Bogoliubov transformation for distinguishable particles

TL;DR

This paper demonstrates that the Bogoliubov transformation, traditionally used for identical bosons, can also be derived for distinguishable particles obeying Boltzmann statistics using dyadic operators, without requiring particle symmetry.

Contribution

It extends the derivation of the Bogoliubov transformation to distinguishable particles, showing it does not depend on particle symmetry or conservation.

Findings

Transformation applies to distinguishable particles

Energy spectrum derivation is analogous to bosonic case

No symmetry breaking needed for derivation

Abstract

The Bogoliubov transformation is generally derived in the context of identical bosons with the use of ``second quantized'' a and $a^{\dagger}$ operators (or, equivalently, in field theory). Here, we show that the transformation, together with its characteristic energy spectrum, can also be derived within the Hilbert space of distinguishable particles, obeying Boltzmann statistics; in this derivation, ordinary dyadic operators play the role usually played by the a and $a^{\dagger}$ operators; therefore, breaking the symmetry of particle conservation is not necessary.