The Resolvent Algebra: A New Approach to Canonical Quantum Systems

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.Comment: 52 pages, no figures; v3: as to appear in Journal of Functional Analysi

Local Quantum Constraints

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.Comment: 52 pages, Latex2e, to appear in Rev. Math. Phys. Corrected a mistake in Sect 5.2,- needed to include additional nonlocal elements in the field algebra which will be eliminated by the constraints. Added Theorem 5.16, proving that the local physical algebras are simple. Added Subsect. 5.6, making explicit the connections between the indefinite metric representation of Gupta-Bleuler electromagnetism, and the C*-algebraic version of it constructed here. Added reference

Amenability of the Gauge Group

Let G be one of the local gauge groups C(X,U(n)), C^\infty(X,U(n)), C(X,SU(n)) or C^\infty(X,SU(n)) where X is a compact Riemannian manifold. We observe that G has a nontrivial group topology, coarser than its natural topology, w.r.t. which it is amenable, viz the relative weak topology of C(X,M(n)). This topology seems more useful than other known amenable topologies for G. We construct a simple fermionic model containing an action of G, continuous w.r.t. this amenable topology.Comment: 8 pages, Late