Quantum many–body systems can be mathematically described by vectors in a certain Hilbert space, the so–called Fock space, whose Schroedinger dynamics are generated by a self–adjoint Hamiltonian operator H. Bogoliubov transformations are a convenient way to manipulate H while keeping the physical predictions in- variant. They have found widespread use for analyzing the dynamics of quantum many–body systems and justifying simplified models that have been heuristically derived by physicists.
In the 1960s, Shale and Stinespring derived a necessary and sufficient condition for when a Bogoliubov transformation is implementable on Fock space, i.e. for when there exists a unitary operator U such that the manipulated Hamiltonian takes the form U*HU. However, non–implementable Bogoliubov transformations appear frequently in the literature for systems of infinite size.
In this talk, we therefore construct two extensions of the Fock space on which certain Bogoliubov transformations become implementable, although they violate the Shale–Stinespring condition.