# A Maxwell Universe – Classical Discreteness ## Sunlight One of the greatest achievements of early quantum theory was predicting the *discrete* energy levels of the hydrogen atom, known as the *Rydberg series*. In 1704, Isaac Newton showed, using a prism, that all the colors of the rainbow are contained in sunlight. When doing the same experiment with a neon light, or with hydrogen—the most abundant element in the known universe—it is easily seen that neither neon light nor hydrogen light contain all the colors as sunlight apparently did. Instead, they emit light only at very specific, sharply defined colors. These gaps in the spectrum constitute the "spectral signature" of the element. ## Color is Energy Here, color is not a representation of energy — it *is* energy, directly perceived as frequency. If we examine closely (naked eye is enough) the spectrum of sunlight, it is readily evident that it is not "continuous", and that there are gaps in the visible spectrum of light. These discrete features were the anomaly that gave rise to "quantum mechanics," as classical electromagnetism seemingly offered no explanation for why an atom should radiate in steps rather than in a continuous sweep. ## The Rydberg Series Long before the internal structure of the atom was understood, experiments showed that glowing hydrogen does not emit a continuous rainbow of light. Instead, it emits light only at very specific, sharply defined colors. In 1888, the Swedish physicist Johannes Rydberg found that these colors—that is, electromagnetic frequencies—follow a simple mathematical pattern involving integers $n = 1, 2, 3, \dots$: $$ E_n \propto \frac{1}{n^2}. $$ The standard explanation, developed later by Bohr and Schrödinger, ties this scaling to the electrostatic interaction between an electron and a proton. In that planetary picture, larger $n$ corresponds to the electron occupying an orbit farther from the nucleus. However, the formula itself contains no reference to distance, radius, or geometry—only to the integer $n$. More precisely, observed spectral lines correspond to transitions between configurations. The emitted radiation carries the exact energy difference: $$ \Delta E \propto \frac{1}{m^2}-\frac{1}{n^2}, \qquad m>n. $$ ## The Geometry of Quantization We interpret the $1/n^2$ factor not as a change in spatial size, but as a reorganization of a fixed total energy into progressively finer internal structure. Imagine a flat rectangular sheet of paper. Draw one vertical line and one horizontal line, each connecting opposite edges. They cross in the middle and divide the sheet into $2\times2=4$ cells. More generally, if we keep adding lines the sheet is then partitioned into $$ n\times n = n^2 \text{ cells}. $$ With no internal lines ($n=1$), the sheet corresponds to a ground configuration with energy $E_1$. As $n$ increases, the total area stays the same, but it is subdivided into smaller and smaller regions. If the total energy is conserved and distributed uniformly across the $n^2$ cells, the energy per cell scales as $$ E_n \propto \frac{E_1}{n^2}. $$ In this abstract but constrained way, the Rydberg scaling appears without invoking particles, wavefunctions, or force balance. Discrete levels are simply discrete global subdivisions of a conserved quantity: energy. ## The Torus To understand the physical basis of this grid, we must look at the topology of confinement. Return to the flat napkin. First, identify and glue together one pair of opposite edges. The flat napkin becomes a tube. Lines that originally ended on one edge now reappear continuously on the opposite edge. Next, take this tube and identify its two circular ends. Gluing these ends together produces a closed surface with no boundary—a **Torus**. Any lines drawn on the original napkin become closed paths on the torus. However, they form closed loops only if they match their own position when crossing an identified edge. This requirement ensures global continuity of the grid. In a source-free Maxwell universe, electromagnetic fields on this surface must satisfy these continuity conditions along the two independent cycles of the torus: the poloidal (around the ring) and toroidal (along the tube) directions. This imposes a discretization condition on the wavelength. Along a closed loop of length $L$, the field must satisfy: $$ n \lambda = L. $$ These are the same conditions that produce standing waves on a string, now applied to a closed surface with two independent winding numbers. ## Energy Reorganization In this view, the Rydberg series does not describe an electron moving to a larger orbit in space. It describes the electromagnetic field reorganizing itself into progressively finer standing-wave patterns. These patterns are self-consistent knots of counter-propagating electromagnetic energy flux, fixed by continuity. Increasing $n$ corresponds to increasing the number of global windings on the surface. More windings impose more nodes on the same conserved topology. Transitions between levels are therefore related to the difference in **cell sizes** (or effective tube widths) between two subdivisions. To move from level $n$ to level $m$, the system must supply exactly the energy difference required to "patch" the geometry from one winding density to another: $$ \Delta E = E_1 \left( \frac{1}{n^2} - \frac{1}{m^2} \right). $$ The photon is the packet of energy that facilitates this topological patching. The ground state ($n=1$) is unique. It represents the configuration where the torus is composed of a single coherent cell—the state where the flux tube is pulled as tight as topologically possible. As we shall see, the geometric limit of this "tightness" is what determines the coupling constant of the universe. ## Charge as Topology Finally, we must account for the appearance of electric charge. In a source-free universe, $$ \nabla \cdot \mathbf{E} = 0 $$ everywhere. No electric field originates from a point. How, then, does a particle appear to have charge? Consider the standing wave on the torus. The field lines wrap around the two independent cycles, characterized by the winding numbers $(m,n)$. These windings represent closed circulations of electromagnetic energy. At any local patch of the surface, the field lines entering and leaving balance so that the net flux vanishes. However, the global circulation -for example, circulation tangent to the surface- does not vanish. Now, enclose this configuration within a spherical surface of radius $r$ much larger than the torus itself. The total electromagnetic circulation (the "topological charge, $(m,n)$") is a conserved quantity fixed by the winding numbers $m$ and $n$. This energy, thought as a bulb turned-off (no radiation, no point source), is constant; so we can think that it's energy is spread evenly around it. This energy, spread accross the area of the sensor we use to measure it (the eye is a sensory organ, "a sensor", as well) is what we measure as a $1/r^2$ dependence. As this fixed quantity is projected through a sphere whose area grows as $4\pi r^2$, the observed field intensity necessarily falls off as: $$ \text{Intensity} \propto \frac{1}{r^2}. $$ This reproduces the phenomenology of charge. In this view, charge is not a primitive substance added to the universe. It is an effective, topological quantity: the far-field signature of closed electromagnetic circulation.