On bogoliubov transformations. II. The general case

Abstract

A rigorous treatment of Bogoliubov transformations is presented along the same lines as in a previous paper, which dealt with a special case. As in the previous paper a formulation in terms of unitary resp. pseudo-unitary operators is used, corresponding to the CAR resp. the CCR. This leads to simple proofs of well-known necessary and sufficient conditions for the transformation to be unitarily implementable in Fock space. The normal form of the implementing operator 1 is studied. It is proved that on the subspace of algebraic tensors 9 equals a strongly convergent intinite series of Wick monomials that sums up to a simple exponential expression. A connection between the fermion and boson transformations studied in the previous paper is established. The analogous correspondence in the general case only hoIds true if the (pseudo) unitary operator equals its own inverse.