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.
Cite this article
Reinhard Schaflitzel, Decompositions of Regular Representations of the Canonical Commutation Relations. Publ. Res. Inst. Math. Sci. 26 (1990), no. 6, pp. 1019–1047DOI 10.2977/PRIMS/1195170574