In quantum mechanics and statistical mechanics, parastatistics is a hypothetical alternative to the established 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. Herbert S. Green is credited with the creation of parastatistics in 1953. The particles predicted by parastatistics have not been experimentally observed.
Formalism Consider the operator algebra of a system of N {\displaystyle N} identical particles. This is a *-algebra. There is an S N {\displaystyle S_{N}} group (symmetric group of order N {\displaystyle N} ) acting upon the operator algebra with the intended interpretation of permuting the N {\displaystyle 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 {\displaystyle N} particles. For example, in the case N = 2 , R 2 − R 1 {\displaystyle N=2,\,R_{2}-R_{1}} cannot be an observable because it changes sign if we switch the two particles, but the distance | R 2 − R 1 | {\displaystyle |R_{2}-R_{1}|} between the two particles is a legitimate observable. In other words, the observable algebra would have to be a *-subalgebra invariant under the action of S N {\displaystyle S_{N}} (noting that this does not mean that every element of the operator algebra invariant under S N {\displaystyle S_{N}} is an observable). This allows different superselection sectors, each parameterized by a Young diagram of S N {\displaystyle S_{N}} . In particular:
For N {\displaystyle N} identical parabosons of order p {\displaystyle p} (where p {\displaystyle p} is a positive integer), permissible Young diagrams are all those with p {\displaystyle p} or fewer rows. For N {\displaystyle N} identical parafermions of order p {\displaystyle p} , permissible Young diagrams are all those with p {\displaystyle p} or fewer columns. If p = 1 {\displaystyle p=1} , this reduces to Bose–Einstein and Fermi–Dirac statistics respectively. If p {\displaystyle p} is arbitrarily large (infinite), this reduces to Maxwell–Boltzmann statistics.
Trilinear relations There are creation and annihilation operators satisfying the trilinear commutation relations
[ a k , [ a l † , a m ] ± ] − = [ a k , a l † ] ∓ a m ± a l † [ a k , a m ] ∓ ± [ a k , a m ] ∓ a l † + a m [ a k , a l † ] ∓ = 2 δ k l a m , {\displaystyle {\big [}a_{k},[a_{l}^{\dagger },a_{m}]_{\pm }{\big ]}_{-}=[a_{k},a_{l}^{\dagger }]_{\mp }a_{m}\pm a_{l}^{\dagger }[a_{k},a_{m}]_{\mp }\pm [a_{k},a_{m}]_{\mp }a_{l}^{\dagger }+a_{m}[a_{k},a_{l}^{\dagger }]_{\mp }=2\delta _{kl}a_{m},}
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