% Maxwell Is All We Need % Anes Palma, An Rodriguez, Max Frre % August 11, 2025 ## Abstract What if all of physics—space, matter, time, causality—could be derived from a single equation? In the Point–Not–Point (PNP) framework, Maxwell’s equations are not just a description of electromagnetism—they are the *only* experimentally grounded starting point we need. By recasting them in a scalar, self-referential form, every observed structure emerges without additional postulates. Fields, quantization, inertia, thermodynamics, and even measurement collapse are deduced as consequences of topology and flow. ## One-Sentence Summary Space, time, matter, and measurement collapse arise naturally from Maxwell’s equations when expressed as the self-referential scalar field $U$. ## Keywords Maxwell; scalar field; topology; causality; quantization; inertia; thermodynamics; time; relational physics ## Introduction Maxwell’s equations, experimentally distilled in the 19th century, describe electric and magnetic fields with unmatched precision. Traditionally, they are written in vector form, on a background of space and time, with particles and charges as inputs. The PNP framework reverses this: there is no space to begin with, no separate particles—only a scalar energy field $U(x,t)$ whose self-referential oscillations and closures produce all observable structure. This approach has **no postulates**. Maxwell’s equations are already abstractions of experimental results; everything else is a logical unfolding from them. ## Theory Let $U:\mathbb{R}^3\times\mathbb{R} \to \mathbb{R}$ be the fundamental scalar energy field. Define: $$ F = d(*dU) $$ Here: - $d$ is the exterior derivative, taking derivatives without coordinates. - $*$ is the **Hodge dual operator**, mapping $p$-forms to $(n-p)$-forms in $n$ dimensions, exchanging “flux” and “circulation” aspects of the field. This definition produces the electromagnetic field tensor directly from $U$, with **no vector potential** $A$ required. In vacuum, Maxwell’s equations emerge: $$ dF = 0, \quad d\!\star F = 0 $$ These are the two homogeneous and two inhomogeneous Maxwell equations in one covariant statement. Gauge redundancy is eliminated—$U$ is gauge-invariant by construction. ### Why not a vector potential? In the standard formalism, one writes $F = dA$, where $A$ is a 1-form (the vector potential). This introduces unphysical degrees of freedom, removed by gauge fixing. In the scalar-first formalism, $U$ already encodes all physical degrees of freedom; $A$ is unnecessary. All measurable predictions match those of the $A$-formalism, but without the intermediate gauge structure. This is more than aesthetic: removing $A$ removes the *assumption* of a background geometry in which $A$ lives. $U$ generates both the fields and the relational structure that appears to us as “space.” ## Energy Flow and Conservation From $F$, electric and magnetic fields appear in the usual way. The energy density $u$ and Poynting vector $\mathbf{S}$ are: $$ u = \frac{\varepsilon_0}{2}(E^2 + c^2 B^2), \quad \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} $$ Poynting’s theorem in vacuum: $$ \partial_t u + \nabla \cdot \mathbf{S} = 0 $$ shows that **energy changes must be accompanied by flux**. This is not heuristic—it is enforced by the field equations themselves. Causality arises here: effects follow causes because flows must carry changes forward. ## The Relational Chain From these foundations, the physical hierarchy unfolds: 1. **Toroidal Modes → Quantization** Closed recurrence of $U$ on two orthogonal loops (toroidal and poloidal) admits only discrete wavelengths, producing quantized energy levels. 2. **Mode Interaction → Inverse-Square Force** Standing-wave interactions decay $\propto 1/r$ in amplitude; energy conservation then yields a force $\propto -1/r^2$—without invoking charges or masses. 3. **Persistent Topology $(1)$ → Causality** The $(1)$ mode (minimal closed oscillation) is self-inverting: inward flow flips phase to outward. This topological invariance defines causal ordering. 4. **Density-Dependent Flow → Cosmic Rotation Curves** Local energy density alters group velocity; Maxwell stresses from this variation explain flat galactic rotation curves—no dark matter required. 5. **Internal Momentum → Effective Mass** Circulating field momentum resists acceleration: $$ m_{\rm eff} = \frac{1}{c^2} \int u\,dV \times \kappa $$ Mass is thus emergent, not fundamental. 6. **Thermodynamics → Non-Fundamental Arrow of Time** Microstates are full field configurations; coarse-graining yields entropy growth. Maxwell’s dynamics remain reversible, but the arrow of time emerges statistically. 7. **Measurement → Reversible Collapse** In finite environments, measurement is reversible. Collapse is a controllable threshold in mode entanglement, not a universal law. ### Master Relational Derivation $$ U \to \text{Maxwell} \to \text{Quantization} \to \frac{1}{r^2} \to \text{Causality} \to \text{Thermodynamics} \to \text{Measurement} \to m_{\rm eff} $$ ## Higher-Order Relations - **First order:** $U$ defines $E,B$ and thus “space.” - **Second order:** Space + fields define reversible time. - **Third order:** Fields + directional flow define structured matter. - **Higher:** Chemistry, biology, self-awareness—any self-sustaining causal loop. Life itself is tied to Maxwell: any system that sustains its own causal loop in field flow meets the same topological persistence criteria. ## Conclusion Space is not a container for fields; it is a relation among them. Time is not fundamental; it is an emergent property of recurrence. Matter is not basic; it is structured, knotted energy flow. We believe it all happens because $(1)$ happened—probably more than once. --- **WIP Note:** Further expansion will include the explicit derivation of density-dependent $v_g(u)$, a full treatment of entropy production from mode statistics, and diagrams of the $(1)$ mode topology for non-specialist audiences.