nLab Wigner classification

not to be confused with Wigner's theorem

---

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 due to

• George Mackey: Induced representations of groups and quantum mechanics, W. A. Benjamin, New York (1968) [ark:/13960/t6841m201]

Review:

• Bert Schroer: Wigner Representation Theory of the Poincaré Group, Localization, Statistics and the S-Matrix, Nucl. Phys. B 499 (1997) 519-546 [arXiv:hep-th/9608092, doi:10.1016/S0550-3213(97)00358-1]

• Norbert Straumann: Unitary Representations of the inhomogeneous Lorentz Group and their Significance in Quantum Physics, talk at Space and Time 100 Years after Minkowski 7-12 September 2008, [arXiv:0809.4942]

• Wikipedia, Wigner’s classification

See also:

• Adam Caulton: Is a particle an irreducible representation of the Poincaré group? [arXiv:2410.02354]