Parastatistics - Citizendia

Particle statistics
Maxwell-Boltzmann statistics
Bose-Einstein statistics
Fermi-Dirac statistics
Parastatistics
Anyonic statistics
Braid statistics
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In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models (Bose-Einstein statistics, Fermi-Dirac statistics and Maxwell-Boltzmann statistics). Particle statistics refers to the particular description of particles in Statistical mechanics. In Statistical mechanics, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in Thermal equilibrium In Statistical mechanics, Bose - Einstein statistics (or more colloquially B-E statistics determines the statistical distribution of In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that In Mathematics and Physics, an anyon is a type of particle that only occurs in two-dimensional systems In Mathematics and Theoretical physics, braid statistics is a generalization of the statistics of Bosons and Fermions based on the concept Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics Particle statistics refers to the particular description of particles in Statistical mechanics. In Statistical mechanics, Bose - Einstein statistics (or more colloquially B-E statistics determines the statistical distribution of In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that In Statistical mechanics, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in Thermal equilibrium Other alternatives include anyonic statistics and braid statistics, both of these involving lower spacetime dimensions. In Mathematics and Physics, an anyon is a type of particle that only occurs in two-dimensional systems In Mathematics and Theoretical physics, braid statistics is a generalization of the statistics of Bosons and Fermions based on the concept

1 Formalism
2 The quantum field theory of parastatistics
3 Explaining Parastatistics
4 See also

Formalism

Consider the operator algebra of a system of N identical particles. In Functional analysis, an operator algebra is an algebra of continuous Linear operators on a Topological vector space with the multiplication This is a *-algebra. -ring In Mathematics, a *-ring is an Associative ring with a map *: A &rarr A which is an Antiautomorphism 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. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In several fields of Mathematics the term permutation is used with different but closely related meanings 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. In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical 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 S_N (noting that this does not mean that every element of the operator algebra invariant under S_N is an observable). In Algebra (mathematics, the word "algebra" usually means a Vector space or module equipped with an additional bilinear operation Therefore we can have different superselection sectors, each parameterized by a Young diagram of S_N. A superselection sector is a concept used in Quantum mechanics when a representation of a *-algebra is decomposed into irreducible components In Mathematics, a Young tableau (pl tableaux) is a combinatorial object useful in Representation theory.

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. In Quantum mechanics and Statistical mechanics, parastatistics is one of several alternatives to the better known Particle statistics models (
If we have N identical parafermions of order p, then the permissible Young diagrams are all those with p or fewer columns. In Quantum mechanics and Statistical mechanics, parastatistics is one of several alternatives to the better known Particle statistics models (
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 parastatistics

A paraboson field of order p, $\phi(x)=\sum_{i=1}^p \phi^{(i)}(x)$ where if x and y are spacelike-separated points, $[φ (i) (x),φ (i) (y)] = 0$ and ${φ (i) (x),φ (j) (y)} = 0$ if $i\neq j$ where [,] is the commutator and {,} is the anticommutator. In quantum field theory (QFT the forces between particles are mediated by other particles SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. Note that this disagrees with the spin-statistics theorem, which is for bosons and not parabosons. The spin-statistics theorem in Quantum mechanics relates the spin of a particle to the statistics obeyed by that particle In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein There might be a group such as the symmetric group S_p acting upon the φ⁽ⁱ⁾s. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying Observables would have to be operators which are invariant under the group in question. In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical 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, ${ψ (i) (x),ψ (i) (y)} = 0$ and $[ψ (i) (x),ψ (j) (y)] = 0$ if $i\neq j$ . SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS 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. In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure

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 x₁, . . . , x_n,

$\phi(x_1)\cdots \phi(x_n)|\Omega\rangle$

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

$\psi(x_1)\cdots \psi(x_n)|\Omega\rangle$

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

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

and

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

gives distinct states for each permutation π in S_n. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying

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

$\mathcal{E}(\pi)\left[\phi(x_1)\cdots \phi(x_n)|\Omega\rangle\right]=\phi(x_{\pi^{-1}(1)})\cdots \phi(x_{\pi^{-1}(n)})|\Omega\rangle$

and

$\mathcal{E}(\pi)\left[\psi(x_1)\cdots \psi(x_n)|\Omega\rangle\right]=\psi(x_{\pi^{-1}(1)})\cdots \psi(x_{\pi^{-1}(n)})|\Omega\rangle$

respectively. This can be shown to be well-defined as long as $\mathcal{E}(\pi)$ is only restricted to states spanned by the vectors given above (essentially the states with n identical particles). It is also unitary. In Functional analysis, a branch of Mathematics, a unitary operator is a Bounded linear operator U H → Moreover, $\mathcal{E}$ 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. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, a unitary representation of a group G is a Linear representation π of G on a complex Hilbert space

QCD can be reformulated using parastatistics with the quarks being parafermions of order 3 and the gluons being parabosons of order 8. Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the Note this is different from the conventional approach where quarks always obey anticommutation relations and gluons commutation relations.

Contents

Formalism

The quantum field theory of parastatistics

Explaining Parastatistics

See also