Parastatistics

Statistical mechanics
Thermodynamics Kinetic theory
Particle statistics Spin–statistics theorem Indistinguishable particles Maxwell–Boltzmann Bose–Einstein Fermi–Dirac Parastatistics Anyonic statistics Braid statistics
Thermodynamic ensembles NVE Microcanonical NVT Canonical µVT Grand canonical NPH Isoenthalpic–isobaric NPT Isothermal–isobaric
Models Debye Einstein Ising Potts
Potentials Internal energy Enthalpy Helmholtz free energy Gibbs free energy Grand potential / Landau free energy
Scientists Maxwell Boltzmann Bose Gibbs Einstein Ehrenfest von Neumann Tolman Debye Fermi
v t e

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In

• 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^[clarification needed].
• 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^[3]

${\displaystyle \left[a_{k},\left[a_{l}^{\dagger },a_{m}\right]_{\pm }\right]_{-}=[a_{k},a_{l}^{\dagger }]_{\mp }a_{m}\pm a_{l}^{\dagger }\left[a_{k},a_{m}\right]_{\mp }\pm [a_{k},a_{m}]_{\mp }a_{l}^{\dagger }+a_{m}\left[a_{k},a_{l}^{\dagger }\right]_{\mp }=2\delta _{kl}a_{m}}$

${\displaystyle \left[a_{k},\left[a_{l}^{\dagger },a_{m}^{\dagger }\right]_{\pm }\right]_{-}=\left[a_{k},a_{l}^{\dagger }\right]_{\mp }a_{m}^{\dagger }\pm a_{l}^{\dagger }\left[a_{k},a_{m}^{\dagger }\right]_{\mp }\pm \left[a_{k},a_{m}^{\dagger }\right]_{\mp }a_{l}^{\dagger }+a_{m}^{\dagger }\left[a_{k},a_{l}^{\dagger }\right]_{\mp }=2\delta _{kl}a_{m}^{\dagger }\pm 2\delta _{km}a_{l}^{\dagger }}$

${\displaystyle \left[a_{k},\left[a_{l},a_{m}\right]_{\pm }\right]_{-}=[a_{k},a_{l}]_{\mp }a_{m}\pm a_{l}\left[a_{k},a_{m}\right]_{\mp }\pm [a_{k},a_{m}]_{\mp }a_{l}+a_{m}\left[a_{k},a_{l}\right]_{\mp }=0}$

Quantum field theory

A paraboson field of order p, ${\textstyle \phi (x)=\sum _{i=1}^{p}\phi ^{(i)}(x)}$ where if x and y are

${\displaystyle [\phi ^{(i)}(x),\phi ^{(i)}(y)]=0}$

${\displaystyle \{\phi ^{(i)}(x),\phi ^{(j)}(y)\}=0}$

${\displaystyle i\neq j}$

under the group in question. However, the existence of such a symmetry is not essential.

A parafermion field ${\textstyle \psi (x)=\sum _{i=1}^{p}\psi ^{(i)}(x)}$ of order p, where if x and y are

spacelike

-separated points, ${\displaystyle \{\psi ^{(i)}(x),\psi ^{(i)}(y)\}=0}$ and ${\displaystyle [\psi ^{(i)}(x),\psi ^{(j)}(y)]=0}$ if ${\displaystyle i\neq j}$. The same comment about

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.[5] 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.^[6]

Explanation

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,

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

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

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

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

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

and

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

gives distinct states for each permutation π in S_n.

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

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

and

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

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

See also

• Klein transformation on how to convert between parastatistics and the more conventional statistics.^[1]

References