Parastatistics

Consider the operator algebra of a system of N identical particles. This is a *-algebra. There is an S_N 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 S_N (noting that this does not mean that every element of the operator algebra invariant under S_N is an observable). This allows different superselection sectors, each parameterized by a Young diagram of S_N.

In particular:

• For N identical parabosons of order p (where p is a positive integer), 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.
• If p is 1, this reduces to Bose–Einstein and Fermi–Dirac statistics respectively.
• If p is arbitrarily large (infinite), this reduces to Maxwell–Boltzmann statistics.

There are creation and annihilation operators satisfying the trilinear commutation relations[2]

${\displaystyle \left[a_{k},\left[a_{l}^{\dagger },a_{m}\right]_{\pm }\right]_{-}=[a_{k},a_{l}^{\dagger }]_{\mp }a_{m}\pm a_{l}^{\dagger }\left[a_{k},a_{m}\right]_{\mp }\pm [a_{k},a_{m}]_{\mp }a_{l}^{\dagger }+a_{m}\left[a_{k},a_{l}^{\dagger }\right]_{\mp }=2\delta _{kl}a_{m}}$

${\displaystyle \left[a_{k},\left[a_{l}^{\dagger },a_{m}^{\dagger }\right]_{\pm }\right]_{-}=\left[a_{k},a_{l}^{\dagger }\right]_{\mp }a_{m}^{\dagger }\pm a_{l}^{\dagger }\left[a_{k},a_{m}^{\dagger }\right]_{\mp }\pm \left[a_{k},a_{m}^{\dagger }\right]_{\mp }a_{l}^{\dagger }+a_{m}^{\dagger }\left[a_{k},a_{l}^{\dagger }\right]_{\mp }=2\delta _{kl}a_{m}^{\dagger }\pm 2\delta _{km}a_{l}^{\dagger }}$

${\displaystyle \left[a_{k},\left[a_{l},a_{m}\right]_{\pm }\right]_{-}=[a_{k},a_{l}]_{\mp }a_{m}\pm a_{l}\left[a_{k},a_{m}\right]_{\mp }\pm [a_{k},a_{m}]_{\mp }a_{l}+a_{m}\left[a_{k},a_{l}\right]_{\mp }=0}$

A paraboson field of order p, ${\textstyle \phi (x)=\sum _{i=1}^{p}\phi ^{(i)}(x)}$ where if x and y are spacelike-separated points, ${\displaystyle [\phi ^{(i)}(x),\phi ^{(i)}(y)]=0}$ and ${\displaystyle \{\phi ^{(i)}(x),\phi ^{(j)}(y)\}=0}$ if ${\displaystyle i\neq j}$ 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 group S_p acting upon the φ⁽ⁱ⁾s. 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 ${\textstyle \psi (x)=\sum _{i=1}^{p}\psi ^{(i)}(x)}$ of order p, where if x and y are spacelike-separated points, ${\displaystyle \{\psi ^{(i)}(x),\psi ^{(i)}(y)\}=0}$ and ${\displaystyle [\psi ^{(i)}(x),\psi ^{(j)}(y)]=0}$ if ${\displaystyle i\neq j}$. 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.[4] 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.[5]

• Klein transformation on how to convert between parastatistics and the more conventional statistics.[7]