Editing Wigner's classification (section)

{{Short description|Classification of irreducible representations of the Poincaré group}}
In [[mathematics]] and [[theoretical physics]], '''[[Eugene Wigner|Wigner's]] classification'''
is a classification of the [[nonnegative]] <math>~ (~E \ge 0~)~</math> [[energy]] [[Irreducible representation|irreducible unitary representation]]s of the [[Poincaré group]] which have either finite or zero mass [[eigenvalue]]s. (These unitary representations are infinite-dimensional; the group is not semisimple and it does not satisfy [[Weyl's theorem on complete reducibility]].) 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 subgroup|stabilizer subgroups]] of that group, dubbed the '''[[Group action (mathematics)#Fixed points and stabilizer subgroups|Wigner little groups]]''' of various mass states.

The [[Casimir invariant]]s of the Poincaré group are <math>~ C_1 = P^\mu \, P_\mu ~ ,</math> ([[Einstein notation]]) where {{mvar|P}} is the [[4-momentum operator]], and <math>~ C_2 = W^\alpha\, W_\alpha ~,</math> where {{mvar|W}} is the [[Pauli&ndash;Lubanski pseudovector]]. The eigenvalues of these operators serve to label the representations. The first is associated with mass-squared and the second with [[helicity (particle physics)|helicity]] or [[Spin (physics)|spin]].

The physically relevant representations may thus be classified according to whether
* <math>~ m > 0 ~;</math>
* <math>~ m = 0 ~</math> but <math>~P_0 > 0 ~; \quad </math> or whether
* <math>~ m = 0 ~</math> with <math>~ P^\mu = 0 ~, \text{ for } \mu = 0, 1, 2, 3 ~.</math>
Wigner found that massless particles are fundamentally different from massive particles.

; For the first case: Note that the [[eigenspace]] (see [[generalized eigenspaces of unbounded operators]]) associated with <math>~P = (m, 0, 0, 0 ) ~</math> is a [[Representations of Lie groups/algebras|representation]] of [[Special orthogonal group|SO(3)]].
In the [[Projective representation|ray interpretation]], one can go over to [[Spin group|Spin(3)]] instead. So, massive states are classified by an irreducible Spin(3) [[unitary representation]] that characterizes their [[Spin (physics)|spin]], and a positive mass, {{mvar|m}}.
; For the second case: Look at the [[stabilizer (group theory)|stabilizer]] of
:<math>~ P = ( k, 0, 0, -k )~.</math>
This is the [[Double covering group|double cover]] of [[Euclidean group|SE(2)]] (see [[projective representation]]). We have two cases, one where [[irrep]]s are described by an integral multiple of {{sfrac|1|2}} called the [[helicity (particle physics)|helicity]], and the other called the "continuous spin" representation.
; For the third case: The only finite-dimensional unitary solution is the [[trivial representation]] called the [[vacuum]].