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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.