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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> multisymplectic geometry </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/790/#Item_7" 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" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="symplectic_geometry">Symplectic geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/almost+symplectic+structure">almost symplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metaplectic+structure">metaplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metalinear+structure">metalinear structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>, <a class="existingWikiWord" href="/nlab/show/n-plectic+form">n-plectic form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> <p><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic infinity-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a>, <a class="existingWikiWord" href="/nlab/show/symplectomorphism+group">symplectomorphism group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+vector+field">symplectic vector field</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+form">Hamiltonian form</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+gradient">symplectic gradient</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+action">Hamiltonian action</a>, <a class="existingWikiWord" href="/nlab/show/moment+map">moment map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+reduction">symplectic reduction</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+formalism">BRST-BV formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isotropic+submanifold">isotropic submanifold</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>, <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></p> </li> </ul> <h2 id="classical_mechanics_and_quantization">Classical mechanics and quantization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></p> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a>,</p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></strong>, <a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+manifold">contact manifold</a>, <a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+form">contact form</a>, <a class="existingWikiWord" href="/nlab/show/Reeb+vector+field">Reeb vector field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a>, <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebra">Poisson bracket Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+n-algebra">Heisenberg Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a></p> </li> </ul> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/symplectic+geometry+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <ul> <li><a href='#traditional'>Traditional</a></li> <li><a href='#from_the_pov'>From the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>POV</a></li> </ul> <li><a href='#extendedphasespace'>Extended phase spaces in covariant field theory</a></li> <ul> <li><a href='#covariant_configuration_bundle'>Covariant configuration bundle</a></li> <li><a href='#covariant_phase_space'>Covariant phase space</a></li> <li><a href='#DonderWeylHamiltonFieldEquations'>De Donder-Weyl-Hamilton field equations</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#FreeFieldTheory'>Free field theory</a></li> <li><a href='#bosonic_particle_propagating_on_a_manifold'>Bosonic particle propagating on a manifold</a></li> <li><a href='#electromagnetism'>Electromagnetism</a></li> <li><a href='#bosonic_string_propagating_on_a_manifold'>Bosonic string propagating on a manifold</a></li> </ul> </ul> <li><a href='#hamiltonian_dimensional_flow'>Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional flow</a></li> <li><a href='#relation_to_symplectic_manifolds'>Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-symplectic manifolds</a></li> <li><a href='#SurveyDevelopments'>Survey of developments in the field</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#on_classical_multisymplectic_geometry'>On classical multisymplectic geometry</a></li> <li><a href='#relation_to_covariant_phase_space_formalism'>Relation to covariant phase space formalism</a></li> <li><a href='#RefsonQuantization'>On quantization of multisymplectic geometry</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <h3 id="traditional">Traditional</h3> <p>Multisymplectic geometry is a generalization of <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> in the context of <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a> and <a class="existingWikiWord" href="/nlab/show/mechanical+systems">mechanical systems</a> in which the <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> is generalized from a closed 2-form to a closed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>-form, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math> – the <a class="existingWikiWord" href="/nlab/show/n-plectic+form">n-plectic form</a>.</p> <p>It is closely related to the <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a> of <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a>. In the context of <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> it is meant to provide a refinement of <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> which is well-adapted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. However, details of the multisymplectic quantization procedure remain under investigation.</p> <h3 id="from_the_pov">From the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>POV</h3> <p>We comment a bit on how to, presumably, think of multisymplectic geometry from the <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a>, in the context of <a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a>. Readers may want to skip ahead to traditional technical discussion at <em><a href="#extendedphasespace">Extended phase space</a></em>.</p> <div class="standout"> <p>Multisymplectic geometry is (or should be) to <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> as <a href="http://ncatlab.org/nlab/show/FQFT#Extended">extended quantum field theory</a> is to non-extended <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>:</p> <p>in the multisymplectic <strong>extended phase space</strong> of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/classical+field+theory">field theory</a> a state is not just a point, but an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional subspace.</p> </div> <p>See also <a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">n-plectic geometry</a>.</p> <p><strong>Multisymplectic geometry</strong> is a generalization of <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> to cases where the symplectic 2-form is generalized to a higher degree <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a>.</p> <p>In as far as <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> encodes <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a>, multisymplectic geometry may be regarded as resolving the symplectic geometry of the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> of <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a>: the kinematics of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional field theory may be encoded in an degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math> symplectic form.</p> <p>In this application to physics, multisymplectic geometry is also known as the <strong>covariant symplectic approach</strong> to field theory (e.g. <a href="http://arxiv.org/PS_cache/physics/pdf/9801/9801019v2.pdf#page=18">section 2 here</a>).</p> <p>The idea is that under a suitable fiber integration multisymplectic geometry becomes ordinary symplectic form on the ordinary phase space of the theory, similar to, and in fact as a special case of, how for instance a <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> on a <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> with a 2-form <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> may arise by <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> from a <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a> down on the original space, with a 3-form class.</p> <p>By effectively undoing this implicit transgression in the ordinary <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> of <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a>, multisymplectic geometry provides a general framework for a geometric, covariant formulation of <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a>, where <em>covariant formulation</em> means that spacelike and timelike directions on a given space-time be treated on equal footing.</p> <h2 id="extendedphasespace">Extended phase spaces in covariant field theory</h2> <p>We discuss here the refinement in multisymplectic geometry of the <a class="existingWikiWord" href="/nlab/show/covariant+phase+spaces">covariant phase spaces</a> of <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a>/<a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a> from (<a class="existingWikiWord" href="/nlab/show/presymplectic+manifold">pre</a>-)<a class="existingWikiWord" href="/nlab/show/symplectic+manifolds">symplectic manifolds</a> of initial value data in a <a class="existingWikiWord" href="/nlab/show/Cauchy+surface">Cauchy surface</a> to multisymplectic manifolds of local initial value data.</p> <p>Recall that an ordinary <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> of a <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a> is a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> whose points correspond to the <em><a class="existingWikiWord" href="/nlab/show/states">states</a></em> of the system. The <strong>extended phase space</strong> of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional quantum field theory is a multisymplectic space whose points correspond to pairs consisting of</p> <ul> <li> <p>a point in the field theory’s parameter space – an “event”;</p> </li> <li> <p>a state of the theory “at that event”.</p> </li> </ul> <p>So <strong>extended phase spaces <em>localizes</em> the information about states</strong> : a point in here encodes not just the entire state of the system, but remembers explicitly what that state is like over any point in parameter space.</p> <h3 id="covariant_configuration_bundle">Covariant configuration bundle</h3> <p>Consider <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a> over a <strong>parameter space</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>. From the point of view of <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> will be one fixed <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> on which we want to understand the (classical) field theory.</p> <p>We assume that a <strong><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field configuration</a></strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/section">section</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>Σ</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\phi : \Sigma \to E</annotation></semantics></math> of some prescribed <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \to \Sigma</annotation></semantics></math>: the <em><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></em>.</p> <div class="num_example"> <h6 id="example">Example</h6> <p>For instance an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> is one whose field configurations on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> are given by maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>Σ</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \phi : \Sigma \longrightarrow X </annotation></semantics></math></div> <p>into some prescribed <strong><a class="existingWikiWord" href="/nlab/show/target+space">target space</a></strong> <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>. This is the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>Σ</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E = \Sigma \times X</annotation></semantics></math> is a <strong>trivial <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a></strong>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Beware of the standard source of confusion here when correlating this formalism with actual physics: the physical <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> that we inhabit may be given either by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> or 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>:</p> <ul> <li> <p>in the description of the <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> of objects propagating <em>in</em> our physical <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, subject to <a class="existingWikiWord" href="/nlab/show/forces">forces</a> exerted by fixed <a class="existingWikiWord" href="/nlab/show/background+gauge+fields">background gauge fields</a> (such as electrons propagating in our particle accelerator, subject to the electromagnetic field in the accelerator tube), physical spacetime is identified with target 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>, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is the <strong>worldvolume</strong> of the object that propagates through <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>. The <em>field configurations</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> are really the maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math> that determine how the object sits in spacetime.</p> </li> <li> <p>in quantum mechanics of fields on spacetime, such as the quantized electromagnetic field in a laser, it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> which represents physical spacetime, and <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 some abstract space, for instance a smooth version of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{B}U(1)</annotation></semantics></math>, so that a field configuration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math> encodes a <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">with connection</a> that encodes a configuration of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>.</p> </li> </ul> </div> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>The <strong>configuration space</strong> of the system is the space of all field configurations, hence the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_\Sigma(E)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of the <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <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>.</p> </div> <p>In the <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> example this is some incarnation of the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math>.</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Beware that in low dimensions one often distinguishes between the space of <em>configurations</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math> and that of <em>trajectories</em> or <em>histories</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>×</mo><mi>ℝ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \times \mathbb{R} \to X</annotation></semantics></math>. This comes from the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\Sigma = *</annotation></semantics></math> where for a particle propagating 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> the maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>≃</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">[*,X] \simeq X</annotation></semantics></math> are the possible configurations of the particle at a given parameter times, while maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>*</mo><mo>×</mo><mi>ℝ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>ℝ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[* \times \mathbb{R}, X] = [\mathbb{R}, X]</annotation></semantics></math> are the trajectories. But for the higher dimensional and <a class="existingWikiWord" href="/nlab/show/FQFT">extended</a> field theories under discussion here, this distinction becomes a bit obsolete and trajectories become just a special case of configurations.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>In the <strong>non-covariant</strong> approach one would try to consider the a <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> of the configuration space <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></mrow><annotation encoding="application/x-tex">\Gamma(E)</annotation></semantics></math> as <em>phase space</em> . Contrary to that, in the <strong>covariant approach</strong> one considers the much smaller space <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> instead. This is then called the <strong>covariant configuration space</strong> or <strong>covariant configuration bundle</strong>.</p> </div> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">J^1 E \to \Sigma</annotation></semantics></math> for the first order <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> of the configuration space bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \to \Sigma</annotation></semantics></math>. Its <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">s \in \Sigma</annotation></semantics></math> are equivalence classses of <a class="existingWikiWord" href="/nlab/show/germs">germs</a> of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> at <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>, where two germs are identified if their first derivatives coincide.</p> </div> <h3 id="covariant_phase_space">Covariant phase space</h3> <div class="num_defn" id="DualFirstJetBundle"> <h6 id="definition_3">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \to \Sigma</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim(\Sigma) = n+1</annotation></semantics></math>, the <strong>affine dual first jet bundle</strong> (or often just <strong>dual first jet bundle</strong> for short) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>J</mi> <mn>1</mn></msub><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">(J_1 E)^\ast \to \Sigma</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> whose <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>∈</mo><msub><mi>E</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">e \in E_s</annotation></semantics></math> is the set of <a class="existingWikiWord" href="/nlab/show/affine+maps">affine maps</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>e</mi> <mn>1</mn></msubsup><mi>E</mi><mo>⟶</mo><msubsup><mi>Λ</mi> <mi>s</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mi>Σ</mi></mrow><annotation encoding="application/x-tex"> J_e^1 E \longrightarrow \Lambda_s^{n+1} \Sigma </annotation></semantics></math></div> <p>from the first <a class="existingWikiWord" href="/nlab/show/jet+bundle">jets</a> at <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> to the degree-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> at <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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>.</p> </div> <div class="num_defn" id="ExtendedCovariantPhaseSpace"> <h6 id="definition_4">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \to \Sigma</annotation></semantics></math>, the <strong>extended covariant phase space</strong> is the multisymplectic manifold</p> <ul> <li> <p>whose underlying manifold is the dual first jet bundle, def. <a class="maruku-ref" href="#DualFirstJetBundle"></a>, of the field bundle</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><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>→</mo><mi>E</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (J^1 E)^* \to E \,, </annotation></semantics></math></div></li> <li> <p>equipped with the canonical degree-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+2)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mi>d</mi><mi>α</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \omega = d \alpha \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> is the canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-form</p> </li> </ul> </div> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\pi \colon E \to \Sigma</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">dim</mi><mi>Σ</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mathrm{dim} \Sigma =n+1</annotation></semantics></math>, the dual jet bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(J^1 E)^*</annotation></semantics></math> is isomorphic to a particular vector sub-bundle of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>-form bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>T</mi> <mo>*</mo></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">\Lambda^{n+1}T^{*}E</annotation></semantics></math>. To see this, first consider the following</p> <div class="num_defn" id="nHorizontalForms"> <h6 id="definition_5">Definition</h6> <p>Given a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">y \in E</annotation></semantics></math>, a tangent vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><msub><mi>T</mi> <mi>y</mi></msub><mi>E</mi></mrow><annotation encoding="application/x-tex">v \in T_{y} E</annotation></semantics></math> is said to be <strong><a class="existingWikiWord" href="/nlab/show/vertical+vector+field">vertical</a></strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>π</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d \pi(v) = 0</annotation></semantics></math>. Define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mn>1</mn> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msup><mi>T</mi> <mo>*</mo></msup><mi>E</mi><mo>↪</mo><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>T</mi> <mo>*</mo></msup><mi>E</mi></mrow><annotation encoding="application/x-tex"> \Lambda^{n+1}_{1}T^{\ast}E \hookrightarrow \Lambda^{n+1} T^{\ast} E </annotation></semantics></math></div> <p>to be the subbundle of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>-form bundle whose fiber at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">y \in E</annotation></semantics></math> consists of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>∈</mo><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msubsup><mi>T</mi> <mi>y</mi> <mo>*</mo></msubsup><mi>E</mi></mrow><annotation encoding="application/x-tex">\beta \in \Lambda^{n+1} T^{*}_{y} E</annotation></semantics></math> 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>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow></msub><mi>β</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \iota_{v_1}\iota_{v_2} \beta =0 </annotation></semantics></math></div> <p>for all <a class="existingWikiWord" href="/nlab/show/vertical+vector+field">vertical vectors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>T</mi> <mi>y</mi></msub><mi>E</mi></mrow><annotation encoding="application/x-tex">v_1,v_2 \in T_{y}E</annotation></semantics></math>. Sections of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mn>1</mn> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msup><mi>T</mi> <mo>*</mo></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">\Lambda^{n+1}_{1}T^{*}E</annotation></semantics></math> are called <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-horizontal</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{n+1}</annotation></semantics></math><strong>-forms</strong>.</p> </div> <p>In words, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-horizontal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-form is one which has at most one “leg” not along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>. This is made very explicit in the proof of the following proposition.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \to \Sigma</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> over a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>Σ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">dim \Sigma = (n+1)</annotation></semantics></math> and assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/orientation">orientable</a>, then there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></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><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>≃</mo><msubsup><mi>Λ</mi> <mn>1</mn> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msup><mi>T</mi> <mo>*</mo></msup><mi>E</mi></mrow><annotation encoding="application/x-tex"> (J^1 E)^{\ast} \simeq \Lambda^{n+1}_{1} T^{\ast}E </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> between the dual first jet bundle 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>, def. <a class="maruku-ref" href="#DualFirstJetBundle"></a>, and the bundle of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-horizontal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-forms on <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>, def. <a class="maruku-ref" href="#nHorizontalForms"></a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>It suffices to work locally with respect to a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a>, so we reduce the statement to the special case of the <a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a> i.e. the trivial bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>Σ</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E = \Sigma \times X</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>. By the assumoption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> admits an <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> we may pick a <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">vol</mo><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>T</mi> <mo>*</mo></msup><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\vol \in \Gamma(\Lambda^{n+1}T^\ast \Sigma)</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>q</mi> <mn>1</mn></msup><mo>,</mo><mi>…</mi><mo>,</mo><msup><mi>q</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">q^1, \dots, q^{n+1}</annotation></semantics></math> be local <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mn>1</mn></msup><mo>,</mo><mi>…</mi><mo>,</mo><msup><mi>u</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">u^1, \dots , u^d</annotation></semantics></math> be local coordinates 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>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mn>1</mn> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msup><mi>T</mi> <mo>*</mo></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">\Lambda_1^{n+1} T^* E</annotation></semantics></math> has a local <a class="existingWikiWord" href="/nlab/show/basis">basis</a> of sections given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-forms of two types: first, the wedge product of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/cotangent+vectors">cotangent vectors</a> of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{d}q^i</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">vol</mo><mo>=</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mn>1</mn></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \vol = \mathbf{d}q^1 \wedge \cdots \wedge \mathbf{d}q^{n+1} </annotation></semantics></math></div> <p>and second, wedge products of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> cotangent vectors of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{d}q^i</annotation></semantics></math> and a single one of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>u</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{d}u^a</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mn>1</mn></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>u</mi> <mi>a</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{d}q^1 \wedge \cdots \wedge \mathbf{d} q^{i-1} \wedge \mathbf{d}q^{i+1} \wedge \cdots \wedge \mathbf{d}q^{n+1} \wedge \mathbf{d}u^a \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Σ</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">y = (p,u) \in \Sigma \times X</annotation></semantics></math>, this basis gives an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mn>1</mn> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msubsup><mi>T</mi> <mi>y</mi> <mo>*</mo></msubsup><mi>E</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msubsup><mi>T</mi> <mi>p</mi> <mo>*</mo></msubsup><mi>Σ</mi><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⊕</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msup><mi>Λ</mi> <mi>n</mi></msup><msubsup><mi>T</mi> <mi>p</mi> <mo>*</mo></msubsup><mi>Σ</mi><mo>⊗</mo><msubsup><mi>T</mi> <mi>u</mi> <mo>*</mo></msubsup><mi>X</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Lambda^{n+1}_1 T^*_y E \;\; \simeq\;\; \left(\Lambda^{n+1} T^*_p \Sigma \right) \; \oplus \; \left(\Lambda^{n} T^*_p \Sigma \otimes T^*_u X \right) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> also determines isomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msubsup><mi>T</mi> <mi>p</mi> <mo>*</mo></msubsup><mi>Σ</mi></mrow><annotation encoding="application/x-tex"> \mathbb{R} \overset{\simeq}{\longrightarrow} \Lambda^{n+1} T^*_p \Sigma </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>c</mi><mo>↦</mo><mi>c</mi><mspace width="thinmathspace"></mspace><msub><mo lspace="0em" rspace="thinmathspace">vol</mo> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex"> c \mapsto c \, \vol_p </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>p</mi></msub><mi>Σ</mi><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>Λ</mi> <mi>n</mi></msup><msubsup><mi>T</mi> <mi>p</mi> <mo>*</mo></msubsup><mi>Σ</mi></mrow><annotation encoding="application/x-tex"> T_p \Sigma \overset{\simeq}{\longrightarrow} \Lambda^{n} T^*_p \Sigma </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>v</mi><mo>↦</mo><msub><mi>ι</mi> <mi>v</mi></msub><msub><mo lspace="0em" rspace="thinmathspace">vol</mo> <mi>p</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> v \mapsto \iota_v \vol_p \,. </annotation></semantics></math></div> <p>We thus have obtained an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mn>1</mn> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msubsup><mi>T</mi> <mi>y</mi> <mo>*</mo></msubsup><mi>E</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≅</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ℝ</mi><mspace width="thickmathspace"></mspace><mo>⊕</mo><mspace width="thickmathspace"></mspace><msub><mi>T</mi> <mi>p</mi></msub><mi>Σ</mi><mo>⊗</mo><msubsup><mi>T</mi> <mi>u</mi> <mo>*</mo></msubsup><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Lambda^{n+1}_1 T^*_y E \;\; \cong \;\; \mathbb{R} \; \oplus \; T_p \Sigma \otimes T^*_u X \,. </annotation></semantics></math></div> <p>On the other hand, the trivialization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>Σ</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E = \Sigma \times X</annotation></semantics></math> gives an isomorphism of <a class="existingWikiWord" href="/nlab/show/affine+spaces">affine spaces</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>y</mi> <mn>1</mn></msubsup><mi>E</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≅</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mi>T</mi> <mi>p</mi> <mo>*</mo></msubsup><mi>Σ</mi><mo>⊗</mo><msub><mi>T</mi> <mi>u</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex"> J^1_y E \; \; \cong \; \; T^*_p \Sigma \otimes T_u X </annotation></semantics></math></div> <p>which has the side-effect of exhibiting on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>y</mi> <mn>1</mn></msubsup><mi>E</mi></mrow><annotation encoding="application/x-tex">J^1_y E</annotation></semantics></math> the structure of a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>. Since we’ve identified</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msubsup><mi>T</mi> <mi>p</mi> <mo>*</mo></msubsup><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Lambda^{n+1} T^*_p \Sigma</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>, an affine map from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>y</mi> <mn>1</mn></msubsup><mi>E</mi></mrow><annotation encoding="application/x-tex">J^1_y E</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msubsup><mi>T</mi> <mi>p</mi> <mo>*</mo></msubsup><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Lambda^{n+1} T^*_p \Sigma</annotation></semantics></math> is just an element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>x</mi></msub><mi>Σ</mi><mo>⊗</mo><msubsup><mi>T</mi> <mi>u</mi> <mo>*</mo></msubsup><mi>X</mi></mrow><annotation encoding="application/x-tex">T_x \Sigma \otimes T^*_u X</annotation></semantics></math> plus a constant. So, we obtain</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><msubsup><mi>J</mi> <mi>y</mi> <mn>1</mn></msubsup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≅</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ℝ</mi><mspace width="thickmathspace"></mspace><mo>⊕</mo><mspace width="thickmathspace"></mspace><msub><mi>T</mi> <mi>p</mi></msub><mi>Σ</mi><mo>⊗</mo><msubsup><mi>T</mi> <mi>u</mi> <mo>*</mo></msubsup><mi>X</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> (J^1_y E)^* \; \; \cong \; \; \mathbb{R} \; \oplus \; T_p \Sigma \otimes T^*_u X . </annotation></semantics></math></div> <p>This gives a specific vector bundle isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>≅</mo><msubsup><mi>Λ</mi> <mn>1</mn> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msup><mi>T</mi> <mo>*</mo></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">(J^1 E)^* \cong \Lambda_1^{n+1} T^* E</annotation></semantics></math>, as desired.</p> </div> <div class="num_remark" id="CanonicalFormInGoodCoordinates"> <h6 id="remark_4">Remark</h6> <p>In practice it is better to use the <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pulled back</a> <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>π</mi> <mo>*</mo></msup><mo lspace="0em" rspace="thinmathspace">vol</mo></mrow><annotation encoding="application/x-tex">\pi^* \vol</annotation></semantics></math> as a substitute for the coordinate-dependent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mn>1</mn></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{d}q^1 \wedge \cdots \wedge \mathbf{d}q^{n+1}</annotation></semantics></math> on <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>. This gives another <a class="existingWikiWord" href="/nlab/show/basis">basis</a> of sections of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mn>1</mn> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msup><mi>T</mi> <mo>*</mo></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">\Lambda_1^{n+1} T^* E</annotation></semantics></math>, whose elements we write suggestively</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>dQ</mi><mo>≔</mo><msup><mi>π</mi> <mo>*</mo></msup><mo lspace="0em" rspace="thinmathspace">vol</mo></mrow><annotation encoding="application/x-tex"> dQ \coloneqq \pi^* \vol </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>dQ</mi> <mi>i</mi> <mi>a</mi></msubsup><mo>≔</mo><mrow><mo>(</mo><msup><mi>π</mi> <mo>*</mo></msup><msub><mi>ι</mi> <mrow><mo>∂</mo><mo stretchy="false">/</mo><mo>∂</mo><msup><mi>q</mi> <mi>i</mi></msup></mrow></msub><mo lspace="0em" rspace="thinmathspace">vol</mo><mo>)</mo></mrow><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>u</mi> <mi>a</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> dQ_i^a \coloneqq \left(\pi^* \iota_{\partial/\partial q^i} \vol\right) \wedge \mathbf{d}u^a \,. </annotation></semantics></math></div> <p>Corresponding to this <a class="existingWikiWord" href="/nlab/show/basis">basis</a> then there are local <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>P</mi> <mi>a</mi> <mi>i</mi></msubsup></mrow><annotation encoding="application/x-tex">P^i_a</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mn>1</mn> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msup><mi>T</mi> <mo>*</mo></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">\Lambda_1^{n+1} T^* E</annotation></semantics></math>, which combined with the coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>q</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">q^i</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex">u^a</annotation></semantics></math> pulled back from <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> give a local coordinate system on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mn>1</mn> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msup><mi>T</mi> <mo>*</mo></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">\Lambda_1^{n+1} T^* E</annotation></semantics></math>.</p> <p>In these <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> the canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>-form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>≅</mo><msubsup><mi>Λ</mi> <mn>1</mn> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msup><mi>T</mi> <mo>*</mo></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">(J^1 E)^* \cong \Lambda_1^{n+1} T^* E</annotation></semantics></math> is:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>=</mo><mi>P</mi><mo>∧</mo><mi>dQ</mi><mo>+</mo><msubsup><mi>P</mi> <mi>a</mi> <mi>i</mi></msubsup><mo>∧</mo><msubsup><mi>dQ</mi> <mi>i</mi> <mi>a</mi></msubsup><mo>,</mo></mrow><annotation encoding="application/x-tex"> \alpha = P \wedge dQ + P^i_a \wedge dQ_i^a, </annotation></semantics></math></div> <p>and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n+2</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/multisymplectic+form">multisymplectic form</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>P</mi><mo>∧</mo><mi>dQ</mi><mo>+</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msubsup><mi>P</mi> <mi>a</mi> <mi>i</mi></msubsup><mo>∧</mo><msubsup><mi>dQ</mi> <mi>i</mi> <mi>a</mi></msubsup><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega = \mathbf{d}P \wedge dQ + \mathbf{d}P^i_a \wedge dQ_i^a. </annotation></semantics></math></div></div> <h3 id="DonderWeylHamiltonFieldEquations">De Donder-Weyl-Hamilton field equations</h3> <p>We discuss the <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations+of+motion">Euler-Lagrange equations of motion</a> of a <a class="existingWikiWord" href="/nlab/show/local+field+theory">local field theory</a> expressed in multisymplectic geoemtry via <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>.</p> <div class="num_defn" id="LocalLagrangian"> <h6 id="definition_6">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \to \Sigma</annotation></semantics></math> as above, a (first order) <a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a> is a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><mo>⟶</mo><msup><mo>∧</mo> <mi>n</mi></msup><msup><mi>T</mi> <mo>*</mo></msup><mi>Σ</mi></mrow><annotation encoding="application/x-tex"> \mathbf{L} \;\colon\; J^1 E \longrightarrow \wedge^n T^\ast \Sigma </annotation></semantics></math></div> <p>on the first <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> 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> with values in <a class="existingWikiWord" href="/nlab/show/densities">densities</a>/<a class="existingWikiWord" href="/nlab/show/volume+forms">volume forms</a>. Equivalently this is a degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,0)</annotation></semantics></math>-form on the jet bundle, in terms of <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a> grading.</p> </div> <div class="num_defn" id="LegendreTransform"> <h6 id="definition_7">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a>, def. <a class="maruku-ref" href="#LocalLagrangian"></a>, its <strong>local <a class="existingWikiWord" href="/nlab/show/Legendre+transform">Legendre transform</a></strong> is the <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔽</mi><mstyle mathvariant="bold"><mi>L</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><mo>⟶</mo><mo stretchy="false">(</mo><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> \mathbb{F}\mathbf{L} \;\colon\; J^1 E \longrightarrow (J^1 E)^\ast </annotation></semantics></math></div> <p>from first <a class="existingWikiWord" href="/nlab/show/jets">jets</a> to the affine dual jet bundle, def. <a class="maruku-ref" href="#DualFirstJetBundle"></a>, which sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{L}</annotation></semantics></math> to its first-order <a class="existingWikiWord" href="/nlab/show/Taylor+series">Taylor series</a>.</p> </div> <p>This definition was proposed in (<a href="#ForgerRomero04">Forger-Romero 04, section 2.5</a>).</p> <div class="num_prop" id="LegendreTransformInLocalCoordinates"> <h6 id="proposition_2">Proposition</h6> <p>In terms of the local <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> of remark <a class="maruku-ref" href="#CanonicalFormInGoodCoordinates"></a> the Legendre transform of def. <a class="maruku-ref" href="#LegendreTransform"></a> is the function with <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>P</mi> <mi>a</mi> <mi>i</mi></msubsup><mo>=</mo><mfrac><mrow><mo>∂</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><mrow><mo>∂</mo><msubsup><mi>q</mi> <mrow><mo>,</mo><mi>i</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac></mrow><annotation encoding="application/x-tex"> P^i_a = \frac{\partial \mathbf{L}}{\partial q^a_{, i}} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>=</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>−</mo><mfrac><mrow><mo>∂</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><mrow><mo>∂</mo><msubsup><mi>q</mi> <mrow><mo>,</mo><mi>i</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><msubsup><mi>q</mi> <mrow><mo>,</mo><mi>i</mi></mrow> <mi>a</mi></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> P = \mathbf{L} - \frac{\partial \mathbf{L}}{\partial q^a_{,i}}q^a_{,i} \,. </annotation></semantics></math></div></div> <p>(<a href="#ForgerRomero04">Forger-Romero 04, section 2.5 (41)</a>).</p> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>The second term in prop. <a class="maruku-ref" href="#LegendreTransformInLocalCoordinates"></a> is what is traditionally called the Legendre transform in multisymplectic geometry/<a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>. Def. <a class="maruku-ref" href="#LegendreTransform"></a> may be regarded as explaining the conceptual role of this expression, in particular in view of the following proposition.</p> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{L}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mstyle mathvariant="bold"><mi>L</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">\omega_{\mathbf{L}}</annotation></semantics></math> of the canonical <a class="existingWikiWord" href="/nlab/show/n-plectic+form">pre-n-plectic form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, def. <a class="maruku-ref" href="#ExtendedCovariantPhaseSpace"></a>, along the <a class="existingWikiWord" href="/nlab/show/Legendre+transform">Legendre transform</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔽</mi><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbb{F}\mathbf{L}</annotation></semantics></math>, def. <a class="maruku-ref" href="#LegendreTransform"></a>, to the first <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> is the sum of the <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>EL</mi> <mstyle mathvariant="bold"><mi>L</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">EL_{\mathbf{L}}</annotation></semantics></math> and the canonical symplectic form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mi>v</mi></msub><msub><mi>θ</mi> <mstyle mathvariant="bold"><mi>L</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">\mathbf{d}_v \theta_{\mathbf{L}}</annotation></semantics></math> from <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> formalism:</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>ω</mi> <mstyle mathvariant="bold"><mi>L</mi></mstyle></msub></mtd> <mtd><mo>≔</mo><mi>𝔽</mi><msup><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mo>*</mo></msup><mi>ω</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>EL</mi> <mstyle mathvariant="bold"><mi>L</mi></mstyle></msub><mo>+</mo><msub><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mi>v</mi></msub><msub><mi>θ</mi> <mstyle mathvariant="bold"><mi>L</mi></mstyle></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \omega_{\mathbf{L}} & \coloneqq \mathbb{F}\mathbf{L}^\ast \omega \\ & = EL_{\mathbf{L}} + \mathbf{d}_v \theta_{\mathbf{L}} \end{aligned} \,. </annotation></semantics></math></div> <p>It follows that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mi>n</mi></msub></mrow></msub><mi>⋯</mi><msub><mi>ι</mi> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">)</mo><msub><mi>ω</mi> <mstyle mathvariant="bold"><mi>L</mi></mstyle></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(\iota_{v_n} \cdots \iota_{v_1}) \omega_{\mathbf{L}} = 0</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation+of+motion">Euler-Lagrange equation of motion</a> in <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl-Hamilton</a>-form;</p> </li> <li> <p>for any <a class="existingWikiWord" href="/nlab/show/Cauchy+surface">Cauchy surface</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\Sigma_{n-1}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>Σ</mi></msub><mo>≔</mo><msub><mo>∫</mo> <mrow><msub><mi>Σ</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><msub><mi>ω</mi> <mstyle mathvariant="bold"><mi>L</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">\omega_\Sigma \coloneqq \int_{\Sigma_{n-1}}\omega_{\mathbf{L}}</annotation></semantics></math> is the canonical pre-symplectic form on <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> (as discussed there).</p> </li> </ol> </div> <p>This statement is essentially the content of (<a href="#ForgerRomero04">Forger-Romero 04, equation (54) and theorem 1</a>). In the above form in terms of <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a> notions this statement has been amplified by <a class="existingWikiWord" href="/nlab/show/Igor+Khavkine">Igor Khavkine</a>.</p> <h3 id="examples">Examples</h3> <h4 id="FreeFieldTheory">Free field theory</h4> <p>We write out the multisymplectic geometry corresponding to a <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>=</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>;</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma = (\mathbb{R}^{d-1;1}, \eta)</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>. Write the canonical <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>σ</mi> <mi>i</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Σ</mi><mo>⟶</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \sigma^i \;\colon\; \Sigma \longrightarrow \mathbb{R} \,. </annotation></semantics></math></div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a>. For simplicity of notation we assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≃</mo><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">X \simeq \mathbb{R}^k</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, too. Write its canonical coordinates as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mi>a</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi^a \;\colon\; X \longrightarrow \mathbb{R} \,. </annotation></semantics></math></div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Σ</mi><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">X \times \Sigma \to \Sigma</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a>. Its first <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> then has canonical coordinates</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><msup><mi>σ</mi> <mi>i</mi></msup><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msup><mi>ϕ</mi> <mi>a</mi></msup><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>i</mi></mrow> <mi>a</mi></msubsup><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>j</mi> <mn>∞</mn> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>Σ</mi><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{ \sigma^i \}, \{\phi^a\}, \{\phi^a_{,i}\} \;\colon\; j_\infty^1(\Sigma \times X) \longrightarrow X \,. </annotation></semantics></math></div> <div class="num_defn"> <h6 id="definition_8">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a> for <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a> with this <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>≔</mo><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><msub><mi>η</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>i</mi></mrow> <mi>a</mi></msubsup><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>j</mi></mrow> <mi>a</mi></msubsup><mo>)</mo></mrow><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>σ</mi> <mn>1</mn></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>σ</mi> <mi>d</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L \coloneqq \left( \frac{1}{2} g^{i j} \eta_{a b} \phi^a_{,i} \phi^a_{,j} \right) \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d \,. </annotation></semantics></math></div></div> <p>The <a class="existingWikiWord" href="/nlab/show/canonical+momenta">canonical momenta</a> are</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><msubsup><mi>p</mi> <mi>a</mi> <mi>i</mi></msubsup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>σ</mi> <mn>1</mn></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>σ</mi> <mi>d</mi></msup></mtd> <mtd><mo>≔</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><msubsup><mi>u</mi> <mi>i</mi> <mi>a</mi></msubsup></mrow></mfrac><mi>L</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><msup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><msub><mi>η</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>j</mi></mrow> <mi>a</mi></msubsup><mo>)</mo></mrow><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>σ</mi> <mn>1</mn></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>σ</mi> <mi>d</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} p_a^i \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d & \coloneqq \frac{\partial}{\partial u^a_i} L \\ & = \left( g^{i j} \eta_{a b} \phi^a_{,j} \right) \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d \end{aligned} </annotation></semantics></math></div> <p>So the boundary term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a>, (see <a href="covariant+phase+space#CanonicalThetaDensityInLocalCoordinates">this remark</a> at <em><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></em> ) is</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><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>u</mi> <mi>a</mi></msup><mo>∧</mo><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>i</mi></msub></mrow></msub><mrow><mo>(</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>i</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>L</mi><mo>)</mo></mrow></mtd> <mtd><mo>=</mo><msubsup><mi>p</mi> <mi>a</mi> <mi>i</mi></msubsup><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mrow><msup><mi>σ</mi> <mi>i</mi></msup></mrow></msub></mrow></msub><mi>vol</mi><mo stretchy="false">)</mo><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>u</mi> <mi>a</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msubsup><mi>p</mi> <mi>a</mi> <mi>i</mi></msubsup><mo>∧</mo><msubsup><mi>dq</mi> <mi>i</mi> <mi>a</mi></msubsup><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{d}u^a \wedge \iota_{\partial_i} \left( \frac{\partial}{\partial \phi^a_{,i}}L \right) & = p^i_a \wedge (\iota_{\partial_{\sigma^i}} vol) \wedge \mathbf{d}u^a \\ & = p^i_a \wedge dq_i^a \,, \end{aligned} </annotation></semantics></math></div> <p>where in the last line we adopted the notation of remark <a class="maruku-ref" href="#CanonicalFormInGoodCoordinates"></a>.</p> <p>This shows that the canonical <a class="existingWikiWord" href="/nlab/show/multisymplectic+form">multisymplectic form</a> is the “covariant symplectic potential current density” which is induced by the free field Lagrangian.</p> <p>See also (<a href="#ForgerRomero04">Forger Romero 04, section 3.2</a>).</p> <h4 id="bosonic_particle_propagating_on_a_manifold">Bosonic particle propagating on a manifold</h4> <blockquote> <p>scratch</p> </blockquote> <p>Ordinary point <a class="existingWikiWord" href="/nlab/show/particle">particle</a> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> on a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> involves <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R} \to X</annotation></semantics></math> in <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 parameter space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\Sigma = \mathbb{R}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, thought of as the abstract “<a class="existingWikiWord" href="/nlab/show/worldline">worldline</a>” of the particle.</p> <ul> <li> <p>parameter space: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\Sigma = \mathbb{R}</annotation></semantics></math>, the <strong>worldline</strong>;</p> </li> <li> <p>target 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>, some <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> – <strong>spacetime</strong>;</p> </li> <li> <p>configuration bundle: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>→</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>ℝ</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E \to \mathbb{R}) = (\mathbb{R}\times X \to \mathbb{R})</annotation></semantics></math>;</p> </li> <li> <p>jet bundle: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><mo>=</mo><mi>ℝ</mi><mo>×</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">J^1 E = \mathbb{R} \times T X</annotation></semantics></math> .</p> </li> </ul> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>E</mi><mo>=</mo><mi>ℝ</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset E = \mathbb{R} \times X</annotation></semantics></math> a local patch with coordinate functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>t</mi><mo>,</mo><msup><mi>q</mi> <mi>i</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{t, q^i\}</annotation></semantics></math>, there are canonically induced coordinates on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">J^1 E</annotation></semantics></math> written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>t</mi><mo>,</mo><msup><mi>q</mi> <mi>i</mi></msup><mo>,</mo><msup><mi>v</mi> <mi>i</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{t,q^i, v^i\}</annotation></semantics></math>.</p> <p>Here a collection of vaues <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>q</mi> <mn>0</mn> <mi>i</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(q^i_0)</annotation></semantics></math> is a <strong>position</strong> of the particle and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>v</mi> <mn>0</mn> <mi>i</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v^i_0)</annotation></semantics></math> is a <strong>velocity</strong> of the particle. Notice that in this covariant approach these are not positions and velocities “at a given time”. Rather, a point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">J^1 E</annotation></semantics></math> specified a parameter time and a corresponding position and velocity.</p> <p>….</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \to X</annotation></semantics></math> be a local patch 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> with canonical coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>x</mi> <mi>i</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x^i\}</annotation></semantics></math>.</p> <p>The canonical 2-form on the extended phase space in this case is traditionally locally written as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo>=</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>α</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo>=</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>i</mi></msub><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>q</mi> <mi>i</mi></msup><mo>+</mo><mi>H</mi><mo>∧</mo><mi>d</mi><mi>t</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega|_U = \mathbf{d} \alpha|_U = \mathbf{d}( p_i \wedge \mathbf{d} q^i + H \wedge d t ) \,. </annotation></semantics></math></div> <p>… blah-blah-blah…</p> <h4 id="electromagnetism">Electromagnetism</h4> <ul> <li>parameter space: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is spacetime;</li> </ul> <p>A field configuration of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> is a <a class="existingWikiWord" href="/nlab/show/line+bundle">line 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>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>. If we assume the corresponding bundle to be trivial, then this is just a <a class="existingWikiWord" href="/nlab/show/differential+form">1-form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>. So in this simplified case we can take</p> <ul> <li>configuration bundle: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mo>*</mo></msup><mi>Σ</mi></mrow><annotation encoding="application/x-tex">T^* \Sigma</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>.</li> </ul> <h4 id="bosonic_string_propagating_on_a_manifold">Bosonic string propagating on a manifold</h4> <ul> <li> <p>parameter space: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> some 2-dimensional manifold – the <strong>worldsheet</strong> – for instance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>=</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\Sigma = S^1 \times \mathbb{R}</annotation></semantics></math> models a closed string propagating without interaction.</p> </li> <li> <p>target 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> spacetime;</p> </li> <li> <p>covariant configuration bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>Σ</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E = \Sigma \times X</annotation></semantics></math>.</p> </li> </ul> <p>We will work out the covariant Hamiltonian formalism (also known as the <strong><a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a></strong>) for this example in detail. We follow here the exposition found in <a href="#Helein02">Hélein 02</a>.</p> <p>For simplicity we will only consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is the cylinder <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}\times S^1</annotation></semantics></math> and <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 <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>-dimensional Minkowski spacetime, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>1</mn><mo>,</mo><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{1,d-1}</annotation></semantics></math>. A solution of the classical bosonic string is then a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi : \Sigma \to X</annotation></semantics></math> which is a critical point of the area subject to certain boundary conditions.</p> <p>Equivalently, by exploiting symmetries in the variational problem, one can describe solutions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> by equipping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R} \times S^{1}</annotation></semantics></math> with its standard Minkowski metric and then solving the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1+1</annotation></semantics></math> dimensional field theory specified by the Lagrangian density</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>g</mi> <mi>ij</mi></msup><msub><mi>η</mi> <mi>ab</mi></msub><mfrac><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow><mrow><mo>∂</mo><msup><mi>q</mi> <mi>i</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>b</mi></msup></mrow><mrow><mo>∂</mo><msup><mi>q</mi> <mi>j</mi></msup></mrow></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{L}=\frac{1}{2} g^{ij}\eta_{ab} \frac{\partial \phi^{a}}{\partial q^i}\frac{\partial \phi^{b}}{\partial q^j}. </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>q</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">q^i</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i = 0,1)</annotation></semantics></math> are standard coordinates on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R} \times S^1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>=</mo><mi mathvariant="normal">diag</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g=\mathrm{diag}(1,-1)</annotation></semantics></math> is the Minkowski metric on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R} \times S^1</annotation></semantics></math>, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex">\phi^a</annotation></semantics></math> are the coordinates of the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>1</mn><mo>,</mo><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{1,d-1}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>=</mo><mi mathvariant="normal">diag</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta = \mathrm{diag}(1,-1,\cdots,-1)</annotation></semantics></math> is the Minkowski metric on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>1</mn><mo>,</mo><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{1,d-1}</annotation></semantics></math>. The corresponding Euler-Lagrange equation is just a version of the wave equation:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mi>ij</mi></msup><msub><mo>∂</mo> <mi>i</mi></msub><msub><mo>∂</mo> <mi>j</mi></msub><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>=</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding="application/x-tex"> g^{ij}\partial_{i} \partial_{j} \phi^a =0. </annotation></semantics></math></div> <p>The space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>Σ</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E=\Sigma \times X</annotation></semantics></math> can be thought of as a trivial bundle over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, and the graph of a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi : \Sigma \to X</annotation></semantics></math> is a smooth section 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>. We write the coordinates of a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">(q,u)\in E</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msup><mi>q</mi> <mi>i</mi></msup><mo>,</mo><msup><mi>u</mi> <mi>a</mi></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(q^i,u^a \right)</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">J^1 E \to E</annotation></semantics></math> be the first jet bundle 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>. We may regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">J^1 E</annotation></semantics></math> as a vector bundle whose fiber over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>q</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">(q,u)\in E</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>T</mi> <mi>q</mi> <mo>*</mo></msubsup><mi>Σ</mi><mo>⊗</mo><msub><mi>T</mi> <mi>u</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">T^*_q \Sigma \otimes T_u X</annotation></semantics></math>.<br />The Lagrangian density for the string can be defined as a smooth function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">J^1 E</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>ℒ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>g</mi> <mi>ij</mi></msup><msub><mi>η</mi> <mi>ab</mi></msub><msubsup><mi>u</mi> <mi>i</mi> <mi>a</mi></msubsup><msubsup><mi>u</mi> <mi>j</mi> <mi>b</mi></msubsup><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{L}=\frac{1}{2} g^{ij}\eta_{ab}u^{a}_{i}u^{b}_{j}, </annotation></semantics></math></div> <p>which depends in this example only on the fiber coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>u</mi> <mi>i</mi> <mi>a</mi></msubsup></mrow><annotation encoding="application/x-tex">u^a_{i}</annotation></semantics></math>.</p> <p>From the Lagrangian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mo>:</mo><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L} : J^{1}E \to \mathbb{R}</annotation></semantics></math>, the <strong><a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+Hamiltonian">de Donder-Weyl Hamiltonian</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi><mo>:</mo><mi>T</mi><mi>Σ</mi><mo>⊗</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H} : T \Sigma \otimes T^*X \to \mathbb{R}</annotation></semantics></math> can be constructed via a Legendre transform. It is given as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi><mo>=</mo><msubsup><mi>p</mi> <mi>a</mi> <mi>i</mi></msubsup><msubsup><mi>u</mi> <mi>i</mi> <mi>a</mi></msubsup><mo>−</mo><mi>ℒ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>η</mi> <mi>ab</mi></msup><msub><mi>g</mi> <mi>ij</mi></msub><msubsup><mi>p</mi> <mi>a</mi> <mi>i</mi></msubsup><msubsup><mi>p</mi> <mi>b</mi> <mi>j</mi></msubsup><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{H}= p^{i}_{a}u^{a}_{i}- \mathcal{L} =\frac{1}{2} \eta^{ab}g_{ij}p_{a}^{i}p_{b}^{j}, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>u</mi> <mi>i</mi> <mi>a</mi></msubsup></mrow><annotation encoding="application/x-tex">u^a_{i}</annotation></semantics></math> are defined implicitly by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>p</mi> <mi>a</mi> <mi>i</mi></msubsup><mo>=</mo><mo>∂</mo><mi>ℒ</mi><mo stretchy="false">/</mo><mo>∂</mo><msubsup><mi>u</mi> <mi>i</mi> <mi>a</mi></msubsup></mrow><annotation encoding="application/x-tex">p_a^{i}=\partial \mathcal{L} / \partial u^{a}_{i}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>p</mi> <mi>a</mi> <mi>i</mi></msubsup></mrow><annotation encoding="application/x-tex">p_a^{i}</annotation></semantics></math> are coordinates on the fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>T</mi> <mi>u</mi> <mo>*</mo></msubsup><mi>X</mi><mo>⊗</mo><msub><mi>T</mi> <mi>q</mi></msub><mi>Σ</mi></mrow><annotation encoding="application/x-tex">T^{*}_{u}X \otimes T_{q}\Sigma</annotation></semantics></math>. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> differs from the standard (non-covariant) Hamiltonian density for a field theory:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>p</mi> <mi>a</mi> <mn>0</mn></msubsup><msubsup><mi>u</mi> <mn>0</mn> <mi>a</mi></msubsup><mo>−</mo><mi>ℒ</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> p^{0}_{a}u^{a}_{0} - \mathcal{L}. </annotation></semantics></math></div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> be a section 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 let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> be a smooth section of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>Σ</mi><mo>⊗</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">T \Sigma \otimes T^*X</annotation></semantics></math> restricted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(\Sigma)</annotation></semantics></math> with fiber coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mi>a</mi> <mi>i</mi></msubsup></mrow><annotation encoding="application/x-tex">\pi_{a}^{i}</annotation></semantics></math>. It is then straightforward to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is a solution of the Euler-Lagrange equations if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> satisfy the following system of equations:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mo>∂</mo><msubsup><mi>π</mi> <mi>a</mi> <mi>i</mi></msubsup></mrow><mrow><mo>∂</mo><msup><mi>q</mi> <mi>i</mi></msup></mrow></mfrac><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mrow><mfrac><mrow><mo>∂</mo><mi>ℋ</mi></mrow><mrow><mo>∂</mo><msup><mi>u</mi> <mi>a</mi></msup></mrow></mfrac><mo>|</mo></mrow> <mrow><mi>u</mi><mo>=</mo><mi>ϕ</mi><mo>,</mo><mi>p</mi><mo>=</mo><mi>π</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \frac{\partial \pi^{i}_{a}}{\partial q^i} = - \left.\frac{\partial \mathcal{H}}{\partial u^{a}} \right \vert_{u=\phi,p=\pi} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow><mrow><mo>∂</mo><msup><mi>q</mi> <mi>i</mi></msup></mrow></mfrac><mo>=</mo><msub><mrow><mfrac><mrow><mo>∂</mo><mi>ℋ</mi></mrow><mrow><mo>∂</mo><msubsup><mi>p</mi> <mi>a</mi> <mi>i</mi></msubsup></mrow></mfrac><mo>|</mo></mrow> <mrow><mi>u</mi><mo>=</mo><mi>ϕ</mi><mo>,</mo><mi>p</mi><mo>=</mo><mi>π</mi></mrow></msub><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{\partial \phi^{a}}{\partial q^i} = \left.\frac{\partial \mathcal{H}}{\partial p^i_{a}} \right \vert_{u=\phi,p=\pi}. </annotation></semantics></math></div> <p>This system of equations is a generalization of Hamilton’s equations for the point particle.</p> <p>As explained above, the covariant phase space for the bosonic string is the dual jet bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(J^1 E)^*</annotation></semantics></math>, and this space is equipped with a canonical 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> whose exterior derivative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mi>d</mi><mi>α</mi></mrow><annotation encoding="application/x-tex">\omega = d \alpha</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/multisymplectic+form">multisymplectic 3-form</a>. Using the isomorphism</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><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>≅</mo><mi>T</mi><mi>Σ</mi><mo>⊗</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi><mo>×</mo><mi>ℝ</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> (J^1 E)^* \cong T \Sigma \otimes T^*X \times \mathbb{R} ,</annotation></semantics></math></div> <p>a point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(J^1 E)^{*}</annotation></semantics></math> gets coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>q</mi> <mi>i</mi></msup><mo>,</mo><msup><mi>u</mi> <mi>a</mi></msup><mo>,</mo><msubsup><mi>p</mi> <mi>a</mi> <mi>i</mi></msubsup><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(q^i,u^a,p^{i}_{a},e)</annotation></semantics></math>. In terms of these coordinates,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>=</mo><mi>e</mi><msup><mi>dq</mi> <mn>0</mn></msup><mo>∧</mo><msup><mi>dq</mi> <mn>1</mn></msup><mo>+</mo><mrow><mo>(</mo><msubsup><mi>p</mi> <mi>a</mi> <mn>0</mn></msubsup><msup><mi>du</mi> <mi>a</mi></msup><mo>∧</mo><msup><mi>dq</mi> <mn>1</mn></msup><mo>−</mo><msubsup><mi>p</mi> <mi>a</mi> <mn>1</mn></msubsup><msup><mi>du</mi> <mi>a</mi></msup><mo>∧</mo><msup><mi>dq</mi> <mn>0</mn></msup><mo>)</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex"> \alpha= e dq^{0} \wedge dq^{1} + \left(p_{a}^{0} du^{a} \wedge dq^{1} - p_{a}^{1} du^{a} \wedge dq^{0} \right) . </annotation></semantics></math></div> <p>The multisymplectic structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(J^1 E)^*</annotation></semantics></math> is thus</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mi>de</mi><mo>∧</mo><msup><mi>dq</mi> <mn>0</mn></msup><mo>∧</mo><msup><mi>dq</mi> <mn>1</mn></msup><mo>+</mo><mrow><mo>(</mo><msubsup><mi>dp</mi> <mi>a</mi> <mn>0</mn></msubsup><mo>∧</mo><msup><mi>du</mi> <mi>a</mi></msup><mo>∧</mo><msup><mi>dq</mi> <mn>1</mn></msup><mo>−</mo><msubsup><mi>dp</mi> <mi>a</mi> <mn>1</mn></msubsup><mo>∧</mo><msup><mi>du</mi> <mi>a</mi></msup><mo>∧</mo><msup><mi>dq</mi> <mn>0</mn></msup><mo>)</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega = de \wedge dq^0 \wedge dq^{1} + \left(dp_a^0 \wedge du^a \wedge dq^{1} - dp_a^1 \wedge du^a \wedge dq^{0} \right) . </annotation></semantics></math></div> <p>So, the variable <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> may be considered as canonically conjugate to the area form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>dq</mi> <mn>0</mn></msup><mo>∧</mo><msup><mi>dq</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">dq^{0} \wedge dq^{1}</annotation></semantics></math>.</p> <p>As before, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> be a section 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 let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> be a smooth section of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>Σ</mi><mo>⊗</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">T \Sigma \otimes T^*X</annotation></semantics></math> restricted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(\Sigma)</annotation></semantics></math>. Consider the submanifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mo stretchy="false">(</mo><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">S \subset (J^1 E)^*</annotation></semantics></math> with coordinates:</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><msup><mi>q</mi> <mi>i</mi></msup><mo>,</mo><msup><mi>ϕ</mi> <mi>a</mi></msup><mo stretchy="false">(</mo><msup><mi>q</mi> <mi>j</mi></msup><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi>π</mi> <mi>a</mi> <mi>i</mi></msubsup><mo stretchy="false">(</mo><msup><mi>q</mi> <mi>j</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>ℋ</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> (q^i,\phi^{a}(q^j),\pi_{a}^{i}(q^j),-\mathcal{H}). </annotation></semantics></math></div> <p>Note that <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> is constructed from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> and from the constraint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>+</mo><mi>ℋ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">e + \mathcal{H}=0</annotation></semantics></math>. This constraint is analogous to the one that is used in finding constant energy solutions in the extended phase space approach to classical mechanics. At each point in <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>, a tangent bivector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>=</mo><msub><mi>v</mi> <mn>0</mn></msub><mo>∧</mo><msub><mi>v</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">v=v_{0} \wedge v_{1}</annotation></semantics></math> can be defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>0</mn></msub><mo>=</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>q</mi> <mn>0</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow><mrow><mo>∂</mo><msup><mi>q</mi> <mn>0</mn></msup></mrow></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>u</mi> <mi>a</mi></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∂</mo><msubsup><mi>π</mi> <mi>a</mi> <mi>i</mi></msubsup></mrow><mrow><mo>∂</mo><msup><mi>q</mi> <mn>0</mn></msup></mrow></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><msubsup><mi>p</mi> <mi>a</mi> <mi>i</mi></msubsup></mrow></mfrac></mrow><annotation encoding="application/x-tex"> v_{0} =\frac{\partial}{\partial q^{0}} + \frac{\partial \phi^a}{\partial q^{0}}\frac{\partial}{\partial u^a} + \frac{\partial \pi_{a}^{i}}{\partial q^{0}} \frac{\partial}{\partial p_{a}^{i}} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>=</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>q</mi> <mn>1</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow><mrow><mo>∂</mo><msup><mi>q</mi> <mn>1</mn></msup></mrow></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>u</mi> <mi>a</mi></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∂</mo><msubsup><mi>π</mi> <mi>a</mi> <mi>i</mi></msubsup></mrow><mrow><mo>∂</mo><msup><mi>q</mi> <mn>1</mn></msup></mrow></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><msubsup><mi>p</mi> <mi>a</mi> <mi>i</mi></msubsup></mrow></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex"> v_{1} = \frac{\partial}{\partial q^{1}} + \frac{\partial \phi^a}{\partial q^{1}} \frac{\partial}{\partial u^a} + \frac{\partial \pi_{a}^{i}}{\partial q^{1}} \frac{\partial}{\partial p_{a}^{i}}. </annotation></semantics></math></div> <p>Explicit computation reveals that the submanifold <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> is generated by solutions to Hamilton’s equations if and only if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega(v_{0},v_{1},\cdot)=0. </annotation></semantics></math></div> <h2 id="hamiltonian_dimensional_flow">Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional flow</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hamiltonian+n-vector+field">Hamiltonian n-vector field</a></li> </ul> <h2 id="relation_to_symplectic_manifolds">Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-symplectic manifolds</h2> <p>There is also the notion of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a>s.</li> </ul> <p>Which is different, but related…</p> <p>….blah-blah….</p> <h2 id="SurveyDevelopments">Survey of developments in the field</h2> <p>There is in this sense a covariant form of the <a class="existingWikiWord" href="/nlab/show/Legendre+transformation">Legendre transformation</a> which associates to every field variable as many conjugated momenta – the multimomenta – as there are space-time dimensions. These, together with the field variables, those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional space-time, and an extra variable, the energy variable, span the multiphase space [1]. For a recent exposition on the differential geometry of this construction, see [2]. Multiphase space, together with a closed, nondegenerate differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-form, the <a class="existingWikiWord" href="/nlab/show/multisymplectic+form">multisymplectic form</a>, is an example of a multisymplectic manifold [3].</p> <p>Among the achievements of the multisymplectic approach is a geometric formulation of the relation of infinitesimal symmetries and covariantly conserved quantities (<a class="existingWikiWord" href="/nlab/show/Noether%27s+theorem">Noether's theorem</a>), see [4] for a recent review, and [5,6] for a clarification of the improvement techniques (“Belinfante-Rosenfeld formula”) of the energy-momentum tensor in classical field theory.</p> <p>Multisymplectic geometry also provides convenient sets of variational integrators for the numerical study of partial differential equations [7].</p> <p>Since in multisymplectic geometry, the symplectic 2-form of classical <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> is replaced by a closed <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a> of higher tensor degree, multivector fields and differential forms have their natural appearance. (See [8] for an interpretation of multivector fields as describing solutions to field equations infinitesimally.) Multivector fields form a <span class="newWikiWord">graded Lie algebra<a href="/nlab/new/graded+Lie+algebra">?</a></span> with the <a class="existingWikiWord" href="/nlab/show/Schouten+bracket">Schouten bracket</a> (see [9] for a review and unified viewpoint). Using the multisymplectic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-form, one can construct a new bracket for the differential forms, the Poisson forms [10], generalizing a well-known formula for the Poisson brackets related to a symplectic 2-form.</p> <p>A remarkable fact is that in order to establish a Jacobi identity, the multisymplectic form has to have a potential, a condition that is not seen in symplectic geometry. Further, the admissible differential forms, the Poisson forms, are subject to the rather strong restrictions on their dependence on the multimomentum variables [11]. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-forms in this context can be shown to arise exactly from the covariantly conserved currents of Noether symmetries [11], which allows their pairing with spacelike hypersurfaces to yield conserved charges in a geometric way.</p> <p>Not much is known about the interpretation of Poisson forms of form degree between zero and n-1. However, as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-forms describe vector fields and hence collections of lines [2, 10], and as (certain) functions describe n-vector fields and hence collections of bundle sections [8], it seems natural to speculate that the intermediate forms may be useful for the <a class="existingWikiWord" href="/nlab/show/brane">brane</a>s of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>.</p> <p>The Hamiltonian, infinite dimensional formulation of <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a> requires the choice of a spacelike hypersurface (“Cauchy surface”) [12] which manifestly breaks the general covariance of the theory at hand. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-forms, the above mentioned new bracket reduces to the Peierls-deWitt bracket after integration over the spacelike hypersurface [13]. With the choice of a hypersurface, a constraint analysis [14] <em>à la</em> Dirac [15,16] can be performed [17]. Again, the necessary breaking of general covariance raises the need for an alternative formulation of all this [18]; first attempts have been made to carry out a <a class="existingWikiWord" href="/nlab/show/Marsden-Weinstein+reduction">Marsden-Weinstein reduction</a> [19] for multisymplectic manifolds with symmetries [20]. However, not very much is known about how to quantize a multisymplectic geometry, see [21] for an approach using a path integral.</p> <p>This discussion so far concerns field theories of first order, i.e. where the Lagrangian depends on the fields and their first derivatives. Higher order theories can be reduced to first order ones for the price of introducing auxiliary fields. A direct treatment would involve higher order jet bundles [22]. A definition of the covariant Legendre transform and the multiphase space has been given for this case [3].</p> <h2 id="references">References</h2> <h3 id="on_classical_multisymplectic_geometry">On classical multisymplectic geometry</h3> <p>A comprehensive source on covariant field theory with plenty of further references is</p> <ul> <li>Mark J. Gotay, James Isenberg, <a class="existingWikiWord" href="/nlab/show/Jerrold+Marsden">Jerrold Marsden</a>, Richard Montgomery, <em>Momentum maps and classical relativistic fields. Part I: covariant field theory</em> (<a href="http://arxiv.org/abs/physics/9801019">arXiv:physics/9801019</a>)</li> </ul> <p>Much of the material in the <a href="#extendedphasespace">section on covariant field theory</a> is based on this.</p> <p>Further discussions:</p> <ul> <li id="Helein02"> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%A9d%C3%A9ric+H%C3%A9lein">Frédéric Hélein</a>, <em>Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory</em> [<a href="http://arxiv.org/abs/math-ph/0212036">arXiv:math-ph/0212036</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Narciso+Rom%C3%A1n-Roy">Narciso Román-Roy</a>, <em>Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories</em>, SIGMA <strong>5</strong> (2009) 100 [<a href="https://doi.org/10.3842/SIGMA.2009.100">doi;10.3842/SIGMA.2009.100</a>, <a href="http://arxiv.org/abs/math-ph/0506022">arXiv:math-ph/0506022</a>]</p> </li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> methods is discussed in</p> <ul> <li id="Hélein11"><a class="existingWikiWord" href="/nlab/show/Fr%C3%A9d%C3%A9ric+H%C3%A9lein">Frédéric Hélein</a>, <em>Multisymplectic formalism and the covariant phase</em>, in <em>Variational Problems in Differential Geometry</em>, Cambridge University Press (2011) 94-126 [<a href="https://doi.org/10.1017/CBO9780511863219.007">doi:10.1017/CBO9780511863219.007</a>, <a href="http://arxiv.org/abs/1106.2086">arXiv:1106.2086</a>]</li> </ul> <p>A discussion of <a class="existingWikiWord" href="/nlab/show/Hamiltonian+n-forms">Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-forms</a> as <a class="existingWikiWord" href="/nlab/show/observables">observables</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Fr%C3%A9d%C3%A9ric+H%C3%A9lein">Frédéric Hélein</a>, <em>The notion of observable in the covariant Hamiltonian formalism for the calculus of variations with several variables</em>, Adv. Theor. Math. Phys. <strong>8</strong> (2004) 735-777 [<a href="https://dx.doi.org/10.4310/ATMP.2004.v8.n4.a4">doi:10.4310/ATMP.2004.v8.n4.a4</a>, <a href="http://arxiv.org/abs/math-ph/0401047">arXiv:math-ph/0401047</a>]</li> </ul> <p>Other texts include</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Carlo+Rovelli">Carlo Rovelli</a>, <em>Covariant hamiltonian formalism for field theory: Hamilton-Jacobi equation on the space <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">\mathcal{G}</annotation></semantics></math></em> (<a href="http://arxiv.org/abs/gr-qc/0207043">arXiv:gr-qc/0207043</a>)</li> </ul> <p>Much of the above <a href="#SurveyDevelopments">survey of recent developments</a> and of the following list of references is reproduced from the web-page</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cornelius+Mertzlufft-Paufler">Cornelius Mertzlufft-Paufler</a>, <em><a href="http://www.mertzlufft-paufler.de/multisymplectic_geometry.html">Multisymplectic geometry</a></em></li> </ul> <p>References mentioned above are</p> <ul> <li> <p>[1] J. Kijowski, W. Szczyrba, <em>A Canonical Structure For Classical Field Theories</em> . Commun. Math. Phys. 46 (1976) 183.</p> </li> <li> <p>[2] M. J. Gotay, J. Isenberg, J. E. Marsden: <em>Momentum maps and classical relativistic fields. I: Covariant field theory</em> <a href="http://arxiv.org/abs/physics/9801019v2">arXiv:physics/9801019v2</a>.</p> </li> <li> <p>[3] M. J. Gotay, <em>A multisymplectic framework for classical field theory and the calculus of variations. I: Covariant Hamiltonian formalism</em> In M. Francaviglia (ed.), <em>Mechanics, analysis and geometry: 200 years after Lagrange</em> Amsterdam etc.: North-Holland (1991), 203–235.</p> </li> <li> <p>[4] M. de Leon, D. Martin de Diego, A. Santamaria-Merino, <em>Symmetries in Classical Field Theory</em> <a href="http://arxiv.org/abs/math-ph/0404013">arXiv:math-ph/0404013</a>.</p> </li> <li> <p>[5] M. J. Gotay, J. E. Marsden: <em>Stress-energy-momentum tensors and the Belinfante–Rosenfeld formula</em> Contemp. Math. vol. 132, AMS, Providence, 1992, 367–392.</p> </li> <li> <p>[6] M. Forger, H. Römer, <em>Currents and the energy-momentum tensor in classical field theory: a fresh look at an old problem</em> Ann. Phys. (N.Y.) 309 (2004) 306–389. <a href="http://arxiv.org/abs/hep-th/0307199">arXiv:hep-th/0307199</a>.</p> </li> <li> <p>[7] A. Lew, J. E. Marsden, M. Ortiz, M. West, <em>An overview of variational integrators</em> In L. P. Franca (ed.), Finite Element Methods: 70’s and Beyond. Barcelona (2003).</p> </li> <li> <p>[8] C. Paufler, H. Römer, <em>The Geometry of Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-vector fields in Multisymplectic Field Theory</em> J. Geom. Phys. 44, No.1(2002), 52–69. <a href="http://arxiv.org/abs/math-ph/0102008">arXiv:math-ph/0102008</a>.</p> </li> <li> <p>[9] Y. Kosmann-Schwarzbach: Derived brackets. Lett. Math. Phys. 69 (2004) 61–87 <a href="http://arxiv.org/abs/math.DG/0312524">arXiv:math.DG/0312524</a>.</p> </li> <li> <p>[10] M. Forger, C. Paufler, H. Römer: <em>The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory</em> Rev. Math. Phys. 15 (2003) 705 <a href="http://arxiv.org/abs/math-ph/0202043">arXiv:math-ph/0202043</a>.</p> </li> <li> <p>[10] M. Forger, C. Paufler, H. Römer, <em>Hamiltonian Multivector Fields and Poisson Forms in Multisymplectic Field Theory</em> <a href="http://arxiv.org/abs/math-ph/0407057">arXiv:math-ph/0407057</a>.</p> </li> <li> <p>[11] M. J. Gotay, <em>A multisymplectic framework for classical field theory and the calculus of variations. II: Space + time decomposition</em> Differ. Geom. Appl. 1(4) (1991), 375–390.</p> </li> <li> <p>[12] M. O. Salles, <em>Campos Hamiltonianos e Colchete de Poisson na Teoria Geométrica dos Campos</em> , PhD thesis, IME-USP, June 2004.</p> </li> <li> <p>[13] M. J. Gotay, J. M. Nester, <em>Generalized constraint algorithm and special presymplectic manifolds</em> In G. E. Kaiser, J. E. Marsden, Geometric methods in mathematical physics, Proc. NSF-CBMS Conf., Lowell/Mass. 1979, Berlin: Springer-Verlag, Lect. Notes Math. 775 (1980) 78–80.</p> </li> <li> <p>[14] P. A. M. Dirac, <em>Lectures on Quantum Mechanic</em> Belfer Graduate School of Science, Yeshiva University, N.Y., 1964.</p> </li> <li> <p>[15] <a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <a class="existingWikiWord" href="/nlab/show/Claudio+Teitelboim">Claudio Teitelboim</a>, <em><a class="existingWikiWord" href="/nlab/show/Quantization+of+Gauge+Systems">Quantization of Gauge Systems</a></em> Princeton University Press, 1992.</p> </li> <li> <p>[16] M. J. Gotay, J. Isenberg, J. E. Marsden, R. Montgomery, <em>Momentum Maps and Classical Relativistic Fields II: Canonical Analysis of Field Theories</em> (2004) <a href="http://arxiv.org/abs/math-ph/0411032">arXiv:math-ph/0411032</a>.</p> </li> <li> <p>[17] N. P. Landsman, <em>Against the Wheeler-DeWitt equation</em> Class. Quan. Grav. 12 (1995) L119-L124. <a href="http://arxiv.org/abs/gr-qc/9510033">arXiv:gr-qc/9510033</a>.</p> </li> <li> <p>[18] J. E. Marsden, A. Weinstein <em>Reduction of symplectic manifolds with symmetry</em> Rept. Math. Phys. 5 (1974) 121–130.</p> </li> <li> <p>[19] F. Munteanu, A. M. Rey, M. Salgado, <em>The Günther’s formalism in classical field theory: momentum map and reduction</em> J. Math. Phys. 45, No. 5 (2004) 1730–1750.</p> </li> <li> <p>[20] D. Bashkirov, G. Sardanashvily, <em>Covariant Hamiltonian Field Theory. Path Integral Quantization</em> <a href="http://arxiv.org/abs/hep-th/0402057">arXiv:hep-th/0402057</a>.</p> </li> <li> <p>[21] D. J. Saunders, <em>The Geometry of Jet Bundles</em> Lond. Math. Soc. Lect. Notes Ser. 142, Cambr. Univ. Pr., Cambridge, 1989.</p> </li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher categorial</a> interpretation of <a class="existingWikiWord" href="/nlab/show/2-plectic+geometry">2-plectic geometry</a> and connection to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras is given in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <em>Categorified symplectic geometry and the string Lie 2-algebra</em>. <a href="http://arxiv.org/abs/0901.4721">arXiv:0901.4721</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, Alexander E. Hoffnung, <a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <em>Categorified symplectic geometry and the classical string</em> (<a href="http://arxiv.org/abs/0808.0246/">arXiv:0808.0246</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <em>Higher symplectic geometry</em> PhD thesis (2011) (<a href="http://arxiv.org/abs/1106.4068">arXiv:1106.4068</a>); <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras from multisymplectic geometry</em>, Lett. Math. Phys., 100(1):29–50 (2012); <em>2-plectic geometry, Courant algebroids, and categorified prequantization</em>, J. Symplectic Geom. 11(1):53–91 (2013)</p> </li> <li> <p>Antonio Michele Miti, <a class="existingWikiWord" href="/nlab/show/Marco+Zambon">Marco Zambon</a>, <em>Observables on multisymplectic manifolds and higher Courant algebroids</em>, <a href="https://arxiv.org/abs/2209.05836">arXiv:2209.05836</a></p> </li> </ul> <p>Higher order <a class="existingWikiWord" href="/nlab/show/moment+maps">moment maps</a> are considered in</p> <ul> <li> <p>Thomas Bruun Madsen, Andrew Swann, <em>Closed forms and multi-moment maps</em> (<a href="http://arxiv.org/abs/1110.6541">arXiv:1110.6541</a>)</p> </li> <li> <p>Martin Callies, <a class="existingWikiWord" href="/nlab/show/Ya%C3%ABl+Fr%C3%A9gier">Yaël Frégier</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <a class="existingWikiWord" href="/nlab/show/Marco+Zambon">Marco Zambon</a>, <em>Homotopy moment maps</em>, Advances in Mathematics <strong>303</strong> (2016) 954–1043 (<a href="https://arxiv.org/abs/1304.2051">arXiv:1304.2051</a>)</p> </li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>-generalization of <a class="existingWikiWord" href="/nlab/show/contact+geometry">contact geometry</a> in line with multisymplectic geometry/<a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">n-plectic geometry</a> is discussed in</p> <ul> <li>Luca Vitagliano, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math> algebras from multicontact geometry</em>, Diff. Geom. Appl. <strong>39</strong> (2015) 147-165 <a href="http://arxiv.org/abs/1311.2751">arXiv:1311.2751</a> <a href="https://doi.org/10.1016/j.difgeo.2015.01.006">doi</a></li> </ul> <h3 id="relation_to_covariant_phase_space_formalism">Relation to covariant phase space formalism</h3> <p>On the relation of multisymplectic formalism to <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> and <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a> methods:</p> <ul> <li id="ForgerRomero04"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Forger">Michael Forger</a>, <a class="existingWikiWord" href="/nlab/show/Sandro+Romero">Sandro Romero</a>, <em>Covariant Poisson brackets in geometric field theory</em>, Commun. Math. Phys. <strong>256</strong> (2005) 375-410 [<a href="http://arxiv.org/abs/math-ph/0408008">arXiv:math-ph/0408008</a>, <a href="https://doi.org/10.1007/s00220-005-1287-8">doi:10.1007/s00220-005-1287-8</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Narciso+Rom%C3%A1n-Roy">Narciso Román-Roy</a>, <em>Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories</em>, SIGMA 5 (2009), 100, 25 pages (<a href="http://arxiv.org/abs/math-ph/0506022">arXiv:math-ph/0506022</a>)</p> </li> </ul> <p>On multisymplectic <a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Zapata">José Zapata</a>, <em>Multisymplectic effective General Boundary Field Theory</em> (2013) [<a href="http://relativity.phys.lsu.edu/ilqgs/zapata090313.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Zapata-MultisymplecticBoundary.pdf" title="pdf">pdf</a>]</li> </ul> <h3 id="RefsonQuantization">On quantization of multisymplectic geometry</h3> <p>The following articles discuss the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> procedure for multisymplectic geometry, generalizing <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> of <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a>.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+V.+Kanatchikov">Igor V. Kanatchikov</a>, <em>De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory</em>, Rept. Math. Phys. <strong>43</strong> (1999) 157-170 [<a href="http://arxiv.org/abs/hep-th/9810165">arXiv:hep-th/9810165</a>, <a href="https://doi.org/10.1016/S0034-4877(99)80024-X">doi:10.1016/S0034-4877(99)80024-X</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+V.+Kanatchikov">Igor V. Kanatchikov</a><em>Geometric (pre)quantization in the polysymplectic approach to field theory</em>, in <em>Differential Geometry and Its Applications</em>, Proc. Conf., Opava (August 2001) [<a href="https://arxiv.org/abs/hep-th/0112263">arXiv:hep-th/0112263</a>, <a href="http://conferences.math.slu.cz/8icdga/PDF/309-322.pdf">pdf</a>]</p> </li> </ul> <p>Kanatchikov’s “algebra of observables” is what he calls a “higher-order Gerstenhaber algebra”. (The “bracket” in this structure fails to satisfy Leibniz’s rule as a derivation of the product.) The relationship between it and the <a class="existingWikiWord" href="/nlab/show/Lie+superalgebra">Lie superalgebra</a> of observables constructed by Forger, Paufler, and Roemer is discussed in this paper:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Forger">Michael Forger</a>, C. Paufler, and H. Roemer, <em>The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory</em> , (<a href="http://arxiv.org/abs/math-ph/0202043v1">arXiv:math-ph/0202043</a>)</li> </ul> <p>and (<a href="#ForgerRomero04">Forger-Romero 04</a>) above.</p> <p>Kanatchikov’s formalism was used by S.P. Hrabak to propose a multisymplectic refinement of <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a>. See there for more details.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on June 19, 2024 at 19:37:14. See the <a href="/nlab/history/multisymplectic+geometry" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/multisymplectic+geometry" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/790/#Item_7">Discuss</a><span class="backintime"><a href="/nlab/revision/multisymplectic+geometry/57" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/multisymplectic+geometry" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/multisymplectic+geometry" accesskey="S" class="navlink" id="history" rel="nofollow">History (57 revisions)</a> <a href="/nlab/show/multisymplectic+geometry/cite" style="color: black">Cite</a> <a href="/nlab/print/multisymplectic+geometry" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/multisymplectic+geometry" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>