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.
See also 
- Induced representation
- Representation theory of the diffeomorphism group
- Representation theory of the Galilean group
- Representation theory of the Poincaré group
- System of imprimitivity