Wigner's classification

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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 $m\equiv \sqrt{P^2}$ is a Casimir invariant of the Poincaré group. So, we can classify the irreps into whether $m > 0$ , $m = 0$ but $P 0 > 0$ and $m = 0$ and $\mathbf{P}=0$ .

For the first case, we note that the eigenspace (see generalized eigenspaces of unbounded operators) associated with $P 0 = m$ 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 stabilizer of $P 0 = k$ , $P 3 = - k$ , $P i = 0$ , $i = 1,2$ . This is the double cover by SU(2) (see again 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.