Representation theory of Lie groups

Wigner's classification

In mathematics and theoretical physics, Wigner's classificationis a classification of the nonnegative energy irreducible unitary representations of the Poincaré group which have either finite or zero 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 (Einstein notation) where P is the 4-momentum operator, and 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 * * but or whether * with Wigner found that massless particles are fundamentally different from massive particles. For the first caseNote that the eigenspace (see generalized eigenspaces of unbounded operators) associated with 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 caseLook at the stabilizer of 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 caseThe only finite-dimensional unitary solution is the trivial representation called the vacuum. (Wikipedia).

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Related pages

Poincaré group | Hyperboloid model | Representation theory of the Poincaré group | Scalar field | Group extension | Representation theory of the Galilean group | Particle physics and representation theory | Pauli–Lubanski pseudovector | Spin group | Lepton | Trivial representation | Spin (physics) | Minkowski space | Hyperbolic space | Mathematics | Induced representation | Energy | Klein–Gordon equation | Euclidean group | Projective representation | System of imprimitivity | Einstein notation | Unitary representation | Irreducible representation