{"id":{"repo_id":"exeter","oai_identifier":"oai:figshare.com:article/31157584"},"canonical_url":"https://search.dev.ndltd.org/etd/exeter/oai:figshare.com:article/31157584","repository":{"repo_id":"exeter","name":"University of Exeter","base_url":"https://api.figshare.com/v2/oai"},"display":{"title":"On the parallels between the Maxwell and Dirac equations","abstract":"Exploring analogies between seemingly disparate physical systems is a powerful research strategy because it allows us to transfer insights, methods, and intuitions from one domain to another, often revealing hidden connections and sparking creative breakthroughs, as exemplified by Hamilton's seminal analogy between classical mechanics and geometric optics---a correspondence that ultimately seeded the development of quantum mechanics. In this work, in the same spirit, we explore deeply the similarity between the Maxwell and Dirac equations. We reformulate the Dirac equation in an \\emph{exact} form of the Maxwell equations, which we refer to as the ``electronic Maxwell equations.'' In this formulation, the standard free Maxwell equations emerge as a special case corresponding to vanishing longitudinal components and zero mass. The presence of a nonzero mass naturally leads to longitudinal components in the matter wave, arising as a consequence of enforcing Lorentz invariance in the electronic Maxwell equations. The physical quantities described by these electronic fields closely parallel their electromagnetic counterparts. We also make a discussion on wave components we may measure if the matter wave is actually measurable. Likewise, we reformulate Maxwell’s equations into an \\emph{exact} form of the Dirac equation, which we term the ``electromagnetic Dirac equation.'' This formulation yields the $8\\times8$ spin operator for spin‑1 particles and is capable of reproducing the Klein--Gordon equation (the wave equation for electromagnetic waves). The absence of longitudinal components characterizes the electromagnetic field as a spin‑1 field. Furthermore, we find that the Zitterbewegung of the photon corresponds to the rapidly oscillating part of the Poynting vector in classical electromagnetism. Both the Maxwell and Dirac equations can be reformulated into closely related quaternionic equations, which directly reveal the difference in how these equations transform under Lorentz transformations. Building on this observation, we propose a possible (spin) geometry underlying both the Maxwell and Dirac equations. Within this new framework, the spacetime itself is a second--order covariant or contravariant spinor. Furthermore, the electronic field is naturally identified as a spinor (first--order spinor), whereas the electromagnetic field emerges as a second--order mixed spinor. We further express both the Maxwell and Dirac equations in terms of these spinors and extend the resulting formalism to curved spacetime. Besides, we explore the parallels between the Feynman path integral---an alternative formalism for the quantum wave equations, and the Rayleigh--Sommerfeld diffraction integral---an alternative formalism for the optical wave equation. We apply the idea developed in the diffraction theory to establish the path integral for the Schr\\\"{o}dinger equation---a generalized Helmholtz equation with an additional temporal derivative term, the Klein--Gordon equation---a generalized optical wave equation with a non--vanishing mass term, and the Dirac equation---the massive Maxwell equations. Based on our results, we dicuss the flaws in the Feynman checkerboard model. Finally, we present two illustrative applications of the analogy between the Maxwell and Dirac equations. The first explores how to assign an effective ``mass'' to the electromagnetic wave. The second examines an intriguing case of a spacetime--varying electromagnetic medium that transforms the Maxwell equations into a massless Dirac equation, with the potentials taking a special form. We derive a general solution for this equation, revealing that such materials can behave like a one--sided mirror.<p></p>","abstract_html":"Exploring analogies between seemingly disparate physical systems is a powerful research strategy because it allows us to transfer insights, methods, and intuitions from one domain to another, often revealing hidden connections and sparking creative breakthroughs, as exemplified by Hamilton&#x27;s seminal analogy between classical mechanics and geometric optics---a correspondence that ultimately seeded the development of quantum mechanics. In this work, in the same spirit, we explore deeply the similarity between the Maxwell and Dirac equations. We reformulate the Dirac equation in an \\emph{exact} form of the Maxwell equations, which we refer to as the ``electronic Maxwell equations.&#x27;&#x27; In this formulation, the standard free Maxwell equations emerge as a special case corresponding to vanishing longitudinal components and zero mass. The presence of a nonzero mass naturally leads to longitudinal components in the matter wave, arising as a consequence of enforcing Lorentz invariance in the electronic Maxwell equations. The physical quantities described by these electronic fields closely parallel their electromagnetic counterparts. We also make a discussion on wave components we may measure if the matter wave is actually measurable. Likewise, we reformulate Maxwell’s equations into an \\emph{exact} form of the Dirac equation, which we term the ``electromagnetic Dirac equation.&#x27;&#x27; This formulation yields the $8\\times8$ spin operator for spin‑1 particles and is capable of reproducing the Klein--Gordon equation (the wave equation for electromagnetic waves). The absence of longitudinal components characterizes the electromagnetic field as a spin‑1 field. Furthermore, we find that the Zitterbewegung of the photon corresponds to the rapidly oscillating part of the Poynting vector in classical electromagnetism. Both the Maxwell and Dirac equations can be reformulated into closely related quaternionic equations, which directly reveal the difference in how these equations transform under Lorentz transformations. Building on this observation, we propose a possible (spin) geometry underlying both the Maxwell and Dirac equations. Within this new framework, the spacetime itself is a second--order covariant or contravariant spinor. Furthermore, the electronic field is naturally identified as a spinor (first--order spinor), whereas the electromagnetic field emerges as a second--order mixed spinor. We further express both the Maxwell and Dirac equations in terms of these spinors and extend the resulting formalism to curved spacetime. Besides, we explore the parallels between the Feynman path integral---an alternative formalism for the quantum wave equations, and the Rayleigh--Sommerfeld diffraction integral---an alternative formalism for the optical wave equation. We apply the idea developed in the diffraction theory to establish the path integral for the Schr\\&quot;{o}dinger equation---a generalized Helmholtz equation with an additional temporal derivative term, the Klein--Gordon equation---a generalized optical wave equation with a non--vanishing mass term, and the Dirac equation---the massive Maxwell equations. Based on our results, we dicuss the flaws in the Feynman checkerboard model. Finally, we present two illustrative applications of the analogy between the Maxwell and Dirac equations. The first explores how to assign an effective ``mass&#x27;&#x27; to the electromagnetic wave. The second examines an intriguing case of a spacetime--varying electromagnetic medium that transforms the Maxwell equations into a massless Dirac equation, with the potentials taking a special form. We derive a general solution for this equation, revealing that such materials can behave like a one--sided mirror.&lt;p&gt;&lt;/p&gt;","abstract_has_math":true,"creators":["Mingjie Li (21042047)"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2026,"date_issued":"2026-02-02T00:00:00Z","date_published":"2026-02-02T00:00:00Z","updated_at":"2026-07-27T19:34:41Z","subjects":["Maxwell's equations","The Dirac equation","Electronic Maxwell's equations","The electromagnetic Dirac equation","Rayleigh–Sommerfeld diffraction integral","Feynman’s path integral","Feynman checker board model","Quaternion","Spin geometry","Lorentz transformation","Spacetime","Spinor","Electromagnetic materials"],"languages":[],"rights":["All rights reserved"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["10779/exe.31157584.v1"],"render_values":[{"text":"10779/exe.31157584.v1","href":null,"code":true}]}]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Mingjie Li (21042047)"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2026-02-02T00:00:00Z"]},{"key":"dc:relation","label":"Dc Relation","values":["https://figshare.com/articles/thesis/On_the_parallels_between_the_Maxwell_and_Dirac_equations/31157584"]},{"key":"dc:type","label":"Dc Type","values":["Text","Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Maxwell's equations","The Dirac equation","Electronic Maxwell's equations","The electromagnetic Dirac equation","Rayleigh–Sommerfeld diffraction integral","Feynman’s path integral","Feynman checker board model","Quaternion","Spin geometry","Lorentz transformation","Spacetime","Spinor","Electromagnetic materials"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["All rights reserved"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["10779/exe.31157584.v1"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Exploring analogies between seemingly disparate physical systems is a powerful research strategy because it allows us to transfer insights, methods, and intuitions from one domain to another, often revealing hidden connections and sparking creative breakthroughs, as exemplified by Hamilton's seminal analogy between classical mechanics and geometric optics---a correspondence that ultimately seeded the development of quantum mechanics. In this work, in the same spirit, we explore deeply the similarity between the Maxwell and Dirac equations. We reformulate the Dirac equation in an \\emph{exact} form of the Maxwell equations, which we refer to as the ``electronic Maxwell equations.'' In this formulation, the standard free Maxwell equations emerge as a special case corresponding to vanishing longitudinal components and zero mass. The presence of a nonzero mass naturally leads to longitudinal components in the matter wave, arising as a consequence of enforcing Lorentz invariance in the electronic Maxwell equations. The physical quantities described by these electronic fields closely parallel their electromagnetic counterparts. We also make a discussion on wave components we may measure if the matter wave is actually measurable. Likewise, we reformulate Maxwell’s equations into an \\emph{exact} form of the Dirac equation, which we term the ``electromagnetic Dirac equation.'' This formulation yields the $8\\times8$ spin operator for spin‑1 particles and is capable of reproducing the Klein--Gordon equation (the wave equation for electromagnetic waves). The absence of longitudinal components characterizes the electromagnetic field as a spin‑1 field. Furthermore, we find that the Zitterbewegung of the photon corresponds to the rapidly oscillating part of the Poynting vector in classical electromagnetism. Both the Maxwell and Dirac equations can be reformulated into closely related quaternionic equations, which directly reveal the difference in how these equations transform under Lorentz transformations. Building on this observation, we propose a possible (spin) geometry underlying both the Maxwell and Dirac equations. Within this new framework, the spacetime itself is a second--order covariant or contravariant spinor. Furthermore, the electronic field is naturally identified as a spinor (first--order spinor), whereas the electromagnetic field emerges as a second--order mixed spinor. We further express both the Maxwell and Dirac equations in terms of these spinors and extend the resulting formalism to curved spacetime. Besides, we explore the parallels between the Feynman path integral---an alternative formalism for the quantum wave equations, and the Rayleigh--Sommerfeld diffraction integral---an alternative formalism for the optical wave equation. We apply the idea developed in the diffraction theory to establish the path integral for the Schr\\\"{o}dinger equation---a generalized Helmholtz equation with an additional temporal derivative term, the Klein--Gordon equation---a generalized optical wave equation with a non--vanishing mass term, and the Dirac equation---the massive Maxwell equations. Based on our results, we dicuss the flaws in the Feynman checkerboard model. Finally, we present two illustrative applications of the analogy between the Maxwell and Dirac equations. The first explores how to assign an effective ``mass'' to the electromagnetic wave. The second examines an intriguing case of a spacetime--varying electromagnetic medium that transforms the Maxwell equations into a massless Dirac equation, with the potentials taking a special form. We derive a general solution for this equation, revealing that such materials can behave like a one--sided mirror.<p></p>"]},{"key":"dc:title","label":"Title","values":["On the parallels between the Maxwell and Dirac equations"]}]}],"canonical_facts":{"dc:creator":["Mingjie Li (21042047)"],"dc:date":["2026-02-02T00:00:00Z"],"dc:description":["Exploring analogies between seemingly disparate physical systems is a powerful research strategy because it allows us to transfer insights, methods, and intuitions from one domain to another, often revealing hidden connections and sparking creative breakthroughs, as exemplified by Hamilton's seminal analogy between classical mechanics and geometric optics---a correspondence that ultimately seeded the development of quantum mechanics. In this work, in the same spirit, we explore deeply the similarity between the Maxwell and Dirac equations. We reformulate the Dirac equation in an \\emph{exact} form of the Maxwell equations, which we refer to as the ``electronic Maxwell equations.'' In this formulation, the standard free Maxwell equations emerge as a special case corresponding to vanishing longitudinal components and zero mass. The presence of a nonzero mass naturally leads to longitudinal components in the matter wave, arising as a consequence of enforcing Lorentz invariance in the electronic Maxwell equations. The physical quantities described by these electronic fields closely parallel their electromagnetic counterparts. We also make a discussion on wave components we may measure if the matter wave is actually measurable. Likewise, we reformulate Maxwell’s equations into an \\emph{exact} form of the Dirac equation, which we term the ``electromagnetic Dirac equation.'' This formulation yields the $8\\times8$ spin operator for spin‑1 particles and is capable of reproducing the Klein--Gordon equation (the wave equation for electromagnetic waves). The absence of longitudinal components characterizes the electromagnetic field as a spin‑1 field. Furthermore, we find that the Zitterbewegung of the photon corresponds to the rapidly oscillating part of the Poynting vector in classical electromagnetism. Both the Maxwell and Dirac equations can be reformulated into closely related quaternionic equations, which directly reveal the difference in how these equations transform under Lorentz transformations. Building on this observation, we propose a possible (spin) geometry underlying both the Maxwell and Dirac equations. Within this new framework, the spacetime itself is a second--order covariant or contravariant spinor. Furthermore, the electronic field is naturally identified as a spinor (first--order spinor), whereas the electromagnetic field emerges as a second--order mixed spinor. We further express both the Maxwell and Dirac equations in terms of these spinors and extend the resulting formalism to curved spacetime. Besides, we explore the parallels between the Feynman path integral---an alternative formalism for the quantum wave equations, and the Rayleigh--Sommerfeld diffraction integral---an alternative formalism for the optical wave equation. We apply the idea developed in the diffraction theory to establish the path integral for the Schr\\\"{o}dinger equation---a generalized Helmholtz equation with an additional temporal derivative term, the Klein--Gordon equation---a generalized optical wave equation with a non--vanishing mass term, and the Dirac equation---the massive Maxwell equations. Based on our results, we dicuss the flaws in the Feynman checkerboard model. Finally, we present two illustrative applications of the analogy between the Maxwell and Dirac equations. The first explores how to assign an effective ``mass'' to the electromagnetic wave. The second examines an intriguing case of a spacetime--varying electromagnetic medium that transforms the Maxwell equations into a massless Dirac equation, with the potentials taking a special form. We derive a general solution for this equation, revealing that such materials can behave like a one--sided mirror.<p></p>"],"dc:identifier":["10779/exe.31157584.v1"],"dc:relation":["https://figshare.com/articles/thesis/On_the_parallels_between_the_Maxwell_and_Dirac_equations/31157584"],"dc:rights":["All rights reserved"],"dc:subject":["Maxwell's equations","The Dirac equation","Electronic Maxwell's equations","The electromagnetic Dirac equation","Rayleigh–Sommerfeld diffraction integral","Feynman’s path integral","Feynman checker board model","Quaternion","Spin geometry","Lorentz transformation","Spacetime","Spinor","Electromagnetic materials"],"dc:title":["On the parallels between the Maxwell and Dirac equations"],"dc:type":["Text","Thesis"]},"updated_at":"2026-07-27T19:34:41Z"}