Dirac and normal states on Weyl-von Neumann algebras

TL;DR

This paper investigates special classes of states on Weyl and von Neumann algebras linked to symplectic spaces, emphasizing Dirac states, their discontinuities, and their connections to harmonic analysis and generalized functions.

Contribution

It characterizes Dirac states on Weyl algebras, analyzes their discontinuities, and explores their relation to harmonic analysis and distributions in quantum physics contexts.

Findings

Non-trivial Dirac states are typically discontinuous.

States on Weyl algebras relate to generalized functions and measures.

Analysis of specific examples illustrates the interplay between states and harmonic analysis.

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 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.