Implementing Bogoliubov Transformations Beyond the Shale-Stinespring Condition.

Area of Science:

• Quantum Field Theory
• Mathematical Physics
• Condensed Matter Physics

Background:

• Bogoliubov transformations are crucial for diagonalizing quadratic Hamiltonians in quantum systems.
• Standard Fock space implementations of Bogoliubov transformations are restricted by the Shale or Shale-Stinespring condition.
• Certain physical models, like BCS theory, involve transformations violating these conditions.

Purpose of the Study:

• To extend the framework for implementing Bogoliubov transformations beyond the limitations of standard Fock space.
• To establish conditions for the extended implementability of Bogoliubov transformations.
• To derive criteria for diagonalizing quadratic Hamiltonians using these extended transformations.

Main Methods:

• Definition of infinite tensor product spaces that generalize Fock spaces.
• Formulation and proof of conditions for the extended implementability of Bogoliubov transformations.
• Derivation of implementability conditions for quadratic Hamiltonians under Bogoliubov transformations.

Main Results:

• Successfully defined extended infinite tensor product spaces accommodating Bogoliubov transformations violating the Shale or Shale-Stinespring condition.
• Provided and proved necessary and sufficient conditions for the extended implementability of such transformations.
• Established criteria for diagonalizing quadratic Hamiltonians via these extended Bogoliubov transformations.

Conclusions:

• The developed framework allows for the implementation of Bogoliubov transformations previously considered impossible on standard Fock spaces.
• Applicability demonstrated for quadratic bosonic interactions and BCS models, showcasing the utility of the extended approach.
• This work broadens the scope of quantum Hamiltonian diagonalization techniques in diverse physical systems.