Implementing Bogoliubov Transformations Beyond the Shale-Stinespring Condition

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.

45 pages, 3 figures

References in corpus (17)

• Functional Evolution of Free Quantum Fields
• The BCS functional of superconductivity and its mathematical properties
• On Nelson-type Hamiltonians and abstract boundary conditions
• Avoiding Ultraviolet Divergence by Means of Interior-Boundary Conditions
• Particle Creation at a Point Source by Means of Interior-Boundary Conditions
• The dilute Fermi gas via Bogoliubov theory
• The mass shell in the semi-relativistic Pauli-Fierz model
• The Renormalised Bogoliubov-Fröhlich Hamiltonian
• A nonrelativistic quantum field theory with point interactions in three dimensions
• Multi-time formulation of particle creation and annihilation via interior-boundary conditions
• On a Direct Description of Pseudorelativistic Nelson Hamiltonians
• Snowmass White Paper: The Quest to Define QFT
• Essential self-adjointness of Wick squares in quasi-free Hadamard representations on curved spacetimes
• Universality in low-dimensional BCS theory
• Bogoliubov Transformations Beyond Shale-Stinespring: Generic $v^{\ast }v$ for bosons
• Extended State Space for describing renormalized Fock spaces in QFT
• Explicit Diagonalization of Pair Interaction Models