Implementation of automorphism groups in certain representations of the canonical commutation relations

Abstract

Necessary and sufficient conditions are given for the unitary implementability of one-parameter unitary groups of one-particle automorphisms of the CCR algebra in representations symplectically related to the Fock representation. The criteria become particularly simple when the one-particle generator of the unitary group is positive and bounded away from zero; in this case the automorphism group is unitarily implementable only in the representations unitarily equivalent to the Fock representation. If the spectrum of the generator includes zero, however, the situation is more complicated; there then exist representations inequivalent to the Fock representation which admit unitary implementation of the automorphism group. It is also shown that whenever implementation of the automorphism group is possible, the implementing operators can be chosen to be a strongly continuous unitary group, guaranteeing the existence of a ‘‘second-quantized’’ self-adjoint generator.