Nonunitary bogoliubov transformations and extension of Wick’s theorem

Linear transformations are considered, which preserve the (anti-) commutation rules, but not the Hermiticity relation, for (fermion) boson creation and annihilation operators; these transformations lead to Fock space representations on biorthogonal bases of the operator algebra. As an application, an extension of Wick’s theorem to matrix elements of an arbitrary operator between two different quasi-particle vacuums is derived. This theorem is useful for calculations which go beyond the variational Hartree-Fock-Bogoliubov methods (H.F.B. with projection, generator co-ordinate method, etc.). A canonical decomposition for Bogoliubov transformations is established, which proves useful, for instance in the calculation of the overlap of two different quasi-particle vacuums.