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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> 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</div> <h2 id="idea">Idea</h2> <p>Fedosov’s method (<a href="#Fedosov94">Fedosov 94</a>) is a prescription for constructing <a class="existingWikiWord" href="/nlab/show/formal+deformation+quantizations">formal deformation quantizations</a> of the <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of</a> <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on any <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> <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>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math>.</p> <p>In particular this establishes the existence of formal deformation quantizations of all <a class="existingWikiWord" href="/nlab/show/symplectic+manifolds">symplectic manifolds</a>. While <a href="deformation+quantization#Kontsevich97">Kontsevich 97</a> more generally proves the existence of deformation quantization of all <a class="existingWikiWord" href="/nlab/show/Poisson+manifolds">Poisson manifolds</a> of finite dimension, Fedosov’s method, in a variant applicable to <a class="existingWikiWord" href="/nlab/show/almost+K%C3%A4hler+structures">almost Kähler structures</a> (<a href="#KarabegovSchlichenmaier01">Karabegov-Schlichenmaier 01</a>) generalizes to infinite-dimensional symplectic manifolds as they appear in <a class="existingWikiWord" href="/nlab/show/local+field+theory">local field theory</a>, where it yields the quantization to <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> equivalent to the method of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> (<a href="#Collini16">Collini 16</a>). (Kontsevich’s deformation quantization is however not compatible with field theory (<a href="#HawkinsRejzner16">Hawkins-Rejzner 16, section 5.3.2</a>).) For more on this see at <em><a class="existingWikiWord" href="/nlab/show/locally+covariant+perturbative+quantum+field+theory">locally covariant perturbative quantum field theory</a></em>.</p> <p>Fedosov’s method proceeds by the following steps:</p> <ol> <li> <p>For each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> regard the <a class="existingWikiWord" href="/nlab/show/tangent+space">tangent space</a> <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>X</mi></mrow><annotation encoding="application/x-tex">T_x X</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>x</mi></msub><mi>X</mi><mo>,</mo><msub><mi>ω</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T_x X, \omega_x)</annotation></semantics></math> and consider its <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒲</mi> <mi>x</mi></msub><mo>≔</mo><msub><mi>A</mi> <mi>Moy</mi></msub><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>x</mi></msub><mi>X</mi><mo>,</mo><msub><mi>ω</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{W}_x \coloneqq A_{Moy}(T_x X, \omega_x)</annotation></semantics></math>. In the context of <a class="existingWikiWord" href="/nlab/show/local+field+theory">local field theory</a> these Moyal algebras are the <a class="existingWikiWord" href="/nlab/show/Wick+algebras">Wick algebras</a> of the underlying <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a>.</p> <p>The union of these yelds an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a>-<a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒲</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{W} \to X</annotation></semantics></math>. From the fiber-wise product in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>Moy</mi></msub><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>x</mi></msub><mi>X</mi><mo>,</mo><msub><mi>ω</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_{Moy}(T_x X, \omega_x)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/space+of+sections">space of sections</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>𝒲</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(\mathcal{W})</annotation></semantics></math> inherits itself the structure of an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a>.</p> </li> <li> <p>Find a <a class="existingWikiWord" href="/nlab/show/flat+connection">flat connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>Fed</mi></msub></mrow><annotation encoding="application/x-tex">\nabla_{Fed}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒲</mi></mrow><annotation encoding="application/x-tex">\mathcal{W}</annotation></semantics></math> which respects the associative algebra structure, and such that there is a <a class="existingWikiWord" href="/nlab/show/linear+isomorphism">linear isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>𝒲</mi><mo stretchy="false">)</mo><mo>⊃</mo><mi>ker</mi><mo stretchy="false">(</mo><msub><mo>∇</mo> <mi>Fed</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X(\mathcal{W}) \supset ker(\nabla_{Fed}) \simeq C^\infty(X) [ [ \hbar ] ] </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/covariantly+constant+sections">covariantly constant sections</a> of the algebra bundle and the algebra of smooth functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal variable</a> <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> adjoined.</p> <p>Such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>Fed</mi></msub></mrow><annotation encoding="application/x-tex">\nabla_{Fed}</annotation></semantics></math> exists, (non-uniquely), it is called a <em>Fedosov connection</em>.</p> </li> <li> <p>By this linear isomorphism the algebra structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>𝒲</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(\mathcal{W})</annotation></semantics></math> induces an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> structure on <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>X</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)[ [ \hbar ] ]</annotation></semantics></math>, and this turns out to be a <a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a> of <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>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math>.</p> <p>In the context of <a class="existingWikiWord" href="/nlab/show/local+field+theory">local field theory</a>, this deformation quantization is equivalent to the result of quantization via <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> (<a href="#Collini16">Collini 16</a>).</p> </li> </ol> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/almost+K%C3%A4hler+geometric+quantization">almost Kähler geometric quantization</a></li> </ul> <h2 id="references">References</h2> <p>The method is due to</p> <ul> <li id="Fedosov94"><a class="existingWikiWord" href="/nlab/show/Boris+Fedosov">Boris Fedosov</a>, <em>A simple geometrical construction of deformation quantization</em> Journal of Differential Geometry, 40(2):213–238, 1994 (<a href="https://projecteuclid.org/euclid.jdg/1214455536">Euclid</a>)</li> </ul> <p>with a monograph account in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Boris+Fedosov">Boris Fedosov</a>, <em>Deformation Quantization and Index Theory</em>, John Wiley & Sons (1995) [ISBN:9783055017162]</li> </ul> <p>Its generalization to <a class="existingWikiWord" href="/nlab/show/almost+K%C3%A4hler+structures">almost Kähler structures</a> is due to</p> <ul> <li id="KarabegovSchlichenmaier01"><a class="existingWikiWord" href="/nlab/show/Alexander+Karabegov">Alexander Karabegov</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Schlichenmaier">Martin Schlichenmaier</a>, <em>Almost-Kähler deformation quantization</em>, Letters in Mathematical Physics, <strong>57</strong> 2 (2001) 135–148 [<a href="https://doi.org/10.1023/A:1017993513935">doi:10.1023/A:1017993513935</a>, <a href="https://arxiv.org/abs/math/0102169">arXiv:0102169</a>]</li> </ul> <p>with a survey in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Martin+Schlichenmaier">Martin Schlichenmaier</a>, <em>Deformation quantization for almost-Kähler manifolds</em>, J. of Nonlinear Math. Phys. <p><strong>11</strong>, Supplement (2004) 49–54 [<a href="https://doi.org/10.2991/jnmp.2004.11.s1.6">doi:10.2991/jnmp.2004.11.s1.6</a>, <a href="http://www.atlantis-press.com/php/download_paper.php?id=540">pdf</a>]</p> </li> </ul> <p>The observation that the construction of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> via <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> is equivalent to Fedosov quantization 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> <p>Discussion showing that this generalization to field theory is not given by Kontsevich deformation quantization:</p> <ul> <li id="HawkinsRejzner16"><a class="existingWikiWord" href="/nlab/show/Eli+Hawkins">Eli Hawkins</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, section 5.3.2 of <em>The Star Product in Interacting Quantum Field Theory</em> (<a href="https://arxiv.org/abs/1612.09157">arXiv:1612.09157</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 8, 2023 at 05:18:26. 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