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 |R₂ − R₁| between the two particles 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{{clarify|date=June 2020}}.
• If p is arbitrarily large (infinite), this reduces to Maxwell–Boltzmann statistics.

There are creation and annihilation operators satisfying the trilinear commutation relations

: $\backslash big[a\_k,\; [a\_l^\backslash dagger,\; a\_m]\_\{\backslash pm\}\backslash big]\_-\; =\; [a\_k,\; a\_l^\backslash dagger]\_\{\backslash mp\}\; a\_m\; \backslash pm\; a\_l^\backslash dagger\; [a\_k,\; a\_m]\_\{\backslash mp\}\; \backslash pm\; [a\_k,\; a\_m]\_\{\backslash mp\}\; a\_l^\backslash dagger\; +\; a\_m\; [a\_k,\; a\_l^\backslash dagger]\_\{\backslash mp\}\; =\; 2\backslash delta\_\{kl\}a\_m,$

: $\backslash big[a\_k,\; [a\_l^\backslash dagger,\; a\_m^\backslash dagger]\_\{\backslash pm\}\backslash big]\_-\; =\; [a\_k,\; a\_l^\backslash dagger]\_\{\backslash mp\}\; a\_m^\backslash dagger\; \backslash pm\; a\_l^\backslash dagger\; [a\_k,\; a\_m^\backslash dagger]\_\{\backslash mp\}\; \backslash pm\; [a\_k,\; a\_m^\backslash dagger]\_\{\backslash mp\}\; a\_l^\backslash dagger\; +\; a\_m^\backslash dagger\; [a\_k,\; a\_l^\backslash dagger]\_\{\backslash mp\}\; =\; 2\backslash delta\_\{kl\}\; a\_m^\backslash dagger\; \backslash pm\; 2\backslash delta\_\{km\}\; a\_l^\backslash dagger,$

: $\backslash big[a\_k,\; [a\_l,\; a\_m]\_\{\backslash pm\}\backslash big]\_-\; =\; [a\_k,\; a\_l]\_\{\backslash mp\}\; a\_m\; \backslash pm\; a\_l\; [a\_k,\; a\_m]\_\{\backslash mp\}\; \backslash pm\; [a\_k,\; a\_m]\_\{\backslash mp\}\; a\_l\; +\; a\_m\; [a\_k,\; a\_l]\_\{\backslash mp\}\; =\; 0.$

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

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,

: $\backslash phi(x\_1)\; \backslash cdots\; \backslash phi(x\_n)\; |\backslash Omega\backslash rangle$

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

: $\backslash psi(x\_1)\; \backslash cdots\; \backslash psi(x\_n)\; |\backslash Omega\backslash rangle$

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

: $\backslash phi(x\_\{\backslash pi(1)\})\; \backslash cdots\; \backslash phi(x\_\{\backslash pi(n)\})\; |\backslash Omega\backslash rangle$

and

: $\backslash psi(x\_\{\backslash pi(1)\})\; \backslash cdots\; \backslash psi(x\_\{\backslash pi(n)\})\; |\backslash Omega\backslash rangle$

give distinct states for each permutation π in S_n.

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

: $\backslash mathcal\{E\}(\backslash pi)\backslash big[\backslash phi(x\_1)\; \backslash cdots\; \backslash phi(x\_n)\; |\backslash Omega\backslash rangle\backslash big]\; =\; \backslash phi(x\_\{\backslash pi^\{-1\}(1)\})\; \backslash cdots\; \backslash phi(x\_\{\backslash pi^\{-1\}(n)\})\; |\backslash Omega\backslash rangle$

and

: $\backslash mathcal\{E\}(\backslash pi)\backslash big[\backslash psi(x\_1)\; \backslash cdots\; \backslash psi(x\_n)\; |\backslash Omega\backslash rangle\backslash big]\; =\; \backslash psi(x\_\{\backslash pi^\{-1\}(1)\})\; \backslash cdots\; \backslash psi(x\_\{\backslash pi^\{-1\}(n)\})\; |\backslash Omega\backslash rangle$

respectively. This can be shown to be well-defined as long as $\backslash mathcal\{E\}(\backslash pi)$ is only restricted to states spanned by the vectors given above (essentially the states with n identical particles). It is also unitary. Moreover, $\backslash mathcal\{E\}$ 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.

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

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Category:Permutations