Parastatistics

Definition

Let $\mathcal H$ be a complex Hilbert space, and let $K$ be an index set of single-particle states. Let $a_k$, $k \in K$, be operators on $\mathcal H$ with adjoints $a_k^\dagger$. Write $[A, B] = AB - BA$, $\set{A, B} = AB + BA$, and $\delta_{kl}$ for the Kronecker delta.

1. The $a_k$ satisfy parabose relations if $[a_k, \set{a_l^\dagger, a_m}] = 2 \delta_{kl} a_m$ and $[a_k, \set{a_l, a_m}] = 0$ for all $k, l, m \in K$.
2. The $a_k$ satisfy parafermi relations if $[a_k, [a_l^\dagger, a_m]] = 2 \delta_{kl} a_m$ and $[a_k, [a_l, a_m]] = 0$ for all $k, l, m \in K$.
3. A representation has order $p$, a positive integer, if there is a unit vector $\lvert 0 \rangle$ with $a_k \lvert 0 \rangle = 0$ and $a_k a_l^\dagger \lvert 0 \rangle = p \, \delta_{kl} \lvert 0 \rangle$ for all $k, l$. Moreover $\mathcal H$ must be spanned by the vectors obtained from $\lvert 0 \rangle$ by applying the $a_l^\dagger$.

Parastatistics of order $p$ is the statistics of particles whose creation and annihilation operators form such a representation of the parabose or parafermi relations.