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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="hamiltonians">Hamiltonians</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#in_mechanical_systems'>In mechanical systems</a></li> <ul> <li><a href='#in_classical_mechanics'>In classical mechanics</a></li> <li><a href='#in_quantum_mechanics'>In quantum mechanics</a></li> <ul> <li><a href='#physical_meaning_and_relation_to_unitary_transformations'>Physical meaning and relation to unitary transformations</a></li> </ul> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <p>Disambiguation: there is an unrelated notion of a Hamilton or Hamiltonian operator also called <a class="existingWikiWord" href="/nlab/show/nabla">nabla</a> in <span class="newWikiWord">vector analysis<a href="/nlab/new/vector+analysis">?</a></span>.</p> <h2 id="definition">Definition</h2> <p>Given a <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson 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><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \{-,-\})</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in \Gamma(T X) </annotation></semantics></math>, a <strong>Hamiltonian</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mi>v</mi></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h_v \in C^\infty(X)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>h</mi> <mi>v</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{h_v,-\}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> corresponding to <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>.</p> <p>Conversely, one says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mi>v</mi></msub></mrow><annotation encoding="application/x-tex">h_v</annotation></semantics></math>.</p> <h2 id="in_mechanical_systems">In mechanical systems</h2> <p>Given a <a class="existingWikiWord" href="/nlab/show/classical+mechanical+system">classical mechanical system</a> evolving in time, there is a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> (or at least <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>) equipped with the <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a> that generates time evolution. Its Hamiltonian is often called <em>the</em> Hamiltonian. This is the concept that Hamilton originally considered and which hence gives the name to the general situaiton.</p> <h3 id="in_classical_mechanics">In classical mechanics</h3> <p>The simplest, so-called “natural”, Hamiltonian (function) of a <a class="existingWikiWord" href="/nlab/show/dynamical+system">dynamical system</a> is the sum of the kinetic and potential energy:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><mi>T</mi><mo>+</mo><mi>V</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> H = T + V. </annotation></semantics></math></div> <p>Knowing only <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> as a function on <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> (so as a function of <a class="existingWikiWord" href="/nlab/show/position">position</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>q</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">q^i</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math>), we can derive other quantities as functions on phase space. In particular, we have: * <a class="existingWikiWord" href="/nlab/show/velocity">velocity</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>v</mi> <mi>i</mi></msup><mo>=</mo><mo>∂</mo><mi>H</mi><mo stretchy="false">/</mo><mo>∂</mo><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow></mrow><annotation encoding="application/x-tex">v^i = \partial{H}/\partial{p_i}</annotation></semantics></math>, * <a class="existingWikiWord" href="/nlab/show/force">force</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>∂</mo><mi>H</mi><mo stretchy="false">/</mo><mo>∂</mo><mrow><msup><mi>q</mi> <mi>i</mi></msup></mrow></mrow><annotation encoding="application/x-tex">f_i = -\partial{H}/\partial{q^i}</annotation></semantics></math>.</p> <p>Setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>v</mi> <mi>i</mi></msup><mo>=</mo><mi mathvariant="normal">d</mi><msup><mi>q</mi> <mi>i</mi></msup><mo stretchy="false">/</mo><mi mathvariant="normal">d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">v^i = \mathrm{d}q^i/\mathrm{d}t</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>=</mo><mi mathvariant="normal">d</mi><msub><mi>p</mi> <mi>i</mi></msub><mo stretchy="false">/</mo><mi mathvariant="normal">d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">f_i = \mathrm{d}p_i/\mathrm{d}t</annotation></semantics></math>, we derive the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> in <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a>.</p> <h3 id="in_quantum_mechanics">In quantum mechanics</h3> <p>The <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> of a point <a class="existingWikiWord" href="/nlab/show/particle">particle</a> in the <em><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a></em> is encoded in a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H} \to \mathbb{R}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">with connection</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> over the real line – the <em>worldline</em> – of the particle.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">t \in \mathbb{R}</annotation></semantics></math> the fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℋ</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{H}_t</annotation></semantics></math> is the <strong>space of <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a>s</strong> of the system, at given parameter time <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>. Since this bundle is necessarily trivializable, we imagine fixing a trivialization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi><mo>≃</mo><msub><mi>ℋ</mi> <mn>0</mn></msub><mo>×</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H} \simeq \mathcal{H}_0 \times \mathbb{R}</annotation></semantics></math>. Then the flat connection on the bundle is canonically a <a class="existingWikiWord" href="/nlab/show/differential+form">1-form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> with values in <a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>H</mi><mspace width="thickmathspace"></mspace><mi>d</mi><mi>t</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>ℝ</mi><mo>,</mo><mi>End</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A = H \;d t \in \Omega^1(\mathbb{R}, End(\mathcal{H})) \,. </annotation></semantics></math></div> <p>The component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><mi>End</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H \in End(\mathcal{H})</annotation></semantics></math> of this canonical 1-form is the <strong>Hamilton(ian) operator</strong> (or the <em>quantum Hamiltonian</em>) of the system.</p> <p>Its <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a> is the <strong>time evolution</strong> of quantum states. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is constant as a function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>, this parallel transport assigns to the path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">t_1</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">t_2</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>:</mo><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>γ</mi></mover><msub><mi>t</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>ℋ</mi> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow></msub><mover><mo>→</mo><mrow><mi>exp</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mi>i</mi><mi>ℏ</mi></mfrac><mi>H</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>t</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow></mover><msub><mi>ℋ</mi> <mrow><msub><mi>t</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> U : (t_1 \stackrel{\gamma}{\to} t_2) \mapsto (\mathcal{H}_{t_1} \stackrel{exp\left(-\frac{i}{\hbar}H (t_2-t_1)\right)}{\to} \mathcal{H}_{t_2}) \,. </annotation></semantics></math></div> <p>If instead <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> does depend on <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> – called the case of <em>time-dependent quantum mechanics</em> – then the full formula for parallel transport applies, which is given by the <span class="newWikiWord">path-ordered exponential<a href="/nlab/new/path-ordered+exponential">?</a></span></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>:</mo><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>γ</mi></mover><msub><mi>t</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>ℋ</mi> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow></msub><mover><mo>→</mo><mrow><mi>P</mi><mi>exp</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mi>i</mi><mi>ℏ</mi></mfrac><msubsup><mo>∫</mo> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow> <mrow><msub><mi>t</mi> <mn>2</mn></msub></mrow></msubsup><mi>H</mi><mi>d</mi><mi>t</mi><mo>)</mo></mrow></mrow></mover><msub><mi>ℋ</mi> <mrow><msub><mi>t</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> U : (t_1 \stackrel{\gamma}{\to} t_2) \mapsto (\mathcal{H}_{t_1} \stackrel{P exp \left(-\frac{i}{\hbar}\int_{t_1}^{t_2}H d t\right)}{\to} \mathcal{H}_{t_2}) \,. </annotation></semantics></math></div> <p>In the physics literature this path-ordered exponential is known as the <strong><a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a></strong> .</p> <h4 id="physical_meaning_and_relation_to_unitary_transformations">Physical meaning and relation to unitary transformations</h4> <p>The <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>s of the Hamiltonian operator for a closed quantum system are exactly the energy eigenvalues of that system. Thus the Hamiltonian is interpreted as being an “energy” operator. Conservation of energy occurs when the Hamiltonian is time-independent.</p> <p>Transformations and evolutions in standard quantum mechanics are represented via <a class="existingWikiWord" href="/nlab/show/unitary+operator">unitary operator</a>s where a time evolving unitary is related to the Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> via</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">U(0,t) = </annotation></semantics></math>exp<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mi>i</mi><mi>ℏ</mi></mfrac><mi>H</mi><mi>t</mi><mo>)</mo></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex">\left(-\frac{i}{\hbar}H t\right),</annotation></semantics></math></p> <p>provided the Hamiltonian is time-independent.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/quadratic+Hamiltonian">quadratic Hamiltonian</a></li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_c2f256a74f9297292524bff7ac0438b8fdc5dbb2_1"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Legendre+transform">Legendre transform</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_c2f256a74f9297292524bff7ac0438b8fdc5dbb2_2"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lagrangian+correspondence">Lagrangian correspondence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/prequantization">prequantization</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/prequantized+Lagrangian+correspondence">prequantized Lagrangian correspondence</a></td></tr> </tbody></table> </div> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamilton%27s+equations">Hamilton's equations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator">quantum operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+action">Hamiltonian action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+form">Hamiltonian form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/propagator">propagator</a></p> </li> </ul> <div> <p><strong>higher and integrated <a class="existingWikiWord" href="/nlab/show/Kostant-Souriau+extensions">Kostant-Souriau extensions</a></strong>:</p> <p>(<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+bisections">∞-group of bisections</a> of <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi></mrow><annotation encoding="application/x-tex">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</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><mi>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>HamSympl</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla) </annotation></semantics></math></div> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></th><th>geometry</th><th>structure</th><th>unextended structure</th><th>extension by</th><th>quantum extension</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+prequantum+geometry">higher prequantum geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+symplectomorphism+%E2%88%9E-group">Hamiltonian symplectomorphism ∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Omega \mathbb{G})</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/flat+%E2%88%9E-connections">flat ∞-connections</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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+%E2%88%9E-group">quantomorphism ∞-group</a></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonians">Hamiltonians</a> under <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+symplectomorphism+group">Hamiltonian symplectomorphism group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-plectic+geometry">2-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie 2-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+2-algebra">Poisson Lie 2-algebra</a></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+n-plectomorphism">Hamiltonian 2-plectomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism 2-group</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">n-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+n-algebra">Poisson Lie n-algebra</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth n-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+n-plectomorphisms">Hamiltonian n-plectomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></td></tr> </tbody></table> <p>(extension are listed for sufficiently connected <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>)</p> </div> <h2 id="references">References</h2> <p>Named after <a class="existingWikiWord" href="/nlab/show/William+Rowan+Hamilton">William Rowan Hamilton</a>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on March 16, 2015 at 14:14:30. See the <a href="/nlab/history/Hamiltonian" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/Hamiltonian" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/Hamiltonian/25" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Hamiltonian" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Hamiltonian" accesskey="S" class="navlink" id="history" rel="nofollow">History (25 revisions)</a> <a href="/nlab/show/Hamiltonian/cite" style="color: black">Cite</a> <a href="/nlab/print/Hamiltonian" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Hamiltonian" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>