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. Explanation 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, corresponds to creating n identical parabosons at x1,..., xn. 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 by and respectively. This can be shown to be well-defined as long as is only restricted to states spanned by the vectors given above (essentially the states with n identical particles). It is also unitary. Moreover, 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. See also Klein transformation on how to convert between parastatistics and the more conventional statistics. References Permutations