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Wigner's classification

A classification of the nonnegative energy unitary irreducible representations of the Poincaré group which have sharp mass eigenvalues.

The double cover of the Poincaré group admits no central extensions.

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

is a Casimir invariant of the Poincaré group. So, we can classify the irreps into whether m>0, m=0 but P0>0 and m=0 and P=0.

For the first case, we note that the eigenspace (see generalized eigenspaces of unbounded continuous operators) associated with P0=m and Pi=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 stabilizer of P0=k, P3=-k, Pi=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 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.

See also the method of induced representations.