Wigner's classification

In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ($E\geq0$) 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 representations according to 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 an irreducible Spin(3) unitary 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 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.

See also

• Induced representation
• Representation theory of the diffeomorphism group
• Representation theory of the Galilean group
• Representation theory of the Poincaré group
• System of imprimitivity

References

• Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics 40 (1): 149–204, doi:10.2307/1968551, MR 1503456 .