  Mean-field methods such as Hartree-Fock (HF) or Hartree-Fock-Bogoliubov (HFB)
constitute the building blocks upon which more elaborate many-body theories are
based on. The HF and HFB wavefunctions are built out of independent
quasi-particles resulting from a unitary linear canonical transformation of the
elementary fermion operators. Here, we discuss the possibility of allowing the
HF transformation to become non-unitary. The properties of such HF vacua are
discussed, as well as the evaluation of matrix elements among such states. We
use a simple ansatz to demonstrate that a non-unitary transformation brings
additional flexibility that can be exploited in variational approximations to
many-fermion wavefunctions. The action of projection operators on non-unitary
based HF states is also discussed and applied to the one-dimensional Hubbard
model with periodic boundary conditions.
