Dirac and normal states on Weyl\u2013von Neumann algebras

Abstract

We study particular classes of states on the Weyl algebra mathcal {W} associated with a symplectic vector space S and on the von Neumann algebras generated in representations of mathcal {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(mathbb {R}^n) or test functions on mathbb {R}^n and relate properties of states on mathcal {W} with those of generalized functions on mathbb {R}^n or with harmonic analysis aspects of corresponding Borel measures on Schwartz functions and on temperate distributions.