Wigner's classification
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. (Since this group is noncompact, these unitary representations are infinite-dimensional.) 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 Casimir invariants of the Poincaré group are C1 = PμPμ, where P is the 4-momentum operator, and C2 = WαWα, where W is the Pauli–Lubanski pseudovector. The eigenvalues of these operators serve to label the representations. The first is associated with mass-squared and the second with helicity or spin.
The physically relevant representations may thus be classified according to whether m > 0 ; m = 0 but P0 > 0; and m = 0 with Pμ = 0. Wigner found that massless particles are fundamentally different from massive particles.
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