On the Reducibility of the Weyl Algebra for a Semibounded Space

When the configuration space of a quantum particle is semibounded, the von Neumann algebra of the observables W(+) is generated by a unitary group {V(beta) = exp(-i beta q)}, beta is an element of R, and a semigroup {U(alpha)}, alpha >= 0, of isometries. We show that when 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 W(+) with the same value of x(0) are equivalent. In the general case W(+) decomposes into the direct integral of factors which correspond to the possible values of x(0).