A Field Representation of Point-Particle Motion and Its Application to Maxwell's Equations

Starting from the continuity equation, we develop a field representation of a conserved quantity carried by amoving point particle, without specifying its physical nature. The Dirac delta distribution and differentialidentities first provide an instantaneous representation for a prescribed trajectory in three-dimensionalEuclidean space. Introducing an auxiliary field with an explicitly chosen evolution law then yields a systemsupporting transverse waves with characteristic speed and helicities +1 and -1. A central result is that, after rescal‐ing the fields, this system takes a form identical to Maxwell’s equations, before any electromagnetic interpreta‐tion is assigned. This formal equivalence follows from the mathematical construction together with the chosenevolution law; it does not follow from conservation alone. Electromagnetism is subsequently identified as aphysical application by specifying the meaning of the sources, fields, and constants. The contribution is a di‐rect point-source calculation that makes the distinction between mathematical representation, selected evolu‐tion, and physical interpretation explicit. Within the stated framework, the method is not restricted to electriccharge, although its application to other physical systems requires their own physical assumptions.