Space-Time Approach to Quantum Electrodynamics

<!DOCTYPE html>     <html class="no-js" lang="en">  <head> <title>Space-Time Approach to Quantum Electrodynamics - NASA/ADS</title>  <link rel="apple-touch-icon" sizes="180x180" href="//styles/favicon/apple-touch-icon.png" /> <link rel="icon" type="image/png" sizes="32x32" href="//styles/favicon/favicon-32x32.png" /> <link rel="icon" type="image/png" sizes="16x16" href="//styles/favicon/favicon-16x16.png" /> <link rel="manifest" href="//styles/favicon/site.webmanifest" /> <link rel="mask-icon" href="//styles/favicon/safari-pinned-tab.svg" color="#5bbad5" /> <meta name="apple-mobile-web-app-title" content="NASA ADS" /> <meta name="application-name" content="NASA ADS" /> <meta name="msapplication-TileColor" content="#ffc40d" /> <meta name="theme-color" content="#ffffff" />  <link rel="stylesheet" href="/styles/css/styles.css"> <meta name="robots" content="noarchive"> <link rel="canonical" href="http://ui.adsabs.harvard.edu/abs/1949PhRv...76..769F/abstract"/> <meta name="description" content="In this paper two things are done. (1) It is shown that a considerable simplification can be attained in writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down directly for any specific problem. Being simply a restatement of conventional electrodynamics, however, the matrix elements diverge for complex processes. (2) Electrodynamics is modified by altering the interaction of electrons at short distances. All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, which gives finite results for these matrices also. The only effects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directly observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modification goes to zero. The results then agree with those of Schwinger. A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons. The simplification in writing the expressions results from an emphasis on the over-all space-time view resulting from a study of the solution of the equations of electrodynamics. The relation of this to the more conventional Hamiltonian point of view is discussed. It would be very difficult to make the modification which is proposed if one insisted on having the equations in Hamiltonian form. The methods apply as well to charges obeying the Klein-Gordon equation, and to the various meson theories of nuclear forces. Illustrative examples are given. Although a modification like that used in electrodynamics can make all matrices finite for all of the meson theories, for some of the theories it is no longer true that all directly observable phenomena are insensitive to the details of the modification used. The actual evaluation of integrals appearing in the matrix elements may be facilitated, in the simpler cases, by methods described in the appendix.">  <meta property="og:type" content="article"> <meta property="og:title" content="Space-Time Approach to Quantum Electrodynamics"> <meta property="og:site_name" content="NASA/ADS"> <meta property="og:description" content="In this paper two things are done. (1) It is shown that a considerable simplification can be attained in writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down directly for any specific problem. Being simply a restatement of conventional electrodynamics, however, the matrix elements diverge for complex processes. (2) Electrodynamics is modified by altering the interaction of electrons at short distances. All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, which gives finite results for these matrices also. The only effects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directly observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modification goes to zero. The results then agree with those of Schwinger. A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons. The simplification in writing the expressions results from an emphasis on the over-all space-time view resulting from a study of the solution of the equations of electrodynamics. The relation of this to the more conventional Hamiltonian point of view is discussed. It would be very difficult to make the modification which is proposed if one insisted on having the equations in Hamiltonian form. The methods apply as well to charges obeying the Klein-Gordon equation, and to the various meson theories of nuclear forces. Illustrative examples are given. Although a modification like that used in electrodynamics can make all matrices finite for all of the meson theories, for some of the theories it is no longer true that all directly observable phenomena are insensitive to the details of the modification used. The actual evaluation of integrals appearing in the matrix elements may be facilitated, in the simpler cases, by methods described in the appendix."> <meta property="og:url" content="https://ui.adsabs.harvard.edu/abs/1949PhRv...76..769F/abstract"> <meta property="og:image" content="https://ui.adsabs.harvard.edu/styles/img/transparent_logo.svg"> <meta property="article:published_time" content="09/1949"> <meta property="article:author" content="Feynman, R. P.">  <meta name="citation_journal_title" content="Physical Review"> <meta name="citation_authors" content="Feynman, R. P."> <meta name="citation_title" content="Space-Time Approach to Quantum Electrodynamics"> <meta name="citation_date" content="09/1949"> <meta name="citation_volume" content="76"> <meta name="citation_issue" content="6"> <meta name="citation_firstpage" content="769"> <meta name="citation_doi" content="10.1103/PhysRev.76.769"> <meta name="citation_issn" content="1536-6065"> <meta name="citation_language" content="en"> <meta name="citation_abstract_html_url" content="https://ui.adsabs.harvard.edu/abs/1949PhRv...76..769F/abstract"> <meta name="citation_publication_date" content="09/1949"> <meta name="citation_lastpage" content="789" /> <link title="schema(PRISM)" rel="schema.prism" href="http://prismstandard.org/namespaces/1.2/basic/" /> <meta name="prism.publicationDate" content="09/1949" /> <meta name="prism.publicationName" content="PhRv" /> <meta name="prism.issn" content="1536-6065" /> <meta name="prism.volume" content="76" /> <meta name="prism.startingPage" content="769" /> <meta name="prism.endingPage" content="789" /> <link title="schema(DC)" rel="schema.dc" href="http://purl.org/dc/elements/1.1/" /> <meta name="dc.identifier" content="doi:10.1103/PhysRev.76.769" /> <meta name="dc.date" content="09/1949" /> <meta name="dc.source" content="PhRv" /> <meta name="dc.title" content="Space-Time Approach to Quantum Electrodynamics" /> <meta name="dc.creator" content="Feynman, R. P.">  <meta name="twitter:card" content="summary_large_image"/> <meta name="twitter:description" content="In this paper two things are done. (1) It is shown that a considerable simplification can be attained in writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down directly for any specific problem. Being simply a restatement of conventional electrodynamics, however, the matrix elements diverge for complex processes. (2) Electrodynamics is modified by altering the interaction of electrons at short distances. All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, which gives finite results for these matrices also. The only effects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directly observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modification goes to zero. The results then agree with those of Schwinger. A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons. The simplification in writing the expressions results from an emphasis on the over-all space-time view resulting from a study of the solution of the equations of electrodynamics. The relation of this to the more conventional Hamiltonian point of view is discussed. It would be very difficult to make the modification which is proposed if one insisted on having the equations in Hamiltonian form. The methods apply as well to charges obeying the Klein-Gordon equation, and to the various meson theories of nuclear forces. Illustrative examples are given. Although a modification like that used in electrodynamics can make all matrices finite for all of the meson theories, for some of the theories it is no longer true that all directly observable phenomena are insensitive to the details of the modification used. The actual evaluation of integrals appearing in the matrix elements may be facilitated, in the simpler cases, by methods described in the appendix."/> <meta name="twitter:title" content="Space-Time Approach to Quantum Electrodynamics"/> <meta name="twitter:site" content="@adsabs"/> <meta name="twitter:domain" content="NASA/ADS"/> <meta name="twitter:image:src" content="https://ui.adsabs.harvard.edu/styles/img/transparent_logo.svg"/> <meta name="twitter:creator" content="@adsabs"/> <meta charset="utf-8"> <meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"> <base href="/"> <style> .btn-full-ads { color: #fff !important; background-color: #1a1a1a !important; border-color: #1a1a1a !important; margin-top: 9px !important; padding-bottom: 10px !important; padding-top: 10px !important; } .btn-full-ads:hover, .btn-full-ads:focus, .btn-full-ads:active, .btn-full-ads.active, .open>.dropdown-toggle.btn-full-ads { color: #000 !important; background-color: #ddd !important; 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b.innerHTML += autoValues[i].match.substr(val.length); } // Insert a input field that will hold the current array item's value: b.innerHTML += "<input type='hidden' value='" + autoValues[i].value + "'>"; // Listen to clicks on the item value (DIV element): b.addEventListener("click", function(e) { var terms = searchBox.value.split(/\s+/); // Remove the current part of the input used for matching terms.pop(); // Insert the value for the autocomplete text field: terms.push(this.getElementsByTagName("input")[0].value); searchBox.value = terms.join(" "); // Move cursor position inside quotes/parenthesis if needed searchBox.focus(); if (searchBox.value[searchBox.value.length-1] === '"' || searchBox.value[searchBox.value.length-1] === ')') { searchBox.setSelectionRange(searchBox.value.length-1, searchBox.value.length-1); } // Close the list of autocompleted values closeAllLists(); }); a.appendChild(b); } } if (a.children.length > 0) { // By default, enter will select the first entry currentFocus = 0; 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// Add class "autocomplete-active": x[currentFocus].classList.add("autocomplete-active"); } function removeActive(x) { // Remove the "active" class from all autocomplete items: for (var i = 0; i < x.length; i++) { x[i].classList.remove("autocomplete-active"); } } function closeAllLists(elmnt) { // Close all autocomplete lists in the document, except the one passed as an argument: var x = document.getElementsByClassName("autocomplete-items"); for (var i = 0; i < x.length; i++) { if (elmnt != x[i] && elmnt != searchBox) { x[i].parentNode.removeChild(x[i]); } } } // Any other clicks in the document: document.addEventListener("click", function (e) { closeAllLists(e.target); }); } var autoList = [ { value: 'author:""', label: 'Author', match: 'author:"' }, { value: 'author:"^"', label: 'First Author', match: 'first author' }, { value: 'author:"^"', label: 'First Author', match: 'author:"^' }, { value: 'bibcode:""', label: 'Bibcode', desc: 'e.g. bibcode:1989ApJ...342L..71R', match: 'bibcode:"' }, { value: 'bibstem:""', label: 'Publication', desc: 'e.g. bibstem:ApJ', match: 'bibstem:"' }, { value: 'bibstem:""', label: 'Publication', desc: 'e.g. bibstem:ApJ', match: 'publication (bibstem)' }, { value: 'arXiv:', label: 'arXiv ID', match: 'arxiv:' }, { value: 'doi:', label: 'DOI', match: 'doi:' }, { value: 'full:""', label: 'Full text search', desc: 'title, abstract, and body', match: 'full:' }, { value: 'full:""', label: 'Full text search', desc: 'title, abstract, and body', match: 'fulltext' }, { value: 'full:""', label: 'Full text search', desc: 'title, abstract, and body', match: 'text' }, { value: 'year:', label: 'Year', match: 'year' }, { value: 'year:1999-2005', label: 'Year Range', desc: 'e.g. 1999-2005', match: 'year range' }, { value: 'aff:""', label: 'Affiliation', match: 'aff:' }, { value: 'abs:""', label: 'Search abstract + title + keywords', match: 'abs:' }, { value: 'database:astronomy', label: 'Limit to papers in the astronomy database', match: 'database:astronomy' }, { value: 'database:physics', label: 'Limit to papers in the physics database', match: 'database:physics' }, { value: 'title:""', label: 'Title', match: 'title:"' }, { value: 'orcid:', label: 'ORCiD identifier', match: 'orcid:' }, { value: 'object:', label: 'SIMBAD object (e.g. object:LMC)', match: 'object:' }, { value: 'property:refereed', label: 'Limit to refereed', desc: '(property:refereed)', match: 'refereed' }, { value: 'property:refereed', label: 'Limit to refereed', desc: '(property:refereed)', match: 'property:refereed' }, { value: 'property:notrefereed', label: 'Limit to non-refereed', desc: '(property:notrefereed)', match: 'property:notrefereed' }, { value: 'property:notrefereed', label: 'Limit to non-refereed', desc: '(property:notrefereed)', match: 'notrefereed' }, { value: 'property:eprint', label: 'Limit to eprints', desc: '(property:eprint)', match: 'eprint' }, { value: 'property:eprint', label: 'Limit to eprints', desc: '(property:eprint)', match: 'property:eprint' }, { value: 'property:openaccess', label: 'Limit to open access', desc: '(property:openaccess)', match: 'property:openaccess' }, { value: 'property:openaccess', label: 'Limit to open access', desc: '(property:openaccess)', match: 'openaccess' }, { value: 'doctype:software', label: 'Limit to software', desc: '(doctype:software)', match: 'software' }, { value: 'doctype:software', label: 'Limit to software', desc: '(doctype:software)', match: 'doctype:software' }, { value: 'property:inproceedings', label: 'Limit to papers in conference proceedings', desc: '(property:inproceedings)', match: 'proceedings' }, { value: 'property:inproceedings', label: 'Limit to papers in conference proceedings', desc: '(property:inproceedings)', match: 'property:inproceedings' }, { value: 'citations()', label: 'Citations', desc: 'Get papers citing your search result set', match: 'citations(' }, { value: 'references()', label: 'References', desc: 'Get papers referenced by your search result set', match: 'references(' }, { value: 'trending()', label: 'Trending', desc: 'Get papers most read by users who recently read your search result set', match: 'trending(' }, { value: 'reviews()', label: 'Review Articles', desc: 'Get most relevant papers that cite your search result set', match: 'reviews(' }, { value: 'useful()', label: 'Useful', desc: 'Get papers most frequently cited by your search result set', match: 'useful(' }, { value: 'similar()', label: 'Similar', desc: 'Get papers that have similar full text to your search result set', match: 'similar(' }, ]; // initiate the autocomplete function on the "q" element, and pass along the operators array as possible autocomplete values: inputBox = document.getElementById("q") if (inputBox) { inputBox.focus() // autofucs inputBox.setSelectionRange(inputBox.value.length, inputBox.value.length); // bring cursor to the end autocomplete(inputBox, autoList); } </script> <script> (function() { // turn off no-js if we have javascript document.documentElement.className = document.documentElement.className.replace("no-js", "js"); function getCookie(cname) { var name = cname + "="; var decodedCookie = decodeURIComponent(document.cookie); var ca = decodedCookie.split(';'); for (var i = 0; i < ca.length; i++) { var c = ca[i]; while (c.charAt(0) == ' ') { c = c.substring(1); } if (c.indexOf(name) == 0) { return c.substring(name.length, c.length); } } return ""; } (function() { // looks for the cookie, and sets true if its 'always' const coreCookie = getCookie('core') === 'always'; // only load bumblebee if we detect the core cookie and we are on abstract page if (coreCookie || (!(/^\/abs\//.test(document.location.pathname)) && !coreCookie)) { return; 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