SherbyNet - /page/wikibook/entry.mu

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Multiplet

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about multiplets in mathematics and physics. For multiplet siblings, see <a href="Multiple_birth" title="Multiple birth">Multiple birth</a>.</div>

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In <a href="Physics" title="Physics">physics</a> and particularly in <a href="Particle_physics" title="Particle physics">particle physics</a>, a multiplet is the <a href="Quantum_state_space" title="Quantum state space">state space</a> for 'internal' <a href="Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a> of a <a href="Subatomic_particle" title="Subatomic particle">particle</a>; that is, degrees of freedom associated to a particle itself, as opposed to 'external' degrees of freedom such as the particle's position in space. Examples of such degrees of freedom are the <a href="Spin_(physics)" title="Spin (physics)">spin state</a> of a particle in <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, or the <a href="Color_charge" title="Color charge">color</a>, <a href="Isospin" title="Isospin">isospin</a> and <a href="Hypercharge" title="Hypercharge">hypercharge</a> state of particles in the <a href="Standard_Model" title="Standard Model">Standard Model</a> of particle physics. Formally, we describe this state space by a <a href="Vector_space" title="Vector space">vector space</a> which carries the action of a group of continuous symmetries.

<div class="mw-heading mw-heading2"><h2 id="Mathematical_formulation">Mathematical formulation</h2></div>

Mathematically, multiplets are described via <a href="Group_representation" title="Group representation">representations</a> of a <a href="Lie_group" title="Lie group">Lie group</a> or its corresponding <a href="Lie_algebra" title="Lie algebra">Lie algebra</a>, and is usually used to refer to <a href="Irreducible_representation" title="Irreducible representation">irreducible representations</a> (irreps, for short).

At the group level, this is a triplet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V,G,\rho )}">

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<annotation encoding="application/x-tex">{\displaystyle (V,G,\rho )}</annotation>

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</math><img src="./3bfff9162dc458681ca53c4585408756c458e70d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.693ex; height:2.843ex;" alt="{\displaystyle (V,G,\rho )}" loading="lazy"> where

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> is a vector space over a <a href="Field_(algebra)" class="mw-redirect" title="Field (algebra)">field</a> (in the algebra sense) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">, generally taken to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {R} }</annotation>

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</math><img src="./a6419d3aa99701ca996737b17a5e1174d53e6c9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {R} }" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is a Lie group. This is often a compact Lie group.</li>

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is a group homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\rightarrow {\text{GL}}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle G\rightarrow {\text{GL}}(V)}</annotation>

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</math><img src="./cca9997dcc417a89e9e4c6e97b054c09be43f612.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.314ex; height:2.843ex;" alt="{\displaystyle G\rightarrow {\text{GL}}(V)}" loading="lazy">, that is, a map from the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> to the space of invertible linear maps on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. This map must preserve the group structure: for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1},g_{2}\in G,}">

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<annotation encoding="application/x-tex">{\displaystyle g_{1},g_{2}\in G,}</annotation>

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</math><img src="./f2e3129a9c3e443368df40bdeb1a7846df5e8e72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.675ex; height:2.509ex;" alt="{\displaystyle g_{1},g_{2}\in G,}" loading="lazy"> we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (g_{1}\cdot g_{2})=\rho (g_{1})\rho (g_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \rho (g_{1}\cdot g_{2})=\rho (g_{1})\rho (g_{2})}</annotation>

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</math><img src="./f0f8cf16e21d52aa659ee36986cfe8f29dd4f18c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.464ex; height:2.843ex;" alt="{\displaystyle \rho (g_{1}\cdot g_{2})=\rho (g_{1})\rho (g_{2})}" loading="lazy">.</li></ul>

At the algebra level, this is a triplet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V,{\mathfrak {g}},\rho )}">

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<annotation encoding="application/x-tex">{\displaystyle (V,{\mathfrak {g}},\rho )}</annotation>

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</math><img src="./63481df8a36067535ab2ef07214af169b82dc876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.038ex; height:2.843ex;" alt="{\displaystyle (V,{\mathfrak {g}},\rho )}" loading="lazy">, where

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> is as before.</li>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation>

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</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> is a Lie algebra. It is often a finite-dimensional Lie algebra over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is an Lie algebra homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}\rightarrow {\text{End}}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}\rightarrow {\text{End}}(V)}</annotation>

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</math><img src="./3e81f51d814cc448bcf1aa2bf9512419a41040f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.55ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {g}}\rightarrow {\text{End}}(V)}" loading="lazy">. This is a linear map which preserves the Lie bracket: for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2}\in {\mathfrak {g}},}">

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<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2}\in {\mathfrak {g}},}</annotation>

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</math><img src="./bc1fe1b2f7c1bb59cf998db7c94edefda5db21e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.65ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2}\in {\mathfrak {g}},}" loading="lazy"> we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ([X_{1},X_{2}])=[\rho (X_{1}),\rho (X_{2})]}">

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<annotation encoding="application/x-tex">{\displaystyle \rho ([X_{1},X_{2}])=[\rho (X_{1}),\rho (X_{2})]}</annotation>

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</math><img src="./473c2b51e20fe08c4501fe50ab665fbdf0600e9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.701ex; height:2.843ex;" alt="{\displaystyle \rho ([X_{1},X_{2}])=[\rho (X_{1}),\rho (X_{2})]}" loading="lazy">.</li></ul>

The symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is used for both Lie algebras and Lie groups as, at least in finite dimension, there is a well understood <a href="Lie_group-Lie_algebra_correspondence" class="mw-redirect" title="Lie group-Lie algebra correspondence">correspondence</a> between Lie groups and Lie algebras.

In mathematics, it is common to refer to the homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> as the representation, for example in the sentence 'consider a representation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy">', and the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> is referred to as the 'representation space'. In physics sometimes the vector space is referred to as the representation, for example in the sentence 'we model the particle as transforming in the singlet representation', or even to refer to a quantum field which takes values in such a representation, and the physical particles which are modelled by such a quantum field.

For an irreducible representation, an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-plet refers to an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> dimensional irreducible representation. Generally, a group may have multiple non-isomorphic representations of the same dimension, so this does not fully characterize the representation. An exception is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SU}}(2)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(2)}</annotation>

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</math><img src="./e1850368e362fa0be265cbb948aca0ff64a83342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.007ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(2)}" loading="lazy"> which has exactly one irreducible representation of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> for each non-negative integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">.

For example, consider real three-dimensional space, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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</math><img src="./f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy">. The group of 3D rotations <a href="SO(3)" class="mw-redirect" title="SO(3)">SO(3)</a> acts naturally on this space as a group of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3}">

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<annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation>

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</math><img src="./ddc0d4d6106875f8006be1d898512ca5843bad8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}" loading="lazy"> matrices. This explicit realisation of the rotation group is known as the fundamental representation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\text{fund}}}">

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<annotation encoding="application/x-tex">{\displaystyle \rho _{\text{fund}}}</annotation>

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</math><img src="./e77298757182641d8937918a56dd7c254af5260d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.679ex; height:2.176ex;" alt="{\displaystyle \rho _{\text{fund}}}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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</math><img src="./f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy"> is a representation space. The full data of the representation is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ^{3},{\text{SO(3)}},\rho _{\text{fund}})}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ^{3},{\text{SO(3)}},\rho _{\text{fund}})}</annotation>

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</math><img src="./1c46d4975c9f8519d3b2db1d0789bfdfe8bd73e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.361ex; height:3.176ex;" alt="{\displaystyle (\mathbb {R} ^{3},{\text{SO(3)}},\rho _{\text{fund}})}" loading="lazy">. Since the dimension of this representation space is 3, this is known as the triplet representation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SO}}(3)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{SO}}(3)}</annotation>

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</math><img src="./76ea569dfd0b7e75a686925ff43ea07e50e79526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle {\text{SO}}(3)}" loading="lazy">, and it is common to denote this as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {3} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {3} }</annotation>

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</math><img src="./c2210f71f47c2aef89cebab1b74ee6b14a05fa4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {3} }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Application_to_theoretical_physics">Application to theoretical physics</h3></div>

For applications to theoretical physics, we can restrict our attention to the representation theory of a handful of physically important groups. Many of these have well understood representation theory:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{U}}(1)}</annotation>

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</math><img src="./799f83909c3169d94893468b6aa2e83b40fee010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(1)}" loading="lazy">: Part of the gauge group of the Standard model, and the gauge group for theories of electromagnetism. Irreps are all 1 dimensional and are indexed by integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

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</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">, given explicitly by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{n}:{\text{U}}(1)\rightarrow {\text{GL}}(\mathbb {C} );e^{i\theta }\mapsto e^{in\theta }}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{n}:{\text{U}}(1)\rightarrow {\text{GL}}(\mathbb {C} );e^{i\theta }\mapsto e^{in\theta }}</annotation>

</semantics>

</math><img src="./14403171ed9deb5490d7966dd5bd2b446ace76c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.394ex; height:3.176ex;" alt="{\displaystyle \rho _{n}:{\text{U}}(1)\rightarrow {\text{GL}}(\mathbb {C} );e^{i\theta }\mapsto e^{in\theta }}" loading="lazy">. The index can be understood as the <a href="Winding_number" title="Winding number">winding number</a> of the map.</li>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(2)\cong {\text{Spin}}(3)}</annotation>

</semantics>

</math><img src="./fe0b4c8b388d6579f38ccb146e93c222df9100ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.602ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(2)\cong {\text{Spin}}(3)}" loading="lazy">: Part of the gauge group of the Standard model. Irreps are indexed by non-negative integers in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} _{\geq 0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} _{\geq 0}}</annotation>

</semantics>

</math><img src="./738026a4c5c22224059037fdcda24a719765497c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.246ex; height:2.676ex;" alt="{\displaystyle n\in \mathbb {N} _{\geq 0}}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> describing the dimension of the representation, or, with appropriate normalisation, the highest weight of the representation. In physics it is common convention to label these by half-integers instead. See <a href="Representation_theory_of_SU(2)" title="Representation theory of SU(2)">Representation theory of SU(2)</a>.</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{SO}}(3)}</annotation>

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</math><img src="./76ea569dfd0b7e75a686925ff43ea07e50e79526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle {\text{SO}}(3)}" loading="lazy">: The group of rotations of 3D space. Irreps are the odd-dimensional irreps of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SU}}(2)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(2)}</annotation>

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</math><img src="./e1850368e362fa0be265cbb948aca0ff64a83342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.007ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(2)}" loading="lazy"></li>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(3)}</annotation>

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</math><img src="./243e935b16d06b5844c21186d129223de42c0e72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.007ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(3)}" loading="lazy">: Part of the gauge group of the Standard model. Irreps are indexed pairs of non-negative integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m,n)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./274d4857135a7d28a94ba9ee8135779615084d43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.278ex; height:2.843ex;" alt="{\displaystyle (m,n)}" loading="lazy">, describing the <a href="Highest_weight" class="mw-redirect" title="Highest weight">highest weight</a> of the representation. See <a href="Clebsch-Gordan_coefficients_for_SU(3)" class="mw-redirect" title="Clebsch-Gordan coefficients for SU(3)">Clebsch-Gordan coefficients for SU(3)</a>.</li>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\text{SO}}(1,3)}</annotation>

</semantics>

</math><img src="./dac29ed3b32e940a2746f4b549fe0742b1d20628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.269ex; height:2.843ex;" alt="{\displaystyle {\text{SO}}(1,3)}" loading="lazy">: The <a href="Lorentz_group" title="Lorentz group">Lorentz group</a>, the linear symmetries of flat spacetime. All representations arise as representations of its corresponding spin group. See <a href="Representation_theory_of_the_Lorentz_group" title="Representation theory of the Lorentz group">Representation theory of the Lorentz group</a>.</li>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{SL}}(2,\mathbb {C} )\cong {\text{Spin}}(1,3)}</annotation>

</semantics>

</math><img src="./7faa829c36b8910594c25463432c7fbc720cd757.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.22ex; height:2.843ex;" alt="{\displaystyle {\text{SL}}(2,\mathbb {C} )\cong {\text{Spin}}(1,3)}" loading="lazy">: The spin group of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SO}}(1,3)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{SO}}(1,3)}</annotation>

</semantics>

</math><img src="./dac29ed3b32e940a2746f4b549fe0742b1d20628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.269ex; height:2.843ex;" alt="{\displaystyle {\text{SO}}(1,3)}" loading="lazy">. Irreps are indexed by pairs of non-negative integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu ,\nu )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mu ,\nu )}</annotation>

</semantics>

</math><img src="./ea1a07b0389d3f118fb3e2e81473a105c9c70705.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.477ex; height:2.843ex;" alt="{\displaystyle (\mu ,\nu )}" loading="lazy">, indexing the dimension of the representation.</li>

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<msup>

</mrow>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle {\text{E}}(1,3)\cong \mathbb {R} ^{1,3}\rtimes {\text{SO}}(1,3)}</annotation>

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</math><img src="./6842ba601dd6bcce10f454ed05b4e189140de4b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.97ex; height:3.176ex;" alt="{\displaystyle {\text{E}}(1,3)\cong \mathbb {R} ^{1,3}\rtimes {\text{SO}}(1,3)}" loading="lazy">: The Poincaré group of isometries of flat spacetime. This can be understood in terms of the representation theory of the groups above. See <a href="Wigner's_classification" title="Wigner's classification">Wigner's classification</a>.</li></ul>

These groups all appear in the theory of the Standard model. For theories which extend these symmetries, the representation theory of some other groups might be considered:

<ul><li>Conformal symmetry: For pseudo-Euclidean space, symmetries are described by the conformal group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Conf}}(p,q)\cong O(p,q)/\mathbb {Z} _{2}}">

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</mrow>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Conf}}(p,q)\cong O(p,q)/\mathbb {Z} _{2}}</annotation>

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</math><img src="./d9f6f1d73913ffef5b9af82a809150750c2cb9e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.648ex; height:2.843ex;" alt="{\displaystyle {\text{Conf}}(p,q)\cong O(p,q)/\mathbb {Z} _{2}}" loading="lazy">.</li>

<li>Supersymmetry: Symmetry described by a supergroup.</li>

<li>Grand unified theories: Gauge groups which contain the Standard model gauge group as a subgroup. Proposed candidates include <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SU}}(5),{\text{SO}}(10)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(5),{\text{SO}}(10)}</annotation>

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</math><img src="./c0f3308adc9f5b9b6fe333b0f9e849b3ef8fae8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.276ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(5),{\text{SO}}(10)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{E}}_{6}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle {\text{E}}_{6}}</annotation>

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</math><img src="./cbfc6b5933b28b922f473e8ba84e93fd6025ec6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle {\text{E}}_{6}}" loading="lazy">.</li></ul>

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

<div class="mw-heading mw-heading3"><h3 id="Quantum_field_theory">Quantum field theory</h3></div>

In quantum physics, the mathematical notion is usually applied to representations of the <a href="Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a>. For example, an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SU}}(2)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(2)}</annotation>

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</math><img src="./e1850368e362fa0be265cbb948aca0ff64a83342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.007ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(2)}" loading="lazy"> gauge theory will have multiplets which are <a href="Gauge_field" class="mw-redirect" title="Gauge field">fields</a> whose representation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SU}}(2)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(2)}</annotation>

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</math><img src="./e1850368e362fa0be265cbb948aca0ff64a83342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.007ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(2)}" loading="lazy"> is determined by the single half-integer number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=:n/2}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle s=:n/2}</annotation>

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</math><img src="./8547d5bd833cccf6536b6570c6e2ee7eac55b5a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.555ex; height:2.843ex;" alt="{\displaystyle s=:n/2}" loading="lazy">, the isospin. Since irreducible <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SU}}(2)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(2)}</annotation>

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</math><img src="./e1850368e362fa0be265cbb948aca0ff64a83342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.007ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(2)}" loading="lazy"> representations are isomorphic to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">th symmetric power of the fundamental representation, every field has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> symmetrized internal indices.

Fields also transform under representations of the Lorentz group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SO}}(1,3)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\text{SO}}(1,3)}</annotation>

</semantics>

</math><img src="./dac29ed3b32e940a2746f4b549fe0742b1d20628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.269ex; height:2.843ex;" alt="{\displaystyle {\text{SO}}(1,3)}" loading="lazy">, or more generally its spin group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spin}}(1,3)}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Spin}}(1,3)}</annotation>

</semantics>

</math><img src="./67f3f9ba5e4cc55d2ca97ffd7f98688607bd1517.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.693ex; height:2.843ex;" alt="{\displaystyle {\text{Spin}}(1,3)}" loading="lazy"> which can be identified with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SL}}(2,\mathbb {C} )}">

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{SL}}(2,\mathbb {C} )}</annotation>

</semantics>

</math><img src="./d8e637053ad8d6d8b35e980698314513fa105c89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.429ex; height:2.843ex;" alt="{\displaystyle {\text{SL}}(2,\mathbb {C} )}" loading="lazy"> due to an <a href="Exceptional_isomorphism" title="Exceptional isomorphism">exceptional isomorphism</a>. Examples include <a href="Scalar_field" title="Scalar field">scalar fields</a>, commonly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy">, which transform in the trivial representation, vector fields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }}">

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle A_{\mu }}</annotation>

</semantics>

</math><img src="./9277f5286335ab99c040c9c9151ab752d3bedc49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.967ex; height:2.843ex;" alt="{\displaystyle A_{\mu }}" loading="lazy"> (strictly, this might be more accurately labelled a covector field), which transforms as a 4-vector, and spinor fields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \psi _{\alpha }}</annotation>

</semantics>

</math><img src="./eae086daf01dbbd5684898cccb1b1b7baf234dde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \psi _{\alpha }}" loading="lazy"> such as <a href="Dirac_spinor" title="Dirac spinor">Dirac</a> or <a href="Weyl_spinor" class="mw-redirect" title="Weyl spinor">Weyl</a> spinors which transform in representations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SL}}(2,\mathbb {C} )}">

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\text{SL}}(2,\mathbb {C} )}</annotation>

</semantics>

</math><img src="./d8e637053ad8d6d8b35e980698314513fa105c89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.429ex; height:2.843ex;" alt="{\displaystyle {\text{SL}}(2,\mathbb {C} )}" loading="lazy">. A right-handed Weyl spinor transforms in the fundamental representation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{2}}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{2}}</annotation>

</semantics>

</math><img src="./6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{2}}" loading="lazy">, of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SL}}(2,\mathbb {C} )}">

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{SL}}(2,\mathbb {C} )}</annotation>

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</math><img src="./d8e637053ad8d6d8b35e980698314513fa105c89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.429ex; height:2.843ex;" alt="{\displaystyle {\text{SL}}(2,\mathbb {C} )}" loading="lazy">.

Beware that besides the Lorentz group, a field can transform under the action of a gauge group. For example, a scalar field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)}</annotation>

</semantics>

</math><img src="./546b660b2f3cfb5f34be7b3ed8371d54f5c74227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.524ex; height:2.843ex;" alt="{\displaystyle \phi (x)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is a spacetime point, might have an isospin state taking values in the fundamental representation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{2}}</annotation>

</semantics>

</math><img src="./6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{2}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SU}}(2)}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(2)}</annotation>

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</math><img src="./e1850368e362fa0be265cbb948aca0ff64a83342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.007ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(2)}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)}</annotation>

</semantics>

</math><img src="./546b660b2f3cfb5f34be7b3ed8371d54f5c74227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.524ex; height:2.843ex;" alt="{\displaystyle \phi (x)}" loading="lazy"> is a vector valued function of spacetime, but is still referred to as a scalar field, as it transforms trivially under Lorentz transformations.

In quantum field theory different particles correspond one to one with gauged fields transforming in irreducible representations of the internal and Lorentz group. Thus, a multiplet has also come to describe a collection of <a href="Subatomic_particle" title="Subatomic particle">subatomic particles</a> described by these representations.

<div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4></div>

The best known example is a <a href="Spin_multiplet" class="mw-redirect" title="Spin multiplet">spin multiplet</a>, which describes symmetries of a <a href="Group_representation" title="Group representation">group representation</a> of an <a href="SU(2)" class="mw-redirect" title="SU(2)">SU(2)</a> <a href="Subgroup" title="Subgroup">subgroup</a> of the <a href="Lorentz_group" title="Lorentz group">Lorentz algebra</a>, which is used to define spin quantization. A <a href="Spin_singlet" class="mw-redirect" title="Spin singlet">spin singlet</a> is a trivial representation, a <a href="Spin_doublet" class="mw-redirect" title="Spin doublet">spin doublet</a> is a <a href="Fundamental_representation" title="Fundamental representation">fundamental representation</a> and a <a href="Spin_triplet" class="mw-redirect" title="Spin triplet">spin triplet</a> is in the vector representation or <a href="Adjoint_representation" title="Adjoint representation">adjoint representation</a>.

In <a href="Quantum_chromodynamics" title="Quantum chromodynamics">QCD</a>, <a href="Quark" title="Quark">quarks</a> are in a multiplet of <a href="SU(3)" class="mw-redirect" title="SU(3)">SU(3)</a>, specifically the three-dimensional fundamental representation.

<div class="mw-heading mw-heading2"><h2 id="Other_uses">Other uses</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Spectroscopy">Spectroscopy</h3></div>

In spectroscopy, particularly <a href="Gamma_spectroscopy" title="Gamma spectroscopy">Gamma spectroscopy</a> and <a href="X-ray_spectroscopy" title="X-ray spectroscopy">X-ray spectroscopy</a>, a multiplet is a group of related or unresolvable <a href="Spectral_line" title="Spectral line">spectral lines</a>. Where the number of unresolved lines is small, these are often referred to specifically as doublet or triplet peaks, while multiplet is used to describe groups of peaks in any number.

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

<ul><li>Georgi, H. (1999). Lie Algebras in Particle Physics: From Isospin to Unified Theories (1st ed.). CRC Press. <a rel="nofollow" class="external free" href="https://doi.org/10.1201/9780429499210">https://doi.org/10.1201/9780429499210</a></li></ul>

<ul><li><a href="Hypercharge" title="Hypercharge">Hypercharge</a></li>

<li><a href="Isospin" title="Isospin">Isospin</a></li>

<li><a href="Spin_(physics)" title="Spin (physics)">Spin (physics)</a></li>

<li><a href="Group_representation" title="Group representation">Group representation</a></li>

<li><a href="Multiplicity_(chemistry)" title="Multiplicity (chemistry)">Multiplicity (chemistry)</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-11-23" href="https://en.wikipedia.org/wiki/?title=Multiplet&oldid=1259058538">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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