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[when defined as?] 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.

Massive scalar fields

As an example, let us visualize the irreducible unitary representation with m > 0 and s = 0. It corresponds to the space of massive scalar fields.

Let M be the hyperboloid sheet defined by:

${\displaystyle P_{0}^{2}-P_{1}^{2}-P_{2}^{2}-P_{3}^{2}=m^{2}}$, ${\displaystyle P_{0}>0}$.

The Minkowski metric restricts to a Riemannian metric on M, giving M the metric structure of a hyperbolic space, in particular it is the hyperboloid model of hyperbolic space, see geometry of Minkowski space for proof. The Poincare group P acts on M because (forgetting the action of the translation subgroup 4 with addition inside P) it preserves the Minkowski inner product, and an element x of the translation subgroup 4 of the Poincare group acts on L2(M) by multiplication by suitable phase multipliers exp(−i p·x), where pM. These two actions can be combined in a clever way using induced representations to obtain an action of P on L2(M) that combines motions of M and phase multiplication.

This yields an action of the Poincare group on the space of square-integrable functions defined on the hypersurface M in Minkowski space. These may be viewed as measures defined on Minkowski space that are concentrated on the set M defined by

${\displaystyle E^{2}-P_{1}^{2}-P_{2}^{2}-P_{3}^{2}=m^{2}}$, ${\displaystyle E\equiv P_{0}>0.}$

The Fourier transform (in all four variables) of such measures yields positive-energy,[clarification needed] finite-energy solutions of the Klein-Gordon equation defined on Minkowski space, namely

${\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}\psi -\nabla ^{2}\psi +m^{2}\psi =0,}$

without physical units. In this way, the m > 0, s = 0 irreducible representation of the Poincare group is realized by its action on a suitable space of solutions of a linear wave equation.

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 transversely 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 ${\displaystyle \epsilon _{T}^{\lambda =1,2}}$ and ${\displaystyle \epsilon _{L}}$ which satisfy ${\displaystyle \epsilon _{T}^{2}=-1}$ and ${\displaystyle \epsilon _{L}^{2}=+1}$, to be compared with the usual case of a free ${\displaystyle Z_{0}}$ boson which has three polarization vectors ${\displaystyle \epsilon _{T}^{\lambda =1,2,3}}$, each of them satisfying ${\displaystyle \epsilon _{T}^{2}=-1}$.