On the Diagonalization of a Bilinear Hamiltonian by a Bogoliubov Transformation

Abstract

Under the condition that a certain hermitian operator has a self-adjoint extension a necessary and sufficient condition that a bilinear Fermion Hamiltonian can be diagonalized by a Bogoliubov transformation is obtained. Under the same assumption, any bilinear Fermion Hamiltonian can be diagonalized in a slightly extended sense by an extended Bogoliubov transformation. The meaning of this diagonalization from the view point of the Clifford C^* algebra is discussed. It is shown that a parallel treatment is possible for a bilinear Boson Hamiltonian (with complications concerning unbounded operators and an indefinite metric) if a spectral theory of pseudo hermitian operator on a Hilbert space of an indefinite metric hold in parallel with that of definite metric Hilbert space.