Implementing Bogoliubov Transformations Beyond the Shale-Stinespring Condition

TL;DR

This paper extends the mathematical framework for implementing Bogoliubov transformations beyond the traditional Shale-Stinespring condition, enabling their application in broader quantum systems for both bosons and fermions.

Contribution

It introduces two extensions of Fock space where Bogoliubov transformations violating the Shale-Stinespring condition become implementable, expanding the scope of quadratic Hamiltonian diagonalization.

Findings

Extensions allow implementation beyond Shale-Stinespring condition

Conditions for quadratic Hamiltonian diagonalization are derived

Examples demonstrate extended implementability despite violations

Abstract

We provide two extensions of a dense subspace of Fock space, such that Bogoliubov transformations become implementable on them, even though they violate the Shale-Stinespring condition, so they are not implementable on Fock space. Both the bosonic and fermionic case are covered. Conditions for implementability in the extended sense are stated and proved. From these, we derive conditions for a quadratic Hamiltonian to be diagonalizable by a Bogoliubov transformation that is implementable in the extended sense. Three examples illustrate situations, in which an implementation in the extended sense is possible although the Shale-Stinespring condition fails to hold.