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<title>Wigner's classification</title>

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Wigner's classification

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In <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Theoretical_physics" title="Theoretical physics">theoretical physics</a>, <a href="Eugene_Wigner" title="Eugene Wigner">Wigner's</a> classification

is a classification of the <a href="Nonnegative" class="mw-redirect" title="Nonnegative">nonnegative</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~(~E\geq 0~)~}">

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<annotation encoding="application/x-tex">{\displaystyle ~(~E\geq 0~)~}</annotation>

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</math><img src="./6c794c85155248583eb416541e79c193c875e5b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.168ex; height:2.843ex;" alt="{\displaystyle ~(~E\geq 0~)~}" loading="lazy"> <a href="Energy" title="Energy">energy</a> <a href="Irreducible_representation" title="Irreducible representation">irreducible unitary representations</a> of the <a href="Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> which have either finite or zero mass <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a>. (These unitary representations are infinite-dimensional; the group is not semisimple and it does not satisfy <a href="Weyl's_theorem_on_complete_reducibility" title="Weyl's theorem on complete reducibility">Weyl's theorem on complete reducibility</a>.) It was introduced by <a href="Eugene_Wigner" title="Eugene Wigner">Eugene Wigner</a>, to classify particles and fields in physics—see the article <a href="Particle_physics_and_representation_theory" title="Particle physics and representation theory">particle physics and representation theory</a>. It relies on the <a href="Stabilizer_subgroup" class="mw-redirect" title="Stabilizer subgroup">stabilizer subgroups</a> of that group, dubbed the <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">Wigner little groups</a> of various mass states.

The <a href="Casimir_invariant" class="mw-redirect" title="Casimir invariant">Casimir invariants</a> of the Poincaré group are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~C_{1}=P^{\mu }\,P_{\mu }~,}">

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<annotation encoding="application/x-tex">{\displaystyle ~C_{1}=P^{\mu }\,P_{\mu }~,}</annotation>

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</math><img src="./647ff9a90f29de34af662d76ef78926899b56d5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.77ex; height:3.009ex;" alt="{\displaystyle ~C_{1}=P^{\mu }\,P_{\mu }~,}" loading="lazy"> (<a href="Einstein_notation" title="Einstein notation">Einstein notation</a>) where P is the <a href="4-momentum_operator" class="mw-redirect" title="4-momentum operator">4-momentum operator</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~C_{2}=W^{\alpha }\,W_{\alpha }~,}">

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<annotation encoding="application/x-tex">{\displaystyle ~C_{2}=W^{\alpha }\,W_{\alpha }~,}</annotation>

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</math><img src="./33a9585bbb854d6e4d6379f073814c11632bd30a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.28ex; height:2.676ex;" alt="{\displaystyle ~C_{2}=W^{\alpha }\,W_{\alpha }~,}" loading="lazy"> where W is the <a href="Pauli%E2%80%93Lubanski_pseudovector" title="Pauli–Lubanski pseudovector">Pauli–Lubanski pseudovector</a>. The eigenvalues of these operators serve to label the representations. The first is associated with mass-squared and the second with <a href="Helicity_(particle_physics)" title="Helicity (particle physics)">helicity</a> or <a href="Spin_(physics)" title="Spin (physics)">spin</a>.

The physically relevant representations may thus be classified according to whether

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<annotation encoding="application/x-tex">{\displaystyle ~m>0~;}</annotation>

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</math><img src="./20ed2fc0d41f0f52fe2d0f88316fe7c6c3faa957.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.109ex; height:2.509ex;" alt="{\displaystyle ~m>0~;}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle ~m=0~}</annotation>

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</math><img src="./55a6a69d3b2b858b2e8021101e98e0a092950551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.463ex; height:2.176ex;" alt="{\displaystyle ~m=0~}" loading="lazy"> but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~P_{0}>0~;\quad }">

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<annotation encoding="application/x-tex">{\displaystyle ~P_{0}>0~;\quad }</annotation>

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</math><img src="./17a5ee47261d0e78ca6493ed9adf81e7201a8d87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.938ex; height:2.509ex;" alt="{\displaystyle ~P_{0}>0~;\quad }" loading="lazy"> or whether</li>

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<annotation encoding="application/x-tex">{\displaystyle ~m=0~}</annotation>

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</math><img src="./55a6a69d3b2b858b2e8021101e98e0a092950551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.463ex; height:2.176ex;" alt="{\displaystyle ~m=0~}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~P^{\mu }=0~,{\text{ for }}\mu =0,1,2,3~.}">

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<annotation encoding="application/x-tex">{\displaystyle ~P^{\mu }=0~,{\text{ for }}\mu =0,1,2,3~.}</annotation>

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</math><img src="./d32750e2517b95213f7180e83024fc44955c70aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.928ex; height:2.843ex;" alt="{\displaystyle ~P^{\mu }=0~,{\text{ for }}\mu =0,1,2,3~.}" loading="lazy"></li></ul>

Wigner found that massless particles are fundamentally different from massive particles.

<dl><dt>For the first case</dt>

<dd>Note that the <a href="Eigenspace" class="mw-redirect" title="Eigenspace">eigenspace</a> (see <a href="Generalized_eigenspaces_of_unbounded_operators" class="mw-redirect" title="Generalized eigenspaces of unbounded operators">generalized eigenspaces of unbounded operators</a>) associated with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~P=(m,0,0,0)~}">

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<annotation encoding="application/x-tex">{\displaystyle ~P=(m,0,0,0)~}</annotation>

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</math><img src="./67bb5ea5e26dcff5b820afed38a69270d5b53e48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.444ex; height:2.843ex;" alt="{\displaystyle ~P=(m,0,0,0)~}" loading="lazy"> is a <a href="Representations_of_Lie_groups/algebras" class="mw-redirect" title="Representations of Lie groups/algebras">representation</a> of <a href="Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">SO(3)</a>.</dd></dl>

In the <a href="Projective_representation" title="Projective representation">ray interpretation</a>, one can go over to <a href="Spin_group" title="Spin group">Spin(3)</a> instead. So, massive states are classified by an irreducible Spin(3) <a href="Unitary_representation" title="Unitary representation">unitary representation</a> that characterizes their <a href="Spin_(physics)" title="Spin (physics)">spin</a>, and a positive mass, m.

<dl><dt>For the second case</dt>

<dd>Look at the <a href="Stabilizer_(group_theory)" class="mw-redirect" title="Stabilizer (group theory)">stabilizer</a> of</dd>

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<annotation encoding="application/x-tex">{\displaystyle ~P=(k,0,0,-k)~.}</annotation>

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</math><img src="./8182c02508e537e96c049277b186382efa679b3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.119ex; height:2.843ex;" alt="{\displaystyle ~P=(k,0,0,-k)~.}" loading="lazy"></dd></dl>

This is the <a href="Double_covering_group" class="mw-redirect" title="Double covering group">double cover</a> of <a href="Euclidean_group" title="Euclidean group">SE(2)</a> (see <a href="Projective_representation" title="Projective representation">projective representation</a>). We have two cases, one where <a href="Irrep" class="mw-redirect" title="Irrep">irreps</a> are described by an integral multiple of <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠1/2⁠ called the <a href="Helicity_(particle_physics)" title="Helicity (particle physics)">helicity</a>, and the other called the "continuous spin" representation.

<dl><dt>For the third case</dt>

<dd>The only finite-dimensional unitary solution is the <a href="Trivial_representation" title="Trivial representation">trivial representation</a> called the <a href="Vacuum" title="Vacuum">vacuum</a>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Massive_scalar_fields">Massive scalar fields</h2></div>

As an example, let us visualize the irreducible unitary representation with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~m>0~,}">

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<annotation encoding="application/x-tex">{\displaystyle ~m>0~,}</annotation>

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</math><img src="./d86db6a4ff73fe2fa4f0b401fde23a3ee71fa09c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.109ex; height:2.509ex;" alt="{\displaystyle ~m>0~,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~s=0~.}">

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<annotation encoding="application/x-tex">{\displaystyle ~s=0~.}</annotation>

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</math><img src="./a8e4541cfe73b20b83150c7b2e895af69b9179c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.16ex; height:2.176ex;" alt="{\displaystyle ~s=0~.}" loading="lazy"> It corresponds to the space of <a href="Scalar_field" title="Scalar field">massive scalar fields</a>.

Let M be the hyperboloid sheet defined by:

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<annotation encoding="application/x-tex">{\displaystyle ~P_{0}^{2}-P_{1}^{2}-P_{2}^{2}-P_{3}^{2}=m^{2}~,\quad }</annotation>

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</math><img src="./43804d6c7f6091c35aad0e97650fc4d2f4be5a12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.348ex; height:3.343ex;" alt="{\displaystyle ~P_{0}^{2}-P_{1}^{2}-P_{2}^{2}-P_{3}^{2}=m^{2}~,\quad }" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~P_{0}>0~.}">

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<annotation encoding="application/x-tex">{\displaystyle ~P_{0}>0~.}</annotation>

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</math><img src="./d20a38f3a08b2800dfb70f9f69830246e5a967e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.616ex; height:2.509ex;" alt="{\displaystyle ~P_{0}>0~.}" loading="lazy"></dd></dl>

The Minkowski metric restricts to a <a href="Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a> on M, giving M the metric structure of a <a href="Hyperbolic_space" title="Hyperbolic space">hyperbolic space</a>, in particular it is the <a href="Hyperboloid_model" title="Hyperboloid model">hyperboloid model</a> of hyperbolic space, see <a href="Minkowski_space#Geometry" title="Minkowski space">geometry of Minkowski space</a> for proof. The Poincare group P acts on M because (forgetting the action of the translation subgroup ℝ4 with addition inside P) it preserves the <a href="Minkowski_inner_product" class="mw-redirect" title="Minkowski inner product">Minkowski inner product</a>, and an element x of the translation subgroup ℝ4 of the Poincare group acts on

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<annotation encoding="application/x-tex">{\displaystyle ~L^{2}(M)~}</annotation>

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</math><img src="./f8946704c88e6e9d35d1bdc9754c52bf0cc3e3c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.05ex; height:3.176ex;" alt="{\displaystyle ~L^{2}(M)~}" loading="lazy">

by multiplication by suitable phase multipliers

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<annotation encoding="application/x-tex">{\displaystyle ~\exp \left(-i{\vec {p}}\cdot {\vec {x}}\right)~,}</annotation>

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</math><img src="./b43d5970d6fe1eb22326ea4e85b64981acef027f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.501ex; height:2.843ex;" alt="{\displaystyle ~\exp \left(-i{\vec {p}}\cdot {\vec {x}}\right)~,}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~p\in M~.}">

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<annotation encoding="application/x-tex">{\displaystyle ~p\in M~.}</annotation>

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</math><img src="./9ca2cf6d8c6519b10fadaee38da8091fdd14f73a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.26ex; height:2.509ex;" alt="{\displaystyle ~p\in M~.}" loading="lazy"> These two actions can be combined in a clever way using <a href="Induced_representations" class="mw-redirect" title="Induced representations">induced representations</a> to obtain an action

of P acting on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~L^{2}(M)~,}">

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<annotation encoding="application/x-tex">{\displaystyle ~L^{2}(M)~,}</annotation>

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</math><img src="./d5dff92fbefd24292e06b849866e549dec1341a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.697ex; height:3.176ex;" alt="{\displaystyle ~L^{2}(M)~,}" loading="lazy"> that combines motions of M and phase multiplication.

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

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<annotation encoding="application/x-tex">{\displaystyle E^{2}-P_{1}^{2}-P_{2}^{2}-P_{3}^{2}=m^{2}~,\quad }</annotation>

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</math><img src="./4a70fe5f1d323cc2a1028694899814b669fa432a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.74ex; height:3.343ex;" alt="{\displaystyle E^{2}-P_{1}^{2}-P_{2}^{2}-P_{3}^{2}=m^{2}~,\quad }" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~E~\equiv ~P_{0}>0~.}">

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<annotation encoding="application/x-tex">{\displaystyle ~E~\equiv ~P_{0}>0~.}</annotation>

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</math><img src="./e285d23e92c479d117ae9ac9b6049a82a9d4a8fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.651ex; height:2.509ex;" alt="{\displaystyle ~E~\equiv ~P_{0}>0~.}" loading="lazy"></dd></dl>

The Fourier transform (in all four variables) of such measures yields positive-energy, finite-energy solutions of the <a href="Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a> defined on Minkowski space, namely

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}\psi -\nabla ^{2}\psi +m^{2}\psi =0,}</annotation>

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</math><img src="./6d628a87cc45583e6e978e60ab4e5b6f0c5b04cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.261ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}\psi -\nabla ^{2}\psi +m^{2}\psi =0,}" loading="lazy"></dd></dl>

without physical units. In this way, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~m>0,\quad s=0~}">

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<annotation encoding="application/x-tex">{\displaystyle ~m>0,\quad s=0~}</annotation>

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</math><img src="./c3b6eb4109d41f6c151035489b69aeb95e4e99eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.171ex; height:2.509ex;" alt="{\displaystyle ~m>0,\quad s=0~}" loading="lazy"> irreducible representation of the Poincare group is realized by its action on a suitable space of solutions of a linear wave equation.

<div class="mw-heading mw-heading2"><h2 id="Theory_of_projective_representations">Theory of projective representations</h2></div>

Physically, one is interested in irreducible <a href="Projective_representation" title="Projective representation">projective unitary representations</a> 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 <a href="Projective_representation#Infinite-dimensional_projective_unitary_representations:_Bargmann's_theorem" title="Projective representation">Bargmann's theorem</a>, 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 <a href="Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> admits no non-trivial one-dimensional <a href="Group_extension#Central_extension" title="Group extension">central extensions</a>.)

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.

<div class="mw-heading mw-heading2"><h2 id="Further_classification">Further classification</h2></div>

Left out from this classification are <a href="Tachyon" title="Tachyon">tachyonic</a> solutions, solutions with no fixed mass, <a href="Infraparticle" title="Infraparticle">infraparticles</a> with no fixed mass, etc. Such solutions are of physical importance, when considering virtual states. A celebrated example is the case of <a href="Deep_inelastic_scattering" title="Deep inelastic scattering">deep inelastic scattering</a>, in which a virtual space-like <a href="Photon" title="Photon">photon</a> is exchanged between the incoming <a href="Lepton" title="Lepton">lepton</a> and the incoming <a href="Hadron" title="Hadron">hadron</a>. 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 <a href="SO(3)" class="mw-redirect" title="SO(3)">SO(3)</a> group encountered in the usual massive case discussed above. This explains the occurrence of two transverse polarization vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\epsilon _{T}^{\lambda =1,2}~}">

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<annotation encoding="application/x-tex">{\displaystyle ~\epsilon _{T}^{\lambda =1,2}~}</annotation>

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</math><img src="./a9a5c6c10683c6bb998a920b953a01dc04b17f35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.676ex; height:3.509ex;" alt="{\displaystyle ~\epsilon _{T}^{\lambda =1,2}~}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\epsilon _{L}~}">

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<annotation encoding="application/x-tex">{\displaystyle ~\epsilon _{L}~}</annotation>

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</math><img src="./97555331749843778b4479e3c869b3c5bf9799dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.457ex; height:2.009ex;" alt="{\displaystyle ~\epsilon _{L}~}" loading="lazy">

which satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\epsilon _{T}^{2}=-1~}">

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<annotation encoding="application/x-tex">{\displaystyle ~\epsilon _{T}^{2}=-1~}</annotation>

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</math><img src="./168950b49ee25586eaf13cc6051e337248e5ce55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.564ex; height:3.176ex;" alt="{\displaystyle ~\epsilon _{T}^{2}=-1~}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\epsilon _{L}^{2}=+1~,}">

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</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ~\epsilon _{L}^{2}=+1~,}</annotation>

</semantics>

</math><img src="./8a1f70bb5727d9231b9d45da865c377aed4146f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.173ex; height:3.176ex;" alt="{\displaystyle ~\epsilon _{L}^{2}=+1~,}" loading="lazy"> to be compared with the usual case of a free <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~Z_{0}~}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ~Z_{0}~}</annotation>

</semantics>

</math><img src="./16d689fb695ce43fa01ad3c8398783924ba74c4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.803ex; height:2.509ex;" alt="{\displaystyle ~Z_{0}~}" loading="lazy"> boson which has three polarization vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\epsilon _{T}^{\lambda }{\text{ for }}\lambda =1,2,3~;}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ~\epsilon _{T}^{\lambda }{\text{ for }}\lambda =1,2,3~;}</annotation>

</semantics>

</math><img src="./4bc5eebe699db8a11de1203f6e14d2754667030a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.098ex; height:3.176ex;" alt="{\displaystyle ~\epsilon _{T}^{\lambda }{\text{ for }}\lambda =1,2,3~;}" loading="lazy"> each of them satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\epsilon _{T}^{2}=-1~.}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ~\epsilon _{T}^{2}=-1~.}</annotation>

</semantics>

</math><img src="./3100896f7f601c1cc0dddf382e0772103593fecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.211ex; height:3.176ex;" alt="{\displaystyle ~\epsilon _{T}^{2}=-1~.}" loading="lazy">

<ul><li><a href="Induced_representation" title="Induced representation">Induced representation</a> – Process of extending a representation of a subgroup to the parent group</li>

<li><a href="Particle_physics_and_representation_theory" title="Particle physics and representation theory">Particle physics and representation theory</a> – Physics-mathematics connection</li>

<li><a href="Pauli%E2%80%93Lubanski_pseudovector" title="Pauli–Lubanski pseudovector">Pauli–Lubanski pseudovector</a> – Operator in quantum field theory</li>

<li><a href="Representation_theory_of_the_diffeomorphism_group" class="mw-redirect" title="Representation theory of the diffeomorphism group">Representation theory of the diffeomorphism group</a> – Representation theory of the symmetries of manifoldsPages displaying short descriptions of redirect targets</li>

<li><a href="Representation_theory_of_the_Galilean_group" title="Representation theory of the Galilean group">Representation theory of the Galilean group</a> – Representation theory of the symmetries of non-relativistic quantum space</li>

<li><a href="Representation_theory_of_the_Poincar%C3%A9_group" title="Representation theory of the Poincaré group">Representation theory of the Poincaré group</a> – Representation theory of an important group in physics</li>

<li><a href="System_of_imprimitivity" title="System of imprimitivity">System of imprimitivity</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBargmannWigner1948" class="citation journal cs1"><a href="Valentine_Bargmann" title="Valentine Bargmann">Bargmann, V.</a>; <a href="Eugene_Wigner" title="Eugene Wigner">Wigner, E.P.</a> (1948). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1079095">"Group theoretical discussion of relativistic wave equations"</a>. <a href="Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America" title="Proceedings of the National Academy of Sciences of the United States of America">Proceedings of the National Academy of Sciences of the United States of America</a>. 34 (5): 211–223. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1948PNAS...34..211B">1948PNAS...34..211B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.34.5.211">10.1073/pnas.34.5.211</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1079095">1079095</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16578292">16578292</a>.</cite></li></ul>

<ul><li><cite id="CITEREFMackey1978" class="citation book cs1"><a href="George_Mackey" title="George Mackey">Mackey, George</a> (1978). Unitary Group Representations in Physics, Probability and Number Theory. Mathematics Lecture Notes Series. Vol. 55. <a href="Benjamin_Cummings" title="Benjamin Cummings">The Benjamin/Cummings Publishing Company</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0805367034</bdi>.</cite></li></ul>

<ul><li><cite id="CITEREFSternberg1994" class="citation book cs1"><a href="Shlomo_Sternberg" title="Shlomo Sternberg">Sternberg, Shlomo</a> (1994). "§3.9. Wigner classification". <a rel="nofollow" class="external text" href="https://archive.org/details/grouptheoryphysi0000ster">Group Theory and Physics</a>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0521248709</bdi>.</cite></li></ul>

<ul><li><cite id="CITEREFTung1985" class="citation book cs1">Tung, Wu-Ki (1985). "Chapter 10. Representations of the Lorentz group and of the Poincare group; Wigner classification". Group Theory in Physics. <a href="World_Scientific_Publishing_Company" class="mw-redirect" title="World Scientific Publishing Company">World Scientific Publishing Company</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-9971966577</bdi>.</cite></li></ul>

<ul><li><cite id="CITEREFWeinberg2002" class="citation book cs1"><a href="Steven_Weinberg" title="Steven Weinberg">Weinberg, S.</a> (2002). "Chapter 2. Relativistic quantum mechanics". The Quantum Theory of Fields. Vol. I. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-55001-7</bdi>.</cite></li></ul>

<ul><li><cite id="CITEREFWigner1939" class="citation journal cs1"><a href="Eugene_Wigner" title="Eugene Wigner">Wigner, E.P.</a> (1939). "On unitary representations of the inhomogeneous Lorentz group". <a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>. 40 (1): 149–204. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1939AnMat..40..149W">1939AnMat..40..149W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968551">10.2307/1968551</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968551">1968551</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1503456">1503456</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121773411">121773411</a>.</cite></li></ul></div><div><div class="zim-footer">

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