Home > 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.

For the first case, we note that the eigenspace (see generalized eigenspaces of unbounded operators) associated with P₀=m and P_i=0 is a representation of SO(3). In the ray interpretation, we can go over to Spin(3) instead. So, massive states are classified by a Spin(3) unitary irrep and a positive mass, m.

For the second case, we look at the stabilizerMathematics: see Group action. In food or chemicals, a stabilizer is a substance added to prevent or retard an unwanted change in physical state. For aircraft, the horizontal stabilizer is a fixed or adjustable surface from which an elevator may be hinged of P₀=k, P₃=-k, P_i=0, i=1,2. This is the double cover of SE(2) (see again unit ray representation ). We have two case, one where irreps are described by an integral multiple of 1/2, called the helicitythis page is about helicity in fluid mechanics. For helicity of magnetic fields, see magnetic helicity). In fluid mechanics, helicity is the extent to which corkscrew-like motion occurs. If a parcel of fluid is moving, undergoing solid body motion rotatin and the other called the "continuous spin" representation.

The last case describes the vacuumThe article on the vacuum cleaner is located elsewhere. In physics, a vacuum is the absence of matter in a volume of space. A partial vacuum is expressed in units of pressure. The SI unit of pressure is the pascal (abbreviated to Pa in usage). It can also. The only finite dimensional unitary solution is the trivial representationIn mathematics, in particular group representation theory, a group representation of the group G is called a trivial representation if (i) it is defined on a one-dimensional vector space V over a field K and (ii) all elements g of G act on V as the identi called the vacuum.

The double cover of the Poincaré group admits no central extensionIn group theory, a central extension of a group G is an exact sequence of groups : such that A is in Z ''E , the center of the group E. Examples of central extensions can be constructed by taking any group G and any abelian group A and setting E to be A 's.

Note: This leaves out tachyonic solutions, solutions with no fixed mass, infraparticle s with no fixed mass, etc..

See also the method of induced representations.