Parastatistics

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, "R"2 − "R"1 cannot be an observable because it changes sign if we switch the two particles, but the distance between the two particles : |"R"2 − "R"1| 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). This allows different superselection sectors, each parameterized by a Young diagram of "SN".
In particular:
The quantum field theory of parastatistics.
A paraboson field of order "p", formula_1 where if "x" and "y" are spacelike-separated points, formula_2 and formula_3 if formula_4 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 formula_5 of order "p", where if "x" and "y" are spacelike-separated points, formula_6 and formula_7 if formula_4. The same comment about observables would apply together with the requirement that they have even grading under the grading where the "ψ"s have odd grading.
The "parafermionic and parabosonic algebras" are generated by elements that obey the commutation and anticommutation relations. They generalize the usual "fermionic algebra" and the "bosonic algebra" of quantum mechanics. The Dirac algebra and the Duffin–Kemmer–Petiau algebra appear as special cases of the parafermionic algebra for order "p" = 1 and "p" = 2, respectively.
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 "x"1, ..., "x""n",
corresponds to creating "n" identical parabosons at "x"1..., "x""n". Similarly,
corresponds to creating "n" identical parafermions. Because these fields neither commute nor anticommute
and
gives distinct states for each permutation π in "Sn".
We can define a permutation operator formula_13 by
and
respectively. This can be shown to be well-defined as long as formula_13 is only restricted to states spanned by the vectors given above (essentially the states with "n" identical particles). It is also unitary. Moreover, formula_17 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.
History of parastatistics.
H. S. (Bert) Green is credited with the creation of parastatistics in 1953.