Quantum-field theories as representations of a single $^\ast$-algebra

We show that many well-known quantum field theories emerge as representations of a single $^\ast$-algebra. These include free quantum field theories in flat and curved space-times, lattice quantum field theories, Wightman quantum field theories, and string theories. We prove that such theories can be approximated on lattices, and we give a rigorous definition of the continuum limit of lattice quantum field theories.


Introduction
The Wightman distributions play a fundamental role in Wightman quantum field theories (Wightman QFTs) [1]. The reconstruction theorem demonstrates that knowledge of the Wightman distributions is sufficient to obtain a unique Wightman QFT. In particular, the Wightman distributions define a state of a Borchers-Uhlmann (BU) algebra, and the associated Wightman QFT is obtained as a representation of a BU-algebra from that state [2,3]. Interestingly, interacting as well as non-interacting Wightman QFTs emerge as representations of the same BU algebra.
Although the Wightman axioms provide a remarkably successful framework in Minkowski space-time, they cannot be generalized to curved space-times in a straightforward manner. This is one of the motivations to study QFT in an algebraic framework, and substantial progress has been achieved with this approach in recent years. An overview of the field can be found in Refs. [4,5], and we will discuss more details of those achievements in Sec. 5. However, the approach taken in Ref. [4] yields an interesting aspect that results from the generalization of the axiomatic approach to curved space-times. Starting point of the axiomatic approach in Ref. [4] is a free * -algebra, Free(M ), of quantum fields on a background structure, M , which, amongst others, refers to a globally hyperbolic space-time. The quantum-field algebra, A(M ), is obtained by factoring Free(M ) by a set of relations, in which coefficients of an operator-product expansion (OPE) play a fundamental role, i.e. there exists a * -homomorphism, π : Free(M ) → A(M ), which essentially is defined by properties of the OPE coefficients. The set of states of the theory, S(M ), is further constrained to support the OPE, positivity, and a microlocal spectrum condition. However, if we construct a representation of A(M ) from a state ω ∈ S(M ), then we basically obtain a representation of Free(M ) with respect to the state ω • π. The set of states of the QFT can therefore also be seen as a subset of the set of states over Free(M ), which is obtained by appropriate constraints. Quite different QFTs can therefore emerge as representations of the same * -algebra, Free(M ), and we note that Free(M ) plays an analogous role as the BU-algebras in Wightman QFTs. We pick up this idea and explore it in a more general approach in this paper.
In Sec. 2.1, we introduce the general setting, in which we define the terms test-function space, Q-map, and Q-theory. Q-maps are mathematical generalizations of quantum fields that lack specific physical properties, and Q-theories are corresponding generalizations of QFTs. We use Q-maps and Q-theories as purely technical devices to construct appropriate * -algebras, from which QFTs arise as representations. In particular, as it is the case for quantum fields in Wightman QFTs, for example, the image of a Q-map is a set of operators in a * -algebra, and a Q-theory is the continuous representation of the polynomial algebra generated by the operators in the image of a Q-map. In Sec. 2.2 we further define in which sense a Q-theory emerges from a representation of a * -algebra (i.e. the polynomial algebra generated by a Q-map), and in Sec. 2.3 we introduce a Q-map whose polynomial algebra, A 0 , can be represented in any Q-theory, so that the Q-theory emerges from that representation. We also prove that Q-theories can be approximated on lattices, and we give a rigorous definition of the continuum limit. In the remaining part of the paper we show that many well-known QFTs are Q-theories that emerge as representations of A 0 . These are the central results of this paper.
We discuss in Sec. 3 the applicability of the approach to hermitian scalar Wightman QFTs, and in Sec. 4 we discuss free scalar QFTs and Dirac QFTs in curved space-times. In Sec. 5, we continue with the discussion of perturbatively interacting quantum fields in curved space-times, and we also relate to recent developments in algebraic QFT in more detail, c.f. Refs. [4,5]. In Sec. 6, we consider string theories and lattice QFTs, and we discuss the continuum limit of lattice QFTs more concretely.
2. Q-maps and Q-theories 2.1. General setting. Let us begin with the definition of some general terms. In Wightman QFTs, quantum fields are defined as operator-valued distributions over a test-function space. We choose a similar approach: (1) A conjugation, C, on a complex vector space, V , is an antilinear map satisfying C 2 = 1 and C(av + bw) =āC(v) +bC(w) for all a, b ∈ l C and all v, w ∈ V . (2) Let V 1 and V 2 be vector spaces with conjugations C 1 and C 2 . A c-homomorphism, A test-function space is a complex separable locally-convex Hausdorff topological vector space on which a conjugation is defined. (4) A Q-map is a complex-linear map from a test-function space into a * -algebra, Φ : V → P (Φ), where P (Φ) is the polynomial algebra generated by the set of (5) A Q-theory is a pair (Φ, ω), where Φ is a Q-map, ω is a state over P (Φ), and, for all n ∈ IN, is a multi-linear continuous functional on V n . We note that we restrict test-function spaces to be separable and that larger test-function spaces will be considered in a subsequent paper. Moreover, we will show in subsequent sections that quantum fields typically are Q-maps and that QFTs typically are Q-theories. We note however that Q-maps and Q-theories lack specific physical properties, like commutation relations, for example, and we emphasize that they serve a purely technical purpose in this paper. In particular, the above definitions are not intended as a set of axioms for QFTs.
Let us further elaborate on our definitions. An example of a test-function space is Schwartz space, S(IR n ). Complex conjugation is given in S(IR n ) by Moreover, for any locally-convex topological vector space, V , a test-function space can be constructed as follows. LetV denote the corresponding complex conjugate vector space (c.f. appendix A.2 in Ref. [6]), and let j : V →V denote the natural anti-linear bijection. The set {j(B) : B open in V } is a locally-convex Hausdorff topology onV . Let W = V ⊕V be equipped with the product topology of V ×V , and define the conjugation C(f, g) = (j −1 (g), j(f )), then W is a test-function space.
A test-function space, V , always has an associated Q-map, which can be constructed as follows. Let denote the tensor algebra of V . For the sake of notational convenience, we denote an element in V ⊗n by Moreover, we define the involution (a n ) * = (C (n) a n ) ((a n ) ∈ A V ), so that A V is a (non-commutative) * -algebra. We further define the complex-linear Q-map, Multiple Q-maps can be combined into a single Q-map as follows. Let Φ i : V i → P (Φ i ) (i ∈ I) be at most countably many Q-maps, for which multiplication and addition of operators are defined. Define and equip V with the product topology, i.e. V is a subspace of i V i . V is a complex separable locally-convex Hausdorff topological vector space, i.e. V is a test-function space. Define further the map then Φ is a Q-map.

2.2.
Quantum theories from Q-map cores. Let (Φ, ω) be a Q-theory. ω induces the representation of P (Φ) on a pre-Hilbert space, D ω , by the GNS construction (for * -algebras). Elements in D ω are given by equivalence classes of operators in P (Φ). The representation is commonly denoted by (H ω , π ω , Ω ω ), where H ω is the completion of D ω , π ω is a * -homomorphism, and Ω ω is the unit vector corresponding to the unity operator, 1. Expectation values of operators in P (Φ) are given by The continuity property in our definition of a Q-theory further guarantees that u, defines a complex-linear multi-continuous functional for all u, v ∈ D ω . This is satisfied in scalar Wightman QFTs, for example. However, let τ ω denote the locally-convex topology generated by the set of semi-norms, { π ω ( · )u : u ∈ D ω }, on P (Φ). Let us further assume that V contains a subset, V s , so that π ω (Φ(f )) is essentially self-adjoint for all f ∈ V s and that P (Φ) is generated by Φ(V s ). We call such a Q-theory regular. Let A be the C * -algebra generated by the set {exp(iπ ω (Φ(f ))) : f ∈ V s }, then π ω (P (Φ)) ′ = A ′ , i.e. the commutants agree, and π ω (P (Φ)) ′′ = A ′′ . A ′′ may be seen as the algebra of observables of the state ω, which contains all projection-valued measures that are relevant in the respective theory (c.f. Def. 2.6.3 in Ref. [7]). If the set {Φ(f ) : f ∈ V s } is irreducible, then the set {exp(iπ ω (Φ(f ))) : f ∈ V s } is also irreducible, and A ′′ equals the set of linear-bounded operators on H ω . However, by von-Neumann's density theorem, A is dense in A ′′ with respect to the weak operator topology, and the restriction of any operator in A ′′ to D ω is the τ ω -limit of a net of polynomials of operators in Φ(V ).

2.3.
A universal Q-map. Let V 0 be the vector space of cofinite complex sequences, and define the conjugation Proof: Let {f n } be a countable, dense subset of V , let W be the linear span of {f n }, and let C denote the conjugation on V . Let further {e n } be a maximal linear-independent subset of is a vector-space homomorphism between V 0 and W , which is compatible with the conjugations, Let V be the class of test-function spaces, and for Note that this is the weakest topology for which all functions h V (V ∈ V) are continuous.
B is therefore absorbing, and V 0 is locally convex. Moreover, V 0 is the union of countably many finite-dimensional spaces, Since each V 0,n is finite-dimensional, the respective subspace topologies are equivalent to the Eucledian topologies, which entails that each V 0,n is separable. Let W 0,n be a countable, dense subset of V 0,n , then n W 0,n is a countable, dense subset of V 0 .
Let Φ 0 : V 0 → A 0 be the Q-map associated with V 0 . Lemma 1 and proposition 1 yield the following theorem.
Considering proposition 2, Φ 0 actually is a universal Q-map, since any Q-theory basically is a Q-theory of Φ 0 , which emerges from a representation of P (Φ 0 ).
Proof: We need to show that, for all n ∈ IN, is a multi-linear continuous functional on V n 0 . Let F be the set of linear functions from . The set of semi-norms, {p f } f ∈F , defines a locally-convex Hausdorff topology on the set V 0 . Let V F 0 denote the corresponding topological space. We choose C 0 as conjugation on V F 0 , so that V F 0 is a test-function space. Each f ∈ F is continuous when considered as a function from V F 0 to l C. Due to the definition of the test-function space V 0 , each open set in V F 0 is also open in V 0 , so that each f ∈ F is also continuous when considered as a function from the test-function space V 0 to l C. Hence, ω (n) is continuous in each argument, and therefore it is continuous on V n 0 . Let S 0 be the set of states over P (Φ 0 ). Each a ∈ P (Φ 0 ) defines a linear functional on S 0 by l a (ω) = ω(a) for ω ∈ S 0 . The corresponding set of semi-norms, p a (ω) = |l a (ω)| = |ω(a)|, defines a topology on S 0 , and a net (ω i ) in S 0 converges to an ω ∈ S 0 with respect to that topology, if lim i ω i (a) = ω(a) for all a ∈ P (Φ 0 ).
Theorem 3: Let (ω i ) be a net of states over P (Φ 0 ). (ω i ) converges to a state ω if and only if (ω i,(n) ) converges to ω (n) for all n ∈ IN.
In the remaining part of this paper, we will show that theorem 1 applies to many wellknown QFTs, and that theorems 2 and 3 define the continuum limit of lattice QFTs in our approach.

Application to Wightman quantum field theories
The relation of our approach to Wightman QFTs can be conveniently discussed with the help of the Wightman reconstruction theorem. For the sake of convenience, let us consider a hermitian scalar Wightman QFT. In the reconstruction theorem, Wightman QFTs are recovered as representations of a Borchers-Uhlmann algebra [1]. The test-function space in Wightman QFTs is Schwartz space, S(IR d ) (d ≥ 2). Using the terminology and the definitions of Sec. 2, the Borchers-Uhlmann algebra is given by A V with V = S(IR d ), and the corresponding Q-map is denoted by Φ V : V → A V . Note that S(IR d ) is separable, and that corollary 1 and theorem 1 apply.
We note that corollary 2 applies to any (hermitian scalar) Wightman QFT involving d ≥ 2 space-time dimensions, and to Wightman QFTs of interacting quantum fields as well as free quantum fields. Moreover, considering proposition 2, we can re-formulate corollary 2 into a looser statement: There exists a * -homomorphism, π, so that (Φ, ω) and (Φ 0 , ω • π) essentially yield the same quantum theory, which emerges from a representation of P (Φ 0 ).
Let us discuss two examples of Wightman QFTs in more detail. In general, the set of field operators in Wightman QFTs is irreducible. Moreover, let V s denote the subset of real functions in S(IR d ), then the set {Φ(f ) : f ∈ V s } is also irreducible, and Φ(V s ) generates P (Φ). In the usual Fock-space representation of free scalar fields [8], the operators Φ(f ) (f ∈ V s ) are essentially self-adjoint, i.e. the QFT is regular. Unfortunately the situation is less straightforward for Wightman QFTs of interacting fields, since there do not exist that many examples. However, let us consider P (ϕ) 2 as presented in Ref. [9]. P (ϕ) 2 is defined in flat space-time with one time dimension and one space dimension. The corresponding Fock space, F, of the free hermitian scalar QFT is the symmetric tensor algebra over L 2 (IR). For each open bounded interval, B ⊂ IR, let A(B) denote the von-Neumann algebra generated by the operators exp(iϕ(f 1 ) + iπ(f 2 )) (f 1 , f 2 ∈ C ∞ 0 (B), f 1 , f 2 real) and let A denote the norm closure of B A(B). P (ϕ) 2 is constructed by considering the GNS representation, (H ω , π ω , Ω ω ), of A with respect to a specific state, ω. In this representation, the unitary groups W t (f 1 , f 2 ) = π ω (exp(itϕ(f 1 ) + itπ(f 2 ))) (f 1 , f 2 ∈ C ∞ 0 (B)) are strongly continuous, and they have self-adjoint generators.

Application to free quantum field theories
In this section, we discuss the relation of our general approach to free QFTs that implement canonical commutation relations (CCRs) or canonical anti-commutation relations (CARs) on Fock space. We first discuss both cases together without specifying if the Fock space, F, is symmetric or anti-symmetric. We assume however that F is constructed over an infinite-dimensional, separable Hilbert space, H.
In conventional representations [10], the annihilation and creation operators, a(f ) and a * (f ), are both defined over H, so that a(f ) is complex anti-linear and that a * (f ) is complex linear. We note that a(f ) and a * (f ) are densely defined, closed, and that a(f ) * = a * (f ). If there is a complex conjugation, C, defined on H, then H and H 2 are test-function spaces, and we can introduce the complex-linear Q-map Φ 1 (f, g) = a * (f ) + a(Cg). Since Φ 1 (f, 0) = a * (f ) and Φ 1 (0, Cf ) = a(f ), P (Φ 1 ) is the polynomial algebra generated by the irreducible set of operators A = {a(f ), a * (f )} f ∈H . In particular, in the symmetric case (CCRs), we obtain Since H 2 is separable, we can apply theorem 1. = a * (f ) + a(g), and define π U (Φ 2 (f, g)) = Φ ′ 2 (U f,Ū g). π U is a * -homomorphism between P (Φ 2 ) and P (Φ ′ 2 ), Φ ′ 2 is a quotient of Φ 2 , and the Fock representation is a * -homomorphic representation of P (Φ 2 ). We summarize.

Corollary 4:
In any free QFT, the polynomial algebra generated by the irreducible set of operators {a(f ), a * (g)} f ∈H,g∈H * is * -homomorphic to P (Φ 2 ), and Φ 0 is a core for Φ 2 .
Adopting the same loose language as after proposition 2, we state that free QFTs emerge from Fock representations of P (Φ 0 ).
Let us discuss two examples of free QFTs. In Ref. [11], CCRs are defined with H = L 2 (S), where S is a Cauchy surface of a globally hyperbolic manifold. Independence of the actual choice of S is due to a * -isomorphism between representations of the CCRs over the same vector space (C ∞ 0 (M )). We note that a more detailed account is given in Ref. [6], where especially the arbitrariness of the choice of scalar product in the definition of H is discussed. However, we note that corollary 3 applies irrespective of the specifically chosen globally hyperbolic manifold and background metric, i.e. the QFTs emerge from representations of P (Φ 0 ) and P (Φ 2 ), respectively.
The second example are Dirac quantum fields on a globally hyperbolic manifold, M , that are constructed in Ref. [12]. The construction is based on the definition of a scalar product on C ∞ 0 (DS), where DS is the Dirac spinor bundle of spinors on a Cauchy surface, S. DS is a vector bundle, and C ∞ 0 (DS) is locally isomorphic to C ∞ 0 (S) 4 . The completion of C ∞ 0 (DS) yields a separable Hilbert space, H. The dual Hilbert space, H * , is the closure of C ∞ 0 (D * S), the space of cross sections with compact support over the dual vector bundle D * S. The representation of CARs over S is further defined as a representation of the CARs over the pair H, H * . We note that corollary 4 applies irrespective of the specifically chosen globally hyperbolic manifold and background metric, i.e. the QFTs emerge from representations of P (Φ 0 ) and P (Φ 2 ), respectively.

Application to perturbative quantum field theory in curved space-times
The perturbative formulation of quantum theories of interacting fields in curved spacetime is closely related to recent achievements in the field of algebraic QFT [4,5]. In conventional approaches (c.f. Sec. 4), free quantum fields in curved space-time are operator-valued distributions over a space of smooth, compactly-supported test functions on a specific space-time. Independence of the specific space-time background can be further achieved in a categorial approach [5], which is also applicable to perturbatively interacting quantum fields. Starting from free QFTs, renormalized perturbation theory on curved space-times was explicitely formulated by methods of microlocal analysis [13,14,15].
Let us consider the example of a QFT of scalar hermitian fields as it is presented in Ref. [14]. The first step is to enlarge the quantum-field algebra of the corresponding free QFT with the help of microlocal analysis. The test function space of a free scalar hermitian QFT on a globally hyperbolic space-time manifold, M , is C ∞ 0 (M ). Assuming a quasi-free Hadamard state, we can represent the the quantum-field algebra by the GNS construction. In such a representation, however, the quantum fields are operator-valued distributions over a larger, distributional test-function space. Let W n (x 1 , .., x n ) = : φ(x 1 )...φ(x n ) : ω denote the (Wick-ordered) operator-valued distribution that is defined over C ∞ 0 (M n ) and let ω denote a quasi-free Hadamard state (n ≥ 1, W 0 = 1). As can be shown by microlocal analysis, the operator-valued distributions are defined on a larger space, E ′ n , which contains C ∞ 0 (M n ) and which is a subspace of the dual space C ∞ 0 (M n ) ′ . Distributions in E ′ n are compactly supported, and they satisfy the wave-front condition WF(t) ⊂ G n (t ∈ E ′ n ), where G n = (T * M ) n \ H n and Note that such distributions can be multiplied with each other, so that local Wick polynomials can rigorously be defined. Let W be the * -algebra of operators generated by 1 and elements {W n (t)} n∈IN,t∈En (c.f definition 2.1 in Ref. [14]). Note also that E ′ n does not have a proper topology in the sense of Hörmander [16], but that it rather has a so-called pseudo topology, see Ref. [14] for more details.
However, let us endow E ′ n with the sub-space topology, which is inherited from the weak * topology on C ∞ 0 (M n ) ′ . E ′ n is separable, and complex conjugation is well-defined since C ∞ 0 (M n ) is a dense subspace, i.e. E ′ n is a test-function space according to our definition in Sec. 2. Each W n is a Q-map, and the corresponding polynomial algebra, P (W n ), is a sub-algebra of W. As outlined in Sec. 2, we can combine the Q-maps by defining the test-function space E ′ = n E ′ n and the Q-map Φ 3 = n W n , so that we obtain P (Φ 3 ) = W. We apply theorem 1.
In the perturbative approach, interacting fields are formally defined by a perturbation series. Series of operators in W form an algebra X = W IN with multiplication (a n ) ⋆ (b n ) = (a 0 b 0 , a 1 b 0 + a 0 b 1 , ...). Note that the product is defined as if one formally multiplies n a n and n b n . Let us consider the test-function space n E ′ , on which we define the Q-map Φ 4 (f ) = (Φ 3 (f n )). Using the CCRs, one can further show that P (Φ 4 ) contains all co-finite sequences in X . Note that the co-finite sequences are dense in X if we consider a weak topology on W and the product topology on X . However, we can again apply theorem 1.
Moreover, the interacting-field algebra is defined in Ref. [17] as a sub-algbera of X as follows. There exists a multi-linear map T (n) is a vector space and L 1 denotes the Lagrangian. V is a vector space, which is generated by a countably infinite (Hamel) basis, and D 1 (M, V) is the space of compactly-supported smooth densities on M with values in V. An element F ∈ D 1 (M, V) can be uniquely expressed as a finite sum, and v i ∈ V. As a vector space, D 1 (M, V) is therefore isomorphic to n C ∞ 0 (M ). The interacting-field algebra is further defined as the algebra generated by the images of the maps T L 1 , then the interacting-field algebra is P (Φ 5 ). We can again apply theorem 1.
Moreover, in Ref. [18], an algorithm is presented to construct the Wilson operator-product expansion (OPE). The algorithm is generally applicable to perturbative interacting QFT in Lorentzian curved space-times, and it is explicitly presented for the example of a scalar hermitian self-interacting field. However, as proposed in Ref. [4], the OPE can actually be elevated to a fundamental level, so that the QFT is determined by its OPE. This yields a general axiomatic framework for QFTs in curved space-times. In particular, the algebra of interacting quantum fields, F i , is obtained by factoring the corresponding free algebra, F 0 , by a set of relations arising from properties of the OPE coefficients. These relations define an ideal, I, in F 0 , and F i = F 0 /I. We note that the corresponding quotient map, π I : F 0 → F i is a * homomorphism. As pointed out in this paper so far, we typically can find a * homomorphism, π 0 : P (Φ 0 ) → F 0 , so that π 0 (P (Φ 0 )) is dense in F 0 with respect to an appropriate topology. Then, π I • π 0 is a * homomorphism that maps P (Φ 0 ) onto a dense set in F i with respect to another topology. Adopting the same loose language as after proposition 2, we then can state that a perturbative interacting QFT emerges from a representation of P (Φ 0 ), if the corresponding free QFT does.
6. Further representations of P (Φ 0 ) 6.1. Lattice quantum field theory. Let us use quantum chromodynamics on a lattice (LQCD) as an example of a lattice QFT. LQCD is a non-perturbative approach to QCD. Calculations usually are performed using the Feynman path-integral approach. Starting point is a set of CCRs and CARs [19] for symmetric operators x a , p a ,x c , andp c : The indexes represent the degrees of freedom of the quantum system, and they consist of a position, x, and a field index. In conventional QFT, x formally is continuous, but in LQCD, x is discrete and only takes finitely many values. In particular, the Feynman integrals are rigorously defined in LQCD, and calculations are performed in exactly the same way as in conventional QFT, where a space-time continuum is considered [20].
Let us assume that a and c take finitely many value, i.e. 1 ≤ a ≤ m and 1 ≤ c ≤ n. Let A m,n be the operator algebra generated by {x a , p a ,x c ,p c : 1 ≤ a ≤ m, 1 ≤ c ≤ n}. A m,n is the operator algebra of the specific LQCD model instance. We define a function x : l C m → A m,n by x((δ ab ) 1≤b≤m ) = x a and by complex-linear extension, and we define x((z a )) * = x ((z a )). x is a Q-map, and we analogously define the Q-maps p, x, andp. We further combine these Q-maps into one Q-map, Φ 6,m,n : V m,n → A m,n , V m,n = l C m ⊗ l C m ⊗ l C n ⊗ l C n = l C 2m+2n . Since V m,n is finite-dimensional, we can apply theorem 1.
In the continuum limit, the lattice spacing is supposed to approach 0, i.e. the indexes a and c take an increasing number of values (n, m → ∞). We note that this does not lead to truly continuous indexes, and that continuous indexes must be differently treated. However, let us assume that for each grid size a state ω m,n is determined. Since Φ 6,m,n is a quotient of Φ 0 , we obtain a series of states over P (Φ 0 ). Let us denote the series by (ω j ) for the sake of notational convenience. Since V m,n = V 0,2m+2n and since ω j,(k) = ω j,(2n+2m) | P (Φ 0,k ) (k ≤ 2n + 2m), we obtain a series of states, ω j,(k) , on each sub-algebra P (Φ 0,k ) of P (Φ 0 ). If each series, (ω j,(k) ), converges on the corresponding sub-algebra P (Φ 0,k ), then there exists a unique limit state ω by theorem 3, and (Φ 0 , ω) is a Q-theory by theorem 2. We note however that (Φ 0 , ω) is not necessarily a QFT, since one has to additonally ensure that essential physical features are retrieved in that limit.
Let us rephrase this result: Increasing grid sizes yield a series of states. Each state of a specific grid defines a so-called reduced state on each sub-grid. A necessary and sufficient criterion for a unique limit state in the continuum limit is that the series of reduced states converges on each sub-grid. If a limit state exists, then we obtain a Q-theory as defined in this paper, and if appropriate physical features are retrieved then the Q-theory is also a reasonable QFT.
6.2. String theory. We base our discussion of string theory on the lecture notes of R. J. Szabo [21]. String theory is still work in progress, and a thorough discussion of the relation of string theory to the approach in this paper is elusive so far. However, there are five different consistent formulations of string theory that are commonly seen as perturbative expansions of a unique underlying theory (M-theory), which is however not well understood yet. The five theories are related by dualities that map perturbative states in one theory to non-perturbative states in another theory.
However, the quantization of the bosonic string yields a countable set of raising and lowering operators that satisfy the relation (a µ n ) * = a µ −n , and, if closed strings are considered, (ã µ n ) * =ã µ −n (0 ≤ µ ≤ d, n ∈ IN). These operators satisfy CCRs. There are also zero-mode operators, x µ 0 and p µ 0 , that are conjugate to each other and that also satisfy the CCRs. The operators act on a Fock space, and we can combine them into an equivalent set of self-adjoint operators as follows: x µ n = a µ n + a µ −n 2 , p µ n = i(a µ n − a µ −n ) 2 x µ n =ã µ n +ã µ −n 2 ,p µ n = i(ã µ n −ã µ −n ) 2 .
We assume that observables in bosonic string theory are contained in the closure of the operator algebra, A b , generated by the operators {x µ n , p µ n ,x µ n ,p µ n } n∈IN with respect to an appropriate topology. However, let v n = (δ n,m ) m∈IN ∈ V 0 and let x µ (v n+1 ) = x µ n and p µ (v n+1 ) = p µ n for n ∈ IN 0 , then, by complex-linear continuation, x and p define Q-maps over the test-function space V 0 . Analogously we can define the additional Q-mapsx andp if we consider closed strings.
Fermions further need to be included in string theory to avoid inconsistencies. Canonical quantization yields another countable set of operators satisfying (ψ µ r ) * = ψ µ −r (r = 0, ±1, ± 1 2 , ±2, ± 3 2 , ...). These operators generate the corresponding operator algebra A f , and we simply denote the total operator algebra generated by operators in A b and A f by A. The Hilbert spaces of the various string theories are subspaces of the full Hilbert space that is obtained by canonical quantization, and there are corresponding representations of A on these sectors. As for the bosonic case, the fermionic operators can be combined into two Q-maps over V 0 . So depending on the case, we obtain four to six Q-maps over V 0 that generate the operator algebra in string theory, A. These Q-maps can be combined into one Q-map, Φ 7,k , over the test-function space V ⊗k 0 (k = 4 or k = 6), i.e. P (Φ 7,k ) is the operator algebra in the respective string theory, and we can apply theorem 1.