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# Parastatistics

Particle statistics
Maxwell-Boltzmann statistics
Bose-Einstein statistics
Fermi-Dirac statistics
Parastatistics
Anyonic statistics
Braid statistics
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In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models (Bose-Einstein statistics, Fermi-Dirac statistics and Maxwell-Boltzmann statistics). Other alternatives include anyonic statistics and braid statistics, both of these involving lower spacetime dimensions.

## Formalism

Consider the operator algebra of a system of N identical particles. This is a *-algebra. There is an SN group (symmetric group of order N) acting upon the operator algebra with the intended interpretation of permuting the N particles. Quantum mechanics requires focus on observables having a physical meaning, and the observables would have to be invariant under all possible permutations of the N particles. For example in the case N=2, R2-R1 cannot be an observable because it changes sign if we switch the two particles, but the distance between the two particules : |R2-R1| is a legitimate observable.

In other words, the observable algebra would have to be a *-subalgebra invariant under the action of SN (noting that this does not mean that every element of the operator algebra invariant under SN is an observable). Therefore we can have different superselection sectors, each parameterized by a Young diagram of SN.

In particular:

• If we have N identical parabosons of order p (where p is a positive integer), then the permissible Young diagrams are all those with p or fewer rows.
• If we have N identical parafermions of order p, then the permissible Young diagrams are all those with p or fewer columns.
• If p is 1, we just have the ordinary cases of Bose-Einstein and Fermi-Dirac statistics respectively.
• If p is infinity (not an integer, but one could also have said arbitrarily large p), we have Maxwell-Boltzmann statistics.

## The quantum field theory of parastatistics

A paraboson field of order p, $\phi(x)=\sum_{i=1}^p \phi^{(i)}(x)$ where if x and y are spacelike-separated points, (i)(x),φ(i)(y)] = 0 and (i)(x),φ(j)(y)} = 0 if $i\neq j$ where [,] is the commutator and {,} is the anticommutator. Note that this disagrees with the spin-statistics theorem, which is for bosons and not parabosons. There might be a group such as the symmetric group Sp acting upon the φ(i)s. Observables would have to be operators which are invariant under the group in question. However, the existence of such a symmetry is not essential.

A parafermion field $\psi(x)=\sum_{i=1}^p \psi^{(i)}(x)$ of order p, where if x and y are spacelike-separated points, (i)(x),ψ(i)(y)} = 0 and (i)(x),ψ(j)(y)] = 0 if $i\neq j$. The same comment about observables would apply together with the requirement that they have even grading under the grading where the ψs have odd grading.

## Explaining Parastatistics

Note that if x and y are spacelike-separated points, φ(x) and φ(y) neither commute nor anticommute unless p=1. The same comment applies to ψ(x) and ψ(y). So, if we have n spacelike separated points x1, ..., xn, $\phi(x_1)\cdots \phi(x_n)|\Omega\rangle$

corresponds to creating n identical parabosons at x1,..., xn. Similarly, $\psi(x_1)\cdots \psi(x_n)|\Omega\rangle$

corresponds to creating n identical parafermions. Because these fields neither commute nor anticommute $\phi(x_{\pi(1)})\cdots \phi(x_{\pi(n)})|\Omega\rangle$

and $\psi(x_{\pi(1)})\cdots \psi(x_{\pi(n)})|\Omega\rangle$

gives distinct states for each permutation π in Sn.

We can define a permutation operator $\mathcal{E}(\pi)$ by $\mathcal{E}(\pi)\left[\phi(x_1)\cdots \phi(x_n)|\Omega\rangle\right]=\phi(x_{\pi^{-1}(1)})\cdots \phi(x_{\pi^{-1}(n)})|\Omega\rangle$

and $\mathcal{E}(\pi)\left[\psi(x_1)\cdots \psi(x_n)|\Omega\rangle\right]=\psi(x_{\pi^{-1}(1)})\cdots \psi(x_{\pi^{-1}(n)})|\Omega\rangle$

respectively. This can be shown to be well-defined as long as $\mathcal{E}(\pi)$ is only restricted to states spanned by the vectors given above (essentially the states with n identical particles). It is also unitary. Moreover, $\mathcal{E}$ is an operator-valued representation of the symmetric group Sn and as such, we can interpret it as the action of Sn upon the n-particle Hilbert space itself, turning it into a unitary representation.

QCD can be reformulated using parastatistics with the quarks being parafermions of order 3 and the gluons being parabosons of order 8. Note this is different from the conventional approach where quarks always obey anticommutation relations and gluons commutation relations.