Consider the operator algebra of a system of N identical particles. This is a *-algebra. There is an SN group 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 particles : |R2 − R1| is a legitimate observable. In other words, the observable algebra would have to be a *-subalgebra invariant under the action of SN. This allows different superselection sectors, each parameterized by a Young diagram of SN. In particular:
For N identical parabosons of order p, permissible Young diagrams are all those with p or fewer rows.
For N identical parafermions of order p, permissible Young diagrams are all those with p or fewer columns.
A paraboson field of order p, where if x and y are spacelike-separated points, and if 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 groupSp acting upon the φs. 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, where if x and y are spacelike-separated points, and if. 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. 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 and φneither commute nor anticommute unless p=1. The same comment applies to ψ and ψ. So, if we have nspacelike separated points x1,..., xn, corresponds to creatingn 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. 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.
History of parastatistics
H. S. Green is credited with the creation of parastatistics in 1953.
