Bogoliubov Transformations Beyond Shale-Stinespring: Generic $ v^* v $ for bosons

TL;DR

This paper extends the mathematical framework for bosonic Bogoliubov transformations, enabling their implementation beyond traditional restrictions by constructing an extended Fock space, thus broadening theoretical applicability.

Contribution

It introduces an extended Fock space construction that allows implementing bosonic Bogoliubov transformations without the Shale-Stinespring trace class condition.

Findings

Extended implementability without restrictions on v* v

Generalization beyond discrete spectrum cases

Broader applicability of bosonic transformations

Abstract

We construct an extension of Fock space and prove that it allows for implementing bosonic Bogoliubov transformations in a certain extended sense. While an implementation in the regular sense on Fock space is only possible if a certain operator $ v^* v $ is trace class (this is the well-known Shale-Stinespring condition), the extended implementation works without any restrictions on this operator. This generalizes a recent result of extended implementability, which required $ v^* v $ to have discrete spectrum.