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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Gross-Neveu-Modell</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Das <b>Gross-Neveu-Modell</b> ist ein Modell aus der <a href="Quantenfeldtheorie" title="Quantenfeldtheorie">Quantenfeldtheorie</a> zur Beschreibung von <a href="Dirac-Fermion" title="Dirac-Fermion">Dirac-Fermionen</a> unter der Vier-Fermionen-Wechselwirkung in einer Zeitdimension und einer Raumdimension, beschrieben durch eine <a href="Nichtlineare_Dirac-Gleichung" title="Nichtlineare Dirac-Gleichung">nichtlineare Dirac-Gleichung</a>. Untersucht wurde das Gross-Neveu-Modell erstmals von <a href="David_Gross" title="David Gross">David Gross</a> und <a href="Andr%C3%A9_Neveu" title="André Neveu">André Neveu</a> im Jahr 1974.
</p>
<div class="mw-heading mw-heading2"><h2 id="Lagrange-Dichte">Lagrange-Dichte</h2></div>
<p>Die <a href="Lagrange-Dichte" title="Lagrange-Dichte">Lagrange-Dichte</a> des Gross-Neveu-Modells verallgemeinert dabei die der <a href="Dirac-Gleichung" title="Dirac-Gleichung">Dirac-Gleichung</a> durch einen zusätzlichen Wechselwirkungsterm:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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}}={\bar {\psi }}_{j}(\mathrm {i} \partial \!\!\!/\,-m)\psi ^{j}+{\frac {g^{2}}{2n}}\left({\bar {\psi }}_{j}\psi ^{j}\right)^{2}.}">
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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\bar {\psi }}_{j}(\mathrm {i} \partial \!\!\!/\,-m)\psi ^{j}+{\frac {g^{2}}{2n}}\left({\bar {\psi }}_{j}\psi ^{j}\right)^{2}.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef1870f2ec444b2013d05c4a75c85947aac2c996.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.666ex; height:5.676ex;" alt="{\displaystyle {\mathcal {L}}={\bar {\psi }}_{j}(\mathrm {i} \partial \!\!\!/\,-m)\psi ^{j}+{\frac {g^{2}}{2n}}\left({\bar {\psi }}_{j}\psi ^{j}\right)^{2}.}" loading="lazy"></span></dd></dl>
<p>Mit der <a href="Euler-Lagrange-Gleichung" class="mw-redirect" title="Euler-Lagrange-Gleichung">Euler-Lagrange-Gleichung</a> ergibt sich daraus:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\bar {\psi }}_{j}}}-\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\bar {\psi }}_{j})}}=(\mathrm {i} \partial \!\!\!/\,-m)\psi _{j}+{\frac {g^{2}}{n}}\left({\bar {\psi }}_{k}\psi ^{k}\right)\psi _{j}=0.}">
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<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\bar {\psi }}_{j}}}-\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\bar {\psi }}_{j})}}=(\mathrm {i} \partial \!\!\!/\,-m)\psi _{j}+{\frac {g^{2}}{n}}\left({\bar {\psi }}_{k}\psi ^{k}\right)\psi _{j}=0.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/654750ef309d777ae005eefff474195013adf136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.819ex; height:7.009ex;" alt="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\bar {\psi }}_{j}}}-\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\bar {\psi }}_{j})}}=(\mathrm {i} \partial \!\!\!/\,-m)\psi _{j}+{\frac {g^{2}}{n}}\left({\bar {\psi }}_{k}\psi ^{k}\right)\psi _{j}=0.}" loading="lazy"></span></dd></dl>
<p>Es lässt sich ebenfalls eine verallgemeinerte Lagrange-Dichte für das Gross-Neveu-Modell betrachten:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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}}={\bar {\psi }}_{j}(\mathrm {i} \partial \!\!\!/\,-m)\psi ^{j}+{\frac {g^{2}}{2n}}\left(\left({\bar {\psi }}_{j}\psi ^{j}\right)^{2}+\left({\bar {\psi }}_{j}\gamma _{5}\psi ^{j}\right)^{2}\right).}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\bar {\psi }}_{j}(\mathrm {i} \partial \!\!\!/\,-m)\psi ^{j}+{\frac {g^{2}}{2n}}\left(\left({\bar {\psi }}_{j}\psi ^{j}\right)^{2}+\left({\bar {\psi }}_{j}\gamma _{5}\psi ^{j}\right)^{2}\right).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/106bb28af9848cfa4ea447b3eff02100e08aa674.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.427ex; height:5.676ex;" alt="{\displaystyle {\mathcal {L}}={\bar {\psi }}_{j}(\mathrm {i} \partial \!\!\!/\,-m)\psi ^{j}+{\frac {g^{2}}{2n}}\left(\left({\bar {\psi }}_{j}\psi ^{j}\right)^{2}+\left({\bar {\psi }}_{j}\gamma _{5}\psi ^{j}\right)^{2}\right).}" loading="lazy"></span></dd></dl>
<p>Für diese folgt aus der Euler-Lagrange-Gleichung:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\bar {\psi }}_{j}}}-\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\bar {\psi }}_{j})}}=(i\partial \!\!\!/\,-m)\psi _{j}+{\frac {g^{2}}{n}}\left({\bar {\psi }}_{k}\psi ^{k}\right)\psi _{j}+{\frac {g^{2}}{n}}\left({\bar {\psi }}_{k}\gamma ^{5}\psi ^{k}\right)\gamma ^{5}\psi _{j}=0.}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\bar {\psi }}_{j}}}-\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\bar {\psi }}_{j})}}=(i\partial \!\!\!/\,-m)\psi _{j}+{\frac {g^{2}}{n}}\left({\bar {\psi }}_{k}\psi ^{k}\right)\psi _{j}+{\frac {g^{2}}{n}}\left({\bar {\psi }}_{k}\gamma ^{5}\psi ^{k}\right)\gamma ^{5}\psi _{j}=0.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fc35646a1662c3969ec94cb28e0e188274ab7fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:76.105ex; height:7.009ex;" alt="{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\bar {\psi }}_{j}}}-\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }{\bar {\psi }}_{j})}}=(i\partial \!\!\!/\,-m)\psi _{j}+{\frac {g^{2}}{n}}\left({\bar {\psi }}_{k}\psi ^{k}\right)\psi _{j}+{\frac {g^{2}}{n}}\left({\bar {\psi }}_{k}\gamma ^{5}\psi ^{k}\right)\gamma ^{5}\psi _{j}=0.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Soler-Modell" title="Soler-Modell">Soler-Modell</a></li>
<li><a href="Thirring-Modell" title="Thirring-Modell">Thirring-Modell</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="David_Gross" title="David Gross">David Gross</a> und <a href="Andr%C3%A9_Neveu" title="André Neveu">André Neveu</a>: <cite style="font-style:italic">Dynamical symmetry breaking in asymptotically free field theories</cite>. In: <cite style="font-style:italic">Phys. Rev. D. 10 (10)</cite>. 1974, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>3235–3253</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1103/PhysRevD.10.3235">10.1103/PhysRevD.10.3235</a></span>, <a href="Bibcode" title="Bibcode">bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974PhRvD..10.3235G">1974PhRvD..10.3235G</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Gross-Neveu-Modell&amp;rft.atitle=Dynamical+symmetry+breaking+in+asymptotically+free+field+theories&amp;rft.au=David+Gross+und+Andr%C3%A9+Neveu&amp;rft.btitle=Phys.+Rev.+D.+10+%2810%29&amp;rft.date=1974&amp;rft.doi=10.1103%2FPhysRevD.10.3235&amp;rft.genre=book&amp;rft.pages=3235-3253" style="display:none">&nbsp;</span></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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