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In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 and statistical mechanics
Statistical mechanics

Statistical mechanics is the application of probability theory, which includes Mathematics tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force....
, parastatistics is one of several alternatives to the better known particle statistics
Particle statistics

Particle statistics refers to the particular description of particles in statistical mechanics....
 models (Bose-Einstein statistics, Fermi-Dirac statistics
Fermi-Dirac statistics

Fermi-Dirac statistics is a part of the science of physics, that applies to a system comprised of many particles that obey the Pauli Exclusion Principle....
 and Maxwell-Boltzmann statistics). Other alternatives include anyonic statistics and braid statistics
Braid statistics

In mathematics and theoretical physics, braid statistics is a generalization of the statistics of bosons and fermions based on the concept of braid group....
, both of these involving lower spacetime dimensions.

ider the operator algebra
Operator algebra

In functional analysis, an operator algebra is an algebra over a field of continuous function linear operators on a topological vector space with the multiplication given by the composition of mappings....
 of a system of N identical particles. This is a *-algebra
Star-algebra

A *-operation on a *-ring is an operation on a ring that behaves similarly to complex conjugation on the complex numbers. A *-operation on a *-algebra is an operation on an algebra over a *-ring that behaves similarly to taking conjugate transpose in ....
. There is an SN group (symmetric group
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
 of order N) acting
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
 upon the operator algebra with the intended interpretation of permuting
Permutation

In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set....
 the N particles.






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In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 and statistical mechanics
Statistical mechanics

Statistical mechanics is the application of probability theory, which includes Mathematics tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force....
, parastatistics is one of several alternatives to the better known particle statistics
Particle statistics

Particle statistics refers to the particular description of particles in statistical mechanics....
 models (Bose-Einstein statistics, Fermi-Dirac statistics
Fermi-Dirac statistics

Fermi-Dirac statistics is a part of the science of physics, that applies to a system comprised of many particles that obey the Pauli Exclusion Principle....
 and Maxwell-Boltzmann statistics). Other alternatives include anyonic statistics and braid statistics
Braid statistics

In mathematics and theoretical physics, braid statistics is a generalization of the statistics of bosons and fermions based on the concept of braid group....
, both of these involving lower spacetime dimensions.

Formalism

Consider the operator algebra
Operator algebra

In functional analysis, an operator algebra is an algebra over a field of continuous function linear operators on a topological vector space with the multiplication given by the composition of mappings....
 of a system of N identical particles. This is a *-algebra
Star-algebra

A *-operation on a *-ring is an operation on a ring that behaves similarly to complex conjugation on the complex numbers. A *-operation on a *-algebra is an operation on an algebra over a *-ring that behaves similarly to taking conjugate transpose in ....
. There is an SN group (symmetric group
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
 of order N) acting
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
 upon the operator algebra with the intended interpretation of permuting
Permutation

In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set....
 the N particles. Quantum mechanics requires focus on observable
Observable

In physics, particularly in quantum physics, a system observable is a property of the State that can be determined by some sequence of physical operational definition....
s having a physical meaning, and the observables would have to be invariant
Invariant

Invariant and invariance may have several meanings, among which are:* Invariant , an expression whose value doesn't change during execution ...
 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 particles : |R2-R1| is a legitimate observable.

In other words, the observable algebra would have to be a *-subalgebra
Subalgebra

In algebra , the word "algebra" usually means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures....
 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 different superselection sector
Superselection sector

A superselection sector is a concept used in quantum mechanics when a representation theory of a star-algebra is decomposed into irreducible representations....
s, each parameterized by a Young 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 identical parafermions 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
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
 of parastatistics


A paraboson field of order p, where if x and y are spacelike-separated points, and if where [,] is the commutator
Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory....
 and is the anticommutator. Note that this disagrees with the spin-statistics theorem
Spin-statistics theorem

In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics obeyed by it. The spin of a particle is its intrinsic angular momentum ....
, which is for boson
Boson

In particle physics, bosons are subatomic particle which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein....
s and not parabosons. There might be a group such as the symmetric group
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
 Sp acting upon the φ(i)s. Observable
Observable

In physics, particularly in quantum physics, a system observable is a property of the State that can be determined by some sequence of physical operational definition....
s would have to be operators which are invariant
Invariant

Invariant and invariance may have several meanings, among which are:* Invariant , an expression whose value doesn't change during execution ...
 under the group in question. However, the existence of such a symmetry is not essential.

A parafermion field of order p, where if x and y are spacelike-separated points, and if . The same comment about observable
Observable

In physics, particularly in quantum physics, a system observable is a property of the State that can be determined by some sequence of physical operational definition....
s would apply together with the requirement that they have even grading
Graded algebra

In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
 under the grading where the ψs 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 x1, ..., xn,

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

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

and

gives distinct states for each permutation π in Sn
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
.

We can define a permutation operator by

and

respectively. This can be shown to be well-defined as long as is only restricted to states spanned by the vectors given above (essentially the states with n identical particles). It is also unitary
Unitary operator

In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H ? H on a Hilbert space H satisfying...
. Moreover, is an operator-valued representation
Group representation

In the mathematics field of representation theory, group representations describe abstract group in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrix so that the group operation can be represented by matrix multiplication....
 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
Unitary representation

In mathematics, a unitary representation of a Group G is a linear representation p of G on a complex Hilbert space V such that p is a unitary operator for every g ? G....
.

QCD
Quantum chromodynamics

Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
 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.

See also

  • Klein transformation
    Klein transformation

    In quantum field theory, the Klein transformation is a redefinition of the fields to patch up the spin-statistics theorem....
     on how to convert between parastatistics and the more conventional statistics.