Parastatistics

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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.

1 Formalism
2 The quantum field theory of parastatistics
3 Explaining Parastatistics
4 See also

[edit] Formalism

Consider the operator algebra of a system of N identical particles. This is a *-algebra. There is an S_N 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 S_N (noting that this does not mean that every element of the operator algebra invariant under S_N is an observable). Therefore we can have different superselection sectors, each parameterized by a Young diagram of S_N.

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.

[edit] 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, $[φ (i) (x),φ (i) (y)] = 0$ and ${φ (i) (x),φ (j) (y)} = 0$ 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 S_p 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, ${ψ (i) (x),ψ (i) (y)} = 0$ and $[ψ (i) (x),ψ (j) (y)] = 0$ 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.

[edit] 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, ..., x_n,

$\phi(x_1)\cdots \phi(x_n)|\Omega\rangle$

corresponds to creating n identical parabosons at x₁,..., x_n. 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 S_n.

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 S_n and as such, we can interpret it as the action of S_n 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.