Wigner's classification
Encyclopedia
In mathematics
and theoretical physics
, Wigner's classification
is a classification of the nonnegative energy
irreducible unitary representations of the Poincaré group
, which have sharp mass eigenvalues. It was proposed by Eugene Wigner, for reasons coming from physics—see the article particle physics and representation theory
.
The mass
is a Casimir invariant
of the Poincaré group. So, we can classify the representations according to whether
, 
but 
and 
and 
.
For the first case, we note that the eigenspace (see generalized eigenspaces of unbounded operators) associated with
and 
is a representation of SO(3). In the ray interpretation, we can go over to Spin(3) instead. So, massive states are classified by an irreducible Spin(3) unitary
and a positive mass,
.
For the second case, we look at the stabilizer of
, 
, 
, 
. This is the double cover of SE(2)
(see unit ray representation). We have two cases, one where irreps are described by an integral multiple of 1/2, called the helicity and the other called the "continuous spin" representation.
The last case describes the vacuum
. The only finite dimensional unitary solution is the trivial representation
called the vacuum.
The double cover of the Poincaré group
admits no central extensions.
Note: This classification leaves out tachyon
ic solutions, solutions with no fixed mass, infraparticle
s with no fixed mass, etc.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, Wigner's classification
is a classification of the nonnegative energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
irreducible unitary representations of the Poincaré group
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
, which have sharp mass eigenvalues. It was proposed by Eugene Wigner, for reasons coming from physics—see the article particle physics and representation theory
Particle physics and representation theory
In physics, the connection between particle physics and representation theory is a natural connection, first noted by Eugene Wigner, between the properties of elementary particles and the representation theory of Lie groups and Lie algebras...
.
The mass
is a Casimir invariantCasimir invariant
In mathematics, a Casimir invariant or Casimir operator is a distinguished element of the centre of the universal enveloping algebra of a Lie algebra...
of the Poincaré group. So, we can classify the representations according to whether
, but and and .For the first case, we note that the eigenspace (see generalized eigenspaces of unbounded operators) associated with
and is a representation of SO(3). In the ray interpretation, we can go over to Spin(3) instead. So, massive states are classified by an irreducible Spin(3) unitaryUnitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
and a positive mass,
.For the second case, we look at the stabilizer of
, , , . This is the double cover of SE(2)Euclidean group
In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...
(see unit ray representation). We have two cases, one where irreps are described by an integral multiple of 1/2, called the helicity and the other called the "continuous spin" representation.
The last case describes the vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
. The only finite dimensional unitary solution is the trivial representation
Trivial representation
In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V...
called the vacuum.
The double cover of the Poincaré group
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
admits no central extensions.
Note: This classification leaves out tachyon
Tachyon
A tachyon is a hypothetical subatomic particle that always moves faster than light. In the language of special relativity, a tachyon would be a particle with space-like four-momentum and imaginary proper time. A tachyon would be constrained to the space-like portion of the energy-momentum graph...
ic solutions, solutions with no fixed mass, infraparticle
Infraparticle
An infraparticle is an electrically charged particle and its surrounding cloud of soft photons—of which there are infinite number, by virtue of the infrared divergence of quantum electrodynamics. That is, it is a dressed particle rather than a bare particle...
s with no fixed mass, etc.
See also
- Induced representationInduced representationIn mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...
- Representation theory of the diffeomorphism group
- Representation theory of the Galilean groupRepresentation theory of the Galilean groupIn nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin as follows:The spacetime symmetry group of nonrelativistic quantum mechanics is the Galilean group...
- Representation theory of the Poincaré groupRepresentation theory of the Poincaré groupIn mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics....
- System of imprimitivitySystem of imprimitivityThe concept of system of imprimitivity is used in mathematics, particularly in algebra and analysis, both within the context of the theory of group representations...