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mw-list-item"><a href="https://ca.wikipedia.org/wiki/C%C3%A0rrega_(f%C3%ADsica)" title="Càrrega (física) – Catalan" lang="ca" hreflang="ca" data-title="Càrrega (física)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BF%D0%BB%D0%B0%D0%BC_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Каплам (физика) – Chuvash" lang="cv" hreflang="cv" data-title="Каплам (физика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/N%C3%A1boj_(fyzika)" title="Náboj (fyzika) – Czech" lang="cs" hreflang="cs" data-title="Náboj (fyzika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gwefr" title="Gwefr – Welsh" lang="cy" hreflang="cy" data-title="Gwefr" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ladung_(Physik)" title="Ladung (Physik) – German" lang="de" hreflang="de" data-title="Ladung (Physik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Laeng" title="Laeng – Estonian" lang="et" hreflang="et" data-title="Laeng" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Carga_(f%C3%ADsica)" title="Carga (física) – Spanish" lang="es" hreflang="es" data-title="Carga (física)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%D8%A7%D8%B1_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="بار (فیزیک) – Persian" lang="fa" hreflang="fa" data-title="بار (فیزیک)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Charge_(physique)" title="Charge (physique) – French" lang="fr" hreflang="fr" data-title="Charge (physique)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%ED%95%98_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="전하 (물리학) – Korean" lang="ko" hreflang="ko" data-title="전하 (물리학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Charjo" title="Charjo – Ido" lang="io" hreflang="io" data-title="Charjo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Carica_(fisica)" title="Carica (fisica) – Italian" lang="it" hreflang="it" data-title="Carica (fisica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/L%C4%81di%C5%86%C5%A1_(fizika)" title="Lādiņš (fizika) – Latvian" lang="lv" hreflang="lv" data-title="Lādiņš (fizika)" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%B6lt%C3%A9s_(fizika)" title="Töltés (fizika) – Hungarian" lang="hu" hreflang="hu" data-title="Töltés (fizika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lading_(natuurkunde)" title="Lading (natuurkunde) – Dutch" lang="nl" hreflang="nl" data-title="Lading (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%81%E3%83%A3%E3%83%BC%E3%82%B8_(%E7%89%A9%E7%90%86%E5%AD%A6)" title="チャージ (物理学) – Japanese" lang="ja" hreflang="ja" data-title="チャージ (物理学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/%C5%81adunek_(fizyka)" title="Ładunek (fizyka) – Polish" lang="pl" hreflang="pl" data-title="Ładunek (fizyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Carga" title="Carga – Portuguese" lang="pt" hreflang="pt" data-title="Carga" 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div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about charge in the most general physics sense. For charge as relates to electrical phenomena, see <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>. For other uses of charge, see <a href="/wiki/Charge_(disambiguation)" class="mw-redirect mw-disambig" title="Charge (disambiguation)">Charge</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Charge_(physics)" title="Special:EditPage/Charge (physics)">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Charge%22+physics">"Charge" physics</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Charge%22+physics+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Charge%22+physics&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Charge%22+physics+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Charge%22+physics">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Charge%22+physics&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">October 2015</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, a <b>charge</b> is any of many different quantities, such as the <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> in <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a> or the <a href="/wiki/Color_charge" title="Color charge">color charge</a> in <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>. Charges correspond to the <a href="/wiki/Time-invariant" class="mw-redirect" title="Time-invariant">time-invariant</a> <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generators</a> of a <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a>, and specifically, to the generators that <a href="/wiki/Commutator_(physics)" class="mw-redirect" title="Commutator (physics)">commute with</a> the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>. Charges are often denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span>, and so the invariance of the charge corresponds to the vanishing <a href="/wiki/Commutator" title="Commutator">commutator</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [Q,H]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>Q</mi> <mo>,</mo> <mi>H</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [Q,H]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/326d24057b764629b9b1d14ca635ca81d721c552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.49ex; height:2.843ex;" alt="{\displaystyle [Q,H]=0}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is the Hamiltonian. Thus, charges are associated with conserved <a href="/wiki/Quantum_number" title="Quantum number">quantum numbers</a>; these are the eigenvalues of the generator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span>. A "charge" can also refer to a point-shaped object with an electric charge and a position, such as in the <a href="/wiki/Method_of_image_charges" title="Method of image charges">method of image charges</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Abstract_definition">Abstract definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Charge_(physics)&action=edit&section=1" title="Edit section: Abstract definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Abstractly, a charge is any generator of a <a href="/wiki/Continuous_symmetry" title="Continuous symmetry">continuous symmetry</a> of the physical system under study. When a physical system has a symmetry of some sort, <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a> implies the existence of a <a href="/wiki/Conserved_current" title="Conserved current">conserved current</a>. The thing that "flows" in the current is the "charge", the charge is the <a href="/wiki/Lie_algebra" title="Lie algebra">generator of the (local) symmetry group</a>. This charge is sometimes called the <b>Noether charge</b>. </p><p>Thus, for example, the <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> is the generator of the <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a> symmetry of <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>. The conserved current is the <a href="/wiki/Electric_current" title="Electric current">electric current</a>. </p><p>In the case of local, dynamical symmetries, associated with every charge is a <a href="/wiki/Gauge_field" class="mw-redirect" title="Gauge field">gauge field</a>; when quantized, the gauge field becomes a <a href="/wiki/Gauge_boson" title="Gauge boson">gauge boson</a>. The charges of the theory "radiate" the gauge field. Thus, for example, the gauge field of electromagnetism is the <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a>; and the gauge boson is the <a href="/wiki/Photon" title="Photon">photon</a>. </p><p>The word "charge" is often used as a synonym for both the generator of a symmetry, and the conserved quantum number (eigenvalue) of the generator. Thus, letting the upper-case letter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ Q\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Q</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ Q\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88494ad054ee0f5339c3b3c564e5f44af71613e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3ex; height:2.509ex;" alt="{\displaystyle \ Q\ }"></span> refer to the generator, one has that the generator <a href="/wiki/Commutator" title="Commutator">commutes</a> with the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \left[\ Q,H\ \right]=0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mi>Q</mi> <mo>,</mo> <mi>H</mi> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left[\ Q,H\ \right]=0~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd0545c9e1d2eb39d6f91910aaadc17b46ecf196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.847ex; height:2.843ex;" alt="{\displaystyle \ \left[\ Q,H\ \right]=0~.}"></span> <a href="/wiki/Commutative_property" title="Commutative property">Commutation</a> implies that the eigenvalues (lower-case) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ q\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>q</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ q\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dca7fa0676c6c83bc15191cc0ffc20320f63562" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.231ex; height:2.009ex;" alt="{\displaystyle \ q\ }"></span> are time-invariant: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\tfrac {~\!\operatorname {d} q~}{~\!\operatorname {d} t~}}=0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mspace width="negativethinmathspace" /> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>q</mi> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <mspace width="negativethinmathspace" /> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>t</mi> <mtext> </mtext> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\tfrac {~\!\operatorname {d} q~}{~\!\operatorname {d} t~}}=0~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf801b79d472333bcccc52095db44e76ac152aeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.124ex; height:4.009ex;" alt="{\displaystyle \ {\tfrac {~\!\operatorname {d} q~}{~\!\operatorname {d} t~}}=0~.}"></span> </p><p>So, for example, when the symmetry group is a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>, then the charge operators correspond to the simple roots of the <a href="/wiki/Root_system" title="Root system">root system</a> of the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>; the <a href="/wiki/Discrete_topology" class="mw-redirect" title="Discrete topology">discreteness</a> of the root system accounting for the quantization of the charge. The simple roots are used, as all the other roots can be obtained as linear combinations of these. The general roots are often called raising and lowering operators, or <a href="/wiki/Ladder_operator" title="Ladder operator">ladder operators</a>. </p><p>The charge quantum numbers then correspond to the weights of the <a href="/wiki/Highest-weight_module" class="mw-redirect" title="Highest-weight module">highest-weight modules</a> of a given <a href="/wiki/Representation_theory" title="Representation theory">representation</a> of the Lie algebra. So, for example, when a particle in a <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> belongs to a symmetry, then it transforms according to a particular representation of that symmetry; the charge quantum number is then the weight of the representation. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Charge_(physics)&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Various charge quantum numbers have been introduced by theories of <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>. These include the charges of the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a>: </p> <ul><li>The <a href="/wiki/Color_charge" title="Color charge">color charge</a> of <a href="/wiki/Quark" title="Quark">quarks</a>. The color charge generates the <a href="/wiki/SU(3)" class="mw-redirect" title="SU(3)">SU(3)</a> color symmetry of <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>.</li> <li>The <a href="/wiki/Weak_isospin" title="Weak isospin">weak isospin</a> quantum numbers of the <a href="/wiki/Electroweak_interaction" title="Electroweak interaction">electroweak interaction</a>. It generates the <a href="/wiki/SU(2)" class="mw-redirect" title="SU(2)">SU(2)</a> part of the electroweak SU(2) × U(1) symmetry. Weak isospin is a local symmetry, whose <a href="/wiki/Gauge_boson" title="Gauge boson">gauge bosons</a> are the <a href="/wiki/W_and_Z_bosons" title="W and Z bosons">W and Z bosons</a>.</li> <li>The <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> for electromagnetic interactions. In mathematics texts, this is sometimes referred to as the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b0c788a124a32684f109737f7cfab7611d6a58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle u_{1}}"></span>-charge of a <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> <a href="/wiki/Weight_(representation_theory)" title="Weight (representation theory)">module</a>.</li></ul> <p>Note that these charge quantum numbers show up in the Lagrangian via the <a href="/wiki/Gauge_covariant_derivative#Standard_Model" title="Gauge covariant derivative">Gauge covariant derivative#Standard_Model</a>. </p><p>Charges of approximate symmetries: </p> <ul><li>The <a href="/wiki/Strong_isospin" class="mw-redirect" title="Strong isospin">strong isospin</a> charges. The symmetry groups is <a href="/wiki/SU(2)" class="mw-redirect" title="SU(2)">SU(2)</a> <a href="/wiki/Flavor_(particle_physics)" class="mw-redirect" title="Flavor (particle physics)">flavor</a> symmetry; the gauge bosons are the <a href="/wiki/Pion" title="Pion">pions</a>. The pions are not <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a>, and the symmetry is only approximate. It is a special case of flavor symmetry.</li> <li>Other <a href="/wiki/Quark" title="Quark">quark</a>-flavor charges, such as <a href="/wiki/Strangeness" title="Strangeness">strangeness</a> or <a href="/wiki/Charm_(quantum_number)" title="Charm (quantum number)">charm</a>. Together with the <span style="white-space:nowrap;"><a href="/wiki/Up_quark" title="Up quark"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>u<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span>–<span style="white-space:nowrap;"><a href="/wiki/Down_quark" title="Down quark"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span>d<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span> isospin mentioned above, these generate the global <a href="/wiki/SU(6)" class="mw-redirect" title="SU(6)">SU(6)</a> flavor symmetry of the fundamental particles; this symmetry is <a href="/wiki/Gell-Mann%E2%80%93Okubo_mass_formula" title="Gell-Mann–Okubo mass formula">badly broken</a> by the masses of the heavy quarks. Charges include the <a href="/wiki/Hypercharge" title="Hypercharge">hypercharge</a>, the <a href="/wiki/X_(charge)" title="X (charge)">X-charge</a> and the <a href="/wiki/Weak_hypercharge" title="Weak hypercharge">weak hypercharge</a>.</li></ul> <p>Hypothetical charges of extensions to the Standard Model: </p> <ul><li>The hypothetical <a href="/wiki/Magnetic_charge" class="mw-redirect" title="Magnetic charge">magnetic charge</a> is another charge in the theory of electromagnetism. Magnetic charges are not seen experimentally in laboratory experiments, but would be present for theories including <a href="/wiki/Magnetic_monopole" title="Magnetic monopole">magnetic monopoles</a>.</li></ul> <p>In <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a>: </p> <ul><li>The <a href="/wiki/Supercharge" title="Supercharge">supercharge</a> refers to the generator that rotates the fermions into bosons, and vice versa, in the supersymmetry.</li></ul> <p>In <a href="/wiki/Conformal_field_theory" title="Conformal field theory">conformal field theory</a>: </p> <ul><li>The <a href="/wiki/Central_charge" title="Central charge">central charge</a> of the <a href="/wiki/Virasoro_algebra" title="Virasoro algebra">Virasoro algebra</a>, sometimes referred to as the <i>conformal central charge</i> or the <a href="/wiki/Conformal_anomaly" title="Conformal anomaly">conformal anomaly</a>. Here, the term 'central' is used in the sense of the <a href="/wiki/Center_(group_theory)" title="Center (group theory)">center</a> in group theory: it is an operator that commutes with all the other operators in the algebra. The central charge is the eigenvalue of the <a href="/wiki/Central_extension_(mathematics)" class="mw-redirect" title="Central extension (mathematics)">central generator</a> of the algebra; here, it is the <a href="/wiki/Energy%E2%80%93momentum_tensor" class="mw-redirect" title="Energy–momentum tensor">energy–momentum tensor</a> of the two-dimensional conformal field theory.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul> <p>In <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitation</a>: </p> <ul><li>Eigenvalues of the energy–momentum tensor correspond to physical <a href="/wiki/Mass" title="Mass">mass</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Charge_conjugation">Charge conjugation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Charge_(physics)&action=edit&section=3" title="Edit section: Charge conjugation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the formalism of particle theories, charge-like quantum numbers can sometimes be inverted by means of a <a href="/wiki/Charge_conjugation" class="mw-redirect" title="Charge conjugation">charge conjugation</a> operator called C. Charge conjugation simply means that a given symmetry group occurs in two inequivalent (but still <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a>) <a href="/wiki/Group_representation" title="Group representation">group representations</a>. It is usually the case that the two charge-conjugate representations are <a href="/wiki/Complex_conjugate_vector_space" class="mw-redirect" title="Complex conjugate vector space">complex conjugate</a> <a href="/wiki/Fundamental_representation" title="Fundamental representation">fundamental representations</a> of the Lie group. Their product then forms the <a href="/wiki/Adjoint_representation_of_a_Lie_group" class="mw-redirect" title="Adjoint representation of a Lie group">adjoint representation</a> of the group. </p><p>Thus, a common example is that the <a href="/wiki/Representation_theory_of_the_Lorentz_group" title="Representation theory of the Lorentz group">product of two charge-conjugate fundamental representations</a> of <a href="/wiki/SL(2,C)" class="mw-redirect" title="SL(2,C)">SL(2,C)</a> (the <a href="/wiki/Spinor" title="Spinor">spinors</a>) forms the adjoint rep of the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a> SO(3,1); abstractly, one writes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\otimes {\overline {2}}=3\oplus 1.\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>3</mn> <mo>⊕</mo> <mn>1.</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\otimes {\overline {2}}=3\oplus 1.\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1aff3010386a028c807c4fd68659452224a202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.772ex; height:3.009ex;" alt="{\displaystyle 2\otimes {\overline {2}}=3\oplus 1.\ }"></span></dd></dl> <p>That is, the product of two (Lorentz) spinors is a (Lorentz) vector and a (Lorentz) scalar. Note that the complex Lie algebra sl(2,C) has a <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Real_form" class="mw-redirect" title="Real form">real form</a> su(2) (in fact, all Lie algebras have a unique compact real form). The same decomposition holds for the compact form as well: the product of two spinors in <a href="/wiki/SU(2)" class="mw-redirect" title="SU(2)">su(2)</a> being a vector in the <a href="/wiki/Rotation_group" class="mw-redirect" title="Rotation group">rotation group</a> <a href="/wiki/O(3)" class="mw-redirect" title="O(3)">O(3)</a> and a singlet. The decomposition is given by the <a href="/wiki/Clebsch%E2%80%93Gordan_coefficient" class="mw-redirect" title="Clebsch–Gordan coefficient">Clebsch–Gordan coefficients</a>. </p><p>A similar phenomenon occurs in the compact group <a href="/wiki/SU(3)" class="mw-redirect" title="SU(3)">SU(3)</a>, where there are two charge-conjugate but inequivalent fundamental representations, dubbed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94d4d5bc734d12c201a23d104e86af5b330cf81c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.277ex; height:2.843ex;" alt="{\displaystyle {\overline {3}}}"></span>, the number 3 denoting the dimension of the representation, and with the quarks transforming under <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"></span> and the antiquarks transforming under <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94d4d5bc734d12c201a23d104e86af5b330cf81c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.277ex; height:2.843ex;" alt="{\displaystyle {\overline {3}}}"></span>. The Kronecker product of the two gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\otimes {\overline {3}}=8\oplus 1.\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>8</mn> <mo>⊕</mo> <mn>1.</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\otimes {\overline {3}}=8\oplus 1.\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69a18603b3fd492e535df0bc5bbac831147ba2b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.772ex; height:3.009ex;" alt="{\displaystyle 3\otimes {\overline {3}}=8\oplus 1.\ }"></span></dd></dl> <p>That is, an eight-dimensional representation, the octet of the <a href="/wiki/Eightfold_Way_(physics)" class="mw-redirect" title="Eightfold Way (physics)">eight-fold way</a>, and a <a href="/wiki/Singlet_state" title="Singlet state">singlet</a>. The decomposition of such products of representations into direct sums of irreducible representations can in general be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda \otimes \Lambda '=\bigoplus _{i}{\mathcal {L}}_{i}\Lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo>⊗</mo> <msup> <mi mathvariant="normal">Λ</mi> <mo>′</mo> </msup> <mo>=</mo> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda \otimes \Lambda '=\bigoplus _{i}{\mathcal {L}}_{i}\Lambda _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c35313a80268497f5a79cc100e470d67d3dc6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.563ex; height:5.509ex;" alt="{\displaystyle \Lambda \otimes \Lambda '=\bigoplus _{i}{\mathcal {L}}_{i}\Lambda _{i}}"></span></dd></dl> <p>for representations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span>. The dimensions of the representations obey the "dimension sum rule": </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\Lambda }\cdot d_{\Lambda '}=\sum _{i}{\mathcal {L}}_{i}d_{\Lambda _{i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Λ</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Λ</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\Lambda }\cdot d_{\Lambda '}=\sum _{i}{\mathcal {L}}_{i}d_{\Lambda _{i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f45501cb81a5448e9ef0c6fc184b80a1bf6d775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.472ex; height:5.509ex;" alt="{\displaystyle d_{\Lambda }\cdot d_{\Lambda '}=\sum _{i}{\mathcal {L}}_{i}d_{\Lambda _{i}}.}"></span></dd></dl> <p>Here, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\Lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Λ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\Lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/428a95e6c9e6942834027701da5e487fc4ca127c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.582ex; height:2.509ex;" alt="{\displaystyle d_{\Lambda }}"></span> is the dimension of the representation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span>, and the integers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span> being the <a href="/wiki/Littlewood%E2%80%93Richardson_coefficient" class="mw-redirect" title="Littlewood–Richardson coefficient">Littlewood–Richardson coefficients</a>. The decomposition of the representations is again given by the Clebsch–Gordan coefficients, this time in the general Lie-algebra setting. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Charge_(physics)&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Casimir_operator" class="mw-redirect" title="Casimir operator">Casimir operator</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Charge_(physics)&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon 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.citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFuchs1992" class="citation cs2">Fuchs, Jurgen (1992), <i>Affine Lie Algebras and Quantum Groups</i>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-48412-X" title="Special:BookSources/0-521-48412-X"><bdi>0-521-48412-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Affine+Lie+Algebras+and+Quantum+Groups&rft.pub=Cambridge+University+Press&rft.date=1992&rft.isbn=0-521-48412-X&rft.aulast=Fuchs&rft.aufirst=Jurgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACharge+%28physics%29" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Charge_(physics)&oldid=1254075788">https://en.wikipedia.org/w/index.php?title=Charge_(physics)&oldid=1254075788</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Electromagnetism" title="Category:Electromagnetism">Electromagnetism</a></li><li><a href="/wiki/Category:Quantum_chromodynamics" title="Category:Quantum chromodynamics">Quantum chromodynamics</a></li><li><a href="/wiki/Category:Physical_quantities" 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