On the Reducibility of the Weyl Algebra for a Semibounded Space

Abstract

When the configuration space of a quantum particle is semibounded, the von Neumann algebra of the observables ${\mathcal{W}_+}$ is generated by a unitary group ${\{V(\beta) = {\rm exp}(-i\beta q)\},\,\beta\in\mathbb{R}}$ , and a semigroup {U(α)}, α ≥ 0, of isometries. We show that when ${\mathcal W_+}$ is a factor it is completely reducible into equivalent components, and that in each component the lower end x ₀ of the spectrum of q is the same. We give an algebraic characterization of x ₀ and also obtain a straightforward new proof that the irreducible representations of ${\mathcal W_+}$ with the same value of x ₀ are equivalent. In the general case ${\mathcal W_+}$ decomposes into the direct integral of factors which correspond to the possible values of x ₀.