replace the operator x with C ∞ c ( R ) , the space ofsmoothly supported compact functions, and the operator i ddx with transla-tions in R , i.e. by the operators U a f ( x ) = f ( x − a ) , a ∈ R , f ∈ L ( R ) , U a f U ∗ a ( x ) = f ( x + a ) . (2)These two formulations of quantum mechanics – let us call them the in-finitesimal and the integrated formulations, respectively – are of course equiv-alent but only as long as the number of degrees of freedom remains finite:as we shall show in this paper, the representation theory of the integratedformulation is, when we consider fields, richer than that of the infinitesi-mal formulation. This realisation is the starting point of the framework ofnon-perturbative quantum field theory, that we present in this paper.Thus, instead of searching for algebraic representations of the canonicalcommutation relations of field operators we shall instead identify algebrasof bounded functions over spaces of field configurations together with trans-lation operators hereon. Such algebras turn out to have nontrivial Hilbertspace representations, which are not accessible via the infinitesimal formula-tion and which allow for a construction of interacting quantum field theorieson curved manifolds.The reason why an approach to non-perturbative quantum field theorybased on the integrated formulation – i.e. on algebras of functions on con-figuration spaces together with translation operators – is advantageous isthat it permits non-local representations, where the canonical commutationrelations are only realised up to a correction at a scale, which we interpret asthe Planck scale. Thus, these representations depend on a scale, which pre-vents arbitrary localisation, and therefore they do not include the operatorvalued distributions known from perturbative quantum field theory.The key step in finding these Hilbert space representations is to constructa measure, where fast oscillating field configurations have smaller probabili-ties than the slow oscillating ones. Concretely, we do this by expanding thefields in terms of eigenvalues of a Laplace operator and then constructing aGaussian measure on the space of the coefficients of this expansion weightedwith the eigenvalues of the Laplace operator. The result is that transitionsbetween field configurations depend on a Sobolev norm, which is inherentlynon-local.In this paper we present both a general method of constructing non-perturbative quantum field theories as well as two concrete examples. Firstwe consider the case of a real scalar field and second we revisit the caseof a gauge field, which was first developed in [1, 2]. In both cases we findthat the operator expansion in terms of weighted eigenvectors of a Laplace3perator coincide in a local and flat limit with the plane wave expansion ofa canonically quantised field. This means that the general framework, thatwe present, includes perturbative quantum field theory as a limiting case.At the heart of our construction is a Hilbert space over R ∞ and a canon-ical Bott-Dirac operator, due to Higson and Kasparov [3], that acts on func-tions on R ∞ . The weighted expansion of fields in eigenfunctions of a Laplaceoperator is an embedding from the space of field configurations into R ∞ , thespace of coefficients in this expansion. The square of the Bott-Dirac operatorgives the Hamilton operator of an infinite-dimensional harmonic oscillator,which in turn gives – both in the case of a real scalar field and in caseof a gauge field – the free Hamiltonian of the corresponding quantum fieldtheory in the flat and local limit. Furthermore, the construction of the Bott-Dirac operator naturally introduces also a fermionic sector to the theorieswe study, where the square of the Bott-Dirac operator gives the Hamiltonoperator for the fermions as well.We also consider the case of interacting quantum field theories. Here themost important feature is that these theories continue to exist also when in-teractions are turned on (irrespectively on whether the fermions are includedor not).The Bott-Dirac operator is a canonical geometrical structure acting ina universal L space over various spaces of field configurations, where itforms a semi-finite spectral triple. The representations, that we have found,involve, however, also an L space over the three-dimensional manifold M and it is therefore natural to consider a also Dirac-type operator, that consistof two parts: the Bott-Dirac operator plus a spatial Dirac operator actingon spinors on the manifold. Such an operator interacts also with the variousrepresentation of the field operators and forms again a spectral triple exceptthat commutators with the algebra are no longer bounded.One particularly interesting construction emerges when one considersthe algebra of field operators for a gauge theory – this is what we call theholonomy-diffeomorphism algebra, denoted HD ( M ) , since it is generatedby holonomies along flows of vector-fields [4] – and let it act on spinorsin L ( M, S ) . This is what we have previously called quantum holonomytheory [2, 5]. Perhaps the most interesting feature of this model is that ithas a possible connection to the standard model of particle physics via itsformulation in terms of noncommutative geometry due to Chamseddine andConnes [6, 7]. The point is that the HD ( M ) algebra produces a so-calledalmost commutative algebra in a semi-classical limit [8], which is the typeof algebra that Chamseddine and Connes have identified as a basic geomet-rical input in their work on the standard model coupled to general relativity.4he idea to construct a Dirac-type operator over a space of field con-figurations and to let it interact with an algebra of field operators was firstproposed in [9] and later developed in [10]-[16]. The HD ( M ) algebra wasfirst introduced in [4, 8] and further studied in [1, 2, 5, 17, 18, 19]. Aftercompleting the papers [1, 2] we became aware of the work by Higson andKasparov [3], where a similar Hilbert space construction was used and wherethe infinite-dimensional Bott-Dirac operator was first introduced – the op-erator, which is the central object in the present paper.We begin in the next section by introducing the infinite-dimensionalBott-Dirac operator acting in a Hilbert space over the projective limit R ∞ .In section 3 we then outline the general construction of a non-perturbativequantum field theory based on the Bott-Dirac operator. Here the key stepis the embedding of the space of field configurations into the space R ∞ .With this we then move on to discuss the first example, in section 4, with areal scalar field. We construct a Hilbert space representation of an algebraof field operators and prove that it exist and is strongly continuous. Wealso show that our construction coincides with perturbative quantum fieldtheory in a flat and local limit. Finally we show that the Bott-Dirac operatortogether with the algebra of field operators form a semi-finite spectral tripleover the space of scalar field configurations. Next, in section 5, we turn togauge theory, where we introduce the HD ( M ) and QHD ( M ) algebras andtheir representations. Again we show that our construction coincides withperturbative quantum field theory in a flat and local limit and that the Bott-Dirac operator together with the HD ( M ) algebra form a semi-finite spectraltriple. In section 6 we show that the construction of the Bott-Dirac operatornaturally introduces a fermionic sector and that its square again gives theHamiltonian. Furthermore, in section 7 we add a spatial Dirac operatorto the Bott-Dirac operator and consider, in section 8, fluctuations of theBott-Dirac operator by inner automorphisms and show how this generates aclass of interacting quantum field theories. In section 9 we then argue thatthe non-local nature of the quantum field theories, that we find, puts intoquestion one of the strongest arguments in favour of a quantum theory ofgravity. We end the paper in section 10 with a discussion.5 An infinite-dimensional Bott-Dirac operator
We begin with a basic geometrical construction, which in the next sectionswill play a key role in the formulation of various quantum field theories. Thefollowing formulation of an infinite-dimensional Bott-Dirac operator is dueto Higson and Kasparov [3].Let H n = L ( R n ) , where the measure is given by the flat metric, andconsider the embedding ϕ n ∶ H n → H n + given by ϕ n ( η )( x , x , . . . x n + ) = η ( x , . . . , x n ) ( s n + τ π ) / e − sn + x n + τ , (3)where { s n } n ∈ N is a monotonously increasing sequence of parameters, whichwe for now leave unspecified . This gives us an inductive system of Hilbertspaces H ϕ Ð→ H ϕ Ð→ . . . ϕ n Ð→ H n + ϕ n + Ð→ . . . and we define L ( R ∞ ) as the Hilbert space direct limit L ( R ∞ ) = lim → L ( R n ) (4)taken over the embeddings { ϕ n } n ∈ N given in (3). We are now going to definethe Bott-Dirac operator on L ( R n ) ⊗ Λ ∗ R n . Denote by ext ( f ) the operatorof external multiplication with f on Λ ∗ R n , where f is a vector in R n , anddenote by int ( f ) its adjoint, i.e. the interior multiplication by f . Denoteby { v i } a set of orthonormal basis vectors on R n and let ¯ c i and c i be theClifford multiplication operators given by c i = ext ( v i ) + int ( v i ) ¯ c i = ext ( v i ) − int ( v i ) (5)that satisfy the relations { c i , ¯ c j } = , { c i , c j } = δ ij , { ¯ c i , ¯ c j } = − δ ij . (6) In [3] these parameters were not included, i.e. s n = ∀ n . The notation L ( R ∞ ) , which we are using here, is somewhat ambiguous. We are hereonly considering functions on R ∞ with a specific tail behaviour, namely the one generatedby (3). We have not included this tail behaviour in the notation. See [1] for further details.
6e shall also use the notation: a † i ∶ = ext ( v i ) a i ∶ = int ( v i ) } with { a i , a † j } = δ ij . (7)The Bott-Dirac operator on L ( R n ) ⊗ Λ ∗ R n is given by B n = n ∑ i = ( τ ¯ c i ∂∂x i + s i c i x i ) , and the square of B n is B n = n ∑ i = (− τ ∂ ∂x i + s i x i ) + τ N − τ ˜ s (8)where ˜ N is the operator ˜ N = n ∑ i = s i a † i a i which would, if we removed the factors s i , be the operator that assignsthe degree to a differential form in Λ ∗ R n , and where ˜ s = ∑ ni = s i . With B n we can then construct the Bott-Dirac operator B on L ( R ∞ ) ⊗ Λ ∗ R ∞ thatcoincides with B n on any finite subspace L ( R n ) . Here we mean by Λ ∗ R ∞ the inductive limit Λ ∗ R ∞ = lim → Λ ∗ R n . For details on the construction of B we refer the reader to [3].We note that when we let the sequence { s i } i ∈ N tend to infinity the eigen-values of the Bott-Dirac operator will tend to infinity. This means that,modulo the infinite degeneracy coming from Λ ( R ∞ ) , the eigen-spaces are fi-nite dimensional. This means that modulo Λ ( R ∞ ) the Bott-Dirac operator B has compact resolvent , i.e. ( B ± i ) − is a compact operator.We shall call the state η gs ( x ) = ∏ i ( s i τ π ) / e − six i τ ∈ L ( R ∞ ) for the ground state. Note that η gs lies in the kernel of B , i.e. B ( η gs ) = . Note that the sequence { s i } i ∈ N was not included in the construction of the Bott-Dirac operator in [3]. Instead the Bott-Dirac operator was there combined with a secondoperator with compact resolvent, the result being again a total operator with compactresolvent.
7f we define the creation and annihilation operators q i = √ s i x i + τ √ s i ∂∂x i , q i † = √ s i x i − τ √ s i ∂∂x i , (9)with x i = √ s i ( q i + q † i ) , ∂∂x i = √ s i τ ( q i − q † i ) , (10)then we can rewrite B as B = ∑ i √ s i ( q † i a i + q i a † i ) and its square as B = ∑ i s i ( q † i q i + a † i a i ) . (11)We then have the relations q i = √ s i { B, a i } , q † i = √ s i { B, a † } , − a i = √ s i [ B, q i ] , a † i = √ s i [ B, q † i ] . (12)With these relations it is easy to see how the Bott-Dirac operator commu-nicates between the CAR and the CCR algebras generated by { a † i , a i } and ( q † i , q i ) respectively.Let us finally introduce an additional piece of notation, that shall becomeuseful later, namely B ∣ b ∶ = ∑ i s i q † i q i , B ∣ f ∶ = ∑ i s i a † i a i . (13) Let M be a compact manifold and let Γ be a general configuration spacewhere each point is given by a field Θ on M . At this point we shall not specifyexactly what type of field Θ is but leave that to the following sections. Weare going to devise a general method of constructing a Hilbert space L ( Γ ) via the Hilbert space construction presented in the previous section. To dothis we first introduce a scalar product ⟨ ⋅ ∣ ⋅ ⟩ s between fields in Γ as well as asystem { ξ i } i ∈ N , ξ i ∈ Γ, with the properties8. that { ξ i } i ∈ N is a real orthonormal basis with respect to ⟨ ⋅ ∣ ⋅ ⟩ s , and2. that ∑ i s − i ∥ ξ i ∥ ∞ < ∞ with ∥ Θ ∥ ∞ = sup m ∈ M ( Θ ( m ) , Θ ( m )) where ( ⋅ , ⋅ ) is an appropriate fiber-wise scalar product that depends on the specifics of the configurationspace Γ.The construction of L ( Γ ) relies on the embeddingΠ ∶ Γ → R ∞ (14)given by Π ( Θ ) = ( x , x , . . . ) where x i = ⟨ Θ ∣ ξ i ⟩ s . This embedding gives us a scalar product on functionson Γ ⟨ η ( Θ )∣ ζ ( Θ )⟩ = ⟨ η ( ∑ i x i ξ i )∣ ζ ( ∑ i x i ξ i )⟩ L ( R ∞ ) , where η and ζ are functions on Γ, that in turn allows us to define L ( Γ ) .With this construction we have a sequence of intermediate Hilbert spaces L ( Γ n ) via the embeddingsΓ n ∋ Θ = n ∑ i = x i ξ i → ( x , x , . . . x n ) with the scalar products ⟨ η ∣ ζ ⟩ Γ n = ∫ R n η ( x ξ + . . . + x n ξ n ) ζ ( x ξ + . . . + x n ξ n ) dx . . . dx n . (15)The Hilbert space L ( Γ ) is then the direct limit of these intermediate spaces L ( Γ ) = lim → L ( Γ n ) . Furthermore, we shall later use the notation Λ ( R ∞ ) = Λ ∗ Γ for the infinite-dimensional Clifford algebra.Next let U ω be the canonical translation operator given by U ω η ( Θ ) = η ( Θ − ω ) In section 4.4 we find that for the commutator between the Bott-Dirac operator anda bounded field operator to be bounded we need the stronger condition ∑ i ∥ ξ i ∥ ∞ < ∞ . Seesection 4.4 for details. ω ∈ Γ and denote by
Alg ( Γ ) an algebra of bounded functions on theconfiguration space Γ. The exact form of Alg ( Γ ) depends on the specifics ofthe configuration space Γ but a general requirement is that a representation ρ ∶ Alg ( Γ ) → B ( L ( Γ ) ⊗ L ( M )) exist. When this is the case we shall interpret the representation of thealgebra generated by Alg ( Γ ) and the translation operators U ω as a kine-matical sector of a quantum field theory over the configuration space Γ andthe square of the Bott-Dirac operator (8) as the Hamiltonian of the freetheory.In the following two sections we shall demonstrate this construction fortwo specific types of field theories, namely a real scalar field theory and aYang-Mills theory, and we shall see that it enables us to define not only thefree theories but also interacting ones. In this section we will define
Alg ( Γ ) in the case of a real scalar field theory,and show that it has a representation on L ( Γ ) ⊗ L ( M ) .For the scalar field theory we will denote the configuration space by S .Furthermore a generic field will be denoted by φ . The kind of elements wewould like have in Alg ( S ) are elements of the form e iφ . We will show thatthe representation is suitably strongly continuous, and thereby ensuring thatwe have self-adjoint operators like ddt e iφ ∣ t = = iφ , and hence also operatorslike φ .Like in the previous section M is a compact manifold. Let C ∞ b ( M × R ) be the bounded smooth functions on M × R , where all the derivatives arealso bounded (We actually only need the first three derivatives). Given f ∈ C ∞ b ( M × R ) and φ ∈ S we define an operator on L ( M ) via ( M f ( φ ) ξ )( m ) = f ( m, φ ( m )) ξ ( m ) . We consider M f as an family of operators over S .Following the general framework presented in the previous section we arenow going to represent C ∞ b ( M × R ) as operators on a Hilbert space L ( S ) ⊗ L ( M ) , which will enable us to include the second ingredient, namely the Note that in the case of a gauge theory these translations are not given by gauge fieldsbut by one-forms. S . To construct L ( S ) we first need to put ameasure on S . We follow the proceeding in the previous section. We choosea metric on M and define the Sobolev scalar product ⟨ φ ∣ φ ⟩ s = ∫ M ( + τ ∆ σ ) φ ( + τ ∆ σ ) φ , φ , φ ∈ S (16)where σ and τ are real numbers and where ∆ is the scalar Laplace operator.Let { ξ i } i ∈ N be an orthonormal basis with respect to (16) where each ξ i is aneigenfunction of ∆. Note that ξ i ( m ) = e i ( m ) + τ λ σi where e i ( m ) is an eigenfunc-tion of the Laplace operator with eigenvalue λ i , i.e. ∆ α i = λ i α i . Given amonotonously increasing sequence { s i } i ∈ N we can choose σ big enough suchthat ∑ i ( s i ) − ∥ ξ i ∥ ∞ < ∞ . (17)We then identify S with a subspace of R ∞ via the map S ∋ φ = ∑ i x i ξ i → ( x , x , . . . ) ∈ R ∞ and subsequently construct the Hilbert space L ( S ) and the action of U ω thereon as outlined in the previous section. The translation operator U ω , ω ∈ S , act by U ω η ( φ ) = η ( φ − ω ) , η ∈ L ( S ) . Like in [1] the action of U ω is strongly continuous and we therefore haveinfinitesimal operators E ω = ddt U tω ∣ t = . Finally, we build the full Hilbert space H scalar = L ( S ) ⊗ L ( M ) , which carries an action of both the M f and U ω operators. We need to show that the operators M f exist on H scalar . This question issimilar to the case considered in [1].We begin by showing 11 heorem 4.1.1. For each f ∈ C ∞ b ( M × R ) and each m ∈ M the limit lim n →∞ ∫ ∞−∞ ⋯ ∫ ∞−∞ f ( m, x ξ ( m ) + . . . x n ξ n ( m )) e − s x + ... + snx nτ dx ⋯ dx n , exists, i.e. the expectation value in a given point m of M f exists on theground state in L ( S ) .Proof. In a given point ( m, r ) ∈ M × R we Taylor-expand f in the r -direction: f ( m, x + r ) = f ( m, r ) + xf ′ ( m, r ) + x f ′′ ( m, r ) + R ( m, x + r ) , with R ( m, x + r ) = ∫ x t f ′′′ ( m, t + r ) dt. Here f ′ = ∂f∂r , f ′′ = ∂ f∂ r , etc. By asumption f ′′′ is bounded, let us say by Bτ − . We hence get the estimate ∣ R ( m, x + r )∣ ≤ ∣ x ∣ Bτ − . We thus have ( sτ π ) / ∫ ∞−∞ f ( m, r + ax ) e − sx τ dx = ( sτ π ) / ∫ ∞−∞ ⎛⎝ f ( m, r ) + axf ′ ( m, r ) + ( ax ) f ′′ ( m, r ) + R ( m, ax + r )⎞⎠ e − sx τ dx = f ( m, r ) + ( sτ π ) / f ′′ ( m, r ) ∫ ∞−∞ ( ax ) e − sx τ dx + ( sτ π ) / ∫ ∞−∞ R ( m, ax + r ) e − sx τ dx, where the first integral can be estimated by τ s ∣ f ′′ ( m, r )∣ a and the secondintegral by s − Ba . Putting in a = ξ k ( m ) and s = s k we get ∣ f ( m, r ) − ( s k τ π ) / ∫ ∞−∞ f ( m, r + x k ξ k ) e − skx kτ dx ∣ ≤ τ ∣ f ′′ ( m, r )∣ s − k ∥ ξ k ∥ ∞ + Bs − k ∥ ξ k ∥ ∞ . f ′′ is bounded, let us say by Cτ , and hence ∣ f ( m, r ) − ( s k τ π ) / ∫ ∞−∞ f ( m, r + x k ξ k ) e − skx kτ dx ∣ ≤ Cs − k ∥ ξ k ∥ ∞ + Bs − k ∥ ξ k ∥ ∞ . We thus have
RRRRRRRRRRR N ( n ) ∫ ∞−∞ ⋯ ∫ ∞−∞ f ( m, n ∑ i = x i ξ i ( m )) e − τ ( ∑ ni = s i x i ) dx ⋯ dx n − N ( n + ) ∫ ∞−∞ ⋯ ∫ ∞−∞ f ( m, n + ∑ i = x i ξ i ( m )) e − τ ( ∑ n + i = s i x i ) dx ⋯ dx n + RRRRRRRRRRR ≤ Cs − n + ∥ eξ n + ∥ ∞ + Bs − n + ∥ ξ n + ∥ ∞ . (18)where N ( n ) = ( πτ ) n / Π ni = s / i is the normalisation factor for the n Gaussianintegrals. The convergence of the expectation value of the M f operators onthe ground state follows from (18) and (17).It follows from the definition of the integrals, that the numerical valueof the limit of the integrals is bounded by ∥ f ∥ ∞ . Like in [1] it follows Theorem 4.1.2.
For all ξ, η ∈ L ( S ) the limit ⟨ ξ ∣ M f ( m, ⋅ ) ∣ η ⟩ = lim n → ∞ ∫ R n ξ ( x , . . . , x n ) f ( m, n ∑ k = x k ξ k ) η ( x , . . . , x n ) dx ⋯ dx n exists. We now turn to strong continuity. We want to show that ⟨ ξ ∣ M f k ( m )∣ η ⟩ converges to ⟨ ξ ∣ M f ( m )∣ η ⟩ when f k → f in a suitable topology. The notionof convergence we choose is the following one: • The sequences ( f k ) , ( f ′ k ) , ( f ′′ k ) and ( f ′′′ k ) are uniformly globally bounded,i.e. there exists a constant K with ∥ f k ∥ ∞ , ∥ f ′ k ∥ ∞ , ∥ f ′′ k ∥ ∞ , ∥ f ′′′ k ∥ ∞ ≤ K . • f k → f , f ′ k → f ′ , f ′′ k → f ′′ and f ′′′ k → f ′′′ locally uniformly, i.e. uni-formly on each compact subset of M × R .We have here chosen local uniformly, and not just uniformly. Uniformconvergence is a too strict condition, since we will later use it for f ( m, x ) = e itx , t → heorem 4.1.3. Let ( f k ) be a sequence converging to f in this topology.We have lim k → ∞ lim n → ∞ ∫ R n ( f ( m, n ∑ l = x l ξ ( m )) − f k ( m, n ∑ l = x l ξ ( m ))) e − τ ( ∑ ni = s i x i ) dx ⋯ dx n = Proof.
The estimates in the proof of theorem 4.1.1 show convergence if f k , f ′′ k and f ′′′ k converges uniformly. We can however refine the argumentslightly:We first choose n big enough with ∞ ∑ n = n s − n ∥ ξ n ∥ ∞ < ε and ∞ ∑ n = n s − n ∥ ξ n ∥ ∞ < ε. We put τ − B k = sup ∣( f − f k ) ′′ ( m, x )∣ and 4 C k τ = sup ∣( f − f k ) ′′′ ( m, x )∣ . According to the estimates in the proof of theorem 4.1.1 we havelim n → ∞ ∫ R n ( f ( m, n ∑ l = x l ξ ( m )) − f k ( m, n ∑ l = x l ξ ( m ))) e − τ ( ∑ ni = s i x i ) dx ⋯ dx n equal to ∫ R n ( f ( m, n ∑ l = x l ξ l ( m )) − f k ( m, n ∑ l = x l ξ l ( m ))) e − τ ( ∑ n i = s i x i ) dx ⋯ dx n up to an error of 2 ( B k + C k ) ε . Since B k and C k are bounded sequences wehave that the error is ε times some global factor. We thus only need to provethatlim k → ∞ ∫ R n ( f ( m, n ∑ l = x l ξ l ( m )) − f k ( m, n ∑ l = x l ξ l ( m ))) e − τ ( ∑ n i = s i x i ) dx ⋯ dx n = f k converges locally uniformly to f , and we can, to agiven error, choose a compact set, such that the integrals outside of this setis smaller that this error. This follows from the uniform global boundednessof the sequence and the properties of the Gaussian integrals.Like previously it is easy to extend this proof to14 heorem 4.1.4. Let f k → f . For all ξ, η ∈ L ( S ) we have lim k → ∞ ⟨ ξ ∣ M f k ( m, ⋅ ) ∣ η ⟩ = ⟨ ξ ∣ M f ( m, ⋅ ) ∣ η ⟩ We thus have strong continuity in each point.
It follows from the argument for strong continuity that m → ⟨ ξ ∣ M f ( m )∣ η ⟩ is continuos. We thus have Theorem 4.1.5.
For all f ∈ C ∞ b ( M × R ) we have well defined boundedoperators M f on L ( S ) × L ( M ) defined by (( M f )( ξ ))( m, φ ) = f ( m, φ ( m )) ξ ( m, φ ) . The action is strongly continuous.Especially we have an unbounded self-adjoint operator iφ ∶ = ddt M f t , where f t ( m, r ) = e itr . Consequently we also have any power of φ acting asself-adjoint unbounded operators on L ( S ) ⊗ L ( M ) . We are now going to compare this construction to the case of a canonicallyquantised real scalar field. In the following we let M = T , the 3-torus.Consider first the infinitesimal translation operators E ω = dds U sω ∣ s = . Ex-panding ω in the Sobolev eigenvectors these can be written as E ω = ∑ i ω i E ξ i , ω = ∑ i ω i ξ i . Note also that there exit a canonical operator E ( m ) = ∑ i ξ i ( m ) E ξ i = ∑ i ∂∂x i ξ i ( m ) where we used the identification E ξ i = ∂∂x i . With this type of operators we15an form an alternative representation given by the linear combinations φ ′ ( m ) = √ ∞ ∑ i = [ x i ( ξ i ( m ) + ξ i ( − m )) + τ s i ∂∂x i ( ξ i ( m ) − ξ i ( − m )) ] = ∞ ∑ i = √ s i ( q i ξ i ( m ) + q † i ξ i ( − m )) ,π ( m ) = √ ∞ ∑ i = [ s i x i ( ξ i ( m ) + ξ i ( − m )) − τ ∂∂x i ( ξ i ( m ) − ξ i ( − m )) ] = ∞ ∑ i = √ s i ( q i ξ i ( m ) − q † i ξ i ( − m )) , (19)which shall shortly be seen to provide a connection to the canonical quanti-zation of a real scalar field.But before we get that far let us first specify the parameters { s i } i ∈ N by s i = √ p i + m ≡ ω p i , where m is a real constant that plays the role of amass and where { p i } i ∈ N is a sequence of parameters that plays the role of amomentum. With this operator B ∣ b in (13) has the form B ∣ b = ∑ i ω p i q † i q i . (20)Let us now compare this construction and in particular equations (19)and (20) with a canonical quantisation of a free, real scalar field. We there-fore let M = R and denote by Π ( m ) the conjugate to the real scalar fieldΦ ( m ) . As is custom we expand Φ ( m ) and Π ( m ) in plane waves accordingto Φ ( m ) = ∫ d p ( π ) √ ω p ( a p e ip ⋅ m + a † p e − ip ⋅ m ) Π ( m ) = ∫ d p ( π ) ( − i )√ ω p ( a p e ip ⋅ m − a † p e − ip ⋅ m ) (21)where ω p = √ p + m and where a p and a † p are the creation and annihilationoperators (indexed by the 3-momentum p ) that act in a corresponding Fockspace. Also, the Hamilton operator for the free scalar field is given by H free = ∫ d p ( π ) ω p a † p a p (22)Already here we see a clear resemblance between the embedding (19)and the plane wave expansion (21) and between the square of the Bott-Dirac operator (20) and the Hamiltonian for the free scalar field (22). If we16ake a limit where M goes from being T to approaching R : p i Ð→ p ∑ i Ð→ ∫ d p ( π ) in which case the Sobolev eigenvectors ξ i can be written ξ i ( m ) Ð→ e − ip ⋅ m + τ λ σi , then it is clear that the framework we have presented in this section is,in the local limit τ →
0, identical to that of a canonically quantised freescalar field. Note that the Hilbert space representation, which we have con-structed in this section, ceases to exist precisely in the limit τ →
0, whichis to be expected as no such representation exist for ordinary (interacting)perturbative quantum field theory. We shall discuss this further in section 9.To construct the Hamilton of an interacting theory we need to considera Hamilton operator of the form H ∣ b + H int , where for example H int = φ . We know that both H ∣ b and H int exists as self-adjoint unbounded operators on H scalar . Strictly speaking we have not provedthat their sum exists but we are certain this will not be hard to prove. Wethink, however, that this should come with a more detailed analysis of thedomains of the operators, as well as the development of a pseudo-differentialcalculus. For instance the natural Sobolev spaces should be given by H k Sobolev ( S ) = Domain of H ∣ k . Another point is that we have in the case of the scalar theory chosena rather minimal algebra. The chosen algebra allows for operators like φ n ,but does not allow derivatives in φ for instance. The situation with theHolonomy-Diffeomorphism algebra, which we shall discuss in the next sec-tion, is different, since this algebra contain enough information to separateeach gauge-orbit.It is illustrative to rewrite the expectation value of the operator M f in L ( S ) with the short-hand notation ⟨ η gs ( φ ) ∣ M f ( φ )∣ η gs ( φ )⟩ L ( S ) ( m ) = ∫ S dφf ( φ ( m )) exp ( − ∥ ω p φ ∥ s ) , (23)17hich has the form of a functional integral where the Sobolev norm ∥ ⋅ ∥ s plays the role of a weight. With this heuristic notation it becomes clear thatthe functional integral is dominated by those field configurations, whichhave a small Sobolev norm and that field configurations, that have a largeSobolev norm, are dampened. In particular, this means that singular fieldconfigurations, i.e. those that are localised in a single point, will have zero weight in this integral. To see this we simply note that the Sobolev normdominates the supremum norm [20] and since the supremum norm of adelta function is infinite so is the Sobolev norm and hence the exponentialfactor in (23) will be equal to zero. This illustrates that the quantum fieldtheories, which we are presenting in this paper, are only local up to the scale τ . Another way to see this non-local aspect is by noting that the planewaves in the operator expansion (21) used in ordinary quantum field theoryand which corresponds to a point-localisation due to ∫ d pe ip ⋅ m ∼ δ ( ) ( m ) ,are in (19) replaced by the Sobolev eigenvectors ξ i , which only correspondto a point-wise localisation up to a correction at the order of the scale τ .Before we end this section let us briefly consider again the embedding(19). The reason why we write down this particular combination is that itmatches the corresponding plane wave expansion in perturbative quantumfield theory. The question is, however, if there exist a deeper reason for thisstructure.Note first that when { ξ i } i ∈ N are given by plane waves (when we set M = T ) , then ξ i ( m ) ± ξ i ( − m ) are their real and imaginary parts respectively.Thus, when ’ x i ’ appears in (19) only in combination with ξ i ( m ) + ξ i ( − m ) ,then it ensures that the expansion ∑ i x i ξ i is well defined, i.e. that the vectorsin the expansion are real. We have assumed that ξ i are real eigenfunctionsfor this reason but it appears that quantum field theory has already takenthis into account.Let us for now no longer assume that { ξ i } i ∈ N are real and rewrite (19)in the form ( φ ′ π ) = ∞ ∑ i = √ ( s − i s i − ) ( x i Re ( ξ i ) τ i ∂∂x i Im ( ξ i ) ) (24)where Re ( ξ i ) = ξ i + ξ ∗ i , Im ( ξ i ) = ξ i − ξ ∗ i . This statement depends on the choice of σ in (16).
18e then find the operators J i = √ ( s − i s i − ) , J i = . It is an interesting question whether there exist a mathematical explanationfor this particular form.
Note that with the construction, that we have presented so far, the space-time metric is not an input, as is the case in most other approaches toquantum field theory. Indeed, the only geometrical input is the metric g on the 3-dimensional manifold M . This raises the interesting question what4-metric will emerge from the construction with a time-evolution given bythe Hamilton operator and whether the quantum theory will be covariantwith respect to this metric.One piece of information about the emergent 4-dimensional metric can bedetermined immediately, namely its signature. The correspondence betweenour construction and a canonically quantised real scalar field, which we havedemonstrated in this section, is compatible with a 4-dimensional metric thathas a Minkowski signature. Since the Bott-Dirac operator is a canonicalstructure it does not seem possible to incorporate any other signature. S So far we have identified three basic ingredients from which we have builta scalar quantum field theory. These are the Bott-Dirac operator B , thealgebra C ∞ b ( M × R ) and the Hilbert space L ( S ) ⊗ Λ ∗ S ⊗ L ( M ) . Since weknow that B has compact resolvent modulo Λ ∗ S when acting on L ( S ) ⊗ Λ ∗ S the question arises whether the commutator [ B, M a ] , a ∈ C ∞ b ( M × R ) , isbounded. The commutator can be computed as [ B, M a ]( m, x ξ ( m ) + x ξ ( m ) + . . . ) = ∞ ∑ k = ¯ c k ξ k ( m ) a ′ ( m, x ξ ( m ) + x ξ ( m ) + . . . ) . We therefore see that the commutator exist and is bounded when ∞ ∑ k = ∥ ξ k ∥ ∞ < ∞ . (25)19he operator B does not have compact resolvent modulo Λ ∗ S when it actson L (S) ⊗ Λ ∗ S ⊗ L ( M ) , due to the L ( M ) factor. We can however repairthis: The algebra C ∞ b ( M, R ) is acting on L (S × M ) = L (S) ⊗ L ( M ) asan algebra of functions. We can therefore consider the trace T r S ⊗ τ ⊗ τ g , where T r S is the ordinary trace on B ( L (S)) , τ is the finite normalised traceon Λ ∗ S , and τ g is the finite trace on functions on M given by τ g ( f ) = ∫ M f ( m ) dg ( m ) . With this trace the triple ( B, C ∞ b ( M × R ) , L (S) ⊗ Λ ∗ S ⊗ L ( M )) (26)becomes a semi-finite spectral triple when the condition (25) is fulfilled. Thisimplies that a scalar quantum field theory can be understood as a geometri-cal construction over the configuration space S of scalar field configurations. We are now going to construct a quantum Yang-Mills theory. Let therefore M be a compact manifold and let A be a configuration space of gaugeconnections that takes values in the Lie-algebra of a compact gauge group G and let Alg (A) = HD ( M ) , where HD ( M ) is an algebra generated byholonomy-diffeomorphisms as will be described next and which was firstdefined in [8, 4].A holonomy-diffeomorphism f e X ∈ HD ( M ) , where f ∈ C ∞ ( M ) , is aparallel transport along the flow t → exp t ( X ) of a vector field X . To seehow this works we first let γ be the path γ ( t ) = exp t ( X )( m ) running from m to m ′ = exp ( X )( m ) . Given a connection ∇ that takesvalues in a n -dimensional representation of the Lie-algebra g of G we thendefine a map e X ∇ ∶ L ( M ) ⊗ C n → L ( M ) ⊗ C n via the holonomy along the flow of X ( e X ∇ ξ )( m ′ ) = Hol ( γ, ∇ ) ξ ( m ) , (27)20here ξ ∈ L ( M, C n ) and where Hol ( γ, ∇ ) denotes the holonomy of ∇ along γ . This map gives rise to an operator valued function on the configurationspace A of G -connections via A ∋ ∇ → e X ∇ , which we denote by e X . For a function f ∈ C ∞ ( M ) we get another operatorvalued function f e X on A , which we call a holonomy-diffeomorphisms .Furthermore, a g valued one-form ω induces a transformation on A andtherefore an operator U ω on functions on A via U ω ( ξ )( ∇ ) = ξ ( ∇ − ω ) , which gives us the quantum holonomy-diffeomorphism algebra, denoted QHD ( M ) , as the algebra generated by HD ( M ) and all the U ω operators(see [19]).To obtain a representation of the QHD ( M ) algebra we let ⟨ ⋅ ∣ ⋅ ⟩ s denotethe Sobolev norm on Ω ( M ⊗ g ) , which has the form ⟨ ω ∣ ω ⟩ s ∶ = ∫ M dx (( + τ ∆ σ ) ω , ( + τ ∆ σ ) ω ) T ∗ x M ⊗ C n (28)where the Hodge-Laplace operator ∆ and the inner product ( , ) T ∗ x M ⊗ C n on T ∗ x M ⊗ C n depend on a metric g and where τ and σ are positive constants.Also, we choose an n -dimensional representation of g .Next, denote by { β i } i ∈ N an orthonormal basis of Ω ( M ⊗ g ) with respectto the scalar product (28). With this we can construct a space L ( A ) asan inductive limit over intermediate spaces L ( A n ) with an inner productgiven by ⟨ η ∣ ζ ⟩ A n = ∫ R n η ( x β + . . . + x n β n ) ζ ( x β + . . . + x n β n ) dx . . . dx n (29)where η and ζ are elements in L ( A ) , as explained in section 3. Finally, wedefine the Hilbert space H YM = L ( A ) ⊗ L ( M, C n ) (30)in which we then construct the following representation of the QHD ( M ) .First, given a smooth one-form ω ∈ Ω ( M, g ) we write ω = ∑ ω i β i . The The holonomy-diffeomorphisms, as presented here, are not a priori unitary, but bymultiplying with a factor that counters the possible change in volume in (27) one canmake them unitary, see [4]. U χ acts by translation in L ( A ) , i.e. U ω ( η )( ω ) = U ω ( η )( x β + x β + . . . ) = η (( x + ω ) β + ( x + ω ) β + . . . ) (31)with η ∈ L ( A ) . Next, we let f e X ∈ HD ( M ) be a holonomy-diffeomorphismand Ψ ( ω, m ) = η ( ω ) ⊗ ψ ( m ) ∈ H YM where ψ ( m ) ∈ L ( M ) ⊗ C n . We write f e X Ψ ( ω, m ′ ) = f ( m ) η ( ω ) Hol ( γ, ω ) ψ ( m ) (32)where γ is again the path generated by the vector field X with m ′ = exp ( X )( m ) . Theorem 5.0.1.
Equations (31) and (32) gives a strongly continuous Hilbertspace representation of the
QHD ( M ) algebra in H YM .Proof. In [1] we prove that (31) and (32) give rise to a strongly continuousHilbert space representation of the
QHD ( M ) algebra in the special casewhere s i = i ∈ N . This proof can be straight forwardly adopted tothe case where { s i } i ∈ N is a monotonously increasing sequence and we leaveit to the reader to check this. Just as we did for the scalar field we now let M = T , the 3-torus, and thenform a linear combination A ( m ) = ∑ i √ s i ( q i β i ( m ) + q † i β i ( − m )) ,E ( m ) = ∑ i √ s i ( q i β i ( m ) − q † i β i ( − m )) , (33)which is similar to (19) but where we must keep in mind that ’ i ’ is a multi-index, that includes also Lie-algebra and vector degrees of freedom. Toclarify this let us split the summations in (33) up by separating out theLie-algebra and spatial part: A a ( m ) = ∑ k,r √ s k ǫ rk + τ λ σk ( q ak,r e ik ⋅ m + q a † k,r e − ik ⋅ m ) ,E a ( m ) = ∑ k,r √ s k ǫ rk + τ λ σk ( q ak,r e ik ⋅ m − q a † k,r e − ik ⋅ m ) , (34)22here r ∈ { , , } are the spatial indices and ’ a ’ a Lie-algebra index. Wehave also assumed that the sequence { s n } n ∈ N only depends on the index k .Let us this time fix the parameters with s k = ω p k = ∣ p k ∣ , which gives B ∣ b in (13) the form B ∣ b = ∑ k,r ∣ p k ∣ q † k,r q k,r (35)Compare this construction to perturbative quantum field theory of ageneral gauge field A ( m ) and its conjugate E ( m ) on M = R in the Coulombgauge [21]. Within the framework of canonical quantisation these fields areexpanded according to A ( m ) = ∫ d p ( π ) √ ∣ p ∣ ∑ r = ǫ r ( p ) ( a rp e ip ⋅ m + a r † p e − ip ⋅ m ) E ( m ) = ∫ d p ( π ) ( − i )√ ∣ p ∣ ∑ r = ǫ r ( p ) ( a rp e ip ⋅ m − a r † p e − ip ⋅ m ) (36)where ǫ r ( p ) is a set of polarisation vectors and where a p and a † p are creationand annihilation operators acting in a corresponding Fock space. Note thatthese are Lie-algebra valued. Also, the Hamiltonian of the free theory is inthe Coulomb gauge given by [21] H free = ∫ d p ( π ) ∣ p ∣ ∑ r = Tr g a r † p a rp where Tr g denotes a trace over the Lie-algebra g .We see that the construction of a Bott-Dirac operator interacting witha representation of the HD ( M ) algebra coincides with the free sector of acanonically quantised gauge field in the Coulomb gauge when we take theflat and local limits M → R , τ →
0. The only discrepancy is that the sumover r in (36) runs only over transversal degrees of freedom where as thesum in (34) runs over all three spatial directions. This is due to the factthat we have not restricted the degrees of freedom in the embedding A ∋ ω = ∑ i ω i β i → ( x , x , ... ) of the configuration space A into R ∞ to only include transversal degrees offreedom. This can be straight forwardly done, however.Thus we conclude that also in the case of a gauge theory does the gen-eral framework of non-perturbative quantum field theory, that we have pre-sented, coincide with that of a canonically quantised gauge field in the flat23nd local limit and with the free Hamiltonian given by the square of theBott-Dirac operator.It is remarkable that it is the same Bott-Dirac operator, that gives riseto the Hamiltonians in both the case of a free scalar field and in the case ofthe free sector of a gauge field. In the next section we will see that the Bott-Dirac operator also gives rise to the Hamiltonian of a quantised fermionicfield.Note again that the signature of the emergent 4-dimensional metric, thatis compatible with the above analysis, will have a Minkowski signature.Next let us briefly consider also the full Yang-Mills Hamiltonian, whichcan be written H YM = ∫ d x (( E aµ ) + ( B aµ ) ) (37)where E is again the conjugate field to the gauge field A and where B µ = ǫ νσµ F νσ with F being the field strength tensor of A . In [1] we showed thatthe operators ( B aµ ) and ( E aµ ) can be constructed within our framework,where they will be local only up to a correction at the order of the scale τ . A We will here consider the ∗ -subalgebra HD ( M )∣ loops of HD ( M ) generatedby the closed flows. The representation of this algebra generated by a givenconnection is an algebra of matrix valued functions over M . We can there-fore, like in the case of the scalar theory, consider the trace T r S ⊗ τ ⊗ τ g , but where τ g ( f ) = ∫ M T r C n ( f ( m )) dg ( m ) . Just as it was the case for the scalar theory we note that we have asemi-finite spectral triple over A consisting of ( B, HD ( M )∣ loops , H ′ YM ) . under the condition (25) and where H ′ YM = H YM ⊗ Λ ∗ A .We have chosen to consider the ∗ -sub-algebra HD ( M )∣ loops , since it isnot clear how one is to define a trace on the L ( M, C n )) part of the Hilbertspace with the full HD ( M ) algebra.24 Fermionic quantum field theory
Since our discussion in the previous sections has been concerned with bosonicquantum field theory only the question arises whether fermionic quantumfield theory has a place in our framework as well. This is the topic of thefollowing discussion, where we will show that this is indeed the case.So far we have found that the free Hamiltonian of a bosonic field theory isgiven by the square of the Bott-Dirac operator (8) acting in the Hilbert space L ( R ∞ ) . The Bott-Dirac operator itself acts, however, in the Hilbert space L ( R ∞ ) ⊗ Λ ∗ R ∞ and the square of the Bott-Dirac operator involves besidesthe harmonic oscillator also the operator B ∣ f , which acts in Λ ∗ R ∞ only.Thus, the additional factor Λ ∗ R ∞ in the Hilbert space involves structurewhich, as we shall see, amounts precise to a fermionic quantum field theory.Let us begin with a spinor ψ and its conjugate iψ † on M = R and itscanonical quantisation in terms of plane waves ψ ( m ) = ∫ d p ( π ) ∑ s = √ ω p [ b sp u s ( p ) e − ip ⋅ x + c s † p v s ( p ) e ip ⋅ x ] ψ † ( m ) = ∫ d p ( π ) ∑ s = √ ω p [ b s † p u s † ( p ) e ip ⋅ x + c sp v s † ( p ) e − ip ⋅ x ] (38)where u and v are spinors and ( b, b † ) and ( c, c † ) are the associated pairsof creation and annihilation operators that satisfy the anti-commutationrelations { b rp , b s † q } = { c rp , c s † q } = ( π ) δ rs δ ( ) ( p − q ) while all other anti-commutators vanish { b rp , b sq } = { b r † p , b s † q } = { c rp , c sq } = { c r † p , c s † q } = . . . = . (39)Also, the Hamilton operator for a quantised spinor field reads H spinor = ∫ d p ( π ) ω p ( b s † p b sp + c s † p c sp ) . (40)Now, we would like to repeat the line of interpretation employed in theprevious sections, where the plane waves were viewed as flat-space and locallimits of eigenfunctions of a Laplace operator and the momentum integralswere viewed as limits of sums over these eigenfunctions. To this end note25rst that the operators ( a i , a † i ) in (7) and the exterior algebra Λ ∗ R ∞ gives usprecisely the CAR algebra of creation and annihilation operators. Further-more, note that the square of the Bott-Dirac operator gives us the fermionicoperator 12 B ∣ f ∶ = ∑ i ω i a † i a i that has precisely the form of the fermionic Hamilton (40) when the limitsmentioned above are taken into consideration and if we identify again s i = ω i .Thus, we see that the construction, that we have presented in this paper,naturally includes fermionic quantum field theories too.There is a caveat, however, which is that the number of degrees of free-dom in the fermionic sector has to match that of the bosonic sector. Whereasit makes sense to consider a bosonic field on its own – we can just considerthe Hilbert space L ( R ∞ ) without the infinite-dimensional exterior algebra –a fermionic field will in this framework always come together with a bosonicfield and have the same number of degrees of freedom. For a scalar fieldtheory this appears to be a problem due to the spin-statistics theorem, buta gauge field, with two transversal degrees of freedom, can match that ofa 2-spinor as long as the gauge field and the spinor field transform in thesame representation of the gauge group. Let us once more consider a real scalar field and its embedding into R ∞ in(19) and let us consider the commutator between the Bott-Dirac operatorand the field φ ′ ( m ) on M = T , i.e. φ ′ ( m ) = ∑ i √ s i ( q i ξ i ( m ) + q † i ξ i ( − m )) ,which reads [ B, φ ′ ( m )] = ∑ i √ ( a † i ξ i ( m ) + a i ξ i ( − m )) (41)Now, if we for a moment ignore the spinors u and v in equation (38) andonce more view the operator expansions in canonical quantisation as a flatand local limit of an expansion in terms of Sobolev eigenvectors ξ i , i.e. ∑ i ←→ ∫ d p ( π ) ξ i ( m ) ←→ e im ⋅ p ,s i ←→ ω p (42)26hen we see a clear resemblance between the commutator (41) and the conju-gate fermionic field operator ψ † in (38), where the operators ( a † i , a i ) generatethe CAR algebra. Schematically, we have the relations [ B, ”boson” ] = ”fermion” , { B, ”fermion” } = ”boson”which corresponds to the relation [ d, f ] = df , where the Bott-Dirac operatoris the differential operator, the ”boson” a zero-form and the ”fermion” aone-form. The only serious discrepancy is that the ”fermion” that the Bott-Dirac operator generates is, in this case, a scalar, which seems to be inviolation with the spin-statistics theorem. Note also that the allocation offactors of s i in (41) is different from that in (38).If we instead consider a gauge field and its expansion (33) [ B, A ( m )] = ∑ i √ ( a i β i ( m ) + a † i β i ( − m )) = ˜ ψ ( m ) (43)then we find ˜ ψ ( m ) , which is a fermionic field that has, once we take thecorrespondence (42) into account, the same structure as ψ ( m ) except thatthe spinors u and v are exchanged with the polarisation vectors ǫ r andexcept for the different allocation of factors of s i . Here fermionic creationand annihilation operators are the ( a i , a † i ) operators from (7), where one hasto remember that the index ’ i ’ is a multi-index that also labels the generatorsof the Lie-group.The point here is that the commutator with the Bott-Dirac operatorshifts between the bosonic and fermionic sectors, i.e. between the CAR al-gebra and the CCR algebra. The commutator with the Bott-Dirac operatorwill play an important role in section 8, where we consider inner fluctuationsof the Bott-Dirac operator. M The algebraic representations related to quantum field theories, which wehave discussed so far, all involve products L ( Γ ) ⊗ Λ ∗ ( Γ ) ⊗ L ( M ) betweena Hilbert space of states on a configuration space Γ and a Hilbert space offunctions on M . Since the Bott-Dirac operator acts only in the first Hilbertspace it is natural to consider also a Dirac type operator, that acts in bothHilbert spaces. Let us therefore introduce a spinor bundle S over M andthe Hilbert space L ( M, S ) and write down the sum D tot = B ⊗ + γ ⊗ D (44)27here D is a Dirac operator acting on spinors in L ( M, S ) and which de-pends on a metric on M and where γ is a suitable grading operator thatsatisfies { B, γ } = γ = D has compact resolvent we can immediately conclude that D tot has compact resolvent too. The question whether the commutator between D tot and elements of the algebra Alg ( Γ ) is bounded or not. We suspect thatthis is not the case. The reason is the following: We can construct a statein L ( Γ ) , which is peaked around a field configuration with fast oscillations.Hence for a suitably element in Alg ( Γ ) the corresponding function on M generated by the representation given by the given field, will be fast oscil-lating as well, and hence have a large commutator with D tot . In particularthe commutator will not be bounded. We thus suspect that ( D tot , Alg ( Γ ) , L ( Γ ) ⊗ Λ ∗ Γ ⊗ L ( M, S )) is not a semi-finite spectral triple. Consider again the HD ( M ) algebra with the gauge group SU ( ) and letit be represented on spinors in L ( M, S ) , i.e. instead of the factor C n in(30) we use the spinor bundle S for the representation. This means that the HD ( M ) algebra (as well as the QHD ( M ) algebra) is represented in theHilbert space H = L (A) ⊗ Λ ∗ A ⊗ L ( M, S ) where it now interacts with D tot and where we by Λ ∗ A again mean the in-ductive limit Λ ∗ R n . This geometrical construction is what we call ’quantumholonomy theory’, which was first proposed in [5] and later developed in[1, 2, 17, 18].The reason why we find this model particularly interesting is that itcomes with a possible link to the formulation of the standard model of parti-cle physics in terms of noncommutative geometry as developed by Chamsed-dine and Connes [6, 7]. The point is that if we restrict the algebra HD ( M ) to loops then it reduces in a classical limit characterised by a single (non-trivial) connection in A to an almost commutative algebra HD ( M )∣ loops classical Ð→ C ∞ ( M ) ⊗ M ( C ) . This correspondence, which was first pointed out in [8], is is fact the mainreason why we first became interested in holonomies, see [9]. Thus, in a28emi-classical limit we have the general structure ( D + D F , C ∞ ( M ) ⊗ M ( C ) , L ( M, S ) ⊗ H F ) where D is again the spatial Dirac operator and where D F is an operatorthat is given by the semi-classical limit of the Bott-Dirac operator and whichwill interact with the matrix factor M ( C ) . Also, by H F we refer to theHilbert space L (A) ⊗ Λ ∗ A in the same limit. Thus, this argument suggestthat something reminiscent of an almost-commutative spectral triple, whichis the backbone of Chamseddine and Connes formulation of the standardmodel, will emerge from this model.Clearly, more analyse is required in order to make this argument rigorous.Also, to fully compare this model with that of Chamseddine and Connes itwould be useful to find a Hamiltonian formulation of the latter. Given a Dirac operator D that interacts with an algebra B it is natural toconsider also the fluctuated Dirac operator [22]˜ D = D + A, A = ∑ i a i [ D, b i ] , a i , b i ∈ B where A in the language of noncommutative geometry is a one-form. Now, inthe present case we have the Bott-Dirac operator and an algebra Alg ( Γ ) offield operators (for example the algebra C ∞ b ( M × R ) or the HD ( M ) algebra).We have already seen that the square of the Bott-Dirac operator gives riseto the free Hamiltonian in the field theories that we have considered. Thequestion is, therefore, what the square of the fluctuated Bott-Dirac operatorgives. As we shall see, the inclusion of the additional term gives rise tointeractions, both bosonic and fermionic.Let us begin with the Bott-Dirac operator and a representation of ageneral algebra
Alg ( Γ ) of field operators where we have the field operatorΘ and the commutator [ B, Θ ] = ˜Ψwhere ˜Ψ is a fermionic field as we discussed in section 6.1. Thus, keepingthe discussion at a general level we find the one-formΘ [ B, Θ ] = Θ ˜Ψ (45)and hence the fluctuated
Bott-Dirac operator has the form˜ B = B + Θ ˜Ψ . B = B + H fluc where the modification H fluc has the form H fluc = ( Θ ˜Ψ ) + { B, Θ ˜Ψ } . Now, the point here is that while B gives the Hamiltonians of the free sys-tem of bosonic and fermionic fields, then H fluc gives us interactions betweenthe bosonic and fermionic sectors. This means that the construction withthe Bott-Dirac operator is a general geometrical structure that based onan embedding of a configuration space Γ produces well-defined interactingquantum field theories of bosonic and fermionic fields.Let us also briefly discuss the more general case with the Dirac type op-erator D tot as we discussed in section 7. The one-form in (45) then becomes Θ [ D tot , Θ ] = Θ ˜Ψ + γ Θ [ D, Θ ] leading to the fluctuated operator˜ D tot = D tot + Θ ˜Ψ + γ Θ [ D, Θ ] . Thus, if we consider the square˜ D tot = D + B + H ′ fluc then we find a Hamilton operator that involves both a gravitational part(which remains classical), the square of the Bott-Dirac operator, that givesthe Hamiltonian of the free bosonic and fermionic sectors, and finally theoperator H ′ fluc , which involves interactions that are both purely bosonic aswell as mixed bosonic and fermionic. It is an interesting question preciselywhat kind of quantum field theories that will emerge from this geometricalsetup, both in the case of a scalar theory and in the case of a gauge theory. In the quantum field theories, that we have discussed so far, the expectationvalues in L ( Γ ) , where Γ is a configuration space, all have the general form ⟨ η ( ψ )∣ Ω ( ψ )∣ η ( ψ )⟩ L ( Γ ) = ∫ Γ dψ Ω ( ψ ) exp ( − ∥ ω p ψ ∥ s ) Note that this one-form is not expected to be bounded. ( ψ ) is some composite field operator. The point here is that theSobolev norm ∥ ⋅ ∥ s plays the role of a weight in the measure over Γ. Thisimplies that field configurations, that have a small Sobolev norm, will havea larger weight compared to field configurations, that have a large Sobolevnorm. Put differently, field configurations, that vary at a large scale havea larger weight than those that vary at a short scale, and in particular itmeans that field configurations, that are singular, have zero measure as suchconfigurations will have infinite Sobolev norm . Thus, extreme situations,such as the initial big bang singularity and big bang singularities appear tobe ruled out within this construction.This means that the quantum field theories, which we have presented,come with an ultra-violet suppression that is enforced by representationtheory. This raises an interesting question concerning the search for a theoryof quantum gravity.It is generally believed that a theory of quantum gravity will in one wayor another quantise space and time and that quantum effects of the gravi-tational fields will play an important role below the Planck length. Simplearguments, that combine quantum mechanics and general relativity, suggestthat measurements below the Planck length are operational meaninglesssince the measuring probe must carry so much energy and momentum thatit will create a black hole, and a theory of quantum gravity is believed to bethe source of such an short scale fix. But if quantum field theory effectivelydampens degrees of freedom below the Planck length – as it happens withinthe framework, that we have presented – then one might ask whether thegravitational field needs to be quantised at all? If quantum field theorysuppresses those very field configurations, that would otherwise probe thequantum domain of the gravitational field, then it seems to us that one of thekey arguments in favour of a theory of quantum gravity would be invalidated.
10 Discussion
One of the most interesting feature of the quantum field theories, that wehave presented in this paper, is that they exist in a rigorous sense. To thebest of our knowledge these are the first examples of interacting quantumfield theories in 3 + This is best seen by the fact that the Sobolev norm, that is relevant for our case,dominates the supremum norm in three dimensions [20].
Alg ( Γ ) of field operators one should choose.Once this choice has been made everything appears to follow essentiallycanonically. We have argued that the HD ( M ) algebra is a particularly nat-ural choice but it may be that other choices will prove even more interesting.Here the ultimate criteria of success must be whether a particular choice ofalgebra is able to connect to and explain the structure of the standard modelof particle physics.What remains now is to understand precisely what these quantum fieldtheories contain and how they incorporate the known features of ordinaryperturbative quantum field theory and where they differ.Here a key question is that of space-time covariance. A (perturbative)quantum field theory on a curved background is usually formulated withrespect to a space-time background metric where the causal structure of themetric is build into the quantum theory via the commutation relations. Inthe present case we start out with a 3-dimensional manifold with a metric.The question is, therefore, whether the quantum theory, that we find, will becovariant with respect to the 4-metric that emerges with time evolution. Itis interesting that this metric appear to automatically have the Minkowskisignature, which suggest that special relativity could be an output of thisframework.Another question is whether the non-locality, that characterises the rep-resentations, which we find, will affect the causal structure of the emergentspace-time manifold, i.e. whether the non-locality will be accompanied witha correction to special relativity.The representations of algebras of field operators, that we find, all dependon a scale τ – we interpret it as the Planck scale – where degrees of freedomare dampened according to their relation to this scale and where field oper-ators can only be localised up to corrections proportional to τ . This meansthat these representations cease to exist in the local limit τ →
0, whichis of course the limit where the UV divergencies known from perturbativequantum field theory arise. This raises the question how renormalisation32heory will emerge in this limit and what role it plays away from τ = HD ( M ) algebra, is not gauge invariant as differentelements in a gauge orbit will have different Sobolev norms. We think thiscan be understood in terms of a gauge fixing procedure but more analysisis required to fully understand this issue.As already mentioned the ultimate test for any candidate for a funda-mental theory must be whether it can explain the structure of the standardmodel of particle physics. Here we believe that the HD ( M ) algebra isparticularly interesting as it produces an almost commutative algebra ina semi-classical limit, which is the type of algebra that Chamseddine andConnes have identified as the cornerstone in their formulation of the stan-dard model [6, 7]. Again, to pursue this line of analysis we will need toemploy the full toolbox of noncommutative geometry as well as carefullyanalysing the semi-classical limit. In respect to the latter it is interestingthat Higson and Kasparov [3] have already developed such an analysis, albeitfor a different purpose.Note that both the CCR and the CAR algebra appear naturally in ourconstruction. Due to the construction of L ( Γ ) we have already chosen arepresentation of the CCR algebra. As for the CAR algebra we have inthis article notationally chosen the Fock space as the representation butwe are free to choose any representation we would like. The CAR algebrahas representations, which in the weak closure give type III von Neumannalgebras. This opens up for applying Tomita-Takesaki theory, where wewould get a one-parameter groups of automorphisms, which would entail atime development. It is interesting to compare this to the dynamics givenby the Hamiltonian.It is striking that the quantum field theories, which we find, appear tosolve the problem of space-time singularities, a problem that is normallyexpected to be solved by a theory of quantum gravity. If states cannot bearbitrarily localised in space it means that those field configurations, whichwould cause space to curve infinitely, cannot form. This seems on the onehand to prevent the singularities purported to exist within black holes andat the initial big bang and on the other hand to remove one of the mostimportant arguments in favour of a theory of quantum gravity. We find thisidea rather interesting.Let us end this paper on a speculative note. One of the most interestingproblems in contemporary theoretical physics is the question of the massgap in Yang-Mills theories. Since the square of the Bott-Dirac operator,which gives the free Hamiltonian for a non-perturbative Yang-Mills theory,33as a discrete spectrum with the lowest eigenvalue being zero, one mightthink that this holds also for a full, interacting theory. If this is the casethen we believe that we will have the interesting situation that the massgap contains information about the size of the universe – in the sense thatthe spectrum of the square of the Bott-Dirac operator becomes continuousin the limit where the manifold is no longer compact. Put differently, theexistence of the mass gap could be read as evidence that our spatial universeis compact. Acknowledgements
We would like to thank Prof. Nigel Higson for bringing his work withProf. Gennadi Kasparov on the Bott-Dirac operator to our attention. JMGwould like to express his gratitude towards Ilyas Khan, United Kingdom,for his generous financial support. JMG would also like to express his grati-tude towards the following sponsors: Ria Blanken, Niels Peter Dahl, SimonKitson, Rita and Hans-Jørgen Mogensen, Tero Pulkkinen and ChristopherSkak for their financial support, as well as all the backers of the 2016 In-diegogo crowdfunding campaign, that has enabled this work. Finally, JMGwould like to thank the mathematical Institute at the Leibniz University inHannover for kind hospitality during numerous visits.
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