On the reducibility of the Weyl algebra for a semibounded space
When the configuration space of a quantum particle is \(\mathbb R^n\), the von Neumann algebra \({\mathcal W}\) of the observables is generated by two strongly continuous unitary groups \(\{V(\vec{\beta})=\mathrm{exp}(-i \vec{\beta}\cdot \vec{q})\}\) and \(\{U(\vec{\alpha})=\mathrm{exp}(-i \vec{\alpha}\cdot \vec{p})\}\), where \(\vec{p}\) and \(\vec{q}\) are the position and momentum operators, respectively. It is well-known that the algebra \({\mathcal W}\) is simply a direct sum of irreducible equivalent representations. However, in a common case when the configuration space is a half-line \([x_0, \infty)\), no selfadjoint momentum operator exists and hence \({\mathcal W}\) has to be replaced by an algebra \({\mathcal W}_+\) generated by \(\{V(\beta)=\mathrm{exp}(-i \beta q)\}\) and a strongly continuous semigroup of isometries \(\{U(\alpha)\}\) satisfying \(U(\alpha)V(\beta)=\mathrm{exp}(i \alpha \beta)V(\beta)U(\alpha)\). This paper investigates the reducibility of \({\mathcal W}_+\). It first shows that, if \({\mathcal W}_+\) is a factor, it is completely reducible into equivalent components having the same lower bound of the spectrum. Applying the fact that any von Neumann algebra in a separable Hilbert space decomposes into the direct integral of factors, it concludes that, in general, \({\mathcal W}_+\) is a direct integral of factors \({\mathcal W}_+(\mu)\) and each factor \({\mathcal W}_+(\mu)\) is characterized by the common lower bound \(x_{\mu}\) of the spectrum.
scientific article; zbMATH DE number 5627095