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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> PCT theorem </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/15727/#Item_3" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="quantum_field_theory">Quantum field theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></strong></p> <h2 id="contents">Contents</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a> / <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a></p> </li> </ul> </li> </ul> </div></div> <h4 id="aqft">AQFT</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></strong> (<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>)</p> <p><a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a></p> <h2 id="concepts">Concepts</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a></strong>:</p> <ul> <li> <p><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></p> </li> <li> <p><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></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+history">field history</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/observables">observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+net">field net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> </ul> </li> <li> <p><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></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a></p> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a></p> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> <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> <p><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a></p> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>)</p> </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#abstract'>Abstract</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#definition_in_the_wightman_approach'>Definition in the Wightman approach</a></li> <li><a href='#definition_in_the_haagkastler_approach'>Definition in the Haag-Kastler approach</a></li> </ul> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>This page is about PCT theorems in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. PCT stands for parity, charge and <a class="existingWikiWord" href="/nlab/show/time-reversal+symmetry">time-reversal symmetry</a> (warning: the order of the letters P, C and T varies, some authors use CPT, for example). The laws of nature as described by <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> are believed to be invariant if one simultaneously reverses the arrow of time, conjugates all charges and reverses all chiral properties. PCT theorems try to make this belief precise by defining the appropriate operators and showing that certain expressions remain constant. Both the statements and the proofs depend on the framework for <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> one uses.</p> <p>Being “invariant” means that every process that can be observerd in our universe can be observed identically in the “mirror” universe, that is there is no experiment in our universe that cannot be duplicated in the mirror universe.</p> <p>For some time physicists believed that subsets of the PCT symmetry are respected by nature, but today there are counterexamples known for every subset, for example:</p> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/weak+nuclear+force">weak nuclear force</a> violates parity symmetry.</p> </li> <li> <p>Charge symmetry would violate the observation that the universe consists mostly of matter and not of matter and antimatter.</p> </li> <li> <p>For CP symmetry violation see Wikipedia: <a href="http://en.wikipedia.org/wiki/CP_violation">CP violation</a></p> </li> <li> <p>All known physical laws are time symmetric, but since PCT symmetry is believed to hold and CP symmetry does not hold, there has to be a process that breaks T symmetry. As of today there is no consenus about what that process may be (there are good reasons to believe that it has nothing to do with the second law of thermodynamics).</p> </li> </ul> <h2 id="abstract">Abstract</h2> <p>…</p> <h2 id="definition">Definition</h2> <h3 id="definition_in_the_wightman_approach">Definition in the Wightman approach</h3> <p>The PCT theorem for Wightman fields (see <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a>) was proved by Res Jost, see references.</p> <p>This proof clarified the different conditions one has to impose, these are:</p> <ol> <li> <p>Covariance of the theory under the (connected part of the) <a class="existingWikiWord" href="/nlab/show/Poincare+group">Poincare group</a>.</p> </li> <li> <p>Positivity of the energy.</p> </li> <li> <p>There are only fields, which transform with respect to finite dimensional representations of the <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a>. (Transformation of the index space.)</p> </li> <li> <p>Locality, which means that for spacelike distances the Bose fields commute with all other fields and the Fermi fields anticommute with each other.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/Minkowski+space">Minkowski space</a> has even dimensions.</p> </li> <li> <p>To every field in the theory appears its conjugate complex partner.</p> </li> </ol> <h3 id="definition_in_the_haagkastler_approach">Definition in the Haag-Kastler approach</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒥</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{J})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/AQFT">Haag-Kastler net</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>.</p> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>A <strong>PCT operator</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi></mrow><annotation encoding="application/x-tex">\Theta</annotation></semantics></math> on the local net is an anti-linear <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a>, that is for every local algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}(\mathcal{O})</annotation></semantics></math>, elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A, B \in \mathcal{M}(\mathcal{O})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\lambda \in \mathbb{C}</annotation></semantics></math> we have the relations</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Θ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>Θ</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(A B) = \Theta(A) \Theta(B)</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mover><mi>λ</mi><mo>¯</mo></mover><mi>Θ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(\lambda A) = \overline \lambda \Theta(A)</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mo stretchy="false">(</mo><mi>ℳ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ℳ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(\mathcal{M}(\mathcal{O})) = \mathcal{M}(-\mathcal{O})</annotation></semantics></math>;</p> </li> </ol> <p>such that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Λ</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>U</mi><mo stretchy="false">(</mo><mi>Λ</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Lambda,a) \mapsto U(\Lambda, a)</annotation></semantics></math> the given <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of the <a class="existingWikiWord" href="/nlab/show/Poincare+group">Poincare group</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}</annotation></semantics></math> we have</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mi>U</mi><mo stretchy="false">(</mo><mi>Λ</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mi>A</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mi>Λ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>a</mi><mo stretchy="false">)</mo><mi>Θ</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\Theta U(\Lambda, a) A = U(\Lambda, -a) \Theta A</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi></mrow><annotation encoding="application/x-tex">\Theta</annotation></semantics></math> transforms every <a class="existingWikiWord" href="/nlab/show/charge+sector">charge sector</a> into its conjugate sector.</p> </li> </ol> </div> <p>A <strong>PCT theorem</strong> in this context is a theorem that states sufficient conditions such that a PCT operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi></mrow><annotation encoding="application/x-tex">\Theta</annotation></semantics></math> exists.</p> <h2 id="properties">Properties</h2> <p>…</p> <h2 id="examples">Examples</h2> <p>…</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antiparticle">antiparticle</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/charge+sector">charge sector</a>, <a class="existingWikiWord" href="/nlab/show/DHR+category">DHR category</a></p> </li> </ul> <h2 id="references">References</h2> <p>Review for the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</p> <ul> <li>Robert S. Thorne, <em>The Standard Model Parity, Charge Conjugation and Time Reversal</em> [<a href="https://www.hep.ucl.ac.uk/postgrad/teaching/sm/SMIIL4.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Thorne-StandardModelPCT.pdf" title="pdf">pdf</a>]</li> </ul> <p>See also:</p> <ul> <li>Wikipedia: <a href="http://en.wikipedia.org/wiki/CPT_symmetry">CPT symmetry</a></li> </ul> <p>Proof from the <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a>:</p> <ul> <li>Res Jost: <em>Eine Bemerkung zum CPT Theorem</em> Helv. Phys. Acta 30 (1957), p.409-416</li> </ul> <p>Monographs on this formulation in <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Raymond+F.+Streater">Raymond F. Streater</a>, <a class="existingWikiWord" href="/nlab/show/Arthur+S.+Wightman">Arthur S. Wightman</a>, <em>PCT, Spin and Statistics, and All That</em>, Princeton University Press (1989, 2000) [<a href="https://press.princeton.edu/books/paperback/9780691070629/pct-spin-and-statistics-and-all-that">ISBN:9780691070629</a>, <a href="https://www.jstor.org/stable/j.ctt1cx3vcq">jstor:j.ctt1cx3vcq</a>]</p> </li> <li id="Strocchi13"> <p><a class="existingWikiWord" href="/nlab/show/Franco+Strocchi">Franco Strocchi</a>, §4.3 in: <em>An Introduction to Non-Perturbative Foundations of Quantum Field Theory</em>, Oxford University Press (2013) [<a href="https://doi.org/10.1093/acprof:oso/9780199671571.001.0001">doi:10.1093/acprof:oso/9780199671571.001.0001</a>]</p> </li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jonathan+Bain">Jonathan Bain</a>: <em>CPT Invariance and the Spin-Statistics Connection</em>, Oxford University Press (2016) [<a href="https://global.oup.com/academic/product/cpt-invariance-and-the-spin-statistics-connection-9780198728801">ISBN:9780198728801</a>, <a href="https://doi.org/10.1093/acprof:oso/9780198728801.001.0001">doi:10.1093/acprof:oso/9780198728801.001.0001</a>]</li> </ul> <p>On the <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> for <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> in <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hans-J%C3%BCrgen+Borchers">Hans-Jürgen Borchers</a>, <a class="existingWikiWord" href="/nlab/show/Jakob+Yngvason">Jakob Yngvason</a>, <em>On the PCT–Theorem in the Theory of Local Observables</em>, in: <em>Mathematical physics in mathematics and physics: Quantum and operator algebraic aspects</em>, Fields Inst. Commun. <strong>30</strong> (2001) 39-64 [<a href="http://arxiv.org/abs/math-ph/0012020">arXiv:math-ph/0012020</a>, <a href="https://inspirehep.net/literature/538524">spire:538524</a>]</li> </ul> <p>Proof for <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> field theory (not falling back to the <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> axiomatics):</p> <ul> <li>Hilary Greaves, <a class="existingWikiWord" href="/nlab/show/Teruji+Thomas">Teruji Thomas</a>, <em>The CPT theorem</em>, Studies in History and Philosophy of Modern Physics 45 (2014) 46-66 (<a href="http://arxiv.org/abs/1204.4674">arXiv:1204.4674</a>)</li> </ul> <p>Discussion for <a class="existingWikiWord" href="/nlab/show/curved+spacetimes">curved spacetimes</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/M.+D.+Pollock">M. D. Pollock</a>, <em>On the Dirac equation in curved space-time</em>, Acta Physica Polonica B <strong>41</strong> (2010) [<a href="https://inspirehep.net/literature/874211">InSpire:874211</a>, <a href="https://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=41&page=1827">pdf</a>]</li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Juven+Wang">Juven Wang</a>, <em>C-P-T Fractionalization</em>, Phys. Rev. D <strong>106</strong> (2022) 105009 [<a href="https://arxiv.org/abs/2109.15320">arXiv:2109.15320</a>, <a href="https://doi.org/10.1103/PhysRevD.106.105009">doi:10.1103/PhysRevD.106.105009</a>]</p> </li> <li> <p>I. P-Castro, J. L. Díaz-Cruz, A. Pérez-Lorenzana: <em>CPT Symmetry and its Breaking in the chiral fermion formalism</em> [<a href="https://arxiv.org/abs/2411.05242">arXiv:2411.05242</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 11, 2024 at 04:12:05. See the <a href="/nlab/history/PCT+theorem" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/PCT+theorem" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/15727/#Item_3">Discuss</a><span class="backintime"><a href="/nlab/revision/PCT+theorem/20" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/PCT+theorem" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/PCT+theorem" accesskey="S" class="navlink" id="history" rel="nofollow">History (20 revisions)</a> <a href="/nlab/show/PCT+theorem/cite" style="color: black">Cite</a> <a href="/nlab/print/PCT+theorem" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/PCT+theorem" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>