nLab Wigner classification

not to be confused with Wigner's theorem

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Idea

The identification of particle species with irreducible unitary representations of the symmetry group acting on matter fields over spacetime. Named after Eugene Wigner.

For details see at

• unitary representation of the Poincaré group.

• unitary representation of the super Poincaré group.

It uses a method that goes back to Frobenius and came to be known as “Wigner’s little group method” or the “Mackey machine” (Mackey 68).

Related concepts

• Doplicher-Roberts reconstruction theorem

• Deligne's theorem on tensor categories– Relevance for Supersymmetry

• supermultiplet

References

The original article is

• Eugene Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals of Mathematics, 40 (1939) 149–204

The observation that Wigner’s representations $U_{(-)}$ are those for which all the maps $g \mapsto \langle \Phi \vert U_g \Psi \rangle$ are measurable and their correspondence of irreducible unitary representations to induced representations of irreps of the stabilizer group of a given momentum is (e.g. Dragon 16) due to

• George Mackey, Induced Representations of Groups and Quantum Mechanics, W. A. Benjamin, New York, 1968

Review includes

• Bert Schroer, Wigner Representation Theory of the Poincaré

  Group, Localization, Statistics and the S-Matrix_, 1996 (pdf)

• Norbert Dragon, Currents for Arbitrary Helicity (arXiv:1601.07825)

• Wikipedia, Wigner’s classification