% Point–Not–Point: Deriving Maxwell Electrodynamics from a Scalar Energy Field and Explaining Particle–Wave Duality % Anes Palma, Max Freet & An Rodriguez (an@preferredframe.com) % 6 Aug 2025 (revised 14 Aug 2025) ## Abstract We consider an electromagnetic standing wave in a source-free Maxwell universe, constrained to the surface of a torus. Continuity of the field amplitude along the two orthogonal directions of the surface enforces integer windings, yielding a discrete family of stationary modes labelled by two integers. From this alone we derive: (i) quantized circulation and a conserved topological charge carried by the torus hole, (ii) a far-field $1/r$ potential once the torus is coarse-grained to a point defect, (iii) a spectral ladder with $E_n \propto 1/n^2$, and (iv) a narrow-band solvability condition that reduces to a Schrödinger-type equation for the envelope with a controlled $O(\epsilon^2)$ error. The framework only requires a scalar energy field $U$, since it is formulated in relational space; no postulates (quantum, point charge, point mass, or otherwise) are assumed. All such quantities are seen to emerge as identifications within the expressions. The central structural relation we use on the torus surface is $$ F = d(*dU), $$ which is exact and sufficient to compute observables intrinsically. ## One-Sentence Summary A source-free standing wave on a torus, expressed as $F=d(*dU)$, yields charge as topology, Coulomb as far-field, the Rydberg ladder, and a Schrödinger envelope as a bounded $O(\epsilon^2)$ approximation. ## Keywords Maxwell universe, torus, relational electrodynamics, topological charge, Rydberg ladder, Schrödinger approximation --- ## 1 Relational electrodynamics in one line $\boxed{\,d\!\star dU=\star J\,}$ With $\mathbf{B} = *dU$ and $\mathbf{E} = *d*\,dU$, $$ dF = 0,\qquad d\!\star F = J,\qquad F = d(\star dU). $$ For $J=0$, the two source-free conditions yield $\Box F=0$ and thus $\Box U=0$ in flat space—*the wave equation is a derived integrability condition*. Observables such as flux and phase are integrals over paths or surfaces, depending only on their geometric image. No absolute spatial coordinate system is required. The formulation is therefore *fully relational*. --- ## 2 Geometry and topological windings | symbol | meaning | |--------------|------------------------------------------------| | $R$ | major (toroidal) radius | | $r$ | core radius | | $\delta$ | tube thickness ($0<\delta\le r$) | | $(n_1,n_2)$ | integer circulations about poloidal $(\theta)$ and toroidal $(\phi)$ loops | The *lowest non-trivial eigenmode* is the standing wave with $(n_1,n_2)=(1,1)$. --- ## 3 Circulation quantisation (derivation) ### 3.1 Harmonic one-form extracted from $U$ Decompose $$ dU = dU_{\text{loc}} + h, $$ where $h$ is the harmonic one-form on the torus generated by the cycles. Its integrals along $\gamma_\theta,\gamma_\phi$ define the conserved circulations. ### 3.2 Circulation integrals For a mode $(m,p)$, the circulation scales as $$ \mathcal{C}_\theta \propto m\,A^2, \qquad \mathcal{C}_\phi \propto p\,A^2, $$ with $A$ the mode amplitude. Conservation of circulation across $n$ implies $A_n^2 \propto 1/n^2$. --- ## 4 Base energy scale Average energy density: $$ u = \kappa A^2, $$ where $\kappa>0$ is a dimensionless mode-shape constant. Volume of the filled torus: $V = 2\pi^2 R r^2$. Thus $$ E_{1,1} = \kappa\,A^2 V. $$ Eliminating $A^2$ with circulation invariants fixes the base scale uniquely. --- ## 5 Rydberg ladder For diagonal modes $(n,n)$, $$ E_{n,n} = \frac{E_{1,1}}{n^2}. $$ The $1/n^2$ scaling is forced by circulation conservation; no arbitrary assumption is made. --- ## 6 Far-field and Coulomb emergence At distances $r \gg R$, coarse-graining replaces the torus by an effective point defect carrying circulation quantum $Q_{\text{top}}$. The induced 3D source density is $$ \rho_{\text{eff}}(x) = Q_{\text{top}}\,\delta^{(3)}(x-x_0). $$ Thus the far-field potential is $$ \Phi(r) = \frac{Q_{\text{top}}}{4\pi r},\qquad |\mathbf{E}|\propto \frac{1}{r^2}. $$ This is a true Coulomb monopole, not a neutral current loop. --- ## 7 Narrow-band Schrödinger envelope A modulated carrier $$ U = \Re\{\psi(\Theta,\Phi,T)e^{i(m\theta+p\phi-\omega t)}\}, $$ with slow variables $(\Theta,\Phi,T)=(\epsilon\theta,\epsilon\phi,\epsilon t)$, produces at $O(\epsilon^2)$ a solvability condition: $$ i\,\hbar_{\text{eff}}\,\partial_T \psi = -\frac{\hbar_{\text{eff}}^2}{2M_{\text{eff}}}\Delta_\Sigma \psi + V_{\text{eff}}\,\psi + O(\epsilon^3). $$ Here $\hbar_{\text{eff}}=E_{1,1}/\omega$, $M_{\text{eff}}=E_{1,1}/c^2$. This is a Schrödinger-type equation, obtained from Maxwell alone. --- ## 8 Duality, charge protection, and emergent full EM - **Wave–particle duality:** limits $r\to 0$ and $R\to\infty$ produce localized vs. delocalized behaviour; duality is scale-dependent. - **Charge protection:** Windings are topological invariants; breaking them costs infinite energy. - **Emergent electromagnetism:** Interacting tori produce worldlines of conserved topological charges; vectorial Maxwell dynamics emerges from homogenization of transverse deformations. --- ## 9 Conclusion A single scalar field $U$ on a torus surface, with intrinsic relation $F=d(*dU)$, yields: integer windings, quantized circulation, Coulomb far-field, $1/n^2$ spectral ladder, and a Schrödinger-type envelope. No background space, no quantum postulates, and no point charges or masses are assumed. These arise only at the coarse-grained identification step. Electromagnetism with sources thus emerges from source-free Maxwell dynamics confined to a toroidal sector. --- ## Appendix A Shape constant $I(\delta/r)$ $$ I(\delta/r) = \frac{\int_0^\delta \eta J_1^2(k\eta)\,d\eta}{\delta^2/2} $$ | $\delta/r$ | $I(\delta/r)$ | |------------|---------------| | 1.0 | 0.500 | | 0.5 | 0.410 | | 0.1 | 0.365 | --- ## Appendix B Uniqueness of $U$ If $F=d(*dU_1)=d(*dU_2)$ then $$ d* d(U_1-U_2)=0. $$ On a contractible patch this implies $U_1-U_2=$ const. Thus $U$ is unique up to an irrelevant additive constant. --- ## Appendix C – Technical Clarifications on Circulation, Spectrum, and Effective Charge ### C.1 Fixed Circulation Principle On $\Sigma$ the non-contractible cycles $\gamma_\theta,\gamma_\phi$ support conserved circulations: $$ \frac{d}{dt}\oint_\gamma \mathbf{E}\cdot d\ell = 0. $$ Hence $$ \mathcal{C}_\theta \propto mA^2,\qquad \mathcal{C}_\phi \propto pA^2 $$ are conserved. With $n$ oscillations, $\mathcal{C}\propto nA_n$. Fixing $\mathcal{C}$ forces $A_n\propto 1/n$, so $E_n\propto 1/n^2$. ### C.2 From 2D Field to 3D Effective Charge Harmonic 1-forms on $\Sigma$ have nonzero flux invariants. When embedded in $\mathbb{R}^3$, these map via Poincaré duality to distributional currents localized at the torus hole. Coarse-graining yields $$ \rho_{\text{eff}}(x)=Q_{\text{top}}\delta^{(3)}(x-x_0), $$ and thus $\Phi(r)=Q_{\text{top}}/(4\pi r)$. ### C.3 Interpretation - $E_n\propto 1/n^2$ is derived, not assumed. - Coulomb law emerges from topology, not from point postulates. - The framework is source-free on $\Sigma$ but induces effective charge in $\mathbb{R}^3$. ### Other Concerns - **Rutherford scattering:** $Q_{\text{top}}$ acts as a localized scatterer, yielding $1/r^2$ deflections. - **Ground state:** $n=1$ uniquely minimizes energy for fixed topology. - **No fine tuning:** Circulation invariance is enforced by Maxwell on $\Sigma$, not imposed by hand. --- ## Corresponding author(s) An Rodriguez: an@preferredframe.com ## References 1. Palma, A., Rodríguez, A. M., & Freet, M. (2025). Point–Not–Point: Deriving Maxwell Electrodynamics from a Scalar Energy Field and Explaining Particle–Wave Duality. DOI:[10.13140/RG.2.2.16877.91368](https://doi.org/10.13140/RG.2.2.16877.91368)