A 256×256 Matrix Form of the Dirac Equation in Curved Spacetime for QED Scattering Without an Explicit Vierbein

A 256×256 Matrix Form of the Dirac Equation in Curved Spacetime for QED Scattering Without an Explicit Vierbein | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A 256×256 Matrix Form of the Dirac Equation in Curved Spacetime for QED Scattering Without an Explicit Vierbein Hirokazu Maruyama This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7747333/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We present a matrix-based formulation of the Dirac equation in curved spacetime that avoids introducing an explicit vierbein and spin connection. The method assembles sixteen two-index gamma matrices into a single 256×256 representation and embeds the metric components directly in the matrix entries. Spacetime remains four dimensional; the number sixteen labels the basis elements of the Dirac algebra. Starting from a Lagrangian identical in form to that of quantum electrodynamics (QED) in flat spacetime, the approach replaces differential-geometric manipulations with matrix products and traces, enabling straightforward symbolic and numerical automation. We develop scattering calculations for Compton, muon–pair production in electron–positron collisions, Møller, and Bhabha processes. Illustrative examples with constant metrics, including off-diagonal components, exhibit characteristic angular modifications of the differential cross sections, whereas in the flat-metric limit the results agree exactly with standard formulas. We delineate the practical scope of the method—constant or slowly varying backgrounds—and identify extensions to coordinate-dependent metrics and loop calculations as priorities for future work, including tests of whether curvature-induced structure can influence high-energy behavior. The formulation thus offers a pragmatic alternative for exploring fermionic processes in gently curved backgrounds while remaining consistent with flat-space quantum electrodynamics. Curved spacetime Dirac equation Quantum electrodynamics Gamma matrices Scattering processes Particle physics High-energy physics Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7747333","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":522858410,"identity":"0791be1b-7a76-4d79-823b-9e57dad78273","order_by":0,"name":"Hirokazu Maruyama","email":"data:image/png;base64,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","orcid":"","institution":"Independent Researcher","correspondingAuthor":true,"prefix":"","firstName":"Hirokazu","middleName":"","lastName":"Maruyama","suffix":""}],"badges":[],"createdAt":"2025-09-30 05:23:22","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7747333/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7747333/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":92561957,"identity":"cb09a16e-f2fb-4eb6-bfcd-c590a9e72457","added_by":"auto","created_at":"2025-10-01 04:51:42","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":3766,"visible":true,"origin":"","legend":"","description":"","filename":"ec8e81fa677d4bdbad75fc81107a4214.json","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/f0850a3a0bbf68ea03709aa3.json"},{"id":92561960,"identity":"5a647df5-b918-4f12-bdfc-f89b0f951fc4","added_by":"auto","created_at":"2025-10-01 04:51:42","extension":"xml","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":82045,"visible":true,"origin":"","legend":"","description":"","filename":"ec8e81fa677d4bdbad75fc81107a42141enriched.xml","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/3120829a5548d0873fe352e4.xml"},{"id":92561965,"identity":"8fbd4079-eb6f-4df7-a493-e5dcd1135e88","added_by":"auto","created_at":"2025-10-01 04:51:42","extension":"pdf","order_by":2,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":651122,"visible":true,"origin":"","legend":"","description":"","filename":"GRGLorentzianMaruyama256x256Dirac20250930v1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/822ec80d22225feaaad612a0.pdf"},{"id":92561958,"identity":"2b0e6d12-6a7c-4331-a5c3-d75dda7d3698","added_by":"auto","created_at":"2025-10-01 04:51:42","extension":"aux","order_by":3,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":15507,"visible":true,"origin":"","legend":"","description":"","filename":"Maruyama256x256DiracCurvedQEDv1.aux","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/5bc1b6dbd494e5f333843734.aux"},{"id":92562667,"identity":"f46afc4b-f482-44e2-9e3b-819e74e88b05","added_by":"auto","created_at":"2025-10-01 05:07:42","extension":"bbl","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":8542,"visible":true,"origin":"","legend":"","description":"","filename":"Maruyama256x256DiracCurvedQEDv1.bbl","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/b077cc4e44db994b2cfdfe9d.bbl"},{"id":92562540,"identity":"bc97550e-1648-4c6d-bc42-a742abbca52d","added_by":"auto","created_at":"2025-10-01 04:59:42","extension":"blg","order_by":5,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":2384,"visible":true,"origin":"","legend":"","description":"","filename":"Maruyama256x256DiracCurvedQEDv1.blg","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/f050c3ef8084d12ee1dbb816.blg"},{"id":92561961,"identity":"332b379d-76a9-4280-ba34-2158aa146567","added_by":"auto","created_at":"2025-10-01 04:51:42","extension":"dvi","order_by":6,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":59016,"visible":true,"origin":"","legend":"","description":"","filename":"Maruyama256x256DiracCurvedQEDv1.dvi","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/ec1aa06f9d4f568daf08d75d.dvi"},{"id":92561959,"identity":"b0ea8efc-8e0a-4721-9b8f-23449cd22a39","added_by":"auto","created_at":"2025-10-01 04:51:42","extension":"log","order_by":7,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":33303,"visible":true,"origin":"","legend":"","description":"","filename":"Maruyama256x256DiracCurvedQEDv1.log","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/2b15580066de06ecfb2276a1.log"},{"id":92561963,"identity":"4d1987d9-bd49-44b8-83de-398b6873fbd9","added_by":"auto","created_at":"2025-10-01 04:51:42","extension":"out","order_by":8,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":8291,"visible":true,"origin":"","legend":"","description":"","filename":"Maruyama256x256DiracCurvedQEDv1.out","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/d42bd190faa0c1db09ebfd03.out"},{"id":92561966,"identity":"6ffbe684-75ee-4635-8cd4-e36a6305f593","added_by":"auto","created_at":"2025-10-01 04:51:42","extension":"pdf","order_by":9,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":647056,"visible":true,"origin":"","legend":"","description":"","filename":"Maruyama256x256DiracCurvedQEDv1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/4faf5969df92b1033f50a725.pdf"},{"id":92561972,"identity":"fd306b9c-a06c-4897-8992-17cca0f89ea9","added_by":"auto","created_at":"2025-10-01 04:51:43","extension":"gz","order_by":10,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":151343,"visible":true,"origin":"","legend":"","description":"","filename":"Maruyama256x256DiracCurvedQEDv1.synctex.gz","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/2e7913890ddf24bc09ba4e17.gz"},{"id":92561969,"identity":"54cfdc50-eb5e-46b1-883e-5e6844c6400a","added_by":"auto","created_at":"2025-10-01 04:51:42","extension":"pdf","order_by":13,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":31850,"visible":true,"origin":"","legend":"","description":"","filename":"coverletterspringerGRGLorentzianwithdisclosure.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/3e5f567a86d1bfa7e4cab204.pdf"},{"id":92562542,"identity":"ee60b385-32a0-4651-9aae-f1d32c9fb343","added_by":"auto","created_at":"2025-10-01 04:59:42","extension":"cls","order_by":16,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":55857,"visible":true,"origin":"","legend":"","description":"","filename":"snjnl.cls","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/8ded75f6ebfffce8949d3e67.cls"},{"id":92561968,"identity":"81e1a600-d4d3-4146-9b05-72baab5d1c28","added_by":"auto","created_at":"2025-10-01 04:51:42","extension":"bst","order_by":17,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":64164,"visible":true,"origin":"","legend":"","description":"","filename":"snmathphysnum.bst","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/af8f8e67c7e255570ee388dd.bst"},{"id":92561970,"identity":"0866d38a-71d8-449f-9e6a-ca5dec748551","added_by":"auto","created_at":"2025-10-01 04:51:43","extension":"xml","order_by":18,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":89320,"visible":true,"origin":"","legend":"","description":"","filename":"ec8e81fa677d4bdbad75fc81107a42141structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/996c349155ea06999c13c53c.xml"},{"id":92562544,"identity":"d33a7f3a-b50e-4160-87db-86eeba1b6105","added_by":"auto","created_at":"2025-10-01 04:59:43","extension":"html","order_by":19,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":99131,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1/47bb45559d5f541286f7ca90.html"},{"id":92600522,"identity":"2a2e4a53-05b1-4516-a743-b1f6613d81df","added_by":"auto","created_at":"2025-10-01 14:24:19","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":524788,"visible":true,"origin":"","legend":"","description":"","filename":"GRGLorentzianMaruyama256x256Dirac20250930v1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7747333/v1_covered_9ac256ab-89ad-442c-bb93-f74c71f8259d.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A 256×256 Matrix Form of the Dirac Equation in Curved Spacetime for QED Scattering Without an Explicit Vierbein","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Curved spacetime, Dirac equation, Quantum electrodynamics, Gamma matrices, Scattering processes, Particle physics, High-energy physics","lastPublishedDoi":"10.21203/rs.3.rs-7747333/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7747333/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe present a matrix-based formulation of the Dirac equation in curved spacetime that avoids introducing an explicit vierbein and spin connection. The method assembles sixteen two-index gamma matrices into a single 256\u0026times;256 representation and embeds the metric components directly in the matrix entries. Spacetime remains four dimensional; the number sixteen labels the basis elements of the Dirac algebra. Starting from a Lagrangian identical in form to that of quantum electrodynamics (QED) in flat spacetime, the approach replaces differential-geometric manipulations with matrix products and traces, enabling straightforward symbolic and numerical automation. We develop scattering calculations for Compton, muon\u0026ndash;pair production in electron\u0026ndash;positron collisions, M\u0026oslash;ller, and Bhabha processes. Illustrative examples with constant metrics, including off-diagonal components, exhibit characteristic angular modifications of the differential cross sections, whereas in the flat-metric limit the results agree exactly with standard formulas. We delineate the practical scope of the method\u0026mdash;constant or slowly varying backgrounds\u0026mdash;and identify extensions to coordinate-dependent metrics and loop calculations as priorities for future work, including tests of whether curvature-induced structure can influence high-energy behavior. The formulation thus offers a pragmatic alternative for exploring fermionic processes in gently curved backgrounds while remaining consistent with flat-space quantum electrodynamics.\u003c/p\u003e","manuscriptTitle":"A 256×256 Matrix Form of the Dirac Equation in Curved Spacetime for QED Scattering Without an Explicit Vierbein","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-01 04:51:37","doi":"10.21203/rs.3.rs-7747333/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"d8e6ecb8-51bc-4427-98ad-999733dfb366","owner":[],"postedDate":"October 1st, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-10-01T04:51:40+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-01 04:51:37","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7747333","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7747333","identity":"rs-7747333","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}