Parastatistics

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.

Formalism

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). Therefore we can have different superselection sectors, each parameterized by a Young diagram of S_N.

In particular:

If we have N identical parabosons of order p (where p is a positive integer), then the permissible Young diagrams are all those with p or fewer rows.
If we have N identical parafermions of order p, then the permissible Young diagrams are all those with p or fewer columns.
If p is 1, we just have the ordinary cases of Bose–Einstein and Fermi–Dirac statistics respectively.
If p is infinity (not an integer, but one could also have said arbitrarily large p), we have Maxwell–Boltzmann statistics.

The quantum field theory of parastatistics

A paraboson field of order p, $\phi(x)=\sum_{i=1}^p \phi^{(i)}(x)$ where if x and y are spacelike-separated points, $[\phi^{(i)}(x),\phi^{(i)}(y)]=0$ and $\{\phi^{(i)}(x),\phi^{(j)}(y)\}=0$ if $i\neq j$ 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 $\psi(x)=\sum_{i=1}^p \psi^{(i)}(x)$ of order p, where if x and y are spacelike-separated points, $\{\psi^{(i)}(x),\psi^{(i)}(y)\}=0$ and $[\psi^{(i)}(x),\psi^{(j)}(y)]=0$ if $i\neq j$ . 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.^[1] 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.^[2]

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, ..., x_n,

\phi(x_1)\cdots \phi(x_n)|\Omega\rangle

corresponds to creating n identical parabosons at x₁,..., x_n. Similarly,

\psi(x_1)\cdots \psi(x_n)|\Omega\rangle

corresponds to creating n identical parafermions. Because these fields neither commute nor anticommute

\phi(x_{\pi(1)})\cdots \phi(x_{\pi(n)})|\Omega\rangle

and

\psi(x_{\pi(1)})\cdots \psi(x_{\pi(n)})|\Omega\rangle

gives distinct states for each permutation π in S_n.

We can define a permutation operator $\mathcal{E}(\pi)$ by

\mathcal{E}(\pi)\left[\phi(x_1)\cdots \phi(x_n)|\Omega\rangle\right]=\phi(x_{\pi^{-1}(1)})\cdots \phi(x_{\pi^{-1}(n)})|\Omega\rangle

and

\mathcal{E}(\pi)\left[\psi(x_1)\cdots \psi(x_n)|\Omega\rangle\right]=\psi(x_{\pi^{-1}(1)})\cdots \psi(x_{\pi^{-1}(n)})|\Omega\rangle

respectively. This can be shown to be well-defined as long as $\mathcal{E}(\pi)$ is only restricted to states spanned by the vectors given above (essentially the states with n identical particles). It is also unitary. Moreover, $\mathcal{E}$ is an operator-valued representation of the symmetric group S_n and as such, we can interpret it as the action of S_n 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. (Bert) Green ^[3] is credited with the invention/discovery of parastatistics in 1953 ^[4]

References