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.
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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, 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 (noting that this does not mean that every element of the operator algebra invariant under SN is an observable). Therefore we can have 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,
A parafermion field
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 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
and
respectively. This can be shown to be well-defined as long as
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 invention/discovery of parastatistics in 1953