Nonlinear Transformation Group of CAR Fermion Algebra

TL;DR

This paper introduces a nonlinear transformation group acting on the CAR fermion algebra, derived from a unitary group action on the Cuntz algebra, encompassing some Bogoliubov transformations as special cases.

Contribution

It establishes a novel connection between unitary group actions on Cuntz algebras and nonlinear transformations of the CAR fermion algebra, expanding the understanding of fermionic symmetries.

Findings

Nonlinear transformations are expressed as finite polynomials in generators.

The transformation group includes some Bogoliubov transformations as special cases.

The approach generalizes previous work on recursive fermion systems.

Abstract

Based on our previous work on the recursive fermion system in the Cuntz algebra, it is shown that a nonlinear transformation group of the CAR fermion algebra is induced from a $U(2^p)$ action on the Cuntz algebra ${\cal O}_{2^p}$ with an arbitrary positive integer $p$. In general, these nonlinear transformations are expressed in terms of finite polynomials in generators. Some Bogoliubov transformations are involved as special cases.