Bogoliubov Transformations Beyond Shale-Stinespring: Generic $v^{\ast }v$ for bosons

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 $v^{\ast }v$ 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 $v^{\ast }v$ to have discrete spectrum.

arXiv comment20 pages, 2 figures, contribution to the Proceedings of the Intensive Period "INdAM Quantum Meetings (IQM22)" at Politecnico di Milano, March - May 2022

References in corpus (7)

• Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime
• Generalized Hartree-Fock Theory and the Hubbard Model
• Functional Evolution of Free Quantum Fields
• Bose-Einstein Condensation Beyond the Gross-Pitaevskii Regime
• Bosonic quadratic Hamiltonians
• Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime
• The dilute Fermi gas via Bogoliubov theory

Cited by in corpus (1)

• Implementing Bogoliubov Transformations Beyond the Shale-Stinespring Condition