Unitary isomorphism of Fock spaces of bosons and fermions arising from a   representation of the Cuntz algebra $\co{2}$
  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 ${\cal H}_{F}$. 
Katsunori Kawamura