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<ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#FormalizaitonInCohesion'>Formalization in cohesive homotopy-type theory</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#scalars_in_mexican_hat_potential'>Scalars in mexican hat potential</a></li> <li><a href='#in_gravity'>In gravity</a></li> <ul> <li><a href='#kaluzaklein_reductions'>Kaluza-Klein reductions</a></li> <li><a href='#super_kaluzaklein_reductions'>Super Kaluza-Klein reductions</a></li> <li><a href='#scherkschwarz_mechanism'>Scherk-Schwarz mechanism</a></li> </ul> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Conf</mi></mrow><annotation encoding="application/x-tex">Conf</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>:</mo><mi>Conf</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> S : Conf \to \mathbb{R} </annotation></semantics></math></div> <p>an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> that is invariant under a <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> of symmetries <a class="existingWikiWord" href="/nlab/show/action">acting</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Conf</mi></mrow><annotation encoding="application/x-tex">Conf</annotation></semantics></math>, in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>g</mi><mo>∈</mo><mi>G</mi><mo>,</mo><mi>ϕ</mi><mo>∈</mo><mi>Conf</mi><mo>:</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>S</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \forall g \in G, \phi \in Conf : \,\,\, S(g(\phi)) = S(\phi) </annotation></semantics></math></div> <p>a solution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\phi_0</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> of motion is said to exhibit <strong>spontaneously broken symmetry</strong> if it is not a fixed-point of that group <a class="existingWikiWord" href="/nlab/show/action">action</a>: if there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>≠</mo><msub><mi>ϕ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">g(\phi_0) \neq \phi_0</annotation></semantics></math>.</p> <p>The “breaking” refers to the fact that the group no longer acts. It is called “spontaneous” because one imagines that by a physical process the theory “finds” one of its solutions. This comes from the class of examples where a statistical system is first considered at high temperature and then cooled down. At some point it will “spontaneously” freeze in one allowed configuration. A standard example is a <span class="newWikiWord">ferromagnet<a href="/nlab/new/ferromagnet">?</a></span>: at high temperature its <span class="newWikiWord">magnetization<a href="/nlab/new/magnetization">?</a></span> vanishes, while at very low temperature it spontaneously finds a direction of magnetization, thus “breaking” rotational symmetry.</p> <p>One calls the subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mrow><msub><mi>ϕ</mi> <mn>0</mn></msub></mrow></msub><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">G_{\phi_0} \subset G</annotation></semantics></math> that fixes the given configuration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\phi_0</annotation></semantics></math> the <em>unbroken symmetry group</em> .</p> <p>In the context of the <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> arising by <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of this action functional one considers the given classical solution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\phi_0</annotation></semantics></math> as a background about which to consider <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbations</a> of the remaining <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a>.</p> <p>The fields in this effective QFT are then small excitations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\delta \phi</annotation></semantics></math> around the given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\phi_0</annotation></semantics></math>. Since the original symmetry group still acts on the full fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>0</mn></msub><mo>+</mo><mi>δ</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi_0 + \delta \phi</annotation></semantics></math>, the remaining symmetry group of the effective field theory is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mrow><msub><mi>ϕ</mi> <mn>0</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">G_{\phi_0}</annotation></semantics></math>, whose elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> send</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mn>0</mn></msub><mo>+</mo><mi>δ</mi><mi>ϕ</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>g</mi><mo stretchy="false">(</mo><mi>δ</mi><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ϕ</mi> <mn>0</mn></msub><mo>+</mo><mi>g</mi><mi>δ</mi><mi>ϕ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g : (\phi_0 + \delta \phi) \mapsto g(\phi_0) + g(\delta \phi) = \phi_0 + g \delta \phi \,. </annotation></semantics></math></div> <p>Since in the effective theory around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\phi_0</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> state where all the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\delta \phi</annotation></semantics></math> have no excitations (or rather: are in their ground state) corresponds to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\phi_0</annotation></semantics></math> itself one says in this context that <em>a quantum theory exhibits spontaneouly broken symmetry if its vacuum state is not invariant under the pertinent symmetries</em> .</p> <h2 id="FormalizaitonInCohesion">Formalization in cohesive homotopy-type theory</h2> <p>We indicate the formalization of the concept in the axiomatics of <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi></mrow><annotation encoding="application/x-tex">\mathbb{G}</annotation></semantics></math> be a cohesive <a class="existingWikiWord" href="/nlab/show/abelian+infinity-group">abelian infinity-group</a> (for instance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi></mrow><annotation encoding="application/x-tex">\mathbb{G}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth cohesion</a>).</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> on a <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is given by a <a class="existingWikiWord" href="/nlab/show/modulating+morphism">modulating morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⟶</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> P \longrightarrow \mathbf{B}\mathbb{G}_{conn} </annotation></semantics></math></div> <p>to the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G}_{conn}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi></mrow><annotation encoding="application/x-tex">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a>.</p> <p>A symmetry of the theory means that there is a cohesive <a class="existingWikiWord" href="/nlab/show/infinity-group">infinity-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinity-action">infinity-acting</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> such that the prequantum bundle descends to the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>P</mi><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P &\longrightarrow& \mathbf{B}\mathbb{G}_{conn} \\ \downarrow & \nearrow \\ P/G } </annotation></semantics></math></div> <p>Now given an <a class="existingWikiWord" href="/nlab/show/infinity-action">infinity-action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi></mrow><annotation encoding="application/x-tex">\mathbb{G}</annotation></semantics></math> on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> (take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} = U(1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">V = \mathbb{C}</annotation></semantics></math> for traditional quantum mechanics) then a <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> (a <a class="existingWikiWord" href="/nlab/show/wavefunction">wavefunction</a>) is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of the <a class="existingWikiWord" href="/nlab/show/associated+infinity-bundle">associated infinity-bundle</a>, hence a diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>⟶</mo><mi>Ψ</mi></mover></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mi>𝔾</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P &\stackrel{\Psi}{\longrightarrow}& V/\mathbb{G} \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{B}\mathbb{G}_{conn} &\longrightarrow& \mathbf{B}\mathbb{G} } </annotation></semantics></math></div> <p>So this is something defined on phase space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>. If that also descends to the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P/G</annotation></semantics></math> (this is hard to draw here, but should be clear) then that makes the wavefunction also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant. If not, then the wavefunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\Psi</annotation></semantics></math> “breaks” the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-symmetry.</p> <p>Now if on top of this we have that the given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\Psi</annotation></semantics></math> is a “ground state”, then if it does not descend to the homotopy quotient we say “the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-symmetry is spontaneously broken”.</p> <p>To axiomatize what “ground state” means: introduce another <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> which is a <a class="existingWikiWord" href="/nlab/show/Hamiltonian+action">Hamiltonian action</a>, i.e. with respect to which the prequantum bundle is required to be equivariant. Then ask <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\Psi</annotation></semantics></math> to (be <a class="existingWikiWord" href="/nlab/show/polarization">polarized</a> and) be a minimal eigenstate of the respective Hamiltonian. That makes it a “ground state”.</p> <p>For more on the general translation between traditional <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> and <a class="existingWikiWord" href="/nlab/show/cohesion">cohesive</a> homotopy theory see at <em><a class="existingWikiWord" href="/schreiber/show/Higher+geometric+prequantum+theory">Higher geometric prequantum theory</a></em>.</p> <h2 id="examples">Examples</h2> <h3 id="scalars_in_mexican_hat_potential">Scalars in mexican hat potential</h3> <p>A standard example which is both very simple but at the same time of central importance in one of the main applications in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> – the <a class="existingWikiWord" href="/nlab/show/electroweak+symmetry+breaking">electroweak symmetry breaking</a> via the <a class="existingWikiWord" href="/nlab/show/Higgs+mechanism">Higgs mechanism</a> – is this:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Conf</mi><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mi>d</mi></msup><mo>,</mo><msup><mi>ℝ</mi> <mi>N</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Conf = C^\infty(X = [0,1]^d, \mathbb{R}^N)</annotation></semantics></math> be the configuration space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> real scalar fields and take the action functional to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>:</mo><mi>ϕ</mi><mo>↦</mo><msub><mo>∫</mo> <mi>X</mi></msub><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">|</mo><mo>∇</mo><mi>ϕ</mi><msup><mo stretchy="false">|</mo> <mn>2</mn></msup><mo>−</mo><mfrac><mi>h</mi><mn>2</mn></mfrac><mo stretchy="false">|</mo><mi>ϕ</mi><msup><mo stretchy="false">|</mo> <mn>2</mn></msup><mo>−</mo><mfrac><mi>g</mi><mn>4</mn></mfrac><mo stretchy="false">|</mo><mi>ϕ</mi><msup><mo stretchy="false">|</mo> <mn>4</mn></msup><mo>)</mo></mrow><mi>d</mi><msub><mi>μ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex"> S : \phi \mapsto \int_X \left( -\frac{1}{2}\vert \nabla \phi \vert^2 - \frac{h}{2} \vert\phi\vert^2 - \frac{g}{4} \vert\phi\vert^4 \right) d\mu_X </annotation></semantics></math></div> <p>for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">h, g \in \mathbb{R}</annotation></semantics></math>. This is manifestly invariant under the canonical action of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = O(N)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Conf</mi></mrow><annotation encoding="application/x-tex">Conf</annotation></semantics></math>.</p> <p>This action functional has a class of critical points given by constant maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>:</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\phi : X \to \mathbb{R}^n : \phi(x) = \Phi</annotation></semantics></math>. These extremize the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> precisely if the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> extremize the potential energy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mi>h</mi><mn>2</mn></mfrac><mo stretchy="false">|</mo><mi>ϕ</mi><msup><mo stretchy="false">|</mo> <mn>2</mn></msup><mo>+</mo><mfrac><mi>g</mi><mn>4</mn></mfrac><mo stretchy="false">|</mo><mi>ϕ</mi><msup><mo stretchy="false">|</mo> <mn>4</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{h}{2} \vert\phi\vert^2 + \frac{g}{4} \vert\phi\vert^4 \,. </annotation></semantics></math></div> <p>If both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> are positive, then there is only one such critical point, given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Phi = 0</annotation></semantics></math>. Therefore in this case the unique constant solution does <em>not</em> break the symmetry, in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>Φ</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>Φ</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g( \Phi = 0) = (\Phi = 0)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g \in O(N)</annotation></semantics></math>.</p> <p>However, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is negative and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> positive, then the solutions are all those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>Φ</mi><msup><mo stretchy="false">|</mo> <mn>2</mn></msup><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mi>h</mi><mi>g</mi></mfrac><mo>></mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \vert \Phi \vert^2 = - \frac{h}{g} \gt 0 \,. </annotation></semantics></math></div> <p>The set of all these is closed under the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = O(N)</annotation></semantics></math> – this group takes one of these solutions into another – but none of these solutions is <em>fixed</em> by this action.</p> <p>One says in this case that any such solution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>x</mi><mo>↦</mo><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\phi : x \mapsto \Phi</annotation></semantics></math> is a solution that <em>spontaneously breaks the symmetry</em> of the theory.</p> <h3 id="in_gravity">In gravity</h3> <p>The theory of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> on a given <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has as <a class="existingWikiWord" href="/nlab/show/configuration+space">configurations</a> <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+metric">pseudo-Riemannian metric</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and its <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> – the <a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a> or one of its variants – is invariant under the <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> group on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>s are <em><a class="existingWikiWord" href="/nlab/show/Einstein%27s+equations">Einstein's equations</a></em>. A given solution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math> <em>breaks</em> the symmetry given by a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : X \to X</annotation></semantics></math> unless <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>. This means that the unbroken symmetries connected to the identity correspond precisely to the <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math>.</p> <h4 id="kaluzaklein_reductions">Kaluza-Klein reductions</h4> <p>Spontaneous symmetry breaking in <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> plays a central role for instance in the context of the <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a>. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>X</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">dim X = 5</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> of gravity around a solution of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>X</mi> <mn>4</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><msub><mi>g</mi> <mn>4</mn></msub><mo>⊗</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X = X_4 \times S^1, g_4 \otimes g_1)</annotation></semantics></math> is 4-dimensional gravity coupled to <a class="existingWikiWord" href="/nlab/show/electromagnetism">electromagnetism</a> (and a <a class="existingWikiWord" href="/nlab/show/dilaton">dilaton</a> field): the components of the field of gravity along the circle transmute into the electromagnetic field. The ansatz <em>breaks</em> all the symmetries that would mix the remaining 4-dimensional gravitational excitations with these new electromagnetic excitations.</p> <p>This is discussed in a bit of detail for instance in (<a href="#Strominger">Strominger, lecture 1</a>).</p> <h4 id="super_kaluzaklein_reductions">Super Kaluza-Klein reductions</h4> <p>The above discussion has a direct analog in theories of higher <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>. By the same logic, one finds that the <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> around classical solutions that are <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein reductions</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>X</mi> <mn>4</mn></msup><mo>×</mo><msup><mi>Y</mi> <mi>d</mi></msup><mo>,</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>4</mn></msub><mo>⊗</mo><msub><mi>g</mi> <mi>d</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X^4 \times Y^d, (g_4 \otimes g_d))</annotation></semantics></math> exhibits as global symmetries all those <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a>s that are not <em>spontaneously broken</em> by this solution.</p> <p>In this case, though, there are also <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> analogs of the plain diffeomorphism action. Such a local supersymmetry remains unbroken in the given solution if it comes from a <a class="existingWikiWord" href="/nlab/show/Killing+spinor+field">Killing spinor field</a>.</p> <p>Therefore KK-reductions to 4-dimensional <a class="existingWikiWord" href="/nlab/show/Minkowski+space">Minkowski space</a> in supergravity that admit precisely four Killing spinors of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ψ</mi> <mn>4</mn></msub><mo>=</mo><mi>const</mi><mo>⊗</mo><msub><mi>ψ</mi> <mi>d</mi></msub><mo>=</mo><mi>covariantly</mi><mi>const</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\psi_4 = const \otimes \psi_d = covariantly const)</annotation></semantics></math> give rise to <a class="existingWikiWord" href="/nlab/show/effective+field+theories">effective field theories</a> with exactly one remaining global <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>. See also at <em><a class="existingWikiWord" href="/nlab/show/supersymmetry+breaking">supersymmetry breaking</a></em>.</p> <p>For more see <a class="existingWikiWord" href="/nlab/show/supersymmetry+and+Calabi-Yau+manifolds">supersymmetry and Calabi-Yau manifolds</a>.</p> <p>This is discussed in a bit of detail for instance in (<a href="#Strominger">Strominger, lecture 2</a>).</p> <h4 id="scherkschwarz_mechanism">Scherk-Schwarz mechanism</h4> <p>Specifically, the <em><a class="existingWikiWord" href="/nlab/show/Scherk-Schwarz+mechanism">Scherk-Schwarz mechanism</a></em> (<a href="#ScherkSchwarz79">Scherk-Schwarz 79</a>) is the <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneous</a> <a class="existingWikiWord" href="/nlab/show/supersymmetry+breaking">supersymmetry breaking</a> by <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> on a <a class="existingWikiWord" href="/nlab/show/circle">circle</a> whose <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> imposes anti-periodic <a class="existingWikiWord" href="/nlab/show/boundary+conditions">boundary conditions</a> for <a class="existingWikiWord" href="/nlab/show/fermion+fields">fermion fields</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/global+gauge+group">global gauge group</a>, <a class="existingWikiWord" href="/nlab/show/local+gauge+group">local gauge group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Goldstone+boson">Goldstone boson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">topological defects</a> in the <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/domain+wall">domain wall</a>, <a class="existingWikiWord" href="/nlab/show/cosmic+string">cosmic string</a>, <a class="existingWikiWord" href="/nlab/show/monopole">monopole</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enhanced+gauge+symmetry">enhanced gauge symmetry</a></p> </li> </ul> <h2 id="references">References</h2> <p>In</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, <em>The quantum theory of fields</em></li> </ul> <p>sponaneously broken <a class="existingWikiWord" href="/nlab/show/global+gauge+group">global gauge group</a> symmetry is discussed in vol I, section 19, and spontaneously broken <a class="existingWikiWord" href="/nlab/show/local+gauge+group">local gauge group</a> symmetry in vol I, section 21.4.</p> <p>Survey:</p> <ul> <li> <p>Jose Bernabeu, <em>Symmetries and their breaking in the fundamental laws of physics</em> (<a href="https://arxiv.org/abs/2006.13996">arXiv:2006.13996</a>)</p> </li> <li> <p>Tomas Brauner, <em>Effective Field Theory for Spontaneously Broken Symmetry</em> [<a href="https://arxiv.org/abs/2404.14518">arXiv:2404.14518</a>]</p> <blockquote> <p>(perspective of <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a>)</p> </blockquote> </li> </ul> <p>Textbook discussion of broken symmetry in <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> and <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> in the context of the <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> is in</p> <ul> <li id="Strominger"><a class="existingWikiWord" href="/nlab/show/Andrew+Strominger">Andrew Strominger</a> (notes by <a class="existingWikiWord" href="/nlab/show/John+Morgan">John Morgan</a>), <em>Kaluza-Klein compactifications, Supersymmetry and Calabi-Yau spaces</em> , volume II, starting on page 1091 in: <em><a class="existingWikiWord" href="/nlab/show/Quantum+Fields+and+Strings">Quantum Fields and Strings</a>, A course for mathematicians</em>, 2 vols. Amer. Math. Soc. Providence 1999. (<a href="http://www.math.ias.edu/qft">web version</a>)</li> </ul> <p>Discussion of spontaneous <a class="existingWikiWord" href="/nlab/show/supersymmetry+breaking">supersymmetry breaking</a> is in</p> <ul> <li> <p>Yael Shadmi, <em>Supersymmetry breaking</em> (<a href="http://arxiv.org/abs/hep-th/0601076">arXiv:hep-th/0601076</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Joseph+Polchinski">Joseph Polchinski</a>, volume II, appendix B of <em><a class="existingWikiWord" href="/nlab/show/String+theory">String theory</a></em></p> </li> </ul> <p>The article</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, Robin Reuvers, <em>A Flea on Schrödinger’s Cat</em>, Found. Phys. 43, 373-407 (2013) (<a href="http://arxiv.org/abs/1210.2353">arXiv:1210.2353</a>, <a href="http://www.math.ru.nl/~landsman/Catresubmission.pdf">pdf</a>)</li> </ul> <p>points out that for symmetric systems with a symmetric ground state, already a tiny perturbation mixing the ground state with the first excited stated causes spontaneous symmetry breaking in the suitable limit, and suggests that this already resolves the <a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a> in <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on April 24, 2024 at 03:41:37. 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