Dirac and normal states on Weyl-von Neumann algebras

We study particular classes of states on the Weyl algebra W associated with a symplectic vector space S and on the von Neumann algebras generated in representations of W. Applications in quantum physics require an implementation of constraint equations, e.g., due to gauge conditions, and can be based on the so-called Dirac states. The states can be characterized by nonlinear functions on S, and it turns out that those corresponding to non-trivial Dirac states are typically discontinuous. We discuss general aspects of this interplay between functions on S and states, but also develop an analysis for a particular example class of non-trivial Dirac states. In the last part, we focus on the specific situation with S = L-2 (R-n) or test functions on R-n and relate properties of states on W with those of generalized functions on R or with harmonic analysis aspects of corresponding Borel measures on Schwartz functions and on temperate distributions.