Wigner's classification

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In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative (E ≥ 0) energy irreducible unitary representations of the Poincaré group, which have sharp mass eigenvalues. It was introduced by Eugene Wigner, to classify particles and fields in physics—see the article particle physics and representation theory. It relies on the stabilizer subgroups of that group, dubbed the Wigner little groups of various mass states.

The mass m ≡ √P ² is a Casimir invariant of the Poincaré group, and may thus serve to label its representations.

The representations may thus be classified according to whether m > 0 ; m = 0 but P0 > 0; and m = 0 with Pμ = 0.

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

For the second case, look at the stabilizer of P0 = k, P3 = –k, Pi = 0, i = 1,2. 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.

The last case describes the vacuum. The only finite-dimensional unitary solution is the trivial representation called the vacuum.

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

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

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