Cocycles for Boson and Fermion Bogoliubov Transformations

TL;DR

This paper derives explicit formulas for cocycles associated with Bogoliubov transformations in quantum field theory, covering both bosonic and fermionic cases, and explores their mathematical structure and phases.

Contribution

It provides new explicit formulas for 2-cocycles in Bogoliubov transformations for charged bosons and fermions, linking to group theory and loop groups.

Findings

Derived formulas for 2-cocycles in fermion and boson cases.

Connected cocycles to loop groups and symplectic operators.

Analyzed phases of unitary propagators and their families.

Abstract

Unitarily implementable Bogoliubov transformations for charged, relativistic bos\-ons and fermions are discussed, and explicit formulas for the 2-cocycles appearing in the group product of their implementers are derived. In the fermion case this provides a simple field theoretic derivation of the well-known cocycle of the group of unitary Hilbert space operators modeled on the Hilbert Schmidt class and closely related to the loop groups. In the boson case the cocycle is obtained for a similar group of pseudo-unitary (symplectic) operators. I also derive explcite formulas for the phases of one-parameter groups of implementers and, more generally, families of implementers which are unitary propagators with parameter dependent generators.