Editing Wigner's classification

{{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]].

== Massive scalar fields ==

As an example, let us visualize the irreducible unitary representation with <math>~ m > 0 ~,</math> and <math>~ s = 0~.</math> It corresponds to the space of [[scalar field|massive scalar field]]s.

Let {{mvar|M}} be the hyperboloid sheet defined by:

:<math>~ P_0^2 - P_1^2 - P_2^2 - P_3^2 = m^2 ~, \quad</math> <math>~P_0 > 0~.</math>

The Minkowski metric restricts to a [[Riemannian metric]] on {{mvar|M}}, giving {{mvar|M}} the metric structure of a [[hyperbolic space]], in particular it is the [[hyperboloid model]] of hyperbolic space,  see [[Minkowski space#Geometry|geometry of Minkowski space]] for proof. The Poincare group {{mvar|''P''}} acts on {{mvar|M}} because (forgetting the action of the translation subgroup {{math|ℝ<sup>4</sup>}} with addition inside {{mvar|P}}) it preserves the [[Minkowski inner product]], and an element {{mvar|x}} of the translation subgroup {{math|ℝ<sup>4</sup>}} of the Poincare group acts on
<math>~ L^2(M) ~</math>
by multiplication by suitable phase multipliers
<math>~ \exp \left( -i \vec{p} \cdot \vec{x} \right) ~,</math>
where <math>~ p \in M ~.</math> These two actions can be combined in a clever way using [[induced representations]] to obtain an action
of {{mvar|P}} acting on <math>~ L^2(M) ~,</math> that combines motions of {{mvar|M}} and phase multiplication.

This yields an action of the Poincare group on the space of square-integrable functions defined on the hypersurface {{mvar|M}} in Minkowski space. These may be viewed as measures defined on Minkowski space that are concentrated on the set {{mvar|M}} defined by

:<math>E^2 - P_1^2 - P_2^2 - P_3^2 = m^2~, \quad </math> <math>~E ~\equiv~ P_0 > 0~.</math>

The Fourier transform (in all four variables) of such measures yields positive-energy,{{clarify|reason=What does this mean?|date=October 2016}} finite-energy solutions of the [[Klein–Gordon equation]] defined on Minkowski space, namely

:<math> \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + m^2 \psi = 0,</math>

without physical units. In this way, the <math>~ m > 0, \quad s = 0 ~</math> irreducible representation of the Poincare group is realized by its action on a suitable space of solutions of a linear wave equation.

==Theory of projective representations==
Physically, one is interested in irreducible [[projective representation|''projective'' unitary representations]] of the Poincaré group. After all, two vectors in the quantum Hilbert space that differ by multiplication by a constant represent the same physical state. Thus, two unitary operators that differ by a multiple of the identity have the same action on physical states. Therefore the unitary operators that represent Poincaré symmetry are only defined up to a constant—and therefore the group composition law need only hold up to a constant.

According to [[Projective representation#Infinite-dimensional projective unitary representations: Bargmann's theorem|Bargmann's theorem]], every projective unitary representation of the Poincaré group comes from an ordinary unitary representation of its universal cover, which is a double cover. (Bargmann's theorem applies because the double cover of the [[Poincaré group]] admits no non-trivial one-dimensional [[Group extension#Central extension|central extension]]s.)

Passing to the double cover is important because it allows for half-odd-integer spin cases. In the positive mass case, for example, the little group is SU(2) rather than SO(3); the representations of SU(2) then include both integer and half-odd-integer spin cases.

Since the general criterion in Bargmann's theorem was not known when Wigner did his classification, he needed to show by hand (§5 of the paper) that the phases can be chosen in the operators to reflect the composition law in the group, up to a sign, which is then accounted for by passing to the double cover of the Poincaré group.

==Further classification==
Left out from this classification are [[tachyon]]ic solutions, solutions with no fixed mass, [[infraparticle]]s with no fixed mass, etc. Such solutions are of physical importance, when considering virtual states. A celebrated example is the case of [[deep inelastic scattering]], in which a virtual space-like [[photon]] is exchanged between the incoming [[lepton]] and the incoming [[hadron]]. This justifies the introduction of transversely and longitudinally-polarized photons, and of the related concept of transverse and longitudinal structure functions, when considering these virtual states as effective probes of the internal quark and gluon contents of the hadrons. From a mathematical point of view, one considers the SO(2,1) group instead of the usual [[SO(3)]] group encountered in the usual massive case discussed above. This explains the occurrence of two transverse polarization vectors <math>~ \epsilon_T^{\lambda=1,2} ~</math> and <math>~ \epsilon_L ~</math>
which satisfy <math>~ \epsilon_T^2 = -1 ~</math> and <math>~ \epsilon_L^2 = +1 ~,</math> to be compared with the usual case of a free <math>~Z_0~</math> boson which has three polarization vectors <math>~\epsilon_T^\lambda \text{ for } \lambda = 1,2,3~;</math> each of them satisfying <math>~ \epsilon_T ^2 = -1 ~.</math>

==See also==
*{{annotated link|Induced representation}}
*{{annotated link|Particle physics and representation theory}}
*{{annotated link|Pauli–Lubanski pseudovector}}
*{{annotated link|Representation theory of the diffeomorphism group}}
*{{annotated link|Representation theory of the Galilean group}}
*{{annotated link|Representation theory of the Poincaré group}}
*{{annotated link|System of imprimitivity}}

== References ==
*{{cite journal
 |last1=Bargmann |first1 = V.   |author1-link=Valentine Bargmann
 |last2=Wigner   |first2 = E.P. |author2-link=Eugene Wigner
 |year=1948
 |title=Group theoretical discussion of relativistic wave equations
 |journal=[[Proceedings of the National Academy of Sciences of the United States of America]]
 |volume=34 |issue=5 |pages=211&ndash;223
 |doi=	10.1073/pnas.34.5.211  |bibcode=1948PNAS...34..211B  |pmc=1079095  |pmid=16578292
|doi-access = free }}

*{{cite book
 |first=George |last=Mackey |author-link=George Mackey
 |year=1978
 |title=Unitary Group Representations in Physics, Probability and Number Theory
 |series=Mathematics Lecture Notes Series
 |volume=55
 |isbn=978-0805367034
 |publisher=[[Benjamin Cummings|The Benjamin/Cummings Publishing Company]]
}}

*{{cite book
 |first=Shlomo |last=Sternberg |author-link=Shlomo Sternberg
 |title=Group Theory and Physics
 |url=https://archive.org/details/grouptheoryphysi0000ster
 |url-access=registration
 |year=1994
 |section=§3.9. Wigner classification 
 |publisher=[[Cambridge University Press]]
 |isbn=978-0521248709
}}

*{{cite book
 |first=Wu-Ki |last=Tung
 |title=Group Theory in Physics
 |year=1985
 |chapter =Chapter 10. Representations of the Lorentz group and of the Poincare group; Wigner classification
 |publisher=[[World Scientific Publishing Company]]
 |isbn=978-9971966577
}}

*{{cite book
 |last=Weinberg |first=S. |author-link=Steven Weinberg
 |year=2002
 |title=The Quantum Theory of Fields
 |volume=I
 |isbn=0-521-55001-7
 |chapter=Chapter 2. Relativistic quantum mechanics
 |publisher=Cambridge University Press
}}

*{{cite journal
 |first=E.P. |last=Wigner |author-link=Eugene Wigner
 |year=1939
 |title=On unitary representations of the inhomogeneous Lorentz group
 |journal=[[Annals of Mathematics]]
 |issue=1 |volume=40 |pages=149–204
 |doi=10.2307/1968551
 |mr=1503456 |bibcode = 1939AnMat..40..149W |jstor=1968551
|s2cid=121773411 }}

[[Category:Representation theory of Lie groups]]
[[Category:Quantum field theory]]
[[Category:Mathematical physics]]