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 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 P^μ = 0. Wigner found that massless particles are fundamentally different than massive particles.

• For the first case, note that the eigenspace (see generalized eigenspaces of unbounded operators) associated with P₀ = m and P_i = 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, spin, and a positive mass, m.

• For the second case, look at the stabilizer of P₀ = k, P₃ = –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.

Left out from this classification are tachyonic solutions, solutions with no fixed mass, infraparticles with no fixed mass, etc. Such solutions are of physical importance, when considering virtual states. A celebrated example is the case of Deep inelastic scattering, in which a virtual space-like photon is exchanged between the incoming lepton and the incoming hadron. This justifies the introduction of transversaly and longitudinally-polarized photons, and of the related concept of transverse and longitudinal structure functions, when considering these virtual states as effective probes of the internal quark and gluon contents of the hadrons. From a mathematical point of view, one considers the SO(2,1) group instead of the usual SO(3) group encountered in the usual massive case discussed above. This explain the occurrence of two transverse polarization vectors $\epsilon_T^{\lambda=1,2}$ and $\epsilon_L$ which satisfy $\epsilon_T^2=-1$ and $\epsilon_L^2=+1$, to be compared with the usual case of a free $Z_0$ boson which has three polarization vectors $\epsilon_T^{\lambda=1,2,3}$, each of them satisfying $\epsilon_T^2=-1$.

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
• Pauli–Lubanski pseudovector

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.
• Bargmann, V., & Wigner, E. P. (1948). "Group theoretical discussion of relativistic wave equations", Proceedings of the National Academy of Sciences of the United States of America 34(5), 211. online