--- This part is not meant to be pedagogical in a mathematical sense. In this spirit, equations will be written both in prose and in symbols, so that a non-technical reader may follow the argument. It is a conceptual draft: a somewhat coherent line of reasoning, stripped of unnecessary assumptions. # Matter and Mass The concept of matter long predates the concept of mass. In antiquity, matter meant simply “that which exists.” The Greek atomists (Democritus) proposed that the world consisted of indivisible units moving through void. Aristotle later formalized matter as *substance*: potentiality waiting to be given shape. Isaac Newton introduced mass not to explain the microscopic constitution of the universe, but as an operational parameter to predict motion. ## The Operational Definition Newton formalized the observation that objects resist changes in motion—a property known as *inertia*. To quantify this, he introduced a parameter called "mass" ($m$). He defined the object's "quantity of motion"—or *momentum* ($p$)—as the product of this intrinsic mass and its velocity ($v$): $$ p = m v . $$ Momentum is taken as a basic fact of the universe. Inertia—the resistance to acceleration—is treated as primitive. As Nobel laureate Richard Feynman emphasized, physics contains no explanation of inertia. We know how to calculate with it, but not what it is. Changes in momentum are explained by the action of a force $F$: $$ F = \frac{dp}{dt} = m \frac{dv}{dt} = m a . $$ This framework works with unprecedented precision, from planetary orbits to rocket science. ## Mass in Quantum Mechanics Newton’s operational concept of mass penetrated even Quantum Mechanics. In Erwin Schrödinger's wave equation, mass appears in the exact same role—as a denominator. $$ i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V \right] \psi . $$ The term inside the brackets is the total energy operator. Since momentum squared ($p^2$) corresponds to $-\hbar^2\nabla^2$, the first term is simply: $$ \text{Kinetic Energy} = \frac{p^2}{2m}. $$ Mass $m$ is still just an external parameter inserted to make the units of momentum work. It describes *how* the wave moves, but not *what* the wave is. ## Light: Momentum Without Mass Crucially, nature provides an example of momentum existing without mass. Light is experimentally observed to have zero mass ($m=0$), yet it exerts pressure. It carries momentum. In classical mechanics ($p=mv$), zero mass would imply zero momentum. But in relativity and electromagnetism, momentum is tied to *energy*, not just mass. For a photon, the relationship connects energy $E$ and momentum $p$: $$ p = \frac{E}{c}. $$ This proves that "stuff" does not need mass to exist or to act dynamically on the world. It only needs energy and movement. If a massless field can carry momentum, perhaps the "mass" of ordinary matter is not a primitive property, but simply trapped energy—light traveling in circles. ---- old PART II This part is not meant to be pedagogical in a mathematical sense. In this spirit, equations will be written both in prose and in symbols, so that a non-technical reader may follow the argument. It is a conceptual draft: a somewhat coherent line of reasoning, stripped of unnecessary assumptions. In this section, we trace the history of "matter" —from the Greek atoms to Newton's mass, and its permeation into Quantum Mechanics— to show that mass has always been a descriptive parameter rather than an explained origin. We conclude by showing that modern physics already allows for momentum without mass (light), setting the stage for an alternate view of matter. # Matter and Mass ## The Ancients The concept of matter long predates the concept of mass. In antiquity, matter meant simply “that which exists.” As early as the 5th century BCE, the Greek atomists Leucippus and his student Democritus proposed that the world consisted solely of indivisible units (atoms) moving through infinite empty space. Living in Abdera, Thrace, they were contemporaries of Socrates, though their works are known to us primarily through the critiques of Aristotle and fragments preserved by later doxographers. Democritus famously noted, "By convention sweet is sweet, bitter is bitter, hot is hot, cold is cold, color is color; but in truth there are only atoms and the void." In this view, matter was defined by its geometric imperishability. Qualities like 'hot' or 'color' were not intrinsic properties of matter itself, but temporary sensations arising from atomic interactions —a sharp distinction between "truth" (structure) and "convention" (perception). A century later, in the 4th century BCE, Aristotle took a different approach. In his *Physics* (Book I), he formalized matter as *substance* (*hylē*): that which underlies form and change. For Aristotle, matter was potentiality waiting to be given shape, distinct from the actualized form it eventually takes. Many centuries later, this philosophical inquiry did not merely shift; it was violently upended. The question changed from the ontological "what is it?" to the operational "how does it move?" In 1633, Galileo Galilei was persecuted by the Roman Inquisition for supporting the Copernican view that the Earth was not the fixed center of the universe. Legend holds that after being forced to recant, he muttered, *"Eppur si muove"* ("And yet it moves"). Galileo laid the groundwork for modern physics in his *Dialogues Concerning Two New Sciences* (1638). He realized that the natural state of an object is not rest, but motion—specifically, that an object on a frictionless plane would move indefinitely. This was a direct, heretical challenge to the Aristotelian view (specifically *Physics*, Book VIII), which held that all objects strive to stop at their "natural place," and that any motion requires a continuous mover to sustain it. In England, during the Renaissance in the 17th century, Isaac Newton introduced mass for the first time as an operational quantity. His goal was not philosophical ontology, but prediction. He needed a parameter to quantify how objects resist the changes in motion that Galileo had observed. In the *Principia* (1687), mass enters as the parameter governing the response to forces. While Newton had developed the calculus (fluxions), the *Principia* was written in the language of classical geometry to be understood by his peers. Mass was defined operationally, not structurally. Newton wrote mass as the “quantity of matter,” but he never defined matter itself. Mass was introduced to quantify motion—from falling objects to the orbits of planets—not to explain microscopic constitution. Newton formalized the observation that matter in motion tends to remain in motion unless acted upon by a force. He called this conserved quantity the "quantity of motion"—mass times velocity—denoted $p$. Because Newton's era was the birth of calculus, he did not write symbolic equations like $p = mv$. instead, he wrote in prose definitions: > "The quantity of motion is the measure of the same, arising from the velocity > and quantity of matter conjointly." Translating his prose into modern symbolism, the momentum $p$ is: $$ p = m v . $$ The momentum $p$ of an object—its tendency to preserve its state of motion—is taken as a basic fact of the universe in all modern theories. This echoes the ancient Greek obsession with the "natural tendency" of objects (which they believed was to fall), but Newton redefined the tendency as *inertia*. Inertia, the resistance to acceleration, is treated as intrinsic. As Nobel laureate Richard Feynman recounted in "What is Science?", his father taught him the difference between knowing the name of something and knowing the thing itself. Using a ball in a wagon, his father explained: > "That is called inertia, but nobody knows why." Physics contains no explanation of inertia. We know how to calculate with it, but not what it is. Changes in momentum over a time interval, $dp/dt$, are expressed symbolically by the action of a force $F$: $$ F = \frac{dp}{dt} = m \frac{dv}{dt} = m a . $$ where $a$ is acceleration, or the change of speed over an interval of time. This was a remarkably accurate empirical framework. Surprisingly, it suddenly allowed us to explain terrestrial mechanics, planetary motion, and celestial dynamics with unprecedented precision. Centuries later, the same mathematics and ideas would literally make rocket science possible, taking humanity to the Moon and beyond. Newton’s operational concept of mass survived even the radical reformulation of space and time introduced by Albert Einstein. In general relativity, it is often said that "mass tells spacetime how to curve, and spacetime tells mass how to move." Any massive object is formally understood as a point rolling along a curve in spacetime. The concept of Newtonian mass $m$ penetrated even Quantum Mechanics. In Erwin Schrödinger's wave equation, mass appears in a familiar position: $$ i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V \right] \psi . $$ The term inside the brackets represents the total energy. Notice the fraction containing mass. In quantum mechanics, the operator corresponding to momentum squared, $p^2$, is $-\hbar^2\nabla^2$. Thus, the first term is exactly the kinetic energy (from the Greek *kinesis*, movement) expression from classical mechanics, re-written in operator form: $$ \text{Kinetic Energy} = \frac{p^2}{2m}. $$ Mass $m$ remains the denominator of motion. It is an external parameter inserted into the theory to make the units of momentum work. It is still nowadays a heavily debated conundrum how to join into a single theory Newton's gravitational mass (extended by Einstein to General Relativity) with Quantum Mechanics. # Light: Momentum Without Mass An important distinction arises with light. Light is experimentally observed to have no mass ($m=0$), yet it strikes objects and exerts pressure. It carries momentum. In classical mechanics, if $m=0$, then $p=mv$ would imply momentum is zero. However, relativity and electromagnetism show that momentum is not strictly tied to mass; it is tied to *energy*. For a photon (a quantum of light), the relationship connects energy $E$, momentum $p$, and the speed of light $c$: $$ E = p c . $$ Or, written for momentum: $$ p = \frac{E}{c}. $$ This suggests that "stuff" does not need mass to exist or to act dynamically on the world. It only needs energy and movement. If a massless field can carry momentum, perhaps the "mass" of ordinary matter is not a collection of hard stones, but simply trapped energy—light traveling in circles. # Charge Follows the Same Pattern Many decades after Newton, electric charge entered physics in an analogous way. Coulomb established experimentally that electric forces obey an inverse-square law: $$ |\mathbf{F}| \propto \frac{1}{r^2}. $$ Charge was introduced as the quantity that sources this force. Following the success of point masses, Maxwell later encoded charge as a source term in the electric field: $$ \nabla \cdot \mathbf{E} = \rho / \epsilon_0 . $$ As with mass, charge is treated as primitive: something localized that produces a field. This treatment mirrors the historical role of mass. It works to an amazing extent, while remaining silent about microscopic structure. # Light in Matter and Linear Response When light enters a material, it is commonly said to slow down. Yet Maxwell’s equations assign a single propagation speed to electromagnetic waves: $$ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}. $$ The observed reduction in speed does not arise because light itself changes its nature. It arises because the material becomes polarized. The incident wave drives bound charges, which emit secondary electromagnetic radiation. The superposition of the incident and secondary fields produces a phase delay. The effective phase velocity becomes $$ v = \frac{c}{n}. $$ This mechanism is fully electromagnetic and requires no modification of Maxwell’s equations. # Polarization Is a Field The linear response of a medium is written as $$ \mathbf{P} = \chi \mathbf{E}, $$ where $\mathbf{P}$ is the polarization field. Polarization is commonly taught as the displacement of physical charges. But mathematically, it is simply an electric field generated in response to another electric field. Nothing in Maxwell’s equations requires $\mathbf{P}$ to originate from "solid" matter. It is simply another electromagnetic vector field. Thus, electromagnetic waves do interact. Superposition and linearity do not imply non-interaction. # Superposition Is Not Non-Interaction Linearity means fields add. It does not mean physics is insensitive to the sum. Electromagnetic energy density is quadratic: $$ u = \tfrac{1}{2}\left( \epsilon_0 |\mathbf{E}|^2 + \mu_0^{-1} |\mathbf{B}|^2 \right). $$ For $\mathbf{E} = \mathbf{E}_1 + \mathbf{E}_2$, the energy density becomes: $$ |\mathbf{E}|^2 = |\mathbf{E}_1|^2 + |\mathbf{E}_2|^2 + 2\,\mathbf{E}_1 \cdot \mathbf{E}_2, $$ and similarly for $\mathbf{B}$. The cross terms ($2\,\mathbf{E}_1 \cdot \mathbf{E}_2$) are unavoidable. They represent real redistribution of energy and momentum. Interaction occurs through the total field configuration, without violating linearity. # Structure Without Sources Consider electromagnetism with no sources: $$ \nabla \cdot \mathbf{E} = 0, \qquad \nabla \cdot \mathbf{B} = 0. $$ This forbids divergence, not structure. Closed electromagnetic configurations remain allowed. Just as a smoke ring is a stable structure in air without being "made" of a different material than the air around it, electromagnetic structures can exist within the field itself. # The Torus and Discretization A torus possesses two independent non-contractible loops. An electromagnetic field defined on a torus must match itself after traversal of each loop. These global continuity conditions discretize the allowed configurations. Let $(m,n)$ denote the integer winding numbers along the two cycles. A standing electromagnetic wave on a torus is therefore characterized by $(m,n)$. # Energy and the Rydberg Structure The total electromagnetic energy is conserved. Increasing the winding numbers subdivides the surface into more regions. Each region carries a smaller fraction of the total energy. For symmetric configurations $(m=n)$, $$ E_n = \frac{E_0}{n^2}. $$ This reproduces the Rydberg series observed in atomic spectra. Asymmetric configurations $(m \neq n)$ carry net circulation and therefore an electromagnetic moment. # Charge as Topology Because $\nabla \cdot \mathbf{E} = 0$ everywhere, no electric field originates from a point. All electromagnetic flow is closed. However, consider the "intensity" of this configuration observed from a distance. Enclose the toroidal configuration within a spherical surface of radius $r$ much larger than the torus. The enclosing area scales as $4\pi r^2$. The total electromagnetic circulation (energy flow) is conserved, much like the total light output of a bulb. As this fixed quantity is distributed over the surface of the growing sphere, the density per unit area must decrease. Because the area grows as $r^2$, the observed density produces an effective $1/r^2$ dependence. "Charge" is thus not primitive, and the quantized electromagnetic flow around the torus' hole can be thought of as a "topological charge". It is an effective, topological quantity: a measure of quantized electromagnetic energy flow, characterized by $(m,n)$.