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((Short description|Notion in statistical mechanics))
((Use American English|date=January 2019))
((Statistical mechanics|cTopic=[[Particle statistics|Particle Statistics]]))

((More citations needed|date=September 2010))

In [[quantum mechanics]] and [[statistical mechanics]], '''parastatistics''' is a hypothetical alternative<ref name="BakerHalvorsonSwanson">((Cite web |last=Baker |first=David John |last2=Halvorson |first2=Hans |last3=Swanson |first3=Noel |date=2015-12-01 |title=The Conventionality of Parastatistics |url=https://www.journals.uchicago.edu/doi/10.1093/bjps/axu018 |access-date=2024-03-17 |website=The British Journal for the Philosophy of Science |publisher=University of Pittsburgh |pages=929–976 |language=en |doi=10.1093/bjps/axu018))</ref> to the established [[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. [[Herbert S. Green]]<ref>((cite web|title=Herbert Sydney (Bert) Green|url=http://www.physics.adelaide.edu.au/mathphysics/hsg_memorial.html|url-status=dead|accessdate=2011-10-30|archiveurl=https://web.archive.org/web/20120418185829/http://www.physics.adelaide.edu.au/mathphysics/hsg_memorial.html|archivedate=2012-04-18))</ref> is credited with the creation of parastatistics in 1953.<ref name=":0">H.S. Green, A Generalized Method of Field Quantization. Phys. Rev. 90, 270–273 (1953).(c)</ref><ref>((Cite arXiv|eprint=0903.4773|class=cond-mat.stat-mech|first1=M.|last1=Cattani|first2=J. M. F.|last2=Bassalo|title=Intermediate Statistics, Parastatistics, Fractionary Statistics and Gentilionic Statistics|year=2009))</ref> The particles predicted by parastatistics have not been experimentally observed.

==Formalism==
Consider the [[operator algebra]] of a system of ''N'' identical particles. This is a [[star-algebra|*-algebra]]. There is an ''S<sub>N</sub>'' group ([[symmetric group]] of order ''N'')  [[Group action (mathematics)|acting]] upon the operator algebra with the intended interpretation of [[permutation|permuting]] the ''N'' particles. Quantum mechanics requires focus on [[observable]]s having a physical meaning, and the observables would have to be [[invariant (mathematics)|invariant]] under all possible permutations of the ''N'' particles. For example, in the case ''N''&nbsp;=&nbsp;2, ''R''<sub>2</sub>&nbsp;&minus;&nbsp;''R''<sub>1</sub> cannot be an observable because it changes sign if we switch the two particles, but the distance between the two particles : |''R''<sub>2</sub>&nbsp;&minus;&nbsp;''R''<sub>1</sub>| is a legitimate observable.

In other words, the observable algebra would have to be a *-[[subalgebra]] invariant under the action of ''S<sub>N</sub>'' (noting that this does not mean that every element of the operator algebra invariant under ''S<sub>N</sub>'' is an observable). This allows different [[superselection sector]]s, each parameterized by a [[Young diagram]] of ''S<sub>N</sub>''.

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((clarify|date=June 2020)).
* If ''p'' is arbitrarily large (infinite), this reduces to Maxwell–Boltzmann statistics.

== Trilinear relations ==
There are [[creation and annihilation operators]] satisfying the trilinear commutation relations<ref name=":0" />

<math>\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</math>

<math>\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</math>

<math>\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</math>

==Quantum field theory==
A paraboson field of order ''p'', <math display="inline">\phi(x)=\sum_{i=1}^p \phi^{(i)}(x)</math> where if ''x'' and ''y'' are [[spacelike]]-separated points, <math>[\phi^{(i)}(x),\phi^{(i)}(y)]=0</math> and <math>\{\phi^{(i)}(x),\phi^{(j)}(y)\}=0</math> if <math>i\neq j</math> where [,] is the [[commutator]] and {,} is the [[anticommutator]]. Note that this disagrees with the [[spin-statistics theorem]], which is for [[boson]]s and not parabosons. There might be a group such as the [[symmetric group]] ''S<sub>p</sub>'' acting upon the ''φ''<sup>(''i'')</sup>s. [[Observable]]s would have to be operators which are [[invariant (mathematics)|invariant]] under the group in question. However, the existence of such a symmetry is not essential.

A parafermion field <math display="inline">\psi(x)=\sum_{i=1}^p \psi^{(i)}(x)</math> of order ''p'', where if ''x'' and ''y'' are [[spacelike]]-separated points, <math>\{\psi^{(i)}(x),\psi^{(i)}(y)\}=0</math> and <math>[\psi^{(i)}(x),\psi^{(j)}(y)]=0</math> if <math>i\neq j</math>. The same comment about [[observable]]s would apply together with the requirement that they have even [[Graded algebra|grading]] under the grading where the ''ψ''s 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.<ref>K. Kanakoglou, C. Daskaloyannis: [https://books.google.com/books?id=KAZL5UBlS4cC&pg=PA207 ''Chapter 18 Bosonisation and Parastatistics'', p. 207 ff.], in: Sergei D. Silvestrov, Eugen Paal, Viktor Abramov, Alexander Stolin (eds.): ''Generalized Lie Theory in Mathematics, Physics and Beyond'', 2008, ((ISBN|978-3-540-85331-2))</ref> The [[Dirac algebra]] and the [[Duffin–Kemmer–Petiau algebra]] appear as special cases of the parafermionic algebra for order ''p''&nbsp;=&nbsp;1 and ''p''&nbsp;=&nbsp;2, respectively.<ref>See citations in ((Cite journal|arxiv=hep-th/0001067|last1=Plyushchay|first1=Mikhail S|title=Cubic root of Klein-Gordon equation|journal=Physics Letters B|volume=477|issue=2000|pages=276–284|author2=Michel Rausch de Traubenberg|year=2000|doi=10.1016/S0370-2693(00)00190-8|bibcode=2000PhLB..477..276P|s2cid=16600516))</ref>

=== Explanation ===
Note that if ''x'' and ''y'' are spacelike-separated points, ''φ''(''x'') and ''&phi;''(''y'') neither commute nor anticommute unless ''p''=1. The same comment applies to ''ψ''(''x'') and ''&psi;''(''y''). So, if we have ''n'' spacelike separated points ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>,

:<math>\phi(x_1)\cdots \phi(x_n)|\Omega\rangle</math>

corresponds to creating ''n'' identical parabosons at ''x''<sub>1</sub>,..., ''x''<sub>''n''</sub>. Similarly,

:<math>\psi(x_1)\cdots \psi(x_n)|\Omega\rangle</math>

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

:<math>\phi(x_{\pi(1)})\cdots \phi(x_{\pi(n)})|\Omega\rangle</math>

and

:<math>\psi(x_{\pi(1)})\cdots \psi(x_{\pi(n)})|\Omega\rangle</math>

gives distinct states for each permutation π in ''[[symmetric group|S<sub>n</sub>]]''.

We can define a permutation operator <math>\mathcal{E}(\pi)</math> by

:<math>\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</math>

and

:<math>\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</math>

respectively. This can be shown to be well-defined as long as <math>\mathcal{E}(\pi)</math> is only restricted to states spanned by the vectors given above (essentially the states with ''n'' identical particles). It is also [[unitary operator|unitary]]. Moreover, <math>\mathcal{E}</math> is an operator-valued [[group representation|representation]] of the symmetric group ''S<sub>n</sub>'' and as such, we can interpret it as the action of ''S<sub>n</sub>'' upon the ''n''-particle Hilbert space itself, turning it into a [[unitary representation]].

==See also==
*[[Klein transformation]] on how to convert between parastatistics and the more conventional statistics.<ref name=BakerHalvorsonSwanson/>

==References==
((reflist))

[[Category:Parastatistics| ]]
[[Category:Permutations]]

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