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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> electromagnetic field </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/15056/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" 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<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> <h4 id="fields_and_quanta">Fields and quanta</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> and <a class="existingWikiWord" href="/nlab/show/fundamental+particle">particles</a> in <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></strong></p> <p><strong>and in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></strong>:</p> <p><strong><a class="existingWikiWord" href="/nlab/show/force">force</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/photon">photon</a> - <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> (<a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/W-boson">W</a>, <a class="existingWikiWord" href="/nlab/show/Z-boson">Z</a>, <a class="existingWikiWord" href="/nlab/show/B-boson">B-boson</a> - <a class="existingWikiWord" href="/nlab/show/electroweak+field">electroweak field</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gluon">gluon</a> - <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong nuclear force</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graviton">graviton</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infraparticle">infraparticle</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/scalar+field">scalar</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Higgs+boson">Higgs boson</a>, <a class="existingWikiWord" href="/nlab/show/inflaton">inflaton</a> (<a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a>)</li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/matter">matter</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a></strong> (<a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>, <a class="existingWikiWord" href="/nlab/show/Dirac+fields">Dirac fields</a>)</p> <div> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> of <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> in the</strong> <br /> <strong><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</strong></th><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a></td><td style="text-align: left;">1st generation</td><td style="text-align: left;">2nd generation</td><td style="text-align: left;">3d generation</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/quarks">quarks</a></strong> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">up-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/up+quark">up quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/charm+quark">charm quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/top+quark">top quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;">down-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/down+quark">down quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/leptons">leptons</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">charged</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron">electron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon">muon</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tauon">tauon</a></td></tr> <tr><td style="text-align: left;">neutral</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron+neutrino">electron neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon+neutrino">muon neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tau+neutrino">tau neutrino</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a>:</strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/mesons">mesons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/light+mesons">light mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/pion">pion</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> <br /> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/strange+quark">strange</a>-mesons</strong>: <br /> <a class="existingWikiWord" href="/nlab/show/%CF%95-meson">ϕ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mover><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">s \bar s</annotation></semantics></math>), <br /> <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a>, <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u s</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">d s</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/eta-meson">eta-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>u</mi><mo>+</mo><mi>d</mi><mi>d</mi><mo>+</mo><mi>s</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u u + d d + s s</annotation></semantics></math>) <br /> <br /> <strong><a class="existingWikiWord" href="/nlab/show/charm+quark">charmed</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>c</mi></mrow><annotation encoding="application/x-tex"> u c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">d c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">s c</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/J%2F%CF%88-meson">J/ψ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mover><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">c \bar c</annotation></semantics></math>)</td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">q b</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mover><mi>b</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">b \bar b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/proton">proton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>u</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u u d)</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u d d)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <p>(also: <a class="existingWikiWord" href="/nlab/show/antiparticles">antiparticles</a>)</p> <p><strong><a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective particles</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Goldstone+bosons">Goldstone bosons</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/hadrons">hadrons</a></strong> (<a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a> of the above <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/meson">meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/scalar+meson">scalar meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%83-meson">σ-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pion">pion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/kaon">kaon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+meson">vector meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/b1-meson">b1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/h1-meson">h1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+meson">tensor meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quarkonium">quarkonium</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/charmonium">charmonium</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+meson">exotic meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/XYZ+meson">XYZ meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tetraquark">tetraquark</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/baryon">baryon</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/proton">proton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/neutron">neutron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chemical+element">chemical element</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/carbon">carbon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nitrogen">nitrogen</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lambda+baryon">Lambda baryon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pentaquark">pentaquark</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/solitons">solitons</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/caloron">caloron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> </ul> <p><strong>in <a class="existingWikiWord" href="/nlab/show/grand+unified+theory">grand unified theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/leptoquark">leptoquark</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Z%27-boson">Z'-boson</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/MSSM">minimally extended supersymmetric standard model</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/superpartners">superpartners</a></strong></p> <p>bosinos:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> (<a class="existingWikiWord" href="/nlab/show/Rarita-Schwinger+field">Rarita-Schwinger field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gaugino">gaugino</a> - <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gluino">gluino</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/sfermions">sfermions</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/squark">squark</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a> candidates</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WIMP">WIMP</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axion">axion</a></p> </li> </ul> <p><strong>Exotica</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/preon">preon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graviphoton">graviphoton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dilaton">dilaton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monopole">monopole</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+graviton">dual graviton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/giant+graviton">giant graviton</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/auxiliary+fields">auxiliary fields</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ghost+field">ghost field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antifield">antifield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antighost+field">antighost field</a>, <a class="existingWikiWord" href="/nlab/show/Nakanishi-Lautrup+field">Nakanishi-Lautrup field</a></p> </li> </ul> </div></div> <h4 id="differential_cohomology">Differential cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="connections_on_bundles">Connections on bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a>, <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> </ul> <h2 id="higher_abelian_differential_cohomology">Higher abelian differential cohomology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+function+complex">differential function complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+orientation">differential orientation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+Thom+class">differential Thom class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characters">differential characters</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe with connection</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> <h2 id="higher_nonabelian_differential_cohomology">Higher nonabelian differential cohomology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">connection on a 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> </li> </ul> <h2 id="fiber_integration">Fiber integration</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+K-theory">fiber integration in differential K-theory</a></p> </li> </ul> </li> </ul> <h2 id="application_to_gauge_theory">Application to gauge theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>/<a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">supergravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/differential+cohomology+-+contents">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='#history'>History</a></li> <li><a href='#mathematical_model_from_physical_input'>Mathematical model from physical input</a></li> <ul> <li><a href='#MaxwellEquations'>Maxwell’s equations</a></li> <ul> <li><a href='#FieldStrengthAsClosed2Form'>Field strength is a closed 2-form</a></li> <li><a href='#vector_potential__as_a_cechdeligne_2cocycle'>Vector potential – as a Cech-Deligne 2-cocycle</a></li> </ul> <li><a href='#kirchhoffs_laws'>Kirchhoff’s laws</a></li> <li><a href='#AsBackgroundGaugeField'>Background gauge field for charge quantum</a></li> </ul> <li><a href='#ChargeQuantization'>Charge quantization</a></li> <ul> <li><a href='#DiracArgument'>Dirac’s original argument</a></li> <li><a href='#electricmagnetic_charge_quantization'>Electric-magnetic charge quantization</a></li> </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>The <strong>electromagnetic field</strong> is is a <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> which unifies the <a class="existingWikiWord" href="/nlab/show/electric+field">electric field</a> and the <a class="existingWikiWord" href="/nlab/show/magnetic+field">magnetic field</a>. A configuration of the <strong>electromagnetic field</strong> on a space <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> in the <em>absence</em> of <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> is modeled by a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat F</annotation></semantics></math> in degree 2 <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>.</p> <p>This may be realized in particular equivalently by</p> <ul> <li> <p>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>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">with connection</a> 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> </li> <li> <p>a degree 2 <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cocycle</a> 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> </li> </ul> <p>In the presence of <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> the electromagnetic field is modeled by a cocycle in differential <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>, where the twist is given by the differential 3-cocycle that models the <a class="existingWikiWord" href="/nlab/show/magnetic+current">magnetic current</a>.</p> <p>The analogous field modeled by a degree 3 <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cocycle</a> is the <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>.</p> <h2 id="history">History</h2> <p>…historical section eventually goes here..</p> <p>…electricity and magnetism were discovered independently, <a class="existingWikiWord" href="/nlab/show/Maxwell%27s+equations">Maxwell's equations</a> in classical vector analysis which allows the formulation as a tensor <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> as below, and “magnetism is a consequence of electrostatics and covariance, hence the composite noun electromagnetism”…</p> <p>(…)</p> <h2 id="mathematical_model_from_physical_input">Mathematical model from physical input</h2> <p>The electromagnetic field is modeled by a <a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a>.</p> <p>We describe how this identification arises from experimental input.</p> <p>The input is two-fold</p> <ol> <li> <p><a href="#MaxwellEquations">Maxwell’s equations</a> say that (using the experimentally observed absence of net <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a>) the <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> of electromagnetism is a closed <a class="existingWikiWord" href="/nlab/show/differential+form">differential 2-form</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>;</p> </li> <li> <p>The <em>Dirac charge quantization argument</em> shows that in order for the electromagnetic field to serve as the <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> to which a charged <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanical</a> <a class="existingWikiWord" href="/nlab/show/particle">particle</a> couples (for instance an <a class="existingWikiWord" href="/nlab/show/electron">electron</a>), it must be true that this 2-form is the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 2-form of a <a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a>.</p> </li> </ol> <p>We say this now in more detail.</p> <h3 id="MaxwellEquations">Maxwell’s equations</h3> <p>We first discuss in <a href="#FieldStrengthAsClosed2Form">Field strength as a closed 2-form</a> how <a class="existingWikiWord" href="/nlab/show/Maxwell%27s+equations">Maxwell's equations</a> state that the electromagnetic <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> is a closed <a class="existingWikiWord" href="/nlab/show/differential+form">differential 2-form</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <h4 id="FieldStrengthAsClosed2Form">Field strength is a closed 2-form</h4> <p>In modern language, the insight of (<a href="#Maxwell">Maxwell, 1865</a>) is that locally, when physical <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is well approximated by a patch of its tangent space, i.e. by a patch of 4-dimensional <a class="existingWikiWord" href="/nlab/show/Minkowski+space">Minkowski space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>4</mn></msup><mo>,</mo><mi>g</mi><mo>=</mo><mi>diag</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \subset (\mathbb{R}^4, g = diag(-1,1,1,1))</annotation></semantics></math>, the electric field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">→</mo></mover><mo>=</mo><mrow><mo>[</mo><mrow><mtable><mtr><mtd><msub><mi>E</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>E</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>E</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex">\vec E = \left[ \array{E_1 \\ E_2 \\ E_3} \right]</annotation></semantics></math> and magnetic field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">→</mo></mover><mo>=</mo><mrow><mo>[</mo><mrow><mtable><mtr><mtd><msub><mi>B</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>B</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>B</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex">\vec B = \left[ \array{B_1 \\ B_2 \\ B_3} \right]</annotation></semantics></math> combine into a differential <a class="existingWikiWord" href="/nlab/show/differential+form">2-form</a> – the <strong>Faraday tensor</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>F</mi></mtd> <mtd><mo>≔</mo><mi>E</mi><mo>∧</mo><mi>d</mi><mi>t</mi><mo>+</mo><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><msub><mi>E</mi> <mn>1</mn></msub><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>∧</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mi>E</mi> <mn>2</mn></msub><mi>d</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>∧</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mi>E</mi> <mn>3</mn></msub><mi>d</mi><msup><mi>x</mi> <mn>3</mn></msup><mo>∧</mo><mi>d</mi><mi>t</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>B</mi> <mn>1</mn></msub><mi>d</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>3</mn></msup><mo>+</mo><msub><mi>B</mi> <mn>2</mn></msub><mi>d</mi><msup><mi>x</mi> <mn>3</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>+</mo><msub><mi>B</mi> <mn>3</mn></msub><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>2</mn></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} F & \coloneqq E \wedge d t + B \\ & \coloneqq E_1 d x^1 \wedge d t + E_2 d x^2 \wedge d t + E_3 d x^3 \wedge d t \\ & + B_1 d x^2 \wedge d x^3 + B_2 d x^3 \wedge d x^1 + B_3 d x^1 \wedge d x^2 \end{aligned} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^2(U)</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> density and current density combine to a differential 3-form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>j</mi> <mi>el</mi></msub></mtd> <mtd><mo>≔</mo><mi>j</mi><mo>∧</mo><mi>dt</mi><mo>−</mo><mi>ρ</mi><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>3</mn></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><msub><mi>j</mi> <mn>1</mn></msub><mi>d</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>3</mn></msup><mo>∧</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mi>j</mi> <mn>2</mn></msub><mi>d</mi><msup><mi>x</mi> <mn>3</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>∧</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mi>j</mi> <mn>3</mn></msub><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>∧</mo><mi>d</mi><mi>t</mi><mo>−</mo><mi>ρ</mi><mspace width="thickmathspace"></mspace><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>3</mn></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} j_{el} & \coloneqq j\wedge dt - \rho d x^1 \wedge d x^2 \wedge d x^3 \\ & \coloneqq j_1 d x^2 \wedge d x^3 \wedge d t + j_2 d x^3 \wedge d x^1 \wedge d t + j_3 d x^1 \wedge d x^2 \wedge d t - \rho \; d x^1 \wedge d x^2 \wedge d x^3 \end{aligned} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^3(U)</annotation></semantics></math> such that the following two equations of differential forms are satisfied</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>d</mi><mi>F</mi><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mi>d</mi><mo>⋆</mo><mi>F</mi><mo>=</mo><msub><mi>j</mi> <mi>el</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} d F = 0 \\ d \star F = j_{el} \end{aligned} \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> is the de Rham differential operator and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Hodge+star">Hodge star</a> operator. If we decompose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋆</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\star F</annotation></semantics></math> into its components as before as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo>⋆</mo><mi>F</mi></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>D</mi><mo>+</mo><mi>H</mi><mo>∧</mo><mi>dt</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>D</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mi>d</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>3</mn></msup><mo>−</mo><msub><mi>D</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mi>d</mi><msup><mi>x</mi> <mn>3</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>−</mo><msub><mi>D</mi> <mn>3</mn></msub><mspace width="thickmathspace"></mspace><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>2</mn></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>H</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>∧</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mi>H</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mi>d</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>∧</mo><mi>d</mi><mi>t</mi><mo>+</mo><msub><mi>H</mi> <mn>3</mn></msub><mspace width="thickmathspace"></mspace><mi>d</mi><msup><mi>x</mi> <mn>3</mn></msup><mo>∧</mo><mi>d</mi><mi>t</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \star F &= -D + H\wedge dt \\ &= -D_1 \; d x^2 \wedge d x^3 -D_2 \; d x^3 \wedge d x^1 -D_3 \; d x^1 \wedge d x^2 \\ & + H_1 \; d x^1 \wedge d t + H_2 \; d x^2 \wedge d t + H_3 \; d x^3 \wedge d t \end{aligned} </annotation></semantics></math></div> <p>then in terms of these components the field equations – called <strong>Maxwell’s equations</strong> – read as follows.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>F</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d F = 0</annotation></semantics></math></p> <ul> <li> <p><strong>magnetic Gauss law</strong>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>div</mi><mi>B</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">div B = 0</annotation></semantics></math></p> </li> <li> <p><strong>Faraday’s law</strong>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mi>B</mi><mo>+</mo><mi>rot</mi><mi>E</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{d}{d t} B + rot E = 0</annotation></semantics></math></p> </li> </ul> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>⋆</mo><mi>F</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d \star F = 0</annotation></semantics></math></p> <ul> <li> <p><strong>Gauss’ law</strong>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>div</mi><mi>D</mi><mo>=</mo><mi>ρ</mi></mrow><annotation encoding="application/x-tex">div D = \rho</annotation></semantics></math></p> </li> <li> <p><strong>Ampère’s law</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mi>D</mi><mo>+</mo><mi>rot</mi><mi>H</mi><mo>=</mo><msub><mi>j</mi> <mi>el</mi></msub></mrow><annotation encoding="application/x-tex">- \frac{d}{d t} D + rot H = j_{el}</annotation></semantics></math></p> </li> </ul> <h4 id="vector_potential__as_a_cechdeligne_2cocycle">Vector potential – as a Cech-Deligne 2-cocycle</h4> <p>By the <a href="#FieldStrengthAsClosed2Form">above</a>, the electromagnetic <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> (in the absence of net <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a>) on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</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> is given by a 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \in \Omega^2(X)</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/electric+current">electric current</a> 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mi>el</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j_{el} \in \Omega^3(X)</annotation></semantics></math> satisfying Maxwell’s equations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>F</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> d F = 0 </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>⋆</mo><mi>F</mi><mo>=</mo><msub><mi>j</mi> <mi>el</mi></msub></mrow><annotation encoding="application/x-tex"> d \star F = j_{el} </annotation></semantics></math></div> <p>The first equation with the <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> implies that one may find</p> <ul> <li> <p>on a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> of <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> by <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math></p> </li> <li> <p>a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">(A_i \in \Omega^1(U))_i</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/differential+form">differential 1-forms</a>, such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>A</mi> <mi>i</mi></msub><mo>=</mo><mi>F</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> d A_i = F|_{U_i} </annotation></semantics></math></div></li> <li> <p>and a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>ℝ</mi><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(\lambda_{i j} \in C^\infty(U_i \cap U_j), \mathbb{R})_{i,j}</annotation></semantics></math> of real valued functions on double overlaps such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>j</mi></msub><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub><mo>−</mo><msub><mi>A</mi> <mi>i</mi></msub><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub><mo>=</mo><mi>d</mi><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A_j|_{U_i \cap U_j} - A_i|_{U_i \cap U_j} = d \lambda_{i j} \,. </annotation></semantics></math></div></li> </ul> <p>The forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{A_i\}</annotation></semantics></math> are called a <strong>vector potential</strong> or the <strong><a class="existingWikiWord" href="/nlab/show/electromagnetic+potential">electromagnetic potential</a></strong> for the electromagnetic field.</p> <p>Notice that it follows that on triple overlaps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j \cap U_k</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>d</mi><msub><mi>λ</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><mi>d</mi><msub><mi>λ</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> d \lambda_{i j} + d \lambda_{j k} = d \lambda_{i k} </annotation></semantics></math></div> <p>which means that on that overlap the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi> <mrow><mi>k</mi><mi>j</mi></mrow></msub><mo>−</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> \lambda_{i j} + \lambda_{k j} - \lambda_{i k} : U_i \cap U_j \cap U_k \to \mathbb{R} </annotation></semantics></math></div> <p>is constant. If one requires these constants all to be inside a discrete subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>↪</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\Gamma \hookrightarrow \mathbb{R}</annotation></semantics></math>, then the data <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mi>mod</mi><mi>Γ</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\{A_i\}, \{\lambda_{i j} mod \Gamma\})</annotation></semantics></math> defines a degree 2-<a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech</a>-<a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> 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> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo stretchy="false">/</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}/\Gamma</annotation></semantics></math>. <a href="#AsBackgroundGaugeField">Below</a> we see that experiment demands that such a subgroup exists and is given by the additive group of <a class="existingWikiWord" href="/nlab/show/integer">integer</a>s.</p> <h3 id="kirchhoffs_laws">Kirchhoff’s laws</h3> <p><a class="existingWikiWord" href="/nlab/show/Kirchhoff%27s+laws">Kirchhoff's laws</a> are a kind of coarse graining of Maxwell’s equations, where instead of infinitesimal quantities one considers actual macroscopic <a class="existingWikiWord" href="/nlab/show/current">current</a> and <a class="existingWikiWord" href="/nlab/show/voltage">voltage</a>.</p> <h3 id="AsBackgroundGaugeField">Background gauge field for charge quantum</h3> <p>The exponentiated <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> for the <a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a> describing a <a class="existingWikiWord" href="/nlab/show/particle">particle</a> on <a class="existingWikiWord" href="/nlab/show/target+space">target space</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> charged under an electromagnetic <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> has to satisfy two properties:</p> <ol> <li> <p>It must be given by the <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> of the 1-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (this encodes, via <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a>, the <a class="existingWikiWord" href="/nlab/show/Lorentz+force">Lorentz force</a> exerted by the electromagnetic field on the particle ):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mover><mo>→</mo><mi>γ</mi></mover><mi>X</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mi>S</mi> <mi>kin</mi></msub><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>hol</mi><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \exp(i S(-)) : (S^1 \stackrel{\gamma}{\to} X) \mapsto \exp(i S_{kin}(\gamma)) hol(\gamma) </annotation></semantics></math></div></li> <li> <p>It must take values in 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> (this is a basic rule of the <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> describing the particle):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \exp( i S ) : C^\infty(S^1, X) \to U(1) \,. </annotation></semantics></math></div></li> </ol> <p>Therefore the above data is subject to the additional constraint that it induces well-defined <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>-valued <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> – this is <strong>Dirac’s quantization condition</strong>, a necessary requirement for the existence of quantum mechanical particles 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> that are charged under the background electromagnetic field.</p> <p>Concretely: for any smooth curve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma : S^1 \to X</annotation></semantics></math> and any <a class="existingWikiWord" href="/nlab/show/cover">cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>V</mi> <mi>i</mi></msub><mo>→</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{V_i \to S^1\}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> refining the pullback of the cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>, and for every triangulation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>v</mi><mo>,</mo><mi>e</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{v, e\}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> subordinate to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math>, i.e. such that there is an index map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>U</mi> <mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\gamma(e) \subset U_{\rho(e)}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>U</mi> <mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\gamma(v) \subset U_{\rho(v)}</annotation></semantics></math></p> <p>the expression</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hol</mi><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>v</mi></munder><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mo>∫</mo> <mi>v</mi></msub><msup><mi>γ</mi> <mo>*</mo></msup><msub><mi>A</mi> <mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>e</mi></munder><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>σ</mi> <mrow><mi>e</mi><mo>,</mo><mi>v</mi></mrow></msub></mrow></msup><msup><mi>γ</mi> <mo>*</mo></msup><msub><mi>λ</mi> <mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> hol(\gamma) \coloneqq \prod_v \exp(i \int_{v} \gamma^* A_{\rho(v)}) \prod_{e} \exp(i (-1)^{\sigma_{e,v}} \gamma^*\lambda_{\rho(e) \rho(v)} (v) </annotation></semantics></math></div> <p>(where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mrow><mi>e</mi><mo>,</mo><mi>v</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\sigma_{e,v} = 1</annotation></semantics></math> if <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> is the final vertex of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math> otherwise)</p> <p>has to be a well defined element in <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> (independent of all the choices made).</p> <p>This implies in particular that cancelling from the triangulation an edge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">e_i</annotation></semantics></math> of vanishing length must have no effect on the formula, which in turn means that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">i,j,k</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msup><mi>γ</mi> <mo>*</mo></msup><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msup><mi>γ</mi> <mo>*</mo></msup><msub><mi>λ</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msup><mi>γ</mi> <mo>*</mo></msup><msub><mi>λ</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \exp(i \gamma^*\lambda_{i j}(v_{i j})) \exp(i \gamma^*\lambda_{j k}(v_{j k})) = \exp(i \gamma^*\lambda_{i k}(v_{i k})) </annotation></semantics></math></div> <p>and hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mi>mod</mi><mn>2</mn><mi>π</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \lambda_{i j} + \lambda_{j k} = \lambda_{i k} mod 2\pi \,. </annotation></semantics></math></div> <p>In short: the <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> of the constant path on a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> must be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1 \in U(1)</annotation></semantics></math>, but if that path sits in a triple intersection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j \cap U_k</annotation></semantics></math> then the holonomy is equivalently given as the exponentiated sum of the three transition functions. This forces the sum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>−</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\lambda_{i j} + \lambda_{j k} - \lambda_{i k}</annotation></semantics></math> to land in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>↪</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \hookrightarrow \mathbb{R}</annotation></semantics></math>.</p> <p>In total this says precisely that the data</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><msub><mi>A</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mi>mod</mi><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (A_i, \lambda_{i j} mod \mathbb{Z}) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cocycle</a> with coefficients in the degree 2 <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne complex</a> whose <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a>2-form is the given Maxwell <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 2-form.</p> <h2 id="ChargeQuantization">Charge quantization</h2> <p>See at <em><a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization">Dirac charge quantization</a></em>.</p> <center> <img src="https://ncatlab.org/nlab/files/DiracChargeQuantizationII.jpg" width="640" /> </center> <h3 id="DiracArgument">Dirac’s original argument</h3> <blockquote> <p>under construction</p> </blockquote> <p>Dirac originally presented the following reasoning, which captures the main point of the above considerations.</p> <p>He considered <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> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^3</annotation></semantics></math> without the origin,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≔</mo><msup><mi>ℝ</mi> <mn>3</mn></msup><mo>\</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \coloneqq \mathbb{R}^3 \backslash \{0\} \,, </annotation></semantics></math></div> <p>which is a manifold of the topology of (<a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weakly homotopy equivalent to</a>) the 2-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math>.</p> <p>He imagined a situation with a magnetic charge supported on the point located at the origin and removed that point in order to keep the field strength <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> to be a <em>closed</em> 2-form on all of <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>(Indeed, if one does not remove the support of magnetic charge, the argument becomes much more sophisticated and involves higher differential cocycles given by <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a>s. This was not understood before Dan Freed’s <a href="http://arxiv.org/abs/hep-th/0011220">Dirac charge quantization and generalized differential cohomology</a>.)</p> <p>Then he considered a single coordinate patch</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>≔</mo><msup><mi>ℝ</mi> <mn>3</mn></msup><mo>\</mo><mo stretchy="false">{</mo><msup><mi>x</mi> <mn>1</mn></msup><mo>≥</mo><mn>0</mn><mo stretchy="false">}</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> U \coloneqq \mathbb{R}^3 \backslash \{x^1 \geq 0\} \subset X </annotation></semantics></math></div> <p>given by <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> minus the right half of the first coordinate axis.</p> <p>Traditionally physicist try to give that half-line a physical interpretation by imagining that it is the body of an idealized infinitely-thin and to one side infinitely-long solenoid. Indeed, such a solenoid would have a magnetic monopole charge on each of its ends, so if the one end is imagined to have disappeared to infinity, then the other one is the magnetic charge that Dirac imagines to sit at the origin of our setup.</p> <p>In this context the half-line <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>1</mn></msup><mo>≥</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x^1 \geq 0\}</annotation></semantics></math> is called a <strong>Dirac string</strong>. While there is the possibility to sensibly discuss the idea that this Dirac string actually models a physical entity like an idealized solenoid, its main purpose historically is to confuse physics students and keep them from understanding the theory of <a class="existingWikiWord" href="/nlab/show/fiber+bundles">fiber bundles</a> (see at <em><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></em>). Therefore here we shall refrain from talking about Dirac strings and consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>≔</mo><msup><mi>ℝ</mi> <mn>3</mn></msup><mo>\</mo><mo stretchy="false">{</mo><msup><mi>x</mi> <mn>1</mn></msup><mo>≥</mo><mn>0</mn><mo stretchy="false">}</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \coloneqq \mathbb{R}^3 \backslash \{x^1 \geq 0\} \subset X</annotation></semantics></math> as exactly what it is, by itself: an open subset that is part of a <a class="existingWikiWord" href="/nlab/show/cover">cover</a> of <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>. Unfortunately, of course, Dirac didn’t mention the other open subsets in that <a class="existingWikiWord" href="/nlab/show/cover">cover</a> (at least one more is needed for a decent discussion), so that the Dirac string keeps haunting physicists.</p> <blockquote> <p>…running out of time…just quickly now…</p> </blockquote> <p>…Dirac effectively considered the overlap cocycle condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>j</mi></msub><mo>−</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>=</mo><mi>something</mi></mrow><annotation encoding="application/x-tex">A_j - A_i = something</annotation></semantics></math>, found that by the requirement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> has well defined holonomy it follows that there must be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>ij</mi></msub></mrow><annotation encoding="application/x-tex">g_{ij}</annotation></semantics></math> a function with values in <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>j</mi></msub><mo>−</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>=</mo><mi>d</mi><mi>log</mi><msub><mi>g</mi> <mi>ij</mi></msub></mrow><annotation encoding="application/x-tex">A_j - A_i = d log g_{ij}</annotation></semantics></math>, then did away with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>-patch (considering a kind of limit as we encircle the half axis) and concluded that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> must be the log-differential of 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>-valued function, whose winding number around the half-axis he identified with the magnetic charge, which in terms of bundles one identifies with the Chern-class of the bundle in question …</p> <blockquote> <p>…have to run…</p> </blockquote> <p>In modern terms:</p> <p>The <a class="existingWikiWord" href="/nlab/show/clutching+construction">clutching construction</a> gives <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>-principal bundle ob <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math> by covering with two hemispheres <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">U_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">U_1</annotation></semantics></math> and picking a transition function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g \colon S^1 \to U(1)</annotation></semantics></math> on the overlap <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>0</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>≃</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_0 \cap U_1 \simeq S^1\times (0,\epsilon)</annotation></semantics></math>. The integral <a class="existingWikiWord" href="/nlab/show/winding+number">winding number</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">g</annotation></semantics></math> represents the <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> of the line bundle.</p> <p>By <a href="connection+on+a+bundle#ExistenceOfConnections">the standard formula</a> for existence of principal connections on given principal bundles, given a choice of <a class="existingWikiWord" href="/nlab/show/partition+of+unity">partition of unity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>ρ</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\rho_i\}</annotation></semantics></math> then the connection on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">U_0</annotation></semantics></math> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>0</mn></msub><mo>=</mo><msub><mi>ρ</mi> <mn>1</mn></msub><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi><mi>g</mi></mrow><annotation encoding="application/x-tex"> A_0 = \rho_1 \mathbf{d} log g </annotation></semantics></math></div> <p>If we think (as we may) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>0</mn></msub><mo>=</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mo>*</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U_0 = S^2 - \{\ast\}</annotation></semantics></math> as covering most of the 2-sphere except one point and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">U_1</annotation></semantics></math> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>-open neighbourhood of that point, then this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">A_0</annotation></semantics></math> vanishes on most of the sphere and close to the point taken out (“the Dirac string”) it becomes non-vanishing and equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>g</mi></mrow><annotation encoding="application/x-tex">g^{-1}\mathbf{d}g</annotation></semantics></math>.</p> <h3 id="electricmagnetic_charge_quantization">Electric-magnetic charge quantization</h3> <p>Suppose we had a <a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a> representing an electromagnetic field with net <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> given by some magnetic <a class="existingWikiWord" href="/nlab/show/current">current</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex">j_{mag}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>F</mi><mo>=</mo><msub><mi>j</mi> <mi>mag</mi></msub></mrow><annotation encoding="application/x-tex"> d F = j_{mag} </annotation></semantics></math></div> <p>that is supported in some compact spatial region <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mi>ℝ</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \times \mathbb{R} \subset X</annotation></semantics></math> with boundary sphere <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>U</mi><mo>≃</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\partial U \simeq S^2</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>=</mo><msub><mo>∫</mo> <mi>U</mi></msub><msub><mi>j</mi> <mi>mag</mi></msub><mo>=</mo><msub><mo>∫</mo> <mi>U</mi></msub><mi>dF</mi><mo>=</mo><msub><mo>∫</mo> <mrow><mo>∂</mo><mi>U</mi></mrow></msub><mi>F</mi><mo>=</mo><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow></msub><mi>F</mi></mrow><annotation encoding="application/x-tex"> q = \int_U j_{mag} = \int_U dF =\int_{\partial U} F = \int_{S^2} F </annotation></semantics></math></div> <p>by the <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a>. It follows from the fact that <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 the curvature 2-form on a circle bundle that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> is integral: it is given by the first <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> of the bundle.</p> <p>(…)</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma : S^1 \to X</annotation></semantics></math> a closed but contractible trajectory of an electrically charged particle, the action functional is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>e</mi><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><msup><mi>γ</mi> <mo>*</mo></msup><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>e</mi><msub><mo>∫</mo> <mrow><msup><mi>D</mi> <mn>2</mn></msup></mrow></msub><msup><mi>γ</mi> <mo>*</mo></msup><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \exp(i e \int_{S^1} \gamma^* A ) = \exp(i e \int_{D^2} \gamma^* F) </annotation></semantics></math></div> <p>by the <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a>, for some 2-disk cobounding the circle. If now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> approaches a constant path and the 2-disk is taken to wrap the 2-cycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">\partial U</annotation></semantics></math>, then this becomes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>e</mi><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow></msub><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>e</mi><mi>q</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 1 = \exp(i e \int_{S^2} F) = \exp(i e q) \,. </annotation></semantics></math></div> <p>Which implies that with the magnetic charge being quantized, also the electric charge is.</p> <p>(…)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/voltage">voltage</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization">Dirac charge quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dipole+moment">dipole moment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-Maxwell+theorem">Hodge-Maxwell theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+electrodynamics">quantum electrodynamics</a>, <a class="existingWikiWord" href="/nlab/show/Lamb+shift">Lamb shift</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Born-Infeld+theory">Born-Infeld theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B-field">B-field</a>, <a class="existingWikiWord" href="/nlab/show/C-field">C-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+origin+of+inhomogeneous+media">geometric origin of inhomogeneous media</a></p> </li> </ul> <h2 id="references">References</h2> <blockquote> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/electromagnetism">electromagnetism</a></em>:</p> </blockquote> <p><a class="existingWikiWord" href="/nlab/show/Maxwell%27s+equations">Maxwell's equations</a> originate in:</p> <ul id="Maxwell"> <li><a class="existingWikiWord" href="/nlab/show/James+Clerk+Maxwell">James Clerk Maxwell</a>, <em><a href="http://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field">A Dynamical Theory of the Electromagnetic Field</a>,</em> Philosophical Transactions of the Royal Society of London 155, 459–512 (1865).</li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Dirac%27s+charge+quantization">Dirac's charge quantization</a> argument appeared in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/P.A.M.+Dirac">P.A.M. Dirac</a>, <em>Quantized Singularities in the Electromagnetic Field</em>, Proceedings of the Royal Society, A133 (1931) pp 60–72.</li> </ul> <p>Review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theodore+Frankel">Theodore Frankel</a>, section 5.5 of: <em><a class="existingWikiWord" href="/nlab/show/The+Geometry+of+Physics+-+An+Introduction">The Geometry of Physics - An Introduction</a></em>, Cambridge University Press (1997, 2004, 2012) [<a href="https://doi.org/10.1017/CBO9781139061377">doi:10.1017/CBO9781139061377</a>, <a href="http://www.math.ucsd.edu/~tfrankel/">webpage</a>]</li> </ul> <p>Discussions of the basic geometry behind Maxwell equations can be found in</p> <ul> <li>Hong-Mo Chan, Sheung Tsun Tsou, <em>Some elementary gauge theory concepts</em>, <a href="http://books.google.hr/books?id=BdB3SqCq33wC">gBooks</a></li> <li>Gregory L. Naber, <em>Topology, geometry, and gauge fields: interactions</em></li> <li><a href="http://eceformsweb.groups.et.byu.net/forms-home.html">Differential Forms in Electromagnetic Theory</a></li> </ul> <p>For undergraduate lectures including experimental material see</p> <ul> <li>Walter Lewin, Electricity and magnetism, 2002, <a href="http://ocw.mit.edu/courses/physics/8-02-electricity-and-magnetism-spring-2002">MIT opencourseware</a></li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz</a>-<a class="existingWikiWord" href="/nlab/show/invariants">invariants</a> of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>:</p> <ul> <li>C. A. Escobar, L. F. Urrutia, <em>The invariants of the electromagnetic field</em>, Journal of Mathematical Physics 55, 032902 (2014) (<a href="https://arxiv.org/abs/1309.4185">arXiv:1309.4185</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 27, 2024 at 12:04:24. 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