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Advances in Maxwell’s Equations: Overview, Applications and Relations with Other...
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Advances in Maxwell’s Equations: Overview, Applications and Relations with Other Foundational Equations of Theoretical Physics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E4: Mathematical Physics".

Deadline for manuscript submissions: 31 December 2026 | Viewed by 1198

Editor

Prof. Dr. David Carfì

E-Mail Website
Guest Editor
1. Department of Mathematics, University of California, Riverside, CA 92521, USA
2. Department of Mathematics and Physical Sciences, MIFT, University of Messina, 98122 Messina, Italy
Interests: Hamiltonian mechanics; Maxwell's electromagnetism; quantum mechanics; Schwartz distribution theory; functional analysis; differential geometry; relativity theory; game theory; decision theory; risk theory; bargaining theory; green economy and sustainability

Special Issue Information

Dear Colleagues,

Maxwell’s equations serve as a foundational bridge between classical field theory and the modern quantum and relativistic landscape. This Special Issue, "Advances in Maxwell’s Equations: Overview, Applications and Relations with Other Foundational Equations of Theoretical Physics," seeks to explore this relationship through the lens of rigorous mathematical frameworks. We aim to highlight how modern geometric methods and functional analysis—including the application of Schwartz distribution theory—provide a deeper understanding of electromagnetic phenomena and their interplay with other fundamental wave equations.

A central theme of this collection is the formal correspondence between Maxwell’s system and Schrödinger-type equations. We encourage submissions that utilize advanced mathematical tools to investigate existence, uniqueness, and the construction of exact solutions. Whether through the study of underlying symmetries, the application of generalized functions, or the exploration of wave–particle dualities in curved spacetimes, we invite contributors to submit original research that bridges the gap between abstract mathematics and theoretical physics, as well as the gap between classical and quantum physics.

By examining Maxwell's theory alongside the foundational equations of quantum and relativistic mechanics, this Special Issue aims to provide a comprehensive overview of the current state of electromagnetic research.

Prof. Dr. David Carfì
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Maxwell’s equations and classic electromagnetism
  • geometric methods in physics
  • functional analysis
  • schwartz distribution theory
  • schrödinger-type equations
  • mathematical physics
  • symmetry and conservation laws
  • generalized functions
  • electromagnetic and quantum wave theory
  • theoretical physics and quantum mechanics foundations

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Published Papers (1 paper)

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Research

67 pages, 531 KB  
Article
Photon Entanglement, Bell Inequality Violation, and Energy Interpretation of the Born Rule in Maxwell–Schwartz Field Theory
by David Carfì
Mathematics 2026, 14(9), 1490; https://doi.org/10.3390/math14091490 - 28 Apr 2026
Viewed by 777
Abstract
In this paper we study photon entanglement in the framework of Maxwell–Schwartz field theory. The ambient state space is the complex Maxwellian distribution space W=S′(M4,C3), whose elements are fields of the form [...] Read more.
In this paper we study photon entanglement in the framework of Maxwell–Schwartz field theory. The ambient state space is the complex Maxwellian distribution space W=S′(M4,C3), whose elements are fields of the form F=E→+icB→. Polarization is realized as a two-dimensional complex subspace of W, generated by suitable linearly polarized Maxwellian solutions associated with opposite propagation directions. This yields canonical polarization sectors PA and PB, each naturally isomorphic to C2. Within this setting, the Bell singlet state is represented by a non-factorizable tensorial Maxwellian field in PA⊗PB⊂W⊗W. By means of the induced rotated polarization bases, the standard joint probabilities of the photon polarization experiment are recovered exactly, and the correlation law E(a,b)=−cos(2(a−b)) is obtained. Consequently, the usual CHSH value 22 is reproduced in the Maxwell–Schwartz framework. To clarify the meaning of this violation, we first formulate the CHSH inequality in a purely measure-theoretic form, as a theorem about four correlators represented on a single probability space by bounded measurable functions. We then show that the correlators produced by the intrinsic Maxwellian Bell state do not admit such a common representation. The obstruction is structural: the ontic state is a global non-product field configuration, and the four correlations arise from different polarization resolutions of the same tensorial Maxwellian state. A second main result concerns the Born rule. For L2 scalar quantum states in the domain of the Maxwellian correspondence, we prove that the squared Hilbert norm, times the constant ε0, coincides with the electromagnetic energy of the associated field. This leads to an energy interpretation of the Born rule: the Born probability density is identified with the normalized electromagnetic energy density up to an interference term depending on the chosen Maxwell–Schwartz isomorphism, which assumes the role of a quantum context. In the context of the Aspect and collaborators’ experiment, we prove that, on the other hand, the polarization probabilities become energy contributions of the corresponding field components. These results show that photon entanglement, Bell inequality violation, and the Born rule admit a coherent interpretation within Maxwell–Schwartz field theory, where the basic ontological objects are electromagnetic-like fields rather than abstract state vectors. Full article
(This article belongs to the Special Issue Advances in Maxwell’s Equations: Overview, Applications and Relations with Other Foundational Equations of Theoretical Physics)
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