Dirac and normal states on Weyl–von Neumann algebras

Abstract

We study particular classes of states on the Weyl algebra $$\mathcal {W}$$ W associated with a symplectic vector space S and on the von Neumann algebras generated in representations of $$\mathcal {W}$$ 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 $$S = L^2(\mathbb {R}^n)$$ S = L 2 ( R n ) or test functions on $$\mathbb {R}^n$$ R n and relate properties of states on $$\mathcal {W}$$ W with those of generalized functions on $$\mathbb {R}^n$$ R n or with harmonic analysis aspects of corresponding Borel measures on Schwartz functions and on temperate distributions.

Functional Integrals in Geometric Approach to Quantum Theory

“…These relations are formally equivalent to (35). We consider the smallest linear subspace of the space of bounded linear operators in F containing all operators V f ; the closure of this space in norm-topology is a C * -algebra that can be regarded as an exponential form of Weyl algebra (see, for example, [14] and references therein for the mathematical theory of Weyl algebra). We will work with this algebra denoted by W. The space of continuous linear functionals on W will be denoted by L. Notice that a functional L ∈ L is determined by its values on operators V f , therefore we can consider L as a non-linear functional L(f ) = L(V f ) on E (the representation of states of Weyl algebra by means of non-linear functionals was rediscovered and studied in [14]).…”