The standard C * -algebraic version of the algebra of canonical commutation relations, the Weyl algebra, frequently causes difficulties in applications since it neither admits the formulation of physically interesting dynamical laws nor does it incorporate pertinent physical observables such as (bounded functions of) the Hamiltonian. Here a novel C * -algebra of the canonical commutation relations is presented which does not suffer from such problems. It is based on the resolvents of the canonical operators and their algebraic relations. The resulting C * -algebra, the resolvent algebra, is shown to have many desirable analytic properties and the regularity structure of its representations is surprisingly simple. Moreover, the resolvent algebra is a convenient framework for applications to interacting and to constrained quantum systems, as we demonstrate by several examples.

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We analyze the situation of a local quantum field theory with constraints, both indexed by the same set of space-time regions. In particular we find "weak" Haag-Kastler axioms which will ensure that the final constrained theory satisfies the usual Haag-Kastler axioms. Gupta-Bleuler electromagnetism is developed in detail as an example of a theory which satisfies the "weak" Haag-Kastler axioms but not the usual ones. This analysis is done by pure C * -algebraic means without employing any indefinite metric representations, and we obtain the same physical algebra and positive energy representation for it than by the usual means. The price for avoiding the indefinite metric, is the use of nonregular representations and complex valued test functions. We also exhibit the precise connection with the usual indefinite metric representation.We conclude the analysis by comparing the final physical algebra produced by a system of local constrainings with the one obtained from a single global constraining and also consider the issue of reduction by stages. For the usual spectral condition on the generators of the translation group, we also find a "weak" version, and show that the Gupta-Bleuler example satisfies it.

The treatment of supersymmetry is known to cause difficulties in the C * -algebraic framework of relativistic quantum field theory; several no-go theorems indicate that super-derivations and super-KMS functionals must be quite singular objects in a C * -algebraic setting. In order to clarify the situation, a simple supersymmetric chiral field theory of a free Fermi and Bose field defined on R is analyzed. It is shown that a meaningful C * -version of this model can be based on the tensor product of a CAR-algebra and a novel version of a CCRalgebra, the "resolvent algebra". The elements of this resolvent algebra serve as mollifiers for the super-derivation. Within this model, unbounded (yet locally bounded) graded KMSfunctionals are constructed and proven to be supersymmetric. From these KMS-functionals, Chern characters are obtained by generalizing formulae of Kastler and of Jaffe, Lesniewski and Osterwalder. The characters are used to define cyclic cocycles in the sense of Connes' noncommutative geometry which are "locally entire". the underlying C * -algebra [9]. Second, there is growing evidence that supersymmetric KMSfunctionals, although being unbounded, are locally bounded, in accordance with the heuristic considerations in [3]. And third, although these functionals generically do not have analytic elements in their domain of definition, they can still be used to define local versions of Connes' entire cyclic cocycles by relying on techniques from complex analysis. Based on these insights, a proper C * -algebraic framework for the formulation of supersymmetry and the analysis of its consequences in quantum field theory seems within reach. We hope to return to this problem elsewhere.The plan of our paper is as follows. We will state our results in the body of the paper, and defer almost all the proofs to the appendix. In Sect. 2 we present in the Wightman framework the basic supersymmetry model which we wish to analyze; in Sect. 3 we prepare for its analysis in a C*-setting by considering algebraic mollifying relations for the quantum fields, which leads to the study of the C*-algebras generated by the resolvents of the fields. In Sect. 4 we use these tools to present the C*-algebraic framework of the model. In Sect. 5 we define (unbounded) graded KMSfunctionals on the model and prove basic properties for them, including their supersymmetry invariance and their local boundedness. In Sect. 6 we use these KMS-functionals to define a Chern character formula (generalizing the construction in [15,11]), from which we obtain a (locally) entire cyclic cocycle in the sense of Connes. This can then be taken as input to an index theory for supersymmetric quantum field theories, of the type proposed by Longo [18]. The modelWe begin by presenting here our model in the Wightman framework, which we would like to model in a C*-algebra setting. It is the the simplest example for supersymmetry on noncompact spacetime, in that we have one dimension, one boson and one fermion.

We construct a mathematically well-defined framework for the kinematics of Hamiltonian QCD on an infinite lattice in R 3 , and it is done in a C*-algebraic context. This is based on the finite lattice model for Hamiltonian QCD developed by Kijowski, Rudolph e.a. (cf. [22,23]). To extend this model to an infinite lattice, we need to take an infinite tensor product of nonunital C*-algebras, which is a nonstandard situation. We use a recent construction for such situations, developed in [10]. Once the field C*-algebra is constructed for the fermions and gauge bosons, we define local and global gauge transformations, and identify the Gauss law constraint. The full field algebra is the crossed product of the previous one with the local gauge transformations. The rest of the paper is concerned with enforcing the Gauss law constraint to obtain the C*-algebra of quantum observables. For this, we use the method of enforcing quantum constraints developed by Grundling and Hurst (cf. [11]). In particular, the natural inductive limit structure of the field algebra is a central component of the analysis, and the constraint system defined by the Gauss law constraint is a system of local constraints in the sense of [15]. Using the techniques developed in that area, we solve the full constraint system by first solving the finite (local) systems and then combining the results appropriately. We do not consider dynamics.