Parastatistics

Other alternatives include anyonic statistics and braid statistics, both of these involving lower spacetime dimensions.

Herbert S. Green[2] is credited with the creation of parastatistics in 1953.

[3][4] The particles predicted by parastatistics have not been experimentally observed.

Consider the operator algebra of a system of N identical particles.

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 |R2 − R1| 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 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: There are creation and annihilation operators satisfying the trilinear commutation relations[3] A paraboson field of order p,

Note that this disagrees with the spin–statistics theorem, which is for bosons and not parabosons.

However, the existence of such a symmetry is not essential.

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.

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 give distinct states for each permutation π in Sn.

is only restricted to states spanned by the vectors given above (essentially the states with n identical particles).

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.