- Parastatistics
In
quantum mechanics andstatistical mechanics , parastatistics is one of several alternatives to the better knownparticle statistics models (Bose-Einstein statistics ,Fermi-Dirac statistics andMaxwell-Boltzmann statistics ). Other alternatives includeanyonic statistics andbraid 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 "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 onobservable s having a physical meaning, and the observables would have to beinvariant 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 particules : |"R2-R1"| 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). Therefore we can have differentsuperselection sector s, each parameterized by aYoung diagram of "SN".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" identicalparafermions 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 parastatisticsA 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 eq j where [,] is thecommutator and {,} is theanticommutator . Note that this disagrees with thespin-statistics theorem , which is forboson s and not parabosons. There might be a group such as thesymmetric group "Sp" acting upon the "φ"("i")s.Observable s would have to be operators which areinvariant 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 eq j. The same comment aboutobservable s would apply together with the requirement that they have even grading under the grading where the "ψ"s have odd grading.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 "x"1, ..., "x""n",
:phi(x_1)cdots phi(x_n)|Omega angle
corresponds to creating "n" identical parabosons at "x"1,..., "x""n". Similarly,
:psi(x_1)cdots psi(x_n)|Omega angle
corresponds to creating "n" identical parafermions. Because these fields neither commute nor anticommute
:phi(x_{pi(1)})cdots phi(x_{pi(n)})|Omega angle
and
:psi(x_{pi(1)})cdots psi(x_{pi(n)})|Omega angle
gives distinct states for each permutation π in "Sn".
We can define a permutation operator mathcal{E}(pi) by
:mathcal{E}(pi)left [phi(x_1)cdots phi(x_n)|Omega angle ight] =phi(x_{pi^{-1}(1)})cdots phi(x_{pi^{-1}(n)})|Omega angle
and
:mathcal{E}(pi)left [psi(x_1)cdots psi(x_n)|Omega angle ight] =psi(x_{pi^{-1}(1)})cdots psi(x_{pi^{-1}(n)})|Omega angle
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 "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 .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.
ee also
*
Klein transformation on how to convert between parastatistics and the more conventional statistics.
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