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17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> nLab Euclidean field theory </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <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/9250/#Item_10" 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> </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="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="measure_and_probability_theory">Measure and probability theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a> <a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a> (<a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a>) <h2 id="measure_theory">Measure theory</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a> </li> </ul> <h2 id="probability_theory">Probability theory</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/state">state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/states+in+AQFT+and+operator+algebra">in AQFT and operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+theormodynamics">generalized second law</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>, <a class="existingWikiWord" href="/nlab/show/neural+networks">neural networks</a></li> </ul> </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'>General</a></li> <li><a href='#wick_rotation_to_relativistic_field_theory'>Wick rotation to Relativistic Field theory</a></li> <li><a href='#TemporalCompactificationToThermalRelativisticFieldTheory'>Temporal compactification to Thermal relativistic field theory</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general_2'>General</a></li> <li><a href='#thermal_quantum_field_theory'>Thermal quantum field theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="general">General</h3> In <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a> one speaks of Euclidean field theory if the underlying <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> on which the <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> are defined are <a class="existingWikiWord" href="/nlab/show/Riemannian+manifolds">Riemannian manifolds</a>, as opposed to <a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian</a> <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a> used in <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic field theory</a>, hence locally are <a class="existingWikiWord" href="/nlab/show/Euclidean+spaces">Euclidean spaces</a> instead of <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetimes">Minkowski spacetimes</a>, whence the name “Euclidean field theory.” Concretely this means that in Euclidean field theory the <a class="existingWikiWord" href="/nlab/show/local+field+theory">locality</a> condition on the <a class="existingWikiWord" href="/nlab/show/net+of+observables">net of</a> <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> requires observables <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A, B</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/commutator">commute</a> as soon as their <a class="existingWikiWord" href="/nlab/show/spacetime+supports">spacetime supports</a> are <a class="existingWikiWord" href="/nlab/show/disjoint+subset">disjoint</a> at all <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>supp</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> supp(A) \cap supp(B) = \emptyset \;\;\;\;\Rightarrow\;\;\;\; [A,B] = 0 \,. </annotation></semantics></math></div> This is in contrast to the analogous condition in <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic field theory</a> whose <a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a> requires this implication only if the two <a class="existingWikiWord" href="/nlab/show/supports">supports</a> are in addition <a class="existingWikiWord" href="/nlab/show/spacelike">spacelike</a>-separated. Equivalently this means that the <a class="existingWikiWord" href="/nlab/show/n-point+functions">n-point functions</a> of Euclidean field theories are <a class="existingWikiWord" href="/nlab/show/distributions+of+several+variables">distributions of several variables</a> with <a class="existingWikiWord" href="/nlab/show/singularities">singularities</a> on the <a class="existingWikiWord" href="/nlab/show/fat+diagonal">fat diagonal</a> (instead of on all of the relative <a class="existingWikiWord" href="/nlab/show/light+cones">light cones</a>, as for <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic field theory</a>). This means that Euclidean <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-point functions <a class="existingWikiWord" href="/nlab/show/restriction+of+distributions">restrict</a> to <a class="existingWikiWord" href="/nlab/show/non-singular+distributions">non-singular distributions</a> on the <a class="existingWikiWord" href="/nlab/show/configuration+spaces+of+points">configuration space of n points</a>, allowing to express <a class="existingWikiWord" href="/nlab/show/correlators+as+differential+forms+on+configuration+spaces+of+points">correlators as differential forms on configuration spaces of points</a>. Systematic discussion of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative</a> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalized</a> Euclidean field theory from this perspective is due to <a href="#BergbauerBrunettiKreimer09">Bergbauer-Brunetti-Kreimer 09</a>, <a href="#Berghoff14a">Berghoff 14a</a>, <a href="#Berghoff14b">Berghoff 14b</a>. This Euclidean locality property applies in particular in <a class="existingWikiWord" href="/nlab/show/statistical+mechanics">statistical mechanics</a>, where the “<a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>” of the field theory are not thought of as encoding the <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>-behaviour of <a class="existingWikiWord" href="/nlab/show/fundamental+particles">fundamental particles</a> as governed by <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a>, but instead the spatial <a class="existingWikiWord" href="/nlab/show/expectation+values">expectation values</a> (at any given time) of <a class="existingWikiWord" href="/nlab/show/equilibrium">equilibrium</a> <a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamic</a>-processes governed by <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a>. An archetypical example of a Euclidean field theory in this thermodynamic sense is the <a class="existingWikiWord" href="/nlab/show/Ising+model">Ising model</a>. Generally, most <a class="existingWikiWord" href="/nlab/show/2d+conformal+field+theories">2d conformal field theories</a> considered are Euclidean field theories and (should) have the interpretation of describing <a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamical</a> systems “at criticality”. However, other Euclidean <a class="existingWikiWord" href="/nlab/show/2d+CFTs">2d CFTs</a> are not necessarily regarded as thermodynamical systems, notably the <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a>-field theories defining a <a class="existingWikiWord" href="/nlab/show/string+perturbation+series">string perturbation series</a> in <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative string theory</a>. Another broad class of examples of Euclidean field theories are <a class="existingWikiWord" href="/nlab/show/topological+quantum+field+theories">topological quantum field theories</a> after their <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a>-<a class="existingWikiWord" href="/nlab/show/symmetry">symmetry</a> is partially <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixed</a> by a choice of <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>. The archetypical example here is <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative</a> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>, see <a href="Chern-Simons+theory#FeynmanPerturbationSeries">there</a> for more. <h3 id="wick_rotation_to_relativistic_field_theory">Wick rotation to Relativistic Field theory</h3> Despite this superficially stark contrast between Euclidean and relativistic field theory, the two turn out to be tightly related to each other, at least under some conditions, in a subtle way that involves and generalizes the concept of <a class="existingWikiWord" href="/nlab/show/analytic+continuation">analytic continuation</a> from <a class="existingWikiWord" href="/nlab/show/complex+analysis">complex analysis</a>, here this is called <a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>. Roughly this says that <a class="existingWikiWord" href="/nlab/show/propagators">propagators</a> and hence <a class="existingWikiWord" href="/nlab/show/n-point+functions">n-point functions</a> of <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> <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> (may) have <a class="existingWikiWord" href="/nlab/show/analytic+continuation">analytic continuation</a> to <a class="existingWikiWord" href="/nlab/show/complex+number">complex values</a> of the <a class="existingWikiWord" href="/nlab/show/time">time</a>-<a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a>, such that replacing <a class="existingWikiWord" href="/nlab/show/real+number">real</a> time with <a class="existingWikiWord" href="/nlab/show/imaginary+number">imaginary</a> time turns these <a class="existingWikiWord" href="/nlab/show/n-point+functions">n-point functions</a> into those of a <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean field theory</a> on <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>, and vice versa. Precise formulation of the conditions that go into this <a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a> between <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic field theory</a> and Euclidean field theory is the content of the <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a>. <h3 id="TemporalCompactificationToThermalRelativisticFieldTheory">Temporal compactification to Thermal relativistic field theory</h3> In fact this relation goes deeper still: Under suitable conditions the Euclidean field theory not on <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> but on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup><mo>×</mo><msubsup><mi>S</mi> <mi>β</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d \times S^1_\beta</annotation></semantics></math>, with the <a class="existingWikiWord" href="/nlab/show/circle">circle</a>-<a class="existingWikiWord" href="/nlab/show/product+space">factor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>β</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_{\beta}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/length">length</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math>, corresponds to relativistic field theory 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> in a <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> that represents <a class="existingWikiWord" href="/nlab/show/thermal+equilibrium">thermal equilibrium</a> at <a class="existingWikiWord" href="/nlab/show/inverse+temperature">inverse temperature</a> <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>. (The previous case of Euclidean field theory on <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> may be thought of as the special case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>→</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">\beta \to \infty</annotation></semantics></math>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">T \to 0</annotation></semantics></math>.) This curious relation of <a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a> with “compact peridodic Euclidean time” makes, when it applies, Euclidean field theory be a unification of <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic field theory</a> with <a class="existingWikiWord" href="/nlab/show/statistical+mechanics">statistical mechanics</a>/<a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a>, then called thermal quantum field theory or quantum statistical field theory or similar. <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><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> Notice that the evident <a class="existingWikiWord" href="/nlab/show/symmetry+breaking">breaking</a> of <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz symmetry</a> on the right side of this correspondence is perfectly consistent with what happens on the left hand: A thermal vacuum state in Minkowski spacetime also singles out a preferred Lorentz frame. The basic idea of this relation seems to go back to <a href="#Bloch58">Bloch 58</a>. The physics literature often states this suggestively but informally in terms of <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>-imagery, see e.g. <a href="#Moore03">Moore 03, section 1.1</a>. The first precise formulation seems to be due to <a href="#HoeghKrohn74">Høegh-Krohn 74</a> (in 1+1 dimensions) and a more comprehensive discussion in view of the <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> for compact Euclidean time is due to <a href="#KleinLandau81">Klein-Landau 81</a>. The use of a “periodic Euclidean time coordinate” is also known as <a class="existingWikiWord" href="/nlab/show/Matsubara+formalism">Matsubara formalism</a> (e.g. <a href="#LandsmanVanWert87">Landsman-vanWert 87, section 2.3.1</a>) and specifically the condition that the periodicity has to be <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 \coloneqq 1/T</annotation></semantics></math> is known as the <a class="existingWikiWord" href="/nlab/show/KMS+conditions">KMS conditions</a> for a <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a> (For Kubo-Martin-Schwinger, due to <a href="#Kubo57">Kubo 57</a>, <a href="#MartinSchwinger59">Martin-Schwinger 59</a> with its final form due to <a href="#HaagHugenholtzWinnink67">Haag-Hugenholtz-Winnink 67</a>, see <a href="#FullingRuijsenaars87">Fulling-Ruijsenaars 87, section 3.1</a>). Beware that literature discussing the KMS-condition often does not make the periodicity of Euclidean time explicit, and vice versa. This is clarified in <a href="#FullingRuijsenaars87">Fulling-Ruijsenaars 87, sections 2 and 3</a>. General introduction to Euclidean and thermal field theory includes <a href="#Thoma00">Thoma 00, section 2.2</a>, <a href="#PeetersZamaklar09">Peeters-Zamaklar 09, section 1.3</a>. <h2 id="examples">Examples</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ising+model">Ising model</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/inverse+temperature">inverse temperature</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a>, <a class="existingWikiWord" href="/nlab/show/caloron">caloron</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/2d+CFT">2d CFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/lattice+gauge+theory">lattice gauge theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinite-temperature+thermal+field+theory">infinite-temperature thermal field theory</a> </li> </ul> <h2 id="references">References</h2> <h3 id="general_2">General</h3> General introduction to Euclidean field theory includes <ul> <li id="PeetersZamaklar09"><a class="existingWikiWord" href="/nlab/show/Kasper+Peeters">Kasper Peeters</a>, <a class="existingWikiWord" href="/nlab/show/Marija+Zamaklar">Marija Zamaklar</a>, Euclidean Field Theory, Lecture notes 2009-2011 (<a href="http://maths.dur.ac.uk/users/kasper.peeters/eft.html">web</a>, <a href="http://maths.dur.ac.uk/users/kasper.peeters/pdf/eft.pdf">pdf</a>)</li> </ul> A systematic development of Euclidean <a class="existingWikiWord" href="/nlab/show/renormalization">renormalized</a> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> with <a class="existingWikiWord" href="/nlab/show/correlators+as+differential+forms+on+configuration+spaces+of+points">correlators as differential forms on configuration spaces of points</a> is described in <ul> <li id="BergbauerBrunettiKreimer09"> <a class="existingWikiWord" href="/nlab/show/Christoph+Bergbauer">Christoph Bergbauer</a>, <a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Dirk+Kreimer">Dirk Kreimer</a>, Renormalization and resolution of singularities (<a href="https://arxiv.org/abs/0908.0633">arXiv:0908.0633</a>) </li> <li id="Bergbauer09"> <a class="existingWikiWord" href="/nlab/show/Christoph+Bergbauer">Christoph Bergbauer</a>, Renormalization and resolution of singularities, talks as IHES and Boston, 2009 (<a href="http://www.emg.uni-mainz.de/Dateien/bergbauer.pdf">pdf</a>) </li> <li id="Berghoff14a"> <a class="existingWikiWord" href="/nlab/show/Marko+Berghoff">Marko Berghoff</a>, Wonderful renormalization, 2014 (<a href="http://www2.mathematik.hu-berlin.de/%7Ekreimer/wp-content/uploads/Berghoff-Marko.pdf">pdf</a>, <a href="https://doi.org/10.18452/17160">doi:10.18452/17160</a>) </li> <li id="Berghoff14b"> <a class="existingWikiWord" href="/nlab/show/Marko+Berghoff">Marko Berghoff</a>, Wonderful compactifications in quantum field theory, Communications in Number Theory and Physics Volume 9 (2015) Number 3 (<a href="https://arxiv.org/abs/1411.5583">arXiv:1411.5583</a>) </li> </ul> <h3 id="thermal_quantum_field_theory">Thermal quantum field theory</h3> Introductions: <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> Alberto Salvio: Introduction to Thermal Field Theory [<a href="https://arxiv.org/abs/2411.02498">arXiv:2411.02498</a>] </li> </ul> The idea of Wick rotating thermal relativistic field theory to compact periodic Euclidean time 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> A systematic discussion of the <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> on <a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a> for the case of thermal field theory/periodic Euclidean time is 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> See also <ul> <li><a href="#FullingRuijsenaars87">Fulling-Ruijsenaars 87, section 2</a></li> </ul> The formulation of the KMS condition is due to <ul> <li id="Kubo57"> R. Kubo Statistical-Mechanical Theory of Irreversible Processes I. General Theory and Simple Applications to Magnetic and Conduction Problems, Journal of the Physical Society of Japan 12, 570-586 1957 </li> <li id="MartinSchwinger59"> Paul C. Martin, <a class="existingWikiWord" href="/nlab/show/Julian+Schwinger">Julian Schwinger</a>, Theory of Many-Particle Systems. I, Physical Review 115, 1342-1373 (1959) </li> </ul> and found its final, now generally accepted, form in <ul> <li id="HaagHugenholtzWinnink67"><a class="existingWikiWord" href="/nlab/show/Rudolf+Haag">Rudolf Haag</a>, N. M. Hugenholtz, M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. Volume 5, Number 3 (1967), 215-236 (<a href="https://projecteuclid.org/euclid.cmp/1103840050">euclid:1103840050</a>)</li> </ul> Review of thermal field theory via Euclidean field theory includes <ul> <li> <a href="#FullingRuijsenaars87">Fulling-Ruijsenaars 87, sections 2 and 3</a> </li> <li id="LandsmanVanWert87"> <a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, Ch.G.van Weert, Real- and imaginary-time field theory at finite temperature and density, Physics Reports Volume 145, Issues 3–4, January 1987, Pages 141-249 (<a href="https://doi.org/10.1016/0370-1573(87)90121-9">doi:10.1016/0370-1573(87)90121-9</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Jean+Zinn-Justin">Jean Zinn-Justin</a>, Quantum Field Theory at Finite Temperature: An Introduction (<a href="https://arxiv.org/abs/hep-ph/0005272">arXiv:hep-ph/0005272</a>) </li> <li id="Thoma00"> Markus H. Thoma, New Developments and Applications of Thermal Field Theory (<a href="https://arxiv.org/abs/hep-ph/0010164">arXiv:hep-ph/0010164</a>) </li> <li> Yuhao Yang, An Introduction to Thermal Field Theory, 2011 (<a href="https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/dissertations/2011/Yuhao-Yang-Dissertation.pdf">pdf</a>) </li> <li> Yi-Cheng Huang, Field Theory in the Imaginary-time Formulation (<a href="https://arxiv.org/abs/1311.1990v4">arXiv:1311.1990v4</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Roberto+Longo">Roberto Longo</a>, Kubo-Martin-Schwinger, Non-Equilibrium Thermal states, and Conformal Field Theory, 2016 (<a href="https://www.mat.uniroma2.it/~longo/Slides_files/Harvard16.pdf">pdf</a>) </li> </ul> Further discussion: <ul> <li> Christian D. Jäkel, The Reeh–Schlieder property for thermal field theories, Journal of Mathematical Physics 41, 1745 2000 (<a href="https://doi.org/10.1063/1.533208">doi:10.1063/1.533208</a>) </li> <li id="JaehelRobl11"> Christian D. Jäkel, Florian Robl, The relativistic KMS condition for the thermal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-point functions of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>ϕ</mi><msub><mo stretchy="false">)</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">P(\phi)_2</annotation></semantics></math> model (<a href="https://arxiv.org/abs/1103.3609">arXiv:1103.3609</a>) </li> </ul> With an eye towards <a class="existingWikiWord" href="/nlab/show/lattice+gauge+theory">lattice gauge theory</a>: <ul> <li id="Moore03">Guy Moore, Informal lectures on lattice gauge theory, 2003 (<a href="https://theorie.ikp.physik.tu-darmstadt.de/qcd/moore/latt_lectures.pdf">pdf</a>)</li> </ul> In application to <a class="existingWikiWord" href="/nlab/show/black+hole+thermodynamics">black hole thermodynamics</a>: <ul> <li> <a href="#FullingRuijsenaars87">Fulling-Ruijsenaars 87, section 4</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> <li id="FrolovZelnikov11"> <a class="existingWikiWord" href="/nlab/show/Valeri+Frolov">Valeri Frolov</a>, Andrei Zelnikov, section F4.4 of Introduction to black hole physics, Oxford 2011 </li> </ul> Discussion of thermal <a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a> 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> The expansion of thermal field theory around the <a class="existingWikiWord" href="/nlab/show/infinite-temperature+thermal+field+theory">infinite-temperature-limit</a> (i.e. around <a class="existingWikiWord" href="/nlab/show/inverse+temperature">inverse temperature</a> <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><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\beta = 1/T = 0</annotation></semantics></math>, i.e. <a class="existingWikiWord" href="/nlab/show/KK-reduction">KK-reduction</a> in compact/periodic Euclidean time) is discussed in <ul> <li id="Ginsparg80"> <a class="existingWikiWord" href="/nlab/show/Paul+Ginsparg">Paul Ginsparg</a>, First and second order phase transitions in gauge theories at finite temperature, Nuclear Physics B Volume 170, Issue 3, 15 December 1980, Pages 388-408 (<a href="https://doi.org/10.1016/0550-3213(80)90418-6">doi:10.1016/0550-3213(80)90418-6</a>) </li> <li id="AppelquistPisarski81"> Thomas Appelquist, <a class="existingWikiWord" href="/nlab/show/Robert+Pisarski">Robert Pisarski</a>, High-temperature Yang-Mills theories and three-dimensional quantum chromodynamics, Phys. Rev. D 23, 2305 (1981) (<a href="https://doi.org/10.1103/PhysRevD.23.2305">doi:10.1103/PhysRevD.23.2305</a>) </li> <li id="Nadkarni83"> Sudhir Nadkarni, Dimensional reduction in finite-temperature quantum chromodynamics, Phys. Rev. D 27, 917 (1983) (<a href="https://doi.org/10.1103/PhysRevD.27.917">doi:10.1103/PhysRevD.27.917</a>) </li> <li id="Nadkarni88"> Sudhir Nadkarni, Dimensional reduction in finite-temperature quantum chromodynamics. II, Phys. Rev. D 38, 3287 (1988) (<a href="https://doi.org/10.1103/PhysRevD.38.3287">doi:10.1103/PhysRevD.38.3287</a>) </li> <li id="Jourjine84"> Alexander N Jourjine, Quantum field theory in the infinite temperature limit, Annals of Physics Volume 155, Issue 2, July 1984, Pages 305-332 (<a href="https://doi.org/10.1016/0003-4916(84)90003-4">doi:10.1016/0003-4916(84)90003-4</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, Limitations to dimensional reduction at high temperature, Nuclear Physics B Volume 322, Issue 2, 14 August 1989, Pages 498-530 (<a href="https://doi.org/10.1016/0550-3213(89)90424-0">doi:10.1016/0550-3213(89)90424-0</a>) </li> <li> T. Reisz, Realization of dimensional reduction at high temperature, Z. Phys. C - Particles and Fields (1992) 53: 169 (<a href="https://doi.org/10.1007/BF01483886">doi:10.1007/BF01483886</a>) </li> <li id="Braaten95"> Eric Braaten, Solution to the Perturbative Infrared Catastrophe of Hot Gauge Theories, Phys. Rev. Lett. 74, 2164 (1995) (<a href="https://doi.org/10.1103/PhysRevLett.74.2164">doi:10.1103/PhysRevLett.74.2164</a>) </li> <li id="KajantieaLaineRummukainenShaposhnikov96"> K. Kajantiea, M. Laine, K. Rummukainen, M. Shaposhnikov, Generic rules for high temperature dimensional reduction and their application to the standard model, Nuclear Physics B Volume 458, Issues 1–2, 1 January 1996, Pages 90-136 (<a href="https://doi.org/10.1016/0550-3213(95)00549-8">doi:10.1016/0550-3213(95)00549-8</a>) </li> </ul> and specifically with an eye to discussion of the <a class="existingWikiWord" href="/nlab/show/quark-gluon+plasma">quark-gluon plasma</a> in <ul> <li id="BlaizotIancuRebhan03">Jean-Paul Blaizot, Edmond Iancu, <a class="existingWikiWord" href="/nlab/show/Anton+Rebhan">Anton Rebhan</a>, Thermodynamics of the high temperature quark gluon plasma, Quark–Gluon Plasma 3, pp. 60-122 (2004) (<a href="https://arxiv.org/abs/hep-ph/0303185">arXiv:hep-ph/0303185</a>, <a href="http://inspirehep.net/record/615570">spire:615570</a>)</li> </ul> General discussion for <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a>: <ul> <li>Jacopo Ghiglieri, Aleksi Kurkela, Michael Strickland, Aleksi Vuorinen, Perturbative Thermal QCD: Formalism and Applications (<a href="https://arxiv.org/abs/2002.10188">arXiv:2002.10188</a>)</li> </ul> See also <ul> <li> Wikipedia, <a href="https://en.wikipedia.org/wiki/Thermal_quantum_field_theory">Thermal quantum field theory</a> </li> <li> Wikipedia, <a href="https://en.wikipedia.org/wiki/Matsubara_frequency">Matsubara frequency</a>, <a href="https://en.wikipedia.org/wiki/Matsubara_formalism">Matsubara formalism</a> </li> <li> Wolfram Math World, <a href="http://mathworld.wolfram.com/KMSCondition.html">KMS condition</a> </li> </ul> </body></html> </div> <div class="revisedby"> Last revised on November 6, 2024 at 05:08:17. 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