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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> <blockquote> <p>this entry needs attention</p> </blockquote> <p>Momentum is, according to <a class="existingWikiWord" href="/nlab/show/Isaac+Newton">Newton</a>, the ‘amount of motion’ in an object. For particles it is proportional to the object's <a class="existingWikiWord" href="/nlab/show/mass">mass</a> and to its <a class="existingWikiWord" href="/nlab/show/velocity">velocity</a>, and thus measured in units of mass times velocity. For photons it is proportional to the frequency of the photon.</p> <p>In <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> (including relativistic and quantum versions), to every (generalized) coordinate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>q</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">q_i</annotation></semantics></math> one associates a (generalized) momentum</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msup><mover><mi>q</mi><mo>˙</mo></mover> <mi>i</mi></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex"> p_i = \frac{\partial L}{\partial \dot{q}^i} </annotation></semantics></math></div> <p>where <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 the <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> of the system. More generally, we can say that <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> is the coordinate of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">d</mi><msup><mi>q</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\mathrm{d}q^i</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/action+functional">action form</a>.</p> <p>If <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> takes the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></msub><msub><mi>m</mi> <mi>i</mi></msub><msubsup><mover><mi>q</mi><mo>˙</mo></mover> <mi>i</mi> <mn>2</mn></msubsup><mo>−</mo><mi>U</mi><mo stretchy="false">(</mo><msub><mi>q</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>q</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2} \sum_i m_i \dot{q}_i^2 - U(q_1,\ldots,q_n)</annotation></semantics></math>, then we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>m</mi> <mi>i</mi></msub><msub><mover><mi>q</mi><mo>˙</mo></mover> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i = m_i \dot{q}_i</annotation></semantics></math>. Thus, the <strong>linear momentum</strong> of a point particle in Newtonian mechanics is traditionally <em>defined</em> as the intrinsic <a class="existingWikiWord" href="/nlab/show/mass">mass</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> times the <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><mover><mover><mi>q</mi><mo>˙</mo></mover><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{\dot{q}}</annotation></semantics></math>. However, this breaks down in some situations:</p> <ul> <li>Relativistically, the velocity factor is more complicated (tending to <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> as velocity tends to the <a class="existingWikiWord" href="/nlab/show/speed+of+light">speed of light</a>). However, it is possible to keep <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mi>m</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">p = m v</annotation></semantics></math> using a velocity-dependent relativistic mass (especially since the mass may depend on other dynamical considerations).</li> <li>In the presence of velocity-dependent forces (notably those produced by a <a class="existingWikiWord" href="/nlab/show/magnetic+field">magnetic field</a>), there is an additional position-dependent term.</li> <li>In nonlinear coordinates (such as for <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>), one not only needs to use a nonlinear velocity (such as <a class="existingWikiWord" href="/nlab/show/angular+velocity">angular velocity</a>) but also replace mass with something more complicated (such as <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a>) which is often position-dependent.</li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a>, the momentum coordinates are half of the basic coordinates on <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>. If phase space is derived as the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> of a <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, then the momentum coordinates are the new coordinates on the cotangent bundle, dual to the original coordinates on configuration space; we have 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><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></msub><msub><mi>p</mi> <mi>i</mi></msub><mi mathvariant="normal">d</mi><msup><mi>q</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\sum_i p_i \mathrm{d}q^i</annotation></semantics></math>. However, Hamiltonian mechanics cares only about phase space as a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> with a <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a>; there is no inherent distinction between position and momentum coordinates, and you can even make a <a class="existingWikiWord" href="/nlab/show/canonical+coordinate+transformation">canonical transformation</a> that swaps them (up to a minus sign).</p> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, the <span class="newWikiWord">canonical quantization<a href="/nlab/new/canonical+quantization">?</a></span> process formally replaces momentum by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ℏ</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>,</mo></mrow><annotation encoding="application/x-tex">-i \hbar \frac{\partial}{\partial x},</annotation></semantics></math> where <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 position. (Equivalently, in <span class="newWikiWord">momentum space<a href="/nlab/new/momentum+space">?</a></span>, canonical quantization replaces <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> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mi>ℏ</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>p</mi></mrow></mfrac><mo>.</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i \hbar \frac{\partial}{\partial p}.)</annotation></semantics></math> In this situation, position and momentum <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relation">fail to commute</a>.</p> <p>There is a generalization of momentum in symplectic geometry, so called <a class="existingWikiWord" href="/nlab/show/moment+map">moment map</a>.</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/plane+waves">plane waves</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_1"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ψ</mi> <mi>k</mi></msub></mrow></mover></mtd> <mtd><mi>ℂ</mi></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>exp</mi><mrow><mo>(</mo><mspace width="thinmathspace"></mspace><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>x</mi> <mi>μ</mi></msup><mspace width="thinmathspace"></mspace><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><msup><mi>x</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>exp</mi><mrow><mo>(</mo><mspace width="thinmathspace"></mspace><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mi>i</mi><msub><mi>k</mi> <mn>0</mn></msub><msup><mi>x</mi> <mn>0</mn></msup><mspace width="thinmathspace"></mspace><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>exp</mi><mrow><mo>(</mo><mspace width="thinmathspace"></mspace><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mi>i</mi><mi>ω</mi><mi>t</mi><mspace width="thinmathspace"></mspace><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{R}^{p,1} &\overset{\psi_k}{\longrightarrow}& \mathbb{C} \\ x &\mapsto& \exp\left( \, i k_\mu x^\mu \, \right) \\ (\vec x, x^0) &\mapsto& \exp\left( \, i \vec k \cdot \vec x + i k_0 x^0 \, \right) \\ (\vec x, c t) &\mapsto& \exp\left( \, i \vec k \cdot \vec x - i \omega t \, \right) } </annotation></semantics></math></div> <table><thead><tr><th>symbol</th><th>name</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_2"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/speed+of+light">speed of light</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_3"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_4"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_5"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_6"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/mass">mass</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_7"><semantics><mrow><mfrac><mi>ℏ</mi><mrow><mi>m</mi><mi>c</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\hbar}{m c}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Compton+wavelength">Compton wavelength</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_8"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_9"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_10"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_11"><semantics><mrow><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec k</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/wave+vector">wave vector</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_12"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>2</mn><mi>π</mi><mo stretchy="false">/</mo><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">\lambda = 2\pi/{\vert \vec k \vert}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/wave+length">wave length</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_13"><semantics><mrow><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>=</mo><mn>2</mn><mi>π</mi><mo stretchy="false">/</mo><mi>λ</mi></mrow><annotation encoding="application/x-tex">{\vert \vec k \vert} = 2\pi/\lambda</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/wave+number">wave number</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_14"><semantics><mrow><mi>ω</mi><mo>≔</mo><msup><mi>k</mi> <mn>0</mn></msup><mi>c</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mn>0</mn></msub><mi>c</mi><mo>=</mo><mn>2</mn><mi>π</mi><mi>ν</mi></mrow><annotation encoding="application/x-tex">\omega \coloneqq k^0 c = -k_0 c = 2\pi \nu</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/angular+frequency">angular frequency</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_15"><semantics><mrow><mi>ν</mi><mo>=</mo><mi>ω</mi><mo stretchy="false">/</mo><mn>2</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">\nu = \omega / 2 \pi</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/frequency">frequency</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_16"><semantics><mrow><mi>p</mi><mo>=</mo><mi>ℏ</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">p = \hbar k</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_17"><semantics><mrow><mover><mi>p</mi><mo stretchy="false">→</mo></mover><mo>=</mo><mi>ℏ</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec p = \hbar \vec k</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/momentum">momentum</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_18"><semantics><mrow><mi>E</mi><mo>=</mo><mi>ℏ</mi><mi>ω</mi></mrow><annotation encoding="application/x-tex">E = \hbar \omega</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/energy">energy</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_19"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><msqrt><mrow><msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover> <mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac><mo>)</mo></mrow> <mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\omega(\vec k) = c \sqrt{ \vec k^2 + \left(\frac{m c}{\hbar}\right)^2 }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon</a> <a class="existingWikiWord" href="/nlab/show/dispersion+relation">dispersion relation</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a5e76c1cf74893ccebfdcb07841f70baee02da66_20"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mover><mi>p</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><msup><mi>c</mi> <mn>2</mn></msup><msup><mover><mi>p</mi><mo stretchy="false">→</mo></mover> <mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>m</mi><msup><mi>c</mi> <mn>2</mn></msup><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">E(\vec p) = \sqrt{ c^2 \vec p^2 + (m c^2)^2 }</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/energy-momentum+relation">energy-momentum relation</a></td></tr> </tbody></table> </div></body></html> </div> <div class="revisedby"> <p> Last revised on November 7, 2017 at 13:46:13. See the <a href="/nlab/history/momentum" 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/momentum" 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/momentum/12" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/momentum" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/momentum" accesskey="S" class="navlink" id="history" rel="nofollow">History (12 revisions)</a> <a href="/nlab/show/momentum/cite" style="color: black">Cite</a> <a href="/nlab/print/momentum" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/momentum" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>