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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="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="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> </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='#examples'>Examples</a></li> <ul> <li><a href='#HarmonicOscillator'>Harmonic oscillator</a></li> </ul> <li><a href='#properties'>Properties</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In the context of <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> of a <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>, a <em>Bohr-Sommerfeld leaf</em> is a <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> on which not only the <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic 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> vanishes, but on which also a given <a class="existingWikiWord" href="/nlab/show/prequantum+bundle">prequantization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> of <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> is trivializable.</p> <p>Therefore given a <a class="existingWikiWord" href="/nlab/show/real+polarization">real polarization</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>, hence a <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a> by <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifolds">Lagrangian submanifolds</a>, the Bohr-Sommerfeld leaves form a discrete subset of the <a class="existingWikiWord" href="/nlab/show/leaf+space">leaf space</a>. The discreteness of this subset is essentially the formal incarnation of “quantization” and this is what <a class="existingWikiWord" href="/nlab/show/Niels+Bohr">Bohr</a> and <a class="existingWikiWord" href="/nlab/show/Arnold+Sommerfeld">Sommerfeld</a> originally considered (in less abstract terms, the archetypical example was the harmonic oscillator as discussed <a href="#HarmonicOscillator">below</a>).</p> <p>(There is a correction to this picture, given by the fact that a <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a>/<a class="existingWikiWord" href="/nlab/show/semiclassical+states">semiclassical states</a>, involve not just Lagrangian submanifolds/Bohr-Sommerfeld leaves, but moreover <a class="existingWikiWord" href="/nlab/show/half-densities">half-densities</a> over these. These are to satisfy an additional condition, encoded by the <em><a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a></em>.)</p> <h2 id="definition">Definition</h2> <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>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> be a (<a class="existingWikiWord" href="/nlab/show/presymplectic+manifold">pre-</a>)<a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/prequantum+bundle">prequatization</a>, hence a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>∇</mo></msub><mo>=</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">F_\nabla = \omega</annotation></semantics></math>.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">L \hookrightarrow X</annotation></semantics></math> is a <strong>Bohr-Sommerfeld leaf</strong> if the restriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><msub><mo stretchy="false">|</mo> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex">\nabla|_L</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/prequantum+connection">prequantum connection</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is trivializable there, hence if its <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> class vanishes in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</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><msub><mo>∇</mo> <mi>L</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn><mo>∈</mo><msubsup><mi>H</mi> <mi>conn</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\nabla_L] = 0 \in H^2_{conn}(X) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>For every <a class="existingWikiWord" href="/nlab/show/isotropic+submanifold">isotropic submanifold</a>, hence in particular every <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">L \hookrightarrow X</annotation></semantics></math> the restriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><msub><mo stretchy="false">|</mo> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex">\nabla|_L</annotation></semantics></math> is necessarily already a <a class="existingWikiWord" href="/nlab/show/flat+connection">flat connection</a>. As discussed there, flat connections are equivalently encoded in the <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> of their <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>: a flat connection is trivializable as a connection precisely if its <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> is trivial. Therefore a Bohr-Sommerfeld leaf is equivalently a Lagrangian submanifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><msub><mo stretchy="false">|</mo> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex">\nabla|_L</annotation></semantics></math> has trivial holonomy. In this form the Bohr-Sommerfeld condition is usually stated in the literature.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The Bohr-Sommerfeld condition is the natural lift of the <a class="existingWikiWord" href="/nlab/show/Lagrangian+subspace">Lagrangian subspace</a>-condition to prequantum geometry:</p> <p>When expressed in terms of <a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a> (see at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em> for background), the (<a class="existingWikiWord" href="/nlab/show/presymplectic+manifold">pre-</a>)<a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic structure</a> is equivalently a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex"> \omega \;\colon\; X \to \Omega^2_{cl} </annotation></semantics></math></div> <p>and a prequantization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> is equivalently a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> in the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∇</mo></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>ω</mi></mover></mtd> <mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}U(1)_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega}{\to}& \Omega^2_{cl}(X) } \,. </annotation></semantics></math></div> <p>The condition on an <a class="existingWikiWord" href="/nlab/show/isotropic+submanifold">isotropic submanifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">L \hookrightarrow X</annotation></semantics></math> is that the composite map</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>L</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>L</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>ω</mi></mover></mtd> <mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \omega|_L \;\colon\; \array{ L &\hookrightarrow & X &\stackrel{\omega}{\to}& \Omega^2_{cl} } </annotation></semantics></math></div> <p>is trivial in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>L</mi><mo>,</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(L,\Omega^2_{cl}) = \Omega^2_{cl}(L)</annotation></semantics></math> (and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> being Lagrangian means that it is maximal with this property). Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is Bohr-Sommerfeld if moreover the restriction of the prequantum lift</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∇</mo><msub><mo stretchy="false">|</mo> <mi>L</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>L</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mo>∇</mo></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \nabla|_L \;\colon\; \array{ L &\hookrightarrow & X &\stackrel{\nabla}{\to}& \mathbf{B}U(1)_{conn} } </annotation></semantics></math></div> <p>is trivial in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>L</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>H</mi> <mi>conn</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(L, \mathbf{B}U(1)_{conn}) = H^2_{conn}(X)</annotation></semantics></math>.</p> </div> <h2 id="examples">Examples</h2> <h3 id="HarmonicOscillator">Harmonic oscillator</h3> <p>For the single 1-dimensional <a class="existingWikiWord" href="/nlab/show/Harmonic+oscillator">Harmonic oscillator</a>, <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> is the <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><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> equipped with the symplectic form</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><msub><mi>d</mi> <mi>dR</mi></msub><mi>t</mi><mo>∧</mo><mi>d</mi><mi>θ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \omega = d_{dR} t \wedge d\theta \,, </annotation></semantics></math></div> <p>where on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>−</mo><msub><mi>ℝ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\mathbb{R}^2- \mathbb{R}_+</annotation></semantics></math> <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><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(t, \theta)</annotation></semantics></math> are the canonical <a class="existingWikiWord" href="/nlab/show/polar+coordinates">polar coordinates</a>.</p> <p>We may choose the trivial <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> given by the globally defined <a class="existingWikiWord" href="/nlab/show/differential+form">differential 1-form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mo>:</mo><mo>=</mo><mi>t</mi><mo>∧</mo><mi>d</mi><mi>θ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Theta := t \wedge d\theta \,. </annotation></semantics></math></div> <p>Then a <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a> is given by the <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a> whose <a class="existingWikiWord" href="/nlab/show/leaves">leaves</a> are the <a class="existingWikiWord" href="/nlab/show/submanifolds">submanifolds</a> of constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a> along any leaf acts 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><msub><mo>∇</mo> <mi>Θ</mi></msub><mi>σ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>θ</mi></mrow></mfrac><mi>σ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>−</mo><mi>i</mi><mi>t</mi><mi>σ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\nabla_\Theta \sigma)(t, \theta) = (\frac{\partial}{\partial \theta} \sigma)(t, \theta) - i t \sigma(t, \theta) \,. </annotation></semantics></math></div> <p>The covariantly sections covariantly constat on a leaf hence must be of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>t</mi><mi>θ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \sigma(t, \theta) = a(t) \exp( i t \theta) \,. </annotation></semantics></math></div> <p>For this to be well-defined as a globally defined section on the whole leaf the condition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>=</mo><mn>2</mn><mi>π</mi><mi>k</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>k</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> t = 2 \pi k \; \; k \in \mathbb{Z} </annotation></semantics></math></div> <p>has to hold. Hence the Bohr-Sommerfeld leaves here are the <a class="existingWikiWord" href="/nlab/show/circles">circles</a> of radius <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>π</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">2 \pi k</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p><strong>(Guillemin-Sternberg)</strong></p> <p>If a <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/regular+fibration">regular fibration</a> with <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> <a class="existingWikiWord" href="/nlab/show/leaves">leaves</a> over a <a class="existingWikiWord" href="/nlab/show/simply+connected">simply connected</a> base <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, then the Bohr-Sommerfeld leaves form a discrete subset given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>F</mi> <mi>BS</mi></msub><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><mi>p</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">|</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℤ</mi> <mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \{F_BS\} = \{ p \in X | (f_1(p), \cdots, f_n(p)) \in \mathbb{Z}^n \} </annotation></semantics></math></div> <p>where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{f_i\}</annotation></semantics></math> are global <span class="newWikiWord">action coordinates<a href="/nlab/new/action+coordinates">?</a></span> on the base space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> </div> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p><strong>(Sniatycki)</strong></p> <p>If the leaf space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a> and the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">X \to B</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> fibers, then the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of the <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of quantum states</a> is given by the number of Bohr-Sommerfeld leaves.</p> </div> <h2 id="references">References</h2> <p>Named after <a class="existingWikiWord" href="/nlab/show/Niels+Bohr">Niels Bohr</a> and <a class="existingWikiWord" href="/nlab/show/Arnold+Sommerfeld">Arnold Sommerfeld</a>.</p> <p>For historical background see also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Old_quantum_theory">Old quantum theory</a></em>.</li> </ul> <p>Discussion in the modern formalism of <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>:</p> <ul> <li id="Śniatycki75"> <p><a class="existingWikiWord" href="/nlab/show/J%C4%99drzej+%C5%9Aniatycki">Jędrzej Śniatycki</a>, <em>Wave functions relative to a real polarization</em>, Internat. J. Theoret. Phys., <strong>14</strong> 4 (1975) 277-288 [<a href="https://doi.org/10.1007/BF01807689">doi:10.1007/BF01807689)</a>]</p> </li> <li id="Śniatycki80"> <p><a class="existingWikiWord" href="/nlab/show/J%C4%99drzej+%C5%9Aniatycki">Jędrzej Śniatycki</a>, <em>Geometric Quantization and Quantum Mechanics</em>, Applied Mathematical Sciences <strong>30</strong>, Springer-Verlag (1980) [<a href="https://doi.org/10.1007/978-1-4612-6066-0">doi:10.1007/978-1-4612-6066-0</a>]</p> </li> <li> <p>Mark Hamilton, <em>Locally toric manifolds and singular Bohr-Sommerfeld leaves</em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eva+Miranda">Eva Miranda</a>: <em>From action-angle coordinates to geometric quantization and back</em> (2011) [<a href="https://ncatlab.org/nlab/files/Miranda-ActionAngleCoordinates.pdf">pdf</a>]</p> </li> </ul> <p>and with application to <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrei+Tyurin">Andrei Tyurin</a>, <em>On Bohr-Sommerfeld bases</em>, Izvestiya: Mathematics <strong>64</strong> 5 (2000) 1033–1064 [<a href="https://arxiv.org/abs/math/9909084">arXiv:math/9909084</a>, <a href="https://iopscience.iop.org/article/10.1070/IM2000v064n05ABEH000308">doi:10.1070/IM2000v064n05ABEH000308</a>, <a href="https://www.mathnet.ru/links/d8ad7540137cf2f33c44ab3e9455d392/im308_eng.pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 2, 2025 at 15:10:55. 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