Decompositions of Regular Representations of the Canonical Commutation Relations

Abstract

Every cyclic regular representation of the canonical commutation relations over any inner product space V can be decomposed into a direct integral of irreducible regular representations, where the fibers are representations over subspaces of V. An example using the so-called direct-product representations shows that generally the irreducible representations cannot be defined over the whole V. So we get a new type of decomposition having no equivalent in the decompositions of locally compact groups.