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Parastatistics
In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models (BoseEinstein statistics, FermiDirac statistics and MaxwellBoltzmann statistics). Other alternatives include anyonic statistics and braid statistics, both of these involving lower spacetime dimensions. Additional recommended knowledge
FormalismConsider 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, R_{2}R_{1} cannot be an observable because it changes sign if we switch the two particles, but the distance between the two particules : R_{2}R_{1} 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:
The quantum field theory of parastatisticsA paraboson field of order p, where if x and y are spacelikeseparated points, [φ^{(i)}(x),φ^{(i)}(y)] = 0 and {φ^{(i)}(x),φ^{(j)}(y)} = 0 if where [,] is the commutator and {,} is the anticommutator. Note that this disagrees with the spinstatistics 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 of order p, where if x and y are spacelikeseparated points, {ψ^{(i)}(x),ψ^{(i)}(y)} = 0 and [ψ^{(i)}(x),ψ^{(j)}(y)] = 0 if . 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. Explaining ParastatisticsNote that if x and y are spacelikeseparated 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 x_{1}, ..., x_{n}, corresponds to creating n identical parabosons at x_{1},..., x_{n}. Similarly, corresponds to creating n identical parafermions. Because these fields neither commute nor anticommute and gives distinct states for each permutation π in S_{n}. We can define a permutation operator by and respectively. This can be shown to be welldefined as long as is only restricted to states spanned by the vectors given above (essentially the states with n identical particles). It is also unitary. Moreover, is an operatorvalued representation of the symmetric group S_{n} and as such, we can interpret it as the action of S_{n} upon the nparticle 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. See also


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Parastatistics". A list of authors is available in Wikipedia. 