Similarity transformations in one- and two-mode Fock space

The normally ordered Hilbert space image of the general (complex) linear similarity transformation in phase space is obtained in a coherent state representation. Although preserving the commutator of a and a+, these quantum mechanical images of classical transformations are generally not unitary. Remarkably, although the kets and bras produced by the non-unitary similarity transformations are not Hermitian conjugates, squeezed state analogues produced using the similarity transformation still satisfy an overcompleteness relation. The results are extended to two-mode Fock space and simple examples of the utility of the similarity transformation are presented. The evaluation of the normally ordered operators is greatly facilitated by the use of integration within ordered products.