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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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="geometric_quantization">Geometric quantization</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></strong> <strong><a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <em><a href="geometry+of+physics#LagrangiansAndActionFunctionals">Lagrangians and Action functionals</a></em> + <em><a href="geometry+of+physics#GeometricQuantization">Geometric Quantization</a></em></p> <h2 id="prerequisites">Prerequisites</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a>, <a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">n-plectic geometry</a>, <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic infinity-groupoid</a></p> </li> </ul> </li> </ul> <h2 id="prequantum_field_theory">Prequantum field theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a> = <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a></p> <ul> <li> <p>prequantum 1-bundle = <a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, regular<a class="existingWikiWord" href="/nlab/show/contact+manifold">contact manifold</a>,<a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> = lift of <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+0-bundle">prequantum 0-bundle</a> = <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</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/symplectomorphism">symplectomorphism</a>, <a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></p> </li> </ul> </li> </ul> <h2 id="geometric_quantization">Geometric quantization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr-Sommerfeld+leaf">Bohr-Sommerfeld leaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></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/geometric+quantization+by+push-forward">geometric quantization by push-forward</a></p> </li> <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/geometric+quantization+of+non-integral+forms">geometric quantization of non-integral forms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+quantization">motivic quantization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/coherent+state+%28in+geometric+quantization%29">coherent state</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil+theorem">Borel-Weil theorem</a>, <a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit+method">orbit method</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schubert+calculus">Schubert calculus</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/geometric+quantization+-+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> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#integral_representation'>Integral representation</a></li> <li><a href='#ViaGeometricQuantization'>Via geometric quantization</a></li> </ul> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>Moyal product</em> is a <a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a> of a linear <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>, hence of a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/Poisson+bivector">Poisson bivector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>∈</mo><mi>V</mi><mo>∧</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">\pi \in V \wedge V</annotation></semantics></math>, regarded as a constant (translation invariant) <a class="existingWikiWord" href="/nlab/show/tensor+field">bivector field</a>.</p> <p>Moyal quantization serves as an intermediate step in quantization of more general situations:</p> <p>Given a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>, then Moyal quantization applies in each <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>. The resulting <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> of Moyal algebras admits a <a class="existingWikiWord" href="/nlab/show/flat+connection">flat connection</a> (non-uniquely) compatible with the algebra structure. The <a class="existingWikiWord" href="/nlab/show/covariantly+constant+sections">covariantly constant sections</a> of this Moyal-algebra bundle constitute a <a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a> of the symplectic manifold, see at <em><a class="existingWikiWord" href="/nlab/show/Fedosov%27s+deformation+quantization">Fedosov's deformation quantization</a></em>.</p> <p>With a little care, the Moyal construction applies also to infinite-dimensional Poisson vector spaces such as appear in <a class="existingWikiWord" href="/nlab/show/local+field+theory">local field theory</a>. Here the Moyal quantization yields <a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a> of <a class="existingWikiWord" href="/nlab/show/free+field+theories">free field theories</a> to <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theories">perturbative quantum field theories</a>, the result are the <em><a class="existingWikiWord" href="/nlab/show/Wick+algebras">Wick algebras</a></em> of free field theory (<a href="#Dito90">Dito 90</a>, <a href="#DutschFredenhagen01">Dütsch-Fredenhagen 01</a>). Combining this this with <a class="existingWikiWord" href="/nlab/show/Fedosov%27s+deformation+quantization">Fedosov's deformation quantization</a> as above yields <a class="existingWikiWord" href="/nlab/show/interaction">interacting</a> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theories">perturbative quantum field theories</a> as constructed via <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> (<a href="#Collini16">Collini 16</a>), see at <em><a class="existingWikiWord" href="/nlab/show/locally+covariant+perturbative+quantum+field+theory">locally covariant perturbative quantum field theory</a></em> for more on this.</p> <h2 id="definition">Definition</h2> <p>The Moyal <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> on <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(V)</annotation></semantics></math> is given on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f,g \in C^\infty(V)</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋆</mo><mi>g</mi><mo>≔</mo><mi>prod</mi><mo>∘</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>ℏ</mi><mi>π</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> f \star g \coloneqq prod \circ \exp(\hbar \pi)(f , g) \,, </annotation></semantics></math></div> <p>where in the exponent we regard <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> as an <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a> on the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(V) \otimes C^\infty(V)</annotation></semantics></math> by <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a> in each argument, where the <a class="existingWikiWord" href="/nlab/show/exponential">exponential</a> denotes the corresponding <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> of iterated applications of this endomorphism, and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>prod</mi><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">prod \colon C^\infty(V) \otimes C^\infty(V) \to C^\infty(V)</annotation></semantics></math> is the usual pointwise product of functions.</p> <p>This means that given a choice of <a class="existingWikiWord" href="/nlab/show/basis">basis</a> <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><msub><mo stretchy="false">}</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\{x^i\}_i</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> such that <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> has components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>π</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{\pi^{i j}\}_{i j}</annotation></semantics></math> in this basis, the resulting <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> in the formal parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a>”) starts out as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>⋆</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo>⋅</mo><mi>g</mi><mo>+</mo><mi>ℏ</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msup><mi>π</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>i</mi></msup></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mo>∂</mo><mi>g</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>j</mi></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>ℏ</mi> <mn>2</mn></msup><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>l</mi></mrow></munder><msup><mi>π</mi> <mrow><mi>k</mi><mi>l</mi></mrow></msup><msup><mi>π</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>k</mi></msup><mo>∂</mo><msup><mi>x</mi> <mi>i</mi></msup></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>g</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>l</mi></msup><mo>∂</mo><msup><mi>x</mi> <mi>j</mi></msup></mrow></mfrac><mo>+</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> (f \star g) = f \cdot g + \hbar \sum_{i,j} \pi^{i j} \frac{\partial f}{\partial x^i}\cdot \frac{\partial g}{\partial x^j} + \frac{1}{2} \hbar^2 \sum_{i,j,k, l} \pi^{k l} \pi^{i j} \frac{\partial^2 f}{\partial x^k\partial x^i}\cdot \frac{\partial^2 g}{\partial x^l \partial x^j} + \cdots </annotation></semantics></math></div> <h2 id="properties">Properties</h2> <h3 id="integral_representation">Integral representation</h3> <div class="num_prop" id="IntegralRepresentationOfStarProduct"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/integral">integral</a> representation of <a class="existingWikiWord" href="/nlab/show/star+product">star product</a>)</strong></p> <p>If the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f,g</annotation></semantics></math> admit <a class="existingWikiWord" href="/nlab/show/Fourier+analysis">Fourier analysis</a> (are <a class="existingWikiWord" href="/nlab/show/functions+with+rapidly+decreasing+partial+derivatives">functions with rapidly decreasing partial derivatives</a>), then their <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> is equivalently given by the following <a class="existingWikiWord" href="/nlab/show/integral">integral</a> expression:</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><mrow><mo>(</mo><mi>f</mi><msub><mo>⋆</mo> <mi>ω</mi></msub><mi>g</mi><mo>)</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>det</mi><mo stretchy="false">(</mo><mi>ω</mi><msup><mo stretchy="false">)</mo> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow></mfrac><mo>∫</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mi>ℏ</mi></mrow></mfrac></mstyle><mi>ω</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mover><mi>y</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mover><mi>y</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mi>y</mi><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mover><mi>y</mi><mo stretchy="false">˜</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left(f \star_\omega g\right)(x) &= \frac{(det(\omega)^{2n})}{(2 \pi \hbar)^{2n} } \int e^{ - \tfrac{1}{i \hbar} \omega((x - \tilde y),(x-y))} f(y) g(\tilde y) \, d^{2 n} y \, d^{2 n} \tilde y \end{aligned} </annotation></semantics></math></div></div> <p>(<a href="#Baker58">Baker 58</a>, see at <em><a class="existingWikiWord" href="/nlab/show/star+product">star product</a></em> <a href="star+product#IntegralRepresentationOfStarProduct">this prop</a>).</p> <h3 id="ViaGeometricQuantization">Via geometric quantization</h3> <p>The Moyal quantization of a Poisson vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,\pi)</annotation></semantics></math> arises equivalently as the canonical <a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a> of the <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> which is the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a> (<a href="#Weinstein91">Weinstein 91, p. 446</a>, <a href="#GBV">Garcia-Bondia & Varilly 94, section V</a>, <a href="#EH">Hawkins 06</a>).</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/star+product">star product</a></em> <a href="star+product#PolarizedSymplecticGroupoidConvolutionProductOfSymplecticVectorSpaceIsMoyalStarProduct">this prop.</a> for the <strong>proof</strong>; and see at <em><a href="geometric+quantization+of+symplectic+groupoids#MoyalQuantizationofPoissonVectorSpace">geometric quantization of symplectic groupoids – Examples – Moyal quantization</a></em> for more.</p> <h2 id="References">References</h2> <p>The Moyal product was introduced independently in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbrand+Groenewold">Hilbrand Groenewold</a>, <em>On the Principles of elementary quantum mechanics</em>, Physica,12 (1946) pp. 405-460.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Moyal">José Moyal</a>, <em>Quantum mechanics as a statistical theory</em>. Mathematical Proceedings of the Cambridge Philosophical Society 45: 99 (1949)</p> </li> </ul> <p>The integral expression (prop. <a class="maruku-ref" href="#IntegralRepresentationOfStarProduct"></a>) is apparently due to</p> <ul> <li id="Baker58">George A. Baker, <em>Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space</em>, Jr. Phys. Rev. 109, 2198 – Published 15 March 1958 (<a href="https://doi.org/10.1103/PhysRev.109.2198">doi:10.1103/PhysRev.109.2198</a>)</li> </ul> <p>General accounts include</p> <ul> <li> <p>D. B. Fairlie, <em>Moyal Brackets, Star Products and the Generalised Wigner Function</em> (<a href="https://arxiv.org/abs/hep-th/9806198">arXiv:hep-th/9806198</a>)</p> </li> <li> <p>Maciej Blaszak, Ziemowit Domanski, <em>Maciej Blaszak, Ziemowit Domanski</em> (<a href="https://arxiv.org/abs/1009.0150">arXiv:1009.0150</a>)</p> </li> </ul> <p>The understanding of the Moyal product as the <a class="existingWikiWord" href="/nlab/show/polarization">polarized</a> <a class="existingWikiWord" href="/nlab/show/groupoid+convolution+algebra">groupoid convolution algebra</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a>, hence as an example of <a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a> had been suggested without proof in</p> <ul> <li id="Weinstein91"><a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>, p. 446 in P. Donato et al. (eds.) <em>Symplectic Geometry and Mathematical Physics, (Birkhäuser, Basel, 1991);</em></li> </ul> <p>and was proven in detail in</p> <ul> <li id="GBV"><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Gracia-Bondia">José Gracia-Bondia</a>, <a class="existingWikiWord" href="/nlab/show/Joseph+Varilly">Joseph Varilly</a>, <em>From geometric quantization to Moyal quantization</em>, J. Math. Phys. 36 (1995) 2691-2701 (<a href="http://arxiv.org/abs/hep-th/9406170">arXiv:hep-th/9406170</a>)</li> </ul> <p>In a broader context this was reconsidered in</p> <ul> <li id="EH"><a class="existingWikiWord" href="/nlab/show/Eli+Hawkins">Eli Hawkins</a>, example 6.2 of <em>A groupoid approach to quantization</em>, J. Symplectic Geom. Volume 6, Number 1 (2008), 61-125. (<a href="http://arxiv.org/abs/math.SG/0612363">arXiv:math.SG/0612363</a>)</li> </ul> <p>The observation that Moyal deformation quantization applied to the <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a> yields the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> quantization of <a class="existingWikiWord" href="/nlab/show/free+field+theories">free field theories</a> is due to</p> <ul> <li id="Dito90">J. Dito, <em>Star-product approach to quantum field theory: The free scalar field</em>. Letters in Mathematical Physics, 20(2):125–134, 1990 (<a href="https://inspirehep.net/record/303898/">spire</a>)</li> </ul> <p>and was amplified in the broader context of <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> in</p> <ul> <li id="DutschFredenhagen01"><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <em>Perturbative algebraic field theory, and deformation quantization</em>, in <a class="existingWikiWord" href="/nlab/show/Roberto+Longo">Roberto Longo</a> (ed.), <em>Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects</em>, volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001</li> </ul> <p>That moreover the corresponding <a class="existingWikiWord" href="/nlab/show/Fedosov+deformation+quantization">Fedosov deformation quantization</a> based on this free field theory star product yields the <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> quantization of interacting field theories is due to</p> <ul> <li id="Collini16"><a class="existingWikiWord" href="/nlab/show/Giovanni+Collini">Giovanni Collini</a>, <em>Fedosov Quantization and Perturbative Quantum Field Theory</em> (<a href="https://arxiv.org/abs/1603.09626">arXiv:1603.09626</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 18, 2020 at 05:43:38. 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