Wigner's classification

In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ${\displaystyle ~(~E\geq 0~)~}$ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (These unitary representations are infinite-dimensional; the group is not semisimple and it does not satisfy Weyl's theorem on complete reducibility.) 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 ${\displaystyle ~C_{1}=P^{\mu }\,P_{\mu }~,}$ (Einstein notation) where P is the 4-momentum operator, and ${\displaystyle ~C_{2}=W^{\alpha }\,W_{\alpha }~,}$ 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

• ${\displaystyle ~m>0~;}$
• ${\displaystyle ~m=0~}$ but ${\displaystyle ~P_{0}>0~;\quad }$ or whether
• ${\displaystyle ~m=0~}$ with ${\displaystyle ~P^{\mu }=0~,{\text{ for }}\mu =0,1,2,3~.}$

Wigner found that massless particles are fundamentally different from massive particles.

For the first case

Note that the eigenspace (see generalized eigenspaces of unbounded operators) associated with ${\displaystyle ~P=(m,0,0,0)~}$ 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 representation that characterizes their spin, and a positive mass, m.

For the second case

Look at the stabilizer of

${\displaystyle ~P=(k,0,0,-k)~.}$

This is the double cover of SE(2) (see projective 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.

For the third case

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