Boson-fermion algebraic mapping in second quantization

TL;DR

This paper introduces an algebraic method for mapping bosonic creation and annihilation operators to fermionic ones, utilizing a deformed Grassmann algebra, with implications for gauge invariance and harmonic oscillator models.

Contribution

It presents a novel algebraic framework for boson-fermion mapping using deformed Grassmann algebra, enhancing understanding of second quantization.

Findings

Mapping corresponds to a deformed Grassmann algebra

The method preserves gauge invariance

Application to harmonic oscillators demonstrates effectiveness

Abstract

We present an algebraic method to derive the structure at the basis of the mapping of bosonic algebras of creation and annihilation operators into fermionic algebras, and vice versa, introducing a suitable identification between bosonic and fermionic generators. The algebraic structure thus obtained corresponds to a deformed Grassmann algebra, involving anticommuting Grassmann-type variables. The role played by the latter in the implementation of gauge invariance in second quantization within our procedure is then discussed, together with the application of the mapping to the case of the bosonic and fermionic harmonic oscillator Hamiltonians.