Wigner classification

A classification of particle types by the unitary irreducible representations of the Poincaré group. Its principal labels are mass and spin, or, for massless particles, helicity.

Poincaré group: The group of spacetime translations and Lorentz transformations in special relativity. Its unitary irreducible representations are the objects being classified.
Unitary representation: A group representation by linear transformations that preserve inner products in a Hilbert space. Particle states must carry symmetry actions that preserve quantum probabilities.
Wigner particle: An idealized stable particle represented by a positive-energy irreducible representation of the Poincaré group. This is the particle concept directly defined by the classification.
Tachyon: A hypothetical particle whose mass squared is negative, implying spacelike energy-momentum. Tachyonic representations differ from the usual positive-energy particle classes.
Wigner's little group: The subgroup of Lorentz transformations that leaves a chosen particle momentum unchanged. Its representations determine the internal labels associated with each momentum orbit.
Irreducible representation: A representation with no nonzero proper invariant subspace. Indecomposable symmetry sectors serve as the classification's particle types.
Relativistic quantum field theory: Quantum field theory formulated consistently with special relativity. Its particle states inherit Poincaré representation labels.
Continuous-spin particle: A massless representation with infinitely many helicity states governed by a continuous parameter. It is an unusual allowed class beyond fixed-helicity massless particles.
Casimir operator: An operator that commutes with every representation of a Lie group and labels its irreducible representations. Poincaré Casimir eigenvalues encode mass and spin-related invariants.
Hilbert space: A complete inner-product space used to represent quantum states. The Poincaré group acts unitarily on the spaces of particle states.

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