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# Parastatistics

In quantum mechanics, despite what many textbooks and articles erronously claim, the Bose-Einstein and Fermi-Dirac statistics (and Maxwell-Boltzmann statistics) are NOT the only alternatives. We can have parastatistics as well. (In lower spacetime dimensions, we can have anyonic statistics and braid statistics as well, but that's an entirely different matter)

Let's look at the operator algebra of a system of N identical particles. This is a *-algebra. There is an SN (symmetric group of order N) group acting upon the operator algebra with the intended interpretation of permuting the N particles. A basic lesson in quantum mechanics is that we have to focus on observables, and the observables in question would have to be invariant under all possible permutations of the N particles. In other words, the observable algebra would have to be a *-subalgebra invariant under the action of SN (of course, this doesn't necessarily mean in general that any element of the operator algebra invariant under SN is an observable). So, we can have different superselection sectors, each parametrized 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 less rows.

If we have N identical parafermions of order p, then the permissible Young diagrams are all those with p or less columns.

If p is 1, we just have the ordinary case of Bose-Einstein and fermi-Dirac statistics respectively.

If p is infinity (not an integer, I know, but I could also have said arbritrarily 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 DOES NOT agree with the spin-statistics theorem, which is really 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 of order p, $\psi(x)=\sum_{i=1}^p \psi^{(i)}(x)$ 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 abservables would apply together with the requirement that they have even grading under the grading where the ψ's have odd grading.

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.

See Klein transformation on how to convert between parastatistics and the more conventional statistics.