Among Quadratic Hamiltonians, Bogoliubov Transformations and Non-Regular States on CCRs *-Algebra. I. Pure and Invariant States. (In the Mood for the Manuceau Verbeure Theorems about Quasi-free States and Automorphisma of the CCR Algebra)

TL;DR

This paper explores quadratic Bose-Hamiltonians, Bogoliubov transformations, and quasi-free states within CCR algebras, establishing a bilingual framework with indefinite inner product spaces to construct invariant states, extending classical theorems.

Contribution

It introduces a comprehensive analysis of quadratic Bose-Hamiltonians and Bogoliubov transformations, including non-diagonalizable cases, and develops a bilingual approach linking CCRs with indefinite inner product spaces.

Findings

Constructed invariant states for quadratic Hamiltonians.

Extended Manuceau-Verbeure theorems to non-regular states.

Established a bilingual framework connecting CCRs and indefinite inner product spaces.

Abstract

The features of the paper are these: 1) we discuss {\bf especially} quadratic (alias bilinear) Bose-Hamiltonians, the related Bogoliubov transformations and {\bf especially} quasi-free-like (alias coherent or Fock-like) states 2) we discuss {\bf any} quadratic Bose-Hamiltonians and Bogoliubov transformations, whether diagonalizable or not, whether proper or improper, and {\bf arbitrary} quasi-free-like states, whether regular or non-regular they are 3) we associate notions and terms of the CCRs theory {CCRs = Canonical Commutation Relations} with notions and terms of the indefinite inner product spaces theory. Then, we apply the corresponding `bilingual dictionary' so as to construct invariant states of some of the quadratic Hamiltonians.