Some particle representations of the canonical commutation relations (Q1116055)

For a given regular representation \(W_ 1\) of the canonical commutation relations we consider the representations obtained by phase displacement, and form their direct integral. The result is a particle representation \(W_ T\) which is examined in this paper (a particle representaion is a representation of the CCR with a number operator in the sense of Chaiken). We give some examples in which the particle representation \(W_ T\) is factorial, although \(W_ 1\) is not a particle representation. In this way also a particle representation can be constructed with the following property: In all its factorial decompositions the representation of almost every fibre has no number operator.