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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> theory (physics) </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4649/#Item_23" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#general_idea'>General idea</a></li> <li><a href='#formalization'>Formalization</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#general_principles'>General principles</a></li> <li><a href='#types_of_theories'>Types of theories</a></li> <li><a href='#the_fundamental_phenomenological_theories'>The fundamental phenomenological theories</a></li> <li><a href='#TheoriesAndTheirModels'>Theories and their models</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="general_idea">General idea</h3> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a> the term <em>theory</em> or <em>physical theory</em> traditionally refers, somewhat vaguely, to a given set of notions and rules, usually formulated in the language of <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a>, that describe how some <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a> or class of physical systems behaves. Typically these systems are highly idealized, in that the theories describe only certain aspects.</p> <p>Often a given such theory depends on many free parameters. When a choice of such parameters is made or the range of the parameters is being restricted one tends to call the result a <em><a class="existingWikiWord" href="/nlab/show/model+%28in+theoretical+physics%29">model (in theoretical physics)</a></em>. For more on this see <em><a href="#TheoriesAndTheirModels">Theories and their Models</a></em> below.</p> <p>The most accurate general theory of fundamental physics known is <em><a class="existingWikiWord" href="/nlab/show/Einstein+gravity">Einstein gravity</a></em> and <em><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></em>. The best available choices of parameters in this general theory that make it fit the specifics of the observed world (<a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a>) are <a class="existingWikiWord" href="/nlab/show/model+%28in+theoretical+physics%29">models</a> which accordingly are called the <em>standard models</em>: there is the <em><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></em> and the <em><a class="existingWikiWord" href="/nlab/show/standard+model+of+cosmology">standard model of cosmology</a></em>.</p> <h3 id="formalization">Formalization</h3> <p>Beware that, therefore, the use of the terms <em>theory</em> and <em>model</em> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is <em>different</em> from the same terms as used in <a class="existingWikiWord" href="/nlab/show/logic">logic</a> (see at <a class="existingWikiWord" href="/nlab/show/theory+%28logic%29">theory (logic)</a> and <a class="existingWikiWord" href="/nlab/show/model+%28logic%29">model (logic)</a>).</p> <p>But most theories of fundamental physics (and many theories of effective physics such as <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a>) fit into a pattern that can be <a class="existingWikiWord" href="/nlab/show/axiom">axiomatized</a> at least to some extent:</p> <p>these physical theories are specified by a (<a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local</a>/<a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended</a>) <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> on a space of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> over a given <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <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>, hence by an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S \;\colon\; [X, \mathbf{Fields}]_{\mathbf{H}} \to U(1) \,. </annotation></semantics></math></div> <p>In particular the corresponding <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a> has as its “space of models” the <a class="existingWikiWord" href="/nlab/show/critical+locus">critical locus</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>ϕ</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mrow></munder><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mi>S</mi> <mi>ϕ</mi></msub><mo>≃</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underset{\phi \in [X, \mathbf{Fields}]_{\mathbf{H}}}{\sum} ( \mathbf{d} S_\phi \simeq 0 ) </annotation></semantics></math></div> <p>of such an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, the space of solutions of the <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a>. A point in this space is a single “physically realizable” configuration of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> in this theory, disregarding <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>-corrections, and a small-parameter subspace is often referred to as “a model” of the theory.</p> <p>In this perspective of classical field theory, two different action functionals on different spaces of fields but with <a class="existingWikiWord" href="/nlab/show/equivalence">equivalent</a> <a class="existingWikiWord" href="/nlab/show/critical+loci">critical loci</a> are regarded as “equivalent physical theories”. One often sees the term “classically equivalent” for this notion used in the literature.</p> <p>But the full <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> determined by a <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a>/<a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> depends on more than just the <a class="existingWikiWord" href="/nlab/show/critical+locus">critical locus</a>, which is just something like the lowest order approximation to the quantum theory (in a sense that can be made precise, for instance in <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> in terms of <a class="existingWikiWord" href="/nlab/show/power+series">power series</a> developments in <a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a>.)</p> <h2 id="examples">Examples</h2> <h3 id="general_principles">General principles</h3> <p>One broad way of classifying physical theories is by the extent to which they take <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> into account. We have</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a></p> </li> <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/quantum+field+theory">quantum field theory</a></p> </li> </ul> <p>Here <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> is the most refined framework, which underlies the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>.</p> <p>The notion of quantum field theory, fundamental as it is, is quite flexible and in particular it naturally captures the concept that a given quantum field theory only describes phenomena that occur below a certain <a class="existingWikiWord" href="/nlab/show/energy">energy</a> range and treats all phenomena at higher energy as the average over an unspecified more refined theory. This is the notion of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a>.</li> </ul> <p>Crucially, <a class="existingWikiWord" href="/nlab/show/Einstein+gravity">Einstein gravity</a> is not known to have a formulation as a <em>fundamental</em> quantum field theory with finitely many unspecified parameters (<a class="existingWikiWord" href="/nlab/show/renormalization">renormalizable</a>). But it may well be a <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a>, the approximation to a more refined physical theory valid at higher energies. (This is the issue of <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a>.) A proposal for a physical theory that achieves this is called <em><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></em>.</p> <h3 id="types_of_theories">Types of theories</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> </li> </ul> <h3 id="the_fundamental_phenomenological_theories">The fundamental phenomenological theories</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Einstein-Maxwell+theory">Einstein-Maxwell theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Einstein-Yang-Mills+theory">Einstein-Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Einstein-Maxwell-Yang-Mills-Dirac-Higgs+theory">Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> <a class="existingWikiWord" href="/nlab/show/standard+model+of+cosmology">and cosmology</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a>:</th><th><a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein</a>-</th><th><a class="existingWikiWord" href="/nlab/show/Einstein-Yang-Mills+theory">Yang-Mills</a>-</th><th><a class="existingWikiWord" href="/nlab/show/Einstein-Yang-Mills-Dirac+theory">Dirac</a>-</th><th><a class="existingWikiWord" href="/nlab/show/Einstein-Yang-Mills-Dirac-Higgs+theory">Higgs</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electroweak+field">electroweak</a> and <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong nuclear force</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fermion">fermionic</a> <a class="existingWikiWord" href="/nlab/show/matter">matter</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> content:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/vielbein+field">vielbein field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_fe1ab21b0c4e83aef0f095e4781c7e48fbcdb78f_1"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_fe1ab21b0c4e83aef0f095e4781c7e48fbcdb78f_2"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spinor">spinor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_fe1ab21b0c4e83aef0f095e4781c7e48fbcdb78f_3"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_fe1ab21b0c4e83aef0f095e4781c7e48fbcdb78f_4"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a>:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/scalar+curvature">scalar curvature</a> density</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> squared</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> component density</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> squared + potential density</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_fe1ab21b0c4e83aef0f095e4781c7e48fbcdb78f_5"><semantics><mrow><mi>L</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">L = </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_fe1ab21b0c4e83aef0f095e4781c7e48fbcdb78f_6"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mi>vol</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>+</mo></mrow><annotation encoding="application/x-tex">R(e) vol(e) + </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_fe1ab21b0c4e83aef0f095e4781c7e48fbcdb78f_7"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>F</mi> <mo>∇</mo></msub><mo>∧</mo><msub><mo>⋆</mo> <mi>e</mi></msub><msub><mi>F</mi> <mo>∇</mo></msub><mo stretchy="false">⟩</mo><mo>+</mo></mrow><annotation encoding="application/x-tex">\langle F_\nabla \wedge \star_e F_\nabla\rangle + </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_fe1ab21b0c4e83aef0f095e4781c7e48fbcdb78f_8"><semantics><mrow><mo stretchy="false">(</mo><mi>ψ</mi><mo>,</mo><msub><mi>D</mi> <mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow></msub><mi>ψ</mi><mo stretchy="false">)</mo><mi>vol</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>+</mo></mrow><annotation encoding="application/x-tex"> (\psi , D_{(e,\nabla)} \psi) vol(e) + </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_fe1ab21b0c4e83aef0f095e4781c7e48fbcdb78f_9"><semantics><mrow><mo>∇</mo><mover><mi>H</mi><mo stretchy="false">¯</mo></mover><mo>∧</mo><msub><mo>⋆</mo> <mi>e</mi></msub><mo>∇</mo><mi>H</mi><mo>+</mo><mrow><mo>(</mo><mi>λ</mi><msup><mrow><mo stretchy="false">|</mo><mi>H</mi><mo stretchy="false">|</mo></mrow> <mn>4</mn></msup><mo>−</mo><msup><mi>μ</mi> <mn>2</mn></msup><msup><mrow><mo stretchy="false">|</mo><mi>H</mi><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><mi>vol</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e) </annotation></semantics></math></td></tr> </tbody></table> </div> <h3 id="TheoriesAndTheirModels">Theories and their models</h3> <div class="num_example"> <h6 id="example">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> is a physical theory which asserts that <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is modeled by a <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo-Riemannian manifold</a> equipped with certain further <a class="existingWikiWord" href="/nlab/show/force">force</a> and <a class="existingWikiWord" href="/nlab/show/matter">matter</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>, such that this data satisfies <a class="existingWikiWord" href="/nlab/show/Einstein+equations">Einstein equations</a>. But if one furthermore specifies a particular such pseudo-Riemannian manifold etc. one may call this a <em>model</em> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>/<a class="existingWikiWord" href="/nlab/show/cosmology">cosmology</a>. The <em><a class="existingWikiWord" href="/nlab/show/FRW+model">FRW model</a></em> is an example: here one specifies that the given pseudo-Riemannian metric and the <a class="existingWikiWord" href="/nlab/show/matter">matter</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> content is <em>homogenous</em> and <em>isotropic</em>. This is highly restrictive but still does not single out a unique solution. The remaining parameter is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">k \in \{-1,0,1\}</annotation></semantics></math>, determining whether in this solution space has negative, positive or vanishing constant curvature.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a>, <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/coordination">coordination</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theory+of+everything">theory of everything</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a></p> </li> <li> <p><a href="http://ncatlab.org/nlab/show/string+theory+FAQ#AsideHowToPhysicalTheorieyGenerallyMakePredictionsAnyway">string theory FAQ – How do physical theories generally make predictions, anyway?</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/James+Wells">James Wells</a>, <em>Lexicon of Theory Qualities</em> (<a href="http://www-personal.umich.edu/~jwells/prms/prm8.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 7, 2018 at 11:02:24. 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