Unitary isomorphism of Fock spaces of bosons and fermions arising from a representation of the Cuntz algebra $\co{2}$

TL;DR

This paper constructs an explicit unitary isomorphism between bosonic and fermionic Fock spaces using representations of the Cuntz algebra, revealing a deep connection between these quantum systems.

Contribution

It introduces a novel explicit unitary operator linking bosonic and fermionic Fock spaces derived from Cuntz algebra representations.

Findings

Explicit formula for the unitary operator $U$ on the standard basis.

The operator $U$ preserves particle number.

Establishment of a unitary isomorphism between bosonic and fermionic Fock spaces.

Abstract

Bosons and fermions are described by using canonical generators of Cuntz algebras on any permutative representation. According to branching laws associated with these descriptions, a certain representation of the Cuntz algebra $\co{2}$ induces Fock representations ${\cal H}_{B}$ and ${\cal H}_{F}$ of bosons and fermions simultaneously. From this, a unitary operator $U$ from ${\cal H}_{B}$ to ${\cal H}_{F}$ is obtained. We show the explicit formula of the action of $U$ on the standard basis of ${\cal H}_{B}$. It is shown that $U$ preserves the particle number of ${\cal H}_{B}$ and ${\cal H}_{F}$.