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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="algebraic_quantum_field_theory">Algebraic Quantum Field Theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative</a>, <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical</a>) <a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a> <h2 id="concepts">Concepts</h2> <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>, <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a>, <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+history">field history</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a><a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> </li> </ul> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>, <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+net">field net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> <a class="existingWikiWord" href="/nlab/show/collapse+of+the+wave+function">collapse of the wave function</a>/<a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>/<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a>, <a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>) </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> </li> </ul> </div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <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> </li> </ul> <hr /> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <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/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> <ul> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <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> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <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> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <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> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <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> </li> <li> examples <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> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <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> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <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> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <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> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </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='#example'>Example</a></li> <li><a href='#method'>Method</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> What is called Wick rotation (after <a class="existingWikiWord" href="/nlab/show/Gian-Carlo+Wick">Gian-Carlo Wick</a>) is a method in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> for finding a construction in <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic field theory</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> or, more generally, on <a class="existingWikiWord" href="/nlab/show/Lorentzian+manifolds">Lorentzian manifolds</a>, from a related construction in <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean field theory</a> on <a class="existingWikiWord" href="/nlab/show/Riemannian+manifolds">Riemannian manifolds</a> in a way that involves or generalizes the idea of <a class="existingWikiWord" href="/nlab/show/analytic+continuation">analytic continuation</a> for the <a class="existingWikiWord" href="/nlab/show/time">time</a> <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a>. <center> <img src="https://ncatlab.org/nlab/files/WickRotation.jpg" width="400" /> </center> <blockquote> The <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a> for a complexified <a class="existingWikiWord" href="/nlab/show/time">time</a> <a class="existingWikiWord" href="/nlab/show/coordinate">coordinate</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>+</mo><mi>i</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">t + i s</annotation></semantics></math>. After Wick rotation it is the <a class="existingWikiWord" href="/nlab/show/imaginary+part">imaginary part</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math> that replaces the “<a class="existingWikiWord" href="/nlab/show/real+part">real</a>” 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>. graphics grabbed from <a href="#FullingRuijsenaars87">Fulling-Ruijsenaars 87</a> </blockquote> This is motivated by the observation that the <a class="existingWikiWord" href="/nlab/show/Minkowski+metric">Minkowski metric</a> (with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1,1,1,1)</annotation></semantics></math> convention for <a class="existingWikiWord" href="/nlab/show/spacetime+signature">spacetime signature</a>) and the four-Euclidean metric are equivalent if the <a class="existingWikiWord" href="/nlab/show/time">time</a> components of either are allowed to have <a class="existingWikiWord" href="/nlab/show/imaginary+part">imaginary</a> values. Hence Wick rotation, when it applies, involves <a class="existingWikiWord" href="/nlab/show/analytic+continuation">analytic continuation</a> of <a class="existingWikiWord" href="/nlab/show/n-point+functions">n-point functions</a> to <a class="existingWikiWord" href="/nlab/show/complex+number">complex valued</a> <a class="existingWikiWord" href="/nlab/show/time">time</a> <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a>. For <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>/<a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d,1}</annotation></semantics></math>/<a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d+1}</annotation></semantics></math> Wick rotation is rigorously understood and takes the form of a <a class="existingWikiWord" href="/nlab/show/theorem">theorem</a> relating the <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> for <a class="existingWikiWord" href="/nlab/show/n-point+functions">n-point functions</a> in <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic field theory</a> to the <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+axioms">Osterwalder-Schrader axioms</a> for <a class="existingWikiWord" href="/nlab/show/correlators">correlators</a> in <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean field theory</a>. See for instance <a href="#FullingRuijsenaars87">Fulling-Ruijsenaars 87, section 2</a> for a clear account. More generally, in this context Wick rotation applies to relativistic field theory in <a class="existingWikiWord" href="/nlab/show/thermal+equilibrium">thermal equilibrium</a> <a class="existingWikiWord" href="/nlab/show/states">states</a> at <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/temperature">temperature</a> <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> (“<a class="existingWikiWord" href="/nlab/show/KMS+states">KMS states</a>”), and then relates this to <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean field theory</a> with compact/periodic Euclidean time of length <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\beta = 1/T</annotation></semantics></math> (see <a href="#FullingRuijsenaars87">Fulling-Ruijsenaars 87, section 3</a>): <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mrow><mrow><mtable><mtr><mtd><mtext>relativistic field theory</mtext></mtd></mtr> <mtr><mtd><mtext>on Minkowski spacetime</mtext></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>,</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><mtext>in a thermal equilibrium state</mtext></mtd></mtr> <mtr><mtd><mtext>at temperature</mtext><mspace width="thickmathspace"></mspace><mi>T</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mover><mo>↔</mo><mtext>Wick rotation</mtext></mover><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mtext>Euclidean field theory</mtext></mtd></mtr> <mtr><mtd><mtext>on Euclidean space</mtext></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>×</mo><msubsup><mi>S</mi> <mi>β</mi> <mn>1</mn></msubsup></mtd></mtr> <mtr><mtd><mtext>with compact/periodic Euclidean time</mtext></mtd></mtr> <mtr><mtd><mtext>of length</mtext><mspace width="thickmathspace"></mspace><mi>β</mi><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mi>T</mi></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd><mphantom><mi>A</mi></mphantom></mtd></mtr> <mtr><mtd><munder><munder><mrow><msub><mrow><mo>⟨</mo><mi>T</mi><mo stretchy="false">|</mo><mo>:</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mi>⋯</mi><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">|</mo><mi>T</mi><mo>⟩</mo></mrow> <mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mtext>equal-time n-point function</mtext></mrow><mrow><mtext>of relativistic fields</mtext></mrow></mfrac></mrow><mrow><mtext> in thermal equilibrium state </mtext><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mi>T</mi><mo stretchy="false">⟩</mo></mrow></mfrac></munder></mtd> <mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><munder><mrow><msub><mrow><mo>⟨</mo><mn>0</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mi>⋯</mi><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mn>0</mn><mo>⟩</mo></mrow> <mrow><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>×</mo><msubsup><mi>S</mi> <mi>β</mi> <mn>1</mn></msubsup></mrow></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mtext>correlator of Euclidean fields</mtext></mrow><mrow><mtext> for "Euclidean time" of periodicity</mtext><mspace width="thickmathspace"></mspace><mi>β</mi><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mi>T</mi></mrow></mfrac></munder></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \left. \array{ \text{relativistic field theory} \\ \text{on Minkowski spacetime} \\ \mathbb{R}^{d,1} \\ \text{in a thermal equilibrium state} \\ \text{at temperature}\; T } \right\} & \;\;\;\; \overset{ \text{Wick rotation} }{\leftrightarrow} \;\;\;\; & \left\{ \array{ \text{Euclidean field theory} \\ \text{on Euclidean space} \\ \mathbb{R}^d \times S^1_{\beta} \\ \text{with compact/periodic Euclidean time} \\ \text{of length} \; \beta = 1/T } \right. \\ \phantom{A} \\ \underset{ { \text{equal-time n-point function} \atop \text{of relativistic fields} } \atop \text{ in thermal equilibrium state } \; \vert T\rangle }{ \underbrace{ \left\langle T\vert :\mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t) : \vert T \right\rangle_{\mathbb{R}^{d,1}} }} &\;=\;& \underset{ \text{correlator of Euclidean fields} \atop \text{ for "Euclidean time" of periodicity}\; \beta = 1/T }{ \underbrace{ \left\langle 0 \vert \mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t) \vert 0 \right\rangle_{\mathbb{R}^{d} \times S^1_{\beta}} }} } </annotation></semantics></math></div> In this form, Wick rotation is also known as <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal quantum field theory</a>. See there for more. <center> <img src="https://ncatlab.org/nlab/files/EuclideanTime.jpg" width="580" /> </center> <blockquote> graphics grabbed form <a href="#FrolovZelnikov11">Frolov-Zelnikov 11</a> </blockquote> Wick rotation also applies on suitable <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a>-<a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a> spring thermodynamics, such as the <a class="existingWikiWord" href="/nlab/show/Bekenstein-Hawking+entropy">Bekenstein-Hawking entropy</a>, find elegant explanations, at least at the level of the manipulation of formulas (see e.g. <a href="#FullingRuijsenaars87">Fulling-Ruijsenaars 87, section 4</a>). <h3 id="example">Example</h3> Consider the Minkowski metric with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1,1,1,1</annotation></semantics></math> convention for the tensor: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msup><mi>s</mi> <mn>2</mn></msup><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>d</mi><mi>t</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>d</mi><mi>x</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>d</mi><mi>y</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>d</mi><mi>z</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">d s^{2}= -(d t)^{2} + (d x)^{2} + (d y)^{2} + (d z)^{2}</annotation></semantics></math> and the four-dimensional Euclidean metric: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msup><mi>s</mi> <mn>2</mn></msup><mo>=</mo><mi>d</mi><msup><mi>τ</mi> <mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>d</mi><mi>x</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>d</mi><mi>y</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>d</mi><mi>z</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">d s^{2}= d \tau^{2} + (d x)^{2} + (d y)^{2} + (d z)^{2}</annotation></semantics></math>. Notice that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>t</mi><mo>=</mo><mi>i</mi><mo>⋅</mo><mi>d</mi><mi>τ</mi></mrow><annotation encoding="application/x-tex">d t = i\cdot d \tau</annotation></semantics></math>, the two are equivalent. <h3 id="method">Method</h3> A typical method for employing Wick rotation would be to make the substitution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>=</mo><mi>i</mi><mi>τ</mi></mrow><annotation encoding="application/x-tex">t=i\tau</annotation></semantics></math> in a problem in Minkowski space. The resulting problem is in Euclidean space and is sometimes easier to solve, after which a reverse substitution can (sometimes) be performed, yielding a solution to the original problem. Technically, this works for any four-vector comparison between Minkowski space and Euclidean space, not just for space-time intervals. <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/reflection+positivity">reflection positivity</a></li> </ul> <h2 id="references">References</h2> Wick rotation between <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic field theory</a> in terms of <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> for <a class="existingWikiWord" href="/nlab/show/n-point+functions">n-point functions</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> and <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean field theory</a> in terms of <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+axioms">Osterwalder-Schrader axioms</a> for <a class="existingWikiWord" href="/nlab/show/correlators">correlators</a> on <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> is due to <ul> <li id="OsterwalderSchrader73"><a class="existingWikiWord" href="/nlab/show/Konrad+Osterwalder">Konrad Osterwalder</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Schrader">Robert Schrader</a>, Axioms for Euclidean Green’s functions, Comm. Math. Phys. Volume 31, Number 2 (1973), 83-112 (<a href="https://projecteuclid.org/euclid.cmp/1103858969">Euclid:1103858969</a>)</li> </ul> The generalization of this for <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/thermal+equilibrium">thermal equilibrium</a> <a class="existingWikiWord" href="/nlab/show/vacuum+states">vacuum states</a>, relating to <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal quantum field theory</a> with compact/periodic Euclidean time is discussed in <ul> <li id="KleinLandau81">Abel Klein, Lawrence Landau, Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle, Pacific Journal of Mathematics, Vol. 94, No. 2, 1981 (<a href="https://msp.org/pjm/1981/94-2/p12.xhtml">DOI: 10.2140/pjm.1981.94.341</a>, <a href="https://msp.org/pjm/1981/94-2/pjm-v94-n2-p12-s.pdf">pdf</a>)</li> </ul> The idea here apparently goes back to <ul> <li id="Bloch58">Claude Bloch, Sur la détermination de l’état fondamental d’un système de particules, Nucl. Phys. 7 (1958) 451</li> </ul> This has maybe first been made precise, for the case of 1+1 dimensions, in <ul> <li id="HoeghKrohn74"><a class="existingWikiWord" href="/nlab/show/Raphael+H%C3%B8egh-Krohn">Raphael Høegh-Krohn</a>, Relativistic Quantum Statistical Mechanics in two-dimensional Space-Time, Communications in Mathematical Physics 38.3 (1974): 195-224 (<a href="https://www.duo.uio.no/bitstream/handle/10852/44072/1973-22.pdf">pdf</a>)</li> </ul> Good review on the relation to <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal quantum field theory</a> and <a class="existingWikiWord" href="/nlab/show/black+hole+thermodynamics">black hole thermodynamics</a> is in <ul> <li id="FullingRuijsenaars87"> S.A. Fulling, S.N.M. Ruijsenaars, Temperature, periodicity and horizons, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 (<a href="https://www1.maths.leeds.ac.uk/~siru/papers/p26.pdf">pdf</a>, <a href="https://doi.org/10.1016/0370-1573(87)90136-0">doi:10.1016/0370-1573(87)90136-0</a>) </li> <li id="GibbonsPerry78"> <a class="existingWikiWord" href="/nlab/show/Gary+Gibbons">Gary Gibbons</a>, Malcolm J. Perry, Black Holes and Thermal Green Functions, Vol. 358, No. 1695 (1978) (<a href="https://www.jstor.org/stable/79482">jstor:79482</a>) </li> </ul> New formalization in terms of complex metrics: <ul> <li><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, Wick rotation and the positivity of energy in quantum field theory, The Quarterly Journal of Mathematics 72 1-2 (2021) 673–699 [<a href="https://arxiv.org/abs/2105.10161">arXiv:2105.10161</a>, <a href="https://doi.org/10.1093/qmath/haab027">doi:10.1093/qmath/haab027</a>]</li> </ul> review talks: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, Wick rotation and the positivity of energy in quantum field theory, talk at Institut des Hautes Études Scientifiques (IHÉS), June 2014 (<a href="https://www.youtube.com/watch?feature=player_embedded&v=vTvXHL6ZJik">video recording</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, Wick Rotation and the Positivity of Energy in Quantum Field Theory, talk at IAS Physics Group Meeting, December 2021 (<a href="https://www.ias.edu/video/wick-rotation-and-positivity-energy-quantum-field-theory">video recording</a>) </li> </ul> Discussion of thermal Wick rotation on global <a class="existingWikiWord" href="/nlab/show/anti-de+Sitter+spacetime">anti-de Sitter spacetime</a> (which is already periodic in real time) is in <ul> <li id="AllenFolacciGibbons87">B. Allen, A. Folacci, <a class="existingWikiWord" href="/nlab/show/Gary+Gibbons">Gary Gibbons</a>, Anti-de Sitter space at finite temperature, Physics Letters B Volume 189, Issue 3, 7 May 1987, Pages 304-310 (<a href="https://doi.org/10.1016/0370-2693(87)91437-7">doi:10.1016/0370-2693(87)91437-7</a>)</li> </ul> See also <ul> <li> Dirk Schlingemann, From euclidean field theory to quantum field theory (<a href="http://arxiv.org/abs/hep-th/9802035">arXiv:hep-th/9802035</a>) </li> <li id="Witten13"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, The Feynman <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">i \epsilon</annotation></semantics></math> in String Theory (<a href="http://arxiv.org/abs/1307.5124">arXiv:1307.5124</a>) </li> </ul> On the (non-)existence of Wick rotation for <a class="existingWikiWord" href="/nlab/show/quantum+field+theory+on+curved+spacetimes">quantum field theory on curved spacetimes</a>: MathOverflow comments: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Igor+Khavkine">Igor Khavkine</a> (<a href="https://mathoverflow.net/a/165317/381">MO:a/165317</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Robert+Bryant">Robert Bryant</a> (<a href="https://mathoverflow.net/a/165345/381">MO:a/165345</a>) </li> </ul> See also: <ul> <li id="HollandsWald14"><a class="existingWikiWord" href="/nlab/show/Stefan+Hollands">Stefan Hollands</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Wald">Robert Wald</a>, p. 7-8 in: Quantum fields in curved spacetime, Physics Reports Volume 574, 16 April 2015, Pages 1-35 (<a href="https://arxiv.org/abs/1401.2026">arXiv:1401.2026</a>, <a href="https://doi.org/10.1016/j.physrep.2015.02.001">doi:10.1016/j.physrep.2015.02.001</a>)</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on November 16, 2023 at 05:04:53. 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