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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> Feynman diagram </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/4702/#Item_34" 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="algebraic_qunantum_field_theory">Algebraic Qunantum Field Theory</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> <h4 id="measure_and_probability_theory">Measure and probability theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a></strong></p> <p>(<a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a>)</p> <h2 id="measure_theory">Measure theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a></p> </li> </ul> <h2 id="probability_theory">Probability theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state">state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/states+in+AQFT+and+operator+algebra">in AQFT and operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a></p> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a></p> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></p> </li> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a></p> </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='#in_perturbative_quantum_field_theory'>In perturbative quantum field theory</a></li> </ul> <li><a href='#details'>Details</a></li> <ul> <li><a href='#ForFinitelyManyDegreesOfFreedom'>For finitely many degrees of freedoms</a></li> <li><a href='#DetailsForPerturbativeQuantumFieldTheory'>In perturbative quantum field theory</a></li> </ul> <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='#ReferencesInCausalPerturbationTheory'>In causal perturbation theory</a></li> <li><a href='#ReferencesInTermsOfMotivicStructures'>In terms of motivic structures</a></li> <li><a href='#ReferencesInHomologicalBVQuantization'>In homological BV-quantization</a></li> <li><a href='#AsStringDiagrams'>As string diagrams (morphisms in monoidal categories)</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="general">General</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\mu_S</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Gaussian+probability+measure">Gaussian probability measure</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>I</mi><mo stretchy="false">)</mo><msub><mi>μ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\exp(-I)\mu_S</annotation></semantics></math> a perturbation with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial</a> at least of degree 3, there is a combinatorial expression for the <a class="existingWikiWord" href="/nlab/show/moments">moments</a>/<a class="existingWikiWord" href="/nlab/show/expectation+values">expectation values</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>I</mi><mo stretchy="false">)</mo><msub><mi>μ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\exp(-I)\mu_S</annotation></semantics></math> as a sum over certain <a class="existingWikiWord" href="/nlab/show/graphs">graphs</a> whose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-ary vertices are labeled by the monomials of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>. (This is such that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">I = 0</annotation></semantics></math> it reduces to <a class="existingWikiWord" href="/nlab/show/Wick%27s+lemma">Wick's lemma</a>.)</p> <p>These graphs are called <em>Feynman diagrams</em>.</p> <h3 id="in_perturbative_quantum_field_theory">In perturbative quantum field theory</h3> <p>Feynman graphs play a central role in <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><msub><mi>μ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\exp(I)\mu_S</annotation></semantics></math> plays the role of an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> on a space of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\mu_S</annotation></semantics></math> is the exponentiated <a class="existingWikiWord" href="/nlab/show/kinetic+action">kinetic action</a> and hence the <a class="existingWikiWord" href="/nlab/show/measure">measure</a> for <a class="existingWikiWord" href="/nlab/show/free+fields">free fields</a>, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(I)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> part of the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>: the order-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> monomials in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> encode an interaction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>. In this context the corresponding Feynman diagrams are traditionally thought of as depicting interaction processes of quanta of these fields, with propagation along the edges and interaction at the vertices. But this interpretation has its limits, which is partly reflected in speaking of “<a class="existingWikiWord" href="/nlab/show/virtual+particles">virtual particles</a>”. A precise interpretation is given by <em><a class="existingWikiWord" href="/nlab/show/worldline+formalism">worldline formalism</a></em>.</p> <h2 id="details">Details</h2> <h3 id="ForFinitelyManyDegreesOfFreedom">For finitely many degrees of freedoms</h3> <p>We discuss Feynman diagrams for a single real <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> on a <a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k\in \mathbb{N}</annotation></semantics></math> points. This contains in it already all the aspects of real Feynman diagrams in <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> and in this context everything is easily well-defined. The generalization to more field components is immediate and simply obtained by thinking of all “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>” in the following as taking values in some appropriate <a class="existingWikiWord" href="/nlab/show/representation">representation</a> space and all products of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>s as given by suitable <a class="existingWikiWord" href="/nlab/show/intertwiners">intertwiners</a> and <a class="existingWikiWord" href="/nlab/show/inner+products">inner products</a>, otherwise the form of the formulas remains the same. Similarly, in generalization to continuous space (non-finitely many degrees of freedom) all the diagrammatics remains the same, the only issue now is to make sense (namely via <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a>) of the numerical value that is assigned to any one Feynman diagram.</p> <p>So a <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> here is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">[k]\to \mathbb{R}</annotation></semantics></math> from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-element <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> to the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>, and hence the space of all field configurations is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math>.</p> <p>Fix then a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>×</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">k \times k</annotation></semantics></math> real-valued <a class="existingWikiWord" href="/nlab/show/matrix">matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>x</mi><mi>y</mi></mrow></msub><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>Mat</mi> <mrow><mi>k</mi><mo>×</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \coloneqq (A_{x y}) \in Mat_{k\times k}(\mathbb{R})</annotation></semantics></math> of non-vanishing <a class="existingWikiWord" href="/nlab/show/determinant">determinant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mi>A</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">det A \neq 0</annotation></semantics></math>.</p> <p>For standard applications this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a discretized version of the <a class="existingWikiWord" href="/nlab/show/Laplacian">Laplacian</a> and then the expression</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>kin</mi></msub><mo>=</mo><msub><mi>E</mi> <mi>kin</mi></msub><mo>≔</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msub><mi>ϕ</mi> <mi>x</mi></msub><msub><mi>A</mi> <mrow><mi>x</mi><mi>y</mi></mrow></msub><msub><mi>ϕ</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex"> S_{kin} = E_{kin} \coloneqq \tfrac{1}{2}\sum_{x,y = 1}^k \phi_x A_{x y} \phi_y </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/kinetic+energy">kinetic energy</a> and <a class="existingWikiWord" href="/nlab/show/kinetic+action">kinetic action</a> of the field configuration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>. More concretely, think of the set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> space points as being a discretization of a circle and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>p</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi_p)</annotation></semantics></math> for the corresponding (<a class="existingWikiWord" href="/nlab/show/discrete+Fourier+transform">discrete</a>) <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi_x)</annotation></semantics></math>. Then the kinetic term of the <a class="existingWikiWord" href="/nlab/show/free+field+theory">free</a> <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> on this space is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> which is the <a class="existingWikiWord" href="/nlab/show/diagonal+matrix">diagonal matrix</a> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-basis with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>p</mi><mo>,</mo><mi>p</mi></mrow></msub><mo>=</mo><msup><mi>p</mi> <mn>2</mn></msup><mo>−</mo><msup><mi>m</mi> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> A_{p,p} = p^2 - m^2 \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> is a constant called the <a class="existingWikiWord" href="/nlab/show/mass">mass</a> of the <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/sum">sum</a> over all values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is the finite (and hence well-defined) analog of a <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>. The <a class="existingWikiWord" href="/nlab/show/Gaussian+integral">Gaussian integral</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is called the <em><a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a></em>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Z</mi> <mn>0</mn></msub></mtd> <mtd><mo>≔</mo><mo>∫</mo><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mi>kin</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>D</mi><mi>ϕ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><msub><mo>∫</mo> <mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow></msub><mspace width="thinmathspace"></mspace><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msub><mi>ϕ</mi> <mi>x</mi></msub><msub><mi>A</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><msub><mi>ϕ</mi> <mi>y</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi>d</mi><msub><mi>ϕ</mi> <mn>1</mn></msub><mi>⋯</mi><mi>d</mi><msub><mi>ϕ</mi> <mi>k</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>k</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>det</mi><mi>A</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Z_0 & \coloneqq \int \exp(- S_{kin}(\phi)) \,D\phi \\ & \coloneqq \int_{\mathbb{R}^k} \, \exp(- \tfrac{1}{2}\sum_{x,y = 1}^k \phi_{x} A_{x,y} \phi_y ) \; d\phi_{1} \cdots d\phi_{k} \\ & = (2\pi)^{k/2} (det A)^{-1/2} \end{aligned} </annotation></semantics></math></div> <p>An <em><a class="existingWikiWord" href="/nlab/show/n-point+function">n-point function</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow></msub><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mi>n</mi></msub></mrow></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle \phi_{x_1} \phi_{x_2} \cdots \phi_{x_n}\rangle</annotation></semantics></math> is an <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>th <a class="existingWikiWord" href="/nlab/show/moment">moment</a> of this Gaussian distribution:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">⟨</mo><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow></msub><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mi>n</mi></msub></mrow></msub><mo stretchy="false">⟩</mo></mtd> <mtd><mo>≔</mo><mfrac><mn>1</mn><mrow><msub><mi>Z</mi> <mn>0</mn></msub></mrow></mfrac><mo>∫</mo><mrow><mo>(</mo><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mi>kin</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow></msub><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mi>n</mi></msub></mrow></msub><mo>)</mo></mrow><mi>D</mi><mi>ϕ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><mfrac><mn>1</mn><mrow><msub><mi>Z</mi> <mn>0</mn></msub></mrow></mfrac><msub><mo>∫</mo> <mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow></msub><mrow><mo>(</mo><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>ϕ</mi> <mi>x</mi></msub><msub><mi>A</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><msub><mi>ϕ</mi> <mi>y</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow></msub><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mi>n</mi></msub></mrow></msub><mo>)</mo></mrow><mi>d</mi><msub><mi>ϕ</mi> <mn>1</mn></msub><mi>⋯</mi><mi>d</mi><msub><mi>ϕ</mi> <mi>k</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \langle \phi_{x_1} \phi_{x_2} \cdots \phi_{x_n}\rangle & \coloneqq \frac{1}{Z_0} \int \left( \exp(-S_{kin}(\phi)) \phi_{x_1}\phi_{x_2}\cdots \phi_{x_n} \right) D \phi \\ & \coloneqq \frac{1}{Z_0} \int_{\mathbb{R}^k} \left( \exp(-\tfrac{1}{2} \phi_x A_{x,y}\phi_y) \, \phi_{x_1}\phi_{x_2} \cdots\phi_{x_n} \right) d\phi_1 \cdots d\phi_k \end{aligned} </annotation></semantics></math></div> <p>In order to compute these conveniently, pass to the <a class="existingWikiWord" href="/nlab/show/generating+function">generating function</a> obtained by adding a <a class="existingWikiWord" href="/nlab/show/source">source</a> <a class="existingWikiWord" href="/nlab/show/variable">variable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>J</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J = (J_x)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>Z</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mo>∫</mo><mo stretchy="false">(</mo><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>S</mi> <mi>kin</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>J</mi><mo>⋅</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>D</mi><mi>ϕ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><msub><mo>∫</mo> <mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msub><mi>ϕ</mi> <mi>x</mi></msub><msub><mi>A</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><msub><mi>ϕ</mi> <mi>y</mi></msub><mo>+</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msub><mi>J</mi> <mi>x</mi></msub><msub><mi>ϕ</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi>d</mi><msub><mi>ϕ</mi> <mn>1</mn></msub><mi>⋯</mi><mi>d</mi><msub><mi>ϕ</mi> <mi>k</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Z</mi> <mn>0</mn></msub><mi>exp</mi><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msub><mi>J</mi> <mi>x</mi></msub><msubsup><mi>A</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>J</mi> <mi>y</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} Z(J) & \coloneqq \int( \exp(- S_{kin}(\phi) + J \cdot\phi) ) D\phi \\ & \coloneqq \int_{\mathbb{R}^k} \exp(- \tfrac{1}{2}\sum_{x = 1}^k \phi_{x} A_{x,y} \phi_y + \sum_{x = 1}^k J_x \phi_x ) \; d\phi_{1} \cdots d\phi_{k} \\ & = Z_0 \exp(\tfrac{1}{2} \sum_{x = 1}^k J_x A^{-1}_{x,y} J_y) \end{aligned} \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>A</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A^{-1} = (A^{-1}_{x,y})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/inverse+matrix">inverse matrix</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, which appears by computing the new integral here again as a <a class="existingWikiWord" href="/nlab/show/Gaussian+integral">Gaussian integral</a> after <a class="existingWikiWord" href="/nlab/show/completing+the+square">completing the square</a> in the exponent.</p> <p>In applications to field theory this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">A^{-1}</annotation></semantics></math> is called the <em><a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></em>. In the standard example where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the kinetic term of the <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field</a> of <a class="existingWikiWord" href="/nlab/show/mass">mass</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> then in Fourier-transformed components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">A^{-1}</annotation></semantics></math> is diagonal with components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>A</mi> <mrow><mi>p</mi><mo>,</mo><mi>p</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>p</mi> <mn>2</mn></msup><mo>−</mo><msup><mi>m</mi> <mn>2</mn></msup></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A^{-1}_{p,p} = \frac{1}{p^2 - m^2} \,. </annotation></semantics></math></div> <p>This means, incidentally. that in the non-finite case of interest in physics, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is not actually naively invertible after all, as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>m</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">p^2 = m^2</annotation></semantics></math> on the mass <a class="existingWikiWord" href="/nlab/show/shell">shell</a>. In this case one has to replace the naive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">A^{-1}</annotation></semantics></math> with its <a class="existingWikiWord" href="/nlab/show/zeta+function+regularization">zeta-function regularized version</a>.</p> <p>The way this works is insightful even when the naive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">A^{-1}</annotation></semantics></math> does exist. Notice that using “<a class="existingWikiWord" href="/nlab/show/Schwinger+parameterization">Schwinger parameterization</a>” the <a class="existingWikiWord" href="/nlab/show/propagator">propagator</a> is equivalently rewritten as a <a class="existingWikiWord" href="/nlab/show/Mellin+transform">Mellin transform</a> <a class="existingWikiWord" href="/nlab/show/integral">integral</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>A</mi> <mrow><mi>p</mi><mo>,</mo><mi>p</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>∞</mn></msubsup><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>τ</mi><msub><mi>A</mi> <mrow><mi>p</mi><mo>,</mo><mi>p</mi></mrow></msub><mo stretchy="false">)</mo><mi>d</mi><mi>τ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A^{-1}_{p,p} = \int_0^\infty \exp(- \tau A_{p,p}) d\tau \,. </annotation></semantics></math></div> <p>Again in the example of the standard <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> kinetic term expressed in Fourier diagonalization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>p</mi><mo>,</mo><mi>p</mi></mrow></msub><mo>=</mo><msup><mi>p</mi> <mn>2</mn></msup><mo>−</mo><msup><mi>m</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">A_{p,p}= p^2 - m^2</annotation></semantics></math> then with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">X \colon [0,\tau] \to \mathbb{R}</annotation></semantics></math> a parameterization of the straight line with slope <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, the exponent is equivalently</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>τ</mi><msub><mi>A</mi> <mrow><mi>p</mi><mo>,</mo><mi>p</mi></mrow></msub></mtd> <mtd><mo>=</mo><mi>τ</mi><mo stretchy="false">(</mo><msup><mi>p</mi> <mn>2</mn></msup><mo>−</mo><msup><mi>m</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mn>0</mn> <mi>τ</mi></msubsup><mo stretchy="false">(</mo><msup><mover><mi>X</mi><mo>˙</mo></mover> <mn>2</mn></msup><mo>−</mo><msup><mi>m</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \tau A_{p,p} & = \tau (p^2 - m^2) \\ & = \int_0^\tau (\dot X^2 - m^2) \end{aligned} \,. </annotation></semantics></math></div> <p>This now happens to be the standard <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> (<a class="existingWikiWord" href="/nlab/show/Polyakov+action">Polyakov action</a>) for a <a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a> describing the propagation of a particle along its <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a>. This means that the propagator of the <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> may be thought of as coming from the <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> of a scalar particle along its <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a>. This perspective is called the “<a class="existingWikiWord" href="/nlab/show/worldline+formalism">worldline formalism</a>”, it is a formalization of <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a>, expressing the dynamics of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> in terms of that of their particle “quanta” running along <a class="existingWikiWord" href="/nlab/show/worldlines">worldlines</a> of the form of the corresponding Feynman diagrams (to which we finally come in a moment).</p> <p>Back to the computation of the <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 function. By construction, it is now equally expressed by <a class="existingWikiWord" href="/nlab/show/partial+derivatives">partial derivatives</a> of the <a class="existingWikiWord" href="/nlab/show/generating+function">generating function</a> with respect to the <a class="existingWikiWord" href="/nlab/show/source">source</a> <a class="existingWikiWord" href="/nlab/show/variable">variable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> and evaluated at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">J = 0</annotation></semantics></math>, and this in turn is a combinatorial expression just in products of the <a class="existingWikiWord" href="/nlab/show/propagator">propagator</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">⟨</mo><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mi>n</mi></msub></mrow></msub><mo stretchy="false">⟩</mo></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>Z</mi> <mn>0</mn></msub></mrow></mfrac><msub><mrow><mo>(</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><msub><mi>J</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow></msub></mrow></mfrac><mi>⋯</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><msub><mi>J</mi> <mrow><msub><mi>x</mi> <mi>n</mi></msub></mrow></msub></mrow></mfrac><mi>Z</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mo>)</mo></mrow> <mrow><mo stretchy="false">|</mo><mi>J</mi><mo>=</mo><mn>0</mn></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mrow><mo>(</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><msub><mi>J</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow></msub></mrow></mfrac><mi>⋯</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><msub><mi>J</mi> <mrow><msub><mi>x</mi> <mi>n</mi></msub></mrow></msub></mrow></mfrac><mi>exp</mi><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msub><mi>J</mi> <mi>x</mi></msub><msubsup><mi>A</mi> <mrow><mi>x</mi><mi>y</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>J</mi> <mi>y</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow> <mrow><mo stretchy="false">|</mo><mi>J</mi><mo>=</mo><mn>0</mn></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>pairings</mi></munder><msubsup><mi>A</mi> <mrow><msub><mi>x</mi> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><msub><mi>x</mi> <mrow><msub><mi>j</mi> <mn>2</mn></msub></mrow></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>⋅</mo><msubsup><mi>A</mi> <mrow><msub><mi>x</mi> <mrow><msub><mi>j</mi> <mn>3</mn></msub></mrow></msub><msub><mi>x</mi> <mrow><msub><mi>j</mi> <mn>4</mn></msub></mrow></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mi>⋯</mi><msubsup><mi>A</mi> <mrow><msub><mi>x</mi> <mrow><msub><mi>j</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><msub><mi>x</mi> <mrow><msub><mi>j</mi> <mi>n</mi></msub></mrow></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \langle \phi_{x_1} \cdots \phi_{x_n} \rangle &= \frac{1}{Z_0} \left( \frac{\partial}{\partial J_{x_1}} \cdots \frac{\partial}{\partial J_{x_n}} Z(J) \right)_{\vert J = 0} \\ & = \left( \frac{\partial}{\partial J_{x_1}} \cdots \frac{\partial}{\partial J_{x_n}} \exp(\tfrac{1}{2} \sum_{x = 1}^k J_{x} A^{-1}_{x y} J_y) \right)_{\vert J = 0} \\ & = \underset{pairings}{\sum} A^{-1}_{x_{j_1} x_{j_2}} \cdot A^{-1}_{x_{j_3} x_{j_4}} \cdots A^{-1}_{x_{j_{n-1}}x_{j_n}} \end{aligned} \,. </annotation></semantics></math></div> <p>Here the last <a class="existingWikiWord" href="/nlab/show/equality">equality</a> – known as <a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a> – comes from simple inspection: take the derivatives inside the <a class="existingWikiWord" href="/nlab/show/exponential+series">exponential series</a> and observe that then the only summands non-vanishing at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">J = 0</annotation></semantics></math> appears for even <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> and are those where all derivatives hit the monomial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></msubsup><msub><mi>J</mi> <mi>x</mi></msub><msubsup><mi>A</mi> <mrow><mi>x</mi><mi>y</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>J</mi> <mi>y</mi></msub><mo>)</mo></mrow> <mrow><mi>n</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\left(\tfrac{1}{2} \sum_{x = 1}^k J_{x} A^{-1}_{x y} J_y\right)^{n/2}</annotation></semantics></math>.</p> <p>Thinking of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>A</mi> <mrow><mi>x</mi><mi>y</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">A^{-1}_{x y}</annotation></semantics></math> here as labeling an <a class="existingWikiWord" href="/nlab/show/edge">edge</a> (a “<a class="existingWikiWord" href="/nlab/show/worldline">worldline</a>”) from <a class="existingWikiWord" href="/nlab/show/vertex">vertex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> to vertex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> This is the source of all Feynman digrammatics.</p> <p>Now consider a <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\phi)</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\geq 3</annotation></semantics></math>. In applications to field theory this represents the <a class="existingWikiWord" href="/nlab/show/potential+energy">potential energy</a> or (self)<a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> of the field configuration. The difference of the <a class="existingWikiWord" href="/nlab/show/kinetic+energy">kinetic energy</a> and the <a class="existingWikiWord" href="/nlab/show/potential+energy">potential energy</a> is called the (here: “Wick rotated”/“Euclidean”) <a class="existingWikiWord" href="/nlab/show/action">action</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>S</mi></mtd> <mtd><mo>=</mo><msub><mi>S</mi> <mi>kin</mi></msub><mo>+</mo><msub><mi>S</mi> <mi>int</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msub><mi>ϕ</mi> <mi>x</mi></msub><msub><mi>A</mi> <mrow><mi>x</mi><mi>y</mi></mrow></msub><msub><mi>ϕ</mi> <mi>y</mi></msub><mo>−</mo><mi>g</mi><mi>V</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} S & = S_{kin} + S_{int} \\ & = \tfrac{1}{2} \sum_{x,y = 1}^k \phi_x A_{x y} \phi_y - g V(\phi) \end{aligned} \,. </annotation></semantics></math></div> <p>The prefactor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is called the <em><a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a></em>.</p> <p>Putting everything together, the integral over the full <a class="existingWikiWord" href="/nlab/show/action">action</a> may be expressed as a <a class="existingWikiWord" href="/nlab/show/power+series">power series</a> in the <a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/moments">moments</a> with respect to the kinetic action of the powers of the interaction term:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>Z</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>Z</mi> <mn>0</mn></msub></mrow></mfrac><mo>∫</mo><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>D</mi><mi>ϕ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><mfrac><mn>1</mn><mrow><msub><mi>Z</mi> <mn>0</mn></msub></mrow></mfrac><mo>∫</mo><mrow><mo>(</mo><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msub><mi>ϕ</mi> <mi>x</mi></msub><msub><mi>A</mi> <mrow><mi>x</mi><mi>y</mi></mrow></msub><msub><mi>ϕ</mi> <mi>y</mi></msub><mo>+</mo><mi>g</mi><mi>V</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>ϕ</mi> <mn>1</mn></msub><mi>⋯</mi><mi>d</mi><msub><mi>ϕ</mi> <mi>k</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">⟨</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>g</mi><mi>V</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>1</mn><mo>+</mo><mi>g</mi><mo stretchy="false">⟨</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo><mo>+</mo><mfrac><mrow><msup><mi>g</mi> <mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo stretchy="false">⟨</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ϕ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">⟩</mo><mo>+</mo><mi>⋯</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Z(g) & = \frac{1}{Z_0}\int \exp(- S(\phi)) \, D\phi \\ & \coloneqq \frac{1}{Z_0} \int \left(\exp(-\tfrac{1}{2} \sum_{x,y = 1}^k \phi_x A_{x y} \phi_y + g V(\phi)) \right) \, d\phi_1 \cdots d\phi_k \\ & = \langle \exp(g V(\phi)) \rangle \\ & = 1 + g \langle V(\phi)\rangle + \frac{g^2}{2} \langle V(\phi) ^2\rangle + \cdots \end{aligned} </annotation></semantics></math></div> <p>By <a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a> stated above, each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ϕ</mi><msup><mo stretchy="false">)</mo> <mi>ℓ</mi></msup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle V(\phi)^\ell\rangle</annotation></semantics></math> is equivalently expressed as a sum over products of components of the <a class="existingWikiWord" href="/nlab/show/propagator">propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>A</mi> <mrow><mi>x</mi><mi>y</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">A^{-1}_{x y}</annotation></semantics></math>. Thinking of each such propagator term as an <a class="existingWikiWord" href="/nlab/show/edge">edge</a> produces a diagram, this is the corresponding Feynman diagram.</p> <p>For instance, for a cubic point interaction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>x</mi></munder><msubsup><mi>ϕ</mi> <mi>x</mi> <mn>3</mn></msubsup></mrow><annotation encoding="application/x-tex"> V(\phi) = \sum_x \phi_x^3 </annotation></semantics></math></div> <p>then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">⟨</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ϕ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">⟩</mo></mtd> <mtd><mo>≔</mo><mo stretchy="false">⟨</mo><mo stretchy="false">(</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msubsup><mi>ϕ</mi> <mi>x</mi> <mn>3</mn></msubsup><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">⟩</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><mo stretchy="false">⟨</mo><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow></msub><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow></msub><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow></msub><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow></msub><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow></msub><msub><mi>ϕ</mi> <mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">⟩</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>prefactor</mi><mspace width="thickmathspace"></mspace><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><munder><munder><mrow><msubsup><mi>A</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub><msub><mi>x</mi> <mn>1</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msubsup><mi>A</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub><msub><mi>x</mi> <mn>2</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msubsup><mi>A</mi> <mrow><msub><mi>x</mi> <mn>2</mn></msub><msub><mi>x</mi> <mn>2</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><mo>⏟</mo></munder><mrow><mi>dumbbell</mi><mspace width="thinmathspace"></mspace><mi>diagram</mi></mrow></munder><mo>+</mo><mi>prefactor</mi><mspace width="thickmathspace"></mspace><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><munder><munder><mrow><msubsup><mi>A</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub><msub><mi>x</mi> <mn>2</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msubsup><mi>A</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub><msub><mi>x</mi> <mn>2</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msubsup><mi>A</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub><msub><mi>x</mi> <mn>2</mn></msub></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><mo>⏟</mo></munder><mrow><mi>theta</mi><mspace width="thinmathspace"></mspace><mi>diagram</mi></mrow></munder></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \langle V(\phi)^2 \rangle &\coloneqq \langle (\sum_{x= 1}^k \phi_x^3)^2 \rangle \\ & = \sum_{x_1, x_2 = 1}^k \langle \phi_{x_1}\phi_{x_1} \phi_{x_1}\phi_{x_2}\phi_{x_2}\phi_{x_2}\rangle \\ & = prefactor \; \sum_{x_1,x_2 = 1}^k \underset{dumbbell\, diagram}{\underbrace{A^{-1}_{x_1 x_1} A^{-1}_{x_1 x_2} A^{-1}_{x_2 x_2}}} + prefactor \; \sum_{x_1, x_2 = 1}^k \underset{theta\, diagram}{\underbrace{A^{-1}_{x_1 x_2} A^{-1}_{x_1 x_2} A^{-1}_{x_1 x_2}}} \end{aligned} </annotation></semantics></math></div> <p>Here the first summand corresponds to the “dumbbell” Feynman diagram of the form</p> <p><img src="http://ncatlab.org/nlab/files/dumbbellFeynmanDiagramm.png" width="200" /></p> <p>and the second summand corresponds to the “theta” Feynman diagram of the form</p> <p><img src="http://ncatlab.org/nlab/files/thetaFeynmanDiagramm.png" width="200" />.</p> <h3 id="DetailsForPerturbativeQuantumFieldTheory">In perturbative quantum field theory</h3> <p>For details see at</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> the section <em><a href="S-matrix#FeynmanDiagrams">Feynman perturbation series</a></em></li> </ul> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/free+field+theory">free field</a> <a class="existingWikiWord" href="/nlab/show/algebra+of+quantum+observables">algebra of quantum observables</a></th><th><a class="existingWikiWord" href="/nlab/show/physics">physics</a> terminology</th><th><a class="existingWikiWord" href="/nlab/show/mathematics">maths</a> terminology</th></tr></thead><tbody><tr><td style="text-align: left;">1)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative product</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_1"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mo>:</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>:</mo></mrow><annotation encoding="application/x-tex">\phantom{AA} :A_1 A_2:</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_2"><semantics><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\phantom{AA} A_1 \cdot A_2</annotation></semantics></math> <br /> pointwise product of functionals</td></tr> <tr><td style="text-align: left;">2)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/non-commutative+algebra">non-commutative product</a> <br /> (<a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation</a> induced by <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_3"><semantics><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\phantom{AA} A_1 A_2</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/operator+product">operator product</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_4"><semantics><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\phantom{AA} A_1 \star_H A_2</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> for <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></td></tr> <tr><td style="text-align: left;">3)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_5"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mi>T</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phantom{AA} T(A_1 A_2)</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_6"><semantics><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\phantom{AA} A_1 \star_F A_2</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> for <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative expansion</a> <br /> of 2) via 1)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Wick%27s+lemma">Wick's lemma</a> <br /> <img src="https://ncatlab.org/nlab/files/WickTheorem.png" width="350" /></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Moyal+star+product">Moyal product</a> for <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_7"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math><br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_8"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>H</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>=</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>exp</mi><mrow><mo>(</mo><mi>ℏ</mi><mo>∫</mo><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>H</mi></msub><msup><mo stretchy="false">)</mo> <mi>ab</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⊗</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>)</mo></mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & A_1 \star_H A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_H)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative expansion</a> <br /> of 3) via 1)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> <br /> <img src="https://ncatlab.org/nlab/files/FeynmanDiagramGlobal.jpg" width="350" /></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Moyal+star+product">Moyal product</a> for <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_9"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_F</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c965b27b918c973b0d6a80ade615bbc435d8f77_10"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⋆</mo> <mi>F</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mo>=</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>exp</mi><mrow><mo>(</mo><mi>ℏ</mi><mo>∫</mo><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>F</mi></msub><msup><mo stretchy="false">)</mo> <mi>ab</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⊗</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>b</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>)</mo></mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & A_1 \star_F A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_F)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}</annotation></semantics></math></td></tr> </tbody></table> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div> <div class="num_example"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a> in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> – example of <a class="existingWikiWord" href="/nlab/show/QED">QED</a>)</strong></p> <p>In <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <strong><a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a></strong> are labeled <a class="existingWikiWord" href="/nlab/show/multigraphs">multigraphs</a> that encode <a class="existingWikiWord" href="/nlab/show/product+of+distributions">products of</a> <a class="existingWikiWord" href="/nlab/show/Feynman+propagators">Feynman propagators</a>, called <em><a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a></em> (<a href="S-matrix#FeynmanPerturbationSeriesAwayFromCoincidingPoints">this prop.</a>) which in turn contribute to <a class="existingWikiWord" href="/nlab/show/probability+amplitudes">probability amplitudes</a> for physical <a class="existingWikiWord" href="/nlab/show/scattering">scattering</a> processes – <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a>:</p> <p>The <a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a> are the summands in the <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a>-expansion of the <em><a class="existingWikiWord" href="/nlab/show/scattering+matrix">scattering matrix</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_1"><semantics><mrow><mi>𝒮</mi><mrow><mo>(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mfrac><mn>1</mn><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mi>k</mi></msup></mrow></mfrac><mi>T</mi><mo stretchy="false">(</mo><munder><munder><mrow><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>S</mi> <mi>int</mi></msub></mrow><mo>⏟</mo></munder><mrow><mi>k</mi><mspace width="thinmathspace"></mspace><mtext>factors</mtext></mrow></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S} \left( S_{int} \right) = \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!} \frac{1}{(i \hbar)^k} T( \underset{k \, \text{factors}}{\underbrace{S_{int}, \cdots , S_{int}}} ) </annotation></semantics></math></div> <p>of a given <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_2"><semantics><mrow><msub><mi>L</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">L_{int}</annotation></semantics></math>.</p> <p>The Feynman amplitudes are the summands in an expansion of the <em><a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_3"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(\cdots)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> with itself, which, away from coincident vertices, is given by the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> of the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_4"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_F</annotation></semantics></math> (<a href="S-matrix#TimeOrderedProductAwayFromDiagonal">this prop.</a>), via the <a class="existingWikiWord" href="/nlab/show/exponential">exponential</a> contraction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_5"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>prod</mi><mo>∘</mo><mi>exp</mi><mrow><mo>(</mo><mi>ℏ</mi><mo>∫</mo><msubsup><mi>Δ</mi> <mi>F</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⊗</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>)</mo></mrow><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>⊗</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> T(S_{int}, S_{int}) \;=\; prod \circ \exp \left( \hbar \int \Delta_{F}^{a b}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}(y)} \right) ( S_{int} \otimes S_{int} ) \,. </annotation></semantics></math></div> <p>Each <a class="existingWikiWord" href="/nlab/show/edge">edge</a> in a <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a> corresponds to a factor of a <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_6"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><munder><munder><mrow><msub><mi>S</mi> <mi>int</mi></msub><mi>⋯</mi><msub><mi>S</mi> <mi>int</mi></msub></mrow><mo>⏟</mo></munder><mrow><mi>k</mi><mspace width="thinmathspace"></mspace><mtext>factors</mtext></mrow></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T( \underset{k \, \text{factors}}{\underbrace{S_{int} \cdots S_{int}}} )</annotation></semantics></math>, being a <a class="existingWikiWord" href="/nlab/show/distribution+of+two+variables">distribution of two variables</a>; and each <a class="existingWikiWord" href="/nlab/show/vertex">vertex</a> corresponds to a factor of the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_7"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math>.</p> <p>For example <a class="existingWikiWord" href="/nlab/show/quantum+electrodynamics">quantum electrodynamics</a> in <a class="existingWikiWord" href="/nlab/show/Gaussian-averaged+Lorenz+gauge">Gaussian-averaged Lorenz gauge</a> involves (via <a href="S-matrix#FieldSpeciesQED">this example</a>):</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a> modelling the <a class="existingWikiWord" href="/nlab/show/electron">electron</a>, with <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> called the <em><a class="existingWikiWord" href="/nlab/show/electron+propagator">electron propagator</a></em> (<a href="Feynman+propagator#FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime">this def.</a>), here to be denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_8"><semantics><mrow><mi>Δ</mi><mphantom><mi>AAAA</mi></mphantom><mtext>electron propagator</mtext></mrow><annotation encoding="application/x-tex"> \Delta \phantom{AAAA} \text{electron propagator} </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> modelling the <a class="existingWikiWord" href="/nlab/show/photon">photon</a>, with <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> called the <em><a class="existingWikiWord" href="/nlab/show/photon+propagator">photon propagator</a></em> (<a href="A+first+idea+of+quantum+field+theory#PhotonPropagatorInGaussianAveragedLorenzGauge">this prop.</a>), here to be denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_9"><semantics><mrow><mi>G</mi><mphantom><mi>AAAA</mi></mphantom><mtext>photon propagator</mtext></mrow><annotation encoding="application/x-tex"> G \phantom{AAAA} \text{photon propagator} </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/electron-photon+interaction">electron-photon interaction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_10"><semantics><mrow><msub><mi>L</mi> <mi>int</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><munder><mrow><mi>i</mi><mi>g</mi><mo stretchy="false">(</mo><msup><mi>γ</mi> <mi>μ</mi></msup><msup><mo stretchy="false">)</mo> <mi>α</mi></msup><msub><mrow></mrow> <mi>β</mi></msub></mrow><mo>⏟</mo></munder><mtext>interaction</mtext></munder><mspace width="thinmathspace"></mspace><munder><munder><mover><mrow><msub><mi>ψ</mi> <mi>α</mi></msub></mrow><mo>¯</mo></mover><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mtext>incoming</mtext></mrow><mrow><mtext>electron</mtext></mrow></mfrac></mrow><mrow><mtext>field</mtext></mrow></mfrac></munder><mspace width="thickmathspace"></mspace><munder><munder><mrow><msub><mi>a</mi> <mi>μ</mi></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mtext>photon</mtext></mrow></mfrac></mrow><mrow><mtext>field</mtext></mrow></mfrac></munder><mspace width="thickmathspace"></mspace><munder><munder><mrow><msup><mi>ψ</mi> <mi>β</mi></msup></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mtext>outgoing</mtext></mrow><mrow><mtext>electron</mtext></mrow></mfrac></mrow><mrow><mtext>field</mtext></mrow></mfrac></munder></mrow><annotation encoding="application/x-tex"> L_{int} \;=\; \underset{ \text{interaction} }{ \underbrace{ i g (\gamma^\mu)^\alpha{}_\beta } } \, \underset{ { \text{incoming} \atop \text{electron} } \atop \text{field} }{\underbrace{\overline{\psi_\alpha}}} \; \underset{ { \, \atop \text{photon} } \atop \text{field} }{\underbrace{a_\mu}} \; \underset{ {\text{outgoing} \atop \text{electron} } \atop \text{field} }{\underbrace{\psi^\beta}} </annotation></semantics></math></div></li> </ol> <p>The <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a> for the <a class="existingWikiWord" href="/nlab/show/electron-photon+interaction">electron-photon interaction</a> alone is</p> <center> <img src="https://ncatlab.org/nlab/files/InteractionVertexOfQED.jpg" width="150" /> </center> <p>where the solid lines correspond to the <a class="existingWikiWord" href="/nlab/show/electron">electron</a>, and the wiggly line to the <a class="existingWikiWord" href="/nlab/show/photon">photon</a>. The corresponding <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> is (written in <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a>-notation)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_11"><semantics><mrow><munder><munder><mrow><msup><mi>ℏ</mi> <mrow><mn>3</mn><mo stretchy="false">/</mo><mn>2</mn><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo>⏟</mo></munder><mtext>loop order</mtext></munder><munder><munder><mrow><mi>i</mi><mi>g</mi><mo stretchy="false">(</mo><msup><mi>γ</mi> <mi>μ</mi></msup><msup><mo stretchy="false">)</mo> <mi>α</mi></msup><msub><mrow></mrow> <mi>β</mi></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mtext>electron-photon</mtext></mrow><mrow><mtext>interaction</mtext></mrow></mfrac></munder><mspace width="thinmathspace"></mspace><mo>.</mo><mspace width="thinmathspace"></mspace><munder><munder><mrow><msub><mover><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>α</mi></mrow></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mtext>incoming</mtext></mrow><mrow><mtext>electron</mtext></mrow></mfrac></mrow><mrow><mtext>propagator</mtext></mrow></mfrac></munder><munder><munder><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mrow><mi>μ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mtext>photon</mtext></mrow></mfrac></mrow><mrow><mtext>propagator</mtext></mrow></mfrac></munder><munder><munder><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mi>β</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></msup></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mtext>outgoing</mtext></mrow><mrow><mtext>electron</mtext></mrow></mfrac></mrow><mrow><mtext>propagator</mtext></mrow></mfrac></munder></mrow><annotation encoding="application/x-tex"> \underset{ \text{loop order} }{ \underbrace{ \hbar^{3/2-1} } } \underset{ \text{electron-photon} \atop \text{interaction} }{ \underbrace{ i g (\gamma^\mu)^\alpha{}_\beta } } \,. \, \underset{ {\text{incoming} \atop \text{electron}} \atop \text{propagator} }{ \underbrace{ \overline{\Delta(-,x)}_{-, \alpha} } } \underset{ { \, \atop \text{photon} } \atop \text{propagator} }{ \underbrace{ G(x,-)_{\mu,-} } } \underset{ { \text{outgoing} \atop \text{electron} } \atop \text{propagator} }{ \underbrace{ \Delta(x,-)^{\beta, -} } } </annotation></semantics></math></div> <p>Hence a typical Feynman diagram in the <a class="existingWikiWord" href="/nlab/show/QED">QED</a> <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> induced by this <a class="existingWikiWord" href="/nlab/show/electron-photon+interaction">electron-photon interaction</a> looks as follows:</p> <center> <img src="https://ncatlab.org/nlab/files/FeynmanDiagramGlobal.jpg" width="560" /> </center> <p>where on the bottom the corresponding <a class="existingWikiWord" href="/nlab/show/Feynman+amplitude">Feynman amplitude</a> <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> is shown; now notationally suppressing the contraction of the internal indices and all prefactors.</p> <p>For instance the two solid <a class="existingWikiWord" href="/nlab/show/edges">edges</a> between the <a class="existingWikiWord" href="/nlab/show/vertices">vertices</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_12"><semantics><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_13"><semantics><mrow><msub><mi>x</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">x_3</annotation></semantics></math> correspond to the two factors of <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_14"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta(x_2,x_2)</annotation></semantics></math>:</p> <center> <img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponent1.jpg" width="560" /> </center> <p>This way each sub-graph encodes its corresponding subset of factors in the <a class="existingWikiWord" href="/nlab/show/Feynman+amplitude">Feynman amplitude</a>:</p> <center> <img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponentTwo.jpg" width="560" /> </center><center> <img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponentThree.jpg" width="560" /> </center> <blockquote> <p>graphics grabbed from <a href="Feynman+diagram#Brouder10">Brouder 10</a></p> </blockquote> <p>A priori this <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> is defined away from coincident vertices: <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_15"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub><mo>≠</mo><msub><mi>x</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">x_i \neq x_j</annotation></semantics></math>. The definition at coincident vertices <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_16"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>x</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">x_i = x_j</annotation></semantics></math> requires a choice of <em><a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a></em> to the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> locus. This choice is the <a class="existingWikiWord" href="/nlab/show/renormalization">("re-")normalization</a> of the <a class="existingWikiWord" href="/nlab/show/Feynman+amplitude">Feynman amplitude</a>.</p> </div></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+series">perturbation series</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral+quantization">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a>, <a class="existingWikiWord" href="/nlab/show/string+scattering+amplitude">string scattering amplitude</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/worldline+formalism">worldline formalism</a></p> </li> <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/tadpole">tadpole</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+order">loop order</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/radiative+correction">radiative correction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graph+complex">graph complex</a>, <a class="existingWikiWord" href="/nlab/show/formality+of+the+little+n-disk+operad">formality of the little n-disk operad</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general_2">General</h3> <p>Traditional review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Radovan+Dermisek">Radovan Dermisek</a>, <em>Path integral for interacting field</em> (<a href="http://www.physics.indiana.edu/~dermisek/QFT_09/qft-I-5-4p.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Weinzierl">Stefan Weinzierl</a>, <em>Introduction to Feynman Integrals</em> [<a href="https://arxiv.org/abs/1005.1855">arXiv:1005.1855</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Weinzierl">Stefan Weinzierl</a>, <em>Feynman Integrals</em>, UNITEXT for Physics, Springer (2022) [<a href="https://arxiv.org/abs/2201.03593">arXiv:2201.03593</a>, <a href="https://doi.org/10.1007/978-3-030-99558-4">doi:10.1007/978-3-030-99558-4</a>] (816 pages)</p> </li> </ul> <p>See also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Matilde+Marcolli">Matilde Marcolli</a>, section 3 of <em><a class="existingWikiWord" href="/nlab/show/Noncommutative+Geometry%2C+Quantum+Fields+and+Motives">Noncommutative Geometry, Quantum Fields and Motives</a></em></p> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Feynman_diagram">Feynman diagram</a></em></p> </li> <li> <p>F. T. Brandt, J. Frenkel, D. G. C. McKeon, <em>Feynman diagrams in terms of on-shell propagators</em> [<a href="https://arxiv.org/abs/2206.14860">arXiv:2206.14860</a>]</p> </li> <li> <p>Zhi-Feng Liu, Yan-Qing Ma, <em>Determining Feynman Integrals with Only Input from Linear Algebra</em>, Physical Review Letters, Volume 129, Issue 22, 23 November 2022 (<a href="
https://doi.org/10.1103/PhysRevLett.129.222001
">doi:10.1103/PhysRevLett.129.222001</a>, <a href="https://arxiv.org/abs/2201.11637">arXiv:2201.11637</a>)</p> </li> </ul> <p>Discussion via <a class="existingWikiWord" href="/nlab/show/D-modules">D-modules</a>:</p> <ul> <li>Johannes Henn, Elizabeth Pratt, Anna-Laura Sattelberger, Simone Zoia, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals</em> [<a href="https://arxiv.org/abs/2303.11105">arXiv:2303.11105</a>]</li> </ul> <h3 id="ReferencesInCausalPerturbationTheory">In causal perturbation theory</h3> <p>Discussion of Feynman diagrams in the rigorous formulation of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> and <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> is due to</p> <ul> <li id="Keller10"><a class="existingWikiWord" href="/nlab/show/Kai+Keller">Kai Keller</a>, chapter IV of <em>Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization</em>, PhD thesis (<a href="https://arxiv.org/abs/1006.2148">arXxiv:1006.2148</a>)</li> </ul> <p>parts of which also appears as</p> <ul> <li id="DuetschFredenhagenKellerRejzner14"><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Kai+Keller">Kai Keller</a>, <a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, <em>Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization</em>, J. Math. Phy. <p>55(12), 122303 (2014) (<a href="https://arxiv.org/abs/1311.5424">arXiv:1311.5424</a>)</p> </li> </ul> <p>An exposition of this is in</p> <ul> <li id="Brouder10"><a class="existingWikiWord" href="/nlab/show/Christian+Brouder">Christian Brouder</a>, <em>Multiplication of distributions</em>, 2010 (<a class="existingWikiWord" href="/nlab/files/BrouderProductOfDistributions.pdf" title="pdf">pdf</a>)</li> </ul> <p>Relation to the <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> structure on Feynman diagrams due to <a href="renormalization#Kreimer97">Kreimer 97</a> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jose+Gracia-Bondia">Jose Gracia-Bondia</a>, S. Lazzarini, <em>Connes-Kreimer-Epstein-Glaser Renormalization</em> (<a href="https://arxiv.org/abs/hep-th/0006106">arXiv:hep-th/0006106</a>)</li> </ul> <p>Relation to <a class="existingWikiWord" href="/nlab/show/periods">periods</a> (in the sense <a href="period#ReferencesInPerturbativeQuantumFieldTheory">here</a>) is discussed in</p> <ul> <li id="Rejzner16"><a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, <em>Renormalization and periods in perturbative Algebraic Quantum Field Theory</em> (<a href="https://arxiv.org/abs/1603.02748">arXiv:1603.02748</a>)</li> </ul> <p>See also</p> <ul> <li id="HawkinsRejzner16"><a class="existingWikiWord" href="/nlab/show/Eli+Hawkins">Eli Hawkins</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, section 5.2 of <em>The Star Product in Interacting Quantum Field Theory</em> (<a href="https://arxiv.org/abs/1612.09157">arXiv:1612.09157</a>)</li> </ul> <p>Review is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, section 6.5.2 of <em>Perturbative Algebraic Quantum Field Theory</em>, Mathematical Physics Studies, Springer 2016 (<a href="https://link.springer.com/book/10.1007%2F978-3-319-25901-7">pdf</a>)</li> </ul> <h3 id="ReferencesInTermsOfMotivicStructures">In terms of motivic structures</h3> <p>In terms of <a class="existingWikiWord" href="/nlab/show/motives">motives</a> (see also <em><a class="existingWikiWord" href="/nlab/show/motives+in+physics">motives in physics</a></em>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Spencer+Bloch">Spencer Bloch</a>, <a class="existingWikiWord" href="/nlab/show/Helene+Esnault">Helene Esnault</a>, <a class="existingWikiWord" href="/nlab/show/Dirk+Kreimer">Dirk Kreimer</a>, <em>On motives associated to graph polynomials, (<a href="http://preprints.ihes.fr/2013/P/P-13-24.pdf">pdf</a>)</em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spencer+Bloch">Spencer Bloch</a>, Pierre Vanhove, The elliptic dilogarithm for the sunset graph, IHES preprint P-13-24, pdf</p> </li> </ul> <h3 id="ReferencesInHomologicalBVQuantization">In homological BV-quantization</h3> <p>A clean derivation of the <a class="existingWikiWord" href="/nlab/show/Feynman+rules">Feynman rules</a> for finite-dimensional spaces of fields in terms of <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> by passing to <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of <a class="existingWikiWord" href="/nlab/show/BV-complexes">BV-complexes</a> (see at <em><a href="geometric%20quantization#AsIndexOfSpinCDiracOperator">BV-formalism – Homological quantization</a>)</em> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Owen+Gwilliam">Owen Gwilliam</a>, <a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, <em>How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism</em> (2011) (<a href="http://arxiv.org/abs/1202.1554">arXiv:1202.1554</a>)</li> </ul> <p>with a review in the broader context of <a class="existingWikiWord" href="/nlab/show/factorization+algebras+of+observables">factorization algebras of observables</a> in section 2.3 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Owen+Gwilliam">Owen Gwilliam</a>, <em>Factorization algebras and free field theories</em> (<a class="existingWikiWord" href="/nlab/files/GwilliamThesis.pdf" title="pdf">pdf</a>)</li> </ul> <h3 id="AsStringDiagrams">As string diagrams (morphisms in monoidal categories)</h3> <p>It has been observed that Feynman diagrams, notably in <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, are in particular <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> (in the sense of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, not here in any sense of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>!) in the given <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a>: the edges are labelled by <a class="existingWikiWord" href="/nlab/show/particle">particle</a> <a class="existingWikiWord" href="/nlab/show/species">species</a>, hence by <a class="existingWikiWord" href="/nlab/show/Wigner+classification">Wigner classification</a> by <a class="existingWikiWord" href="/nlab/show/irreps">irreps</a>, and the vertices are labeled by representation <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> (“<a class="existingWikiWord" href="/nlab/show/intertwiners">intertwiners</a>”) which, indeed, label the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> of particles in the Feynman diagram.</p> <p>A review of this formulation is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Aaron+Lauda">Aaron Lauda</a>, <em>A Prehistory of n-Categorical Physics</em>, Deep beauty, 13-128, Cambridge Univ. Press, Cambridge, 2011 (<a href="http://arxiv.org/abs/0908.2469">arXiv:0908.2469</a>)</li> </ul> <p>Moreover, one may think of <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> in <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> as providing <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> for <a class="existingWikiWord" href="/nlab/show/proof+nets">proof nets</a> (see there for more) in multiplicative <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>. Under this identification then Feynman diagrams have a relation to <a class="existingWikiWord" href="/nlab/show/proof+nets">proof nets</a>. Something like this is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Richard+Blute">Richard Blute</a>, <a class="existingWikiWord" href="/nlab/show/Prakash+Panangaden">Prakash Panangaden</a>, <em>Proof nets as formal Feynman diagrams</em> (<a href="http://www.indiana.edu/~iulg/qliqc/phi.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 21, 2023 at 03:44:05. See the <a href="/nlab/history/Feynman+diagram" 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/Feynman+diagram" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4702/#Item_34">Discuss</a><span class="backintime"><a href="/nlab/revision/Feynman+diagram/45" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Feynman+diagram" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Feynman+diagram" accesskey="S" class="navlink" id="history" rel="nofollow">History (45 revisions)</a> <a href="/nlab/show/Feynman+diagram/cite" style="color: black">Cite</a> <a href="/nlab/print/Feynman+diagram" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Feynman+diagram" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>