Smooth and point-source constructions: distinct for nonzero charge, both zero at zero charge

This example is conditional on the definitions Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, The electric and magnetic fields as fixed-time spatial vector fields and the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches, and assumes Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ both for the Lebesgue-volume source branch and for the injectivity of the regular-distribution embedding used below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

(a) [Smooth source] Let $f\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ be nonnegative with Lebesgue normalization ${\int }_{{\mathbb{R}}^3}fd{\lambda }_3=1$ (the Riemann integral over any compact box containing $\mathrm{supp}f$ has the same value by Riemann–Lebesgue comparison for distribution test integrands) (for instance any nonzero nonnegative smooth compactly supported bump divided by its finite nonzero integral), let $q\in \mathbb{R}$, and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be a nonempty open set containing $\mathrm{supp}f$. Then $\rho :=qf$ and $J:=0$ is a smooth source field of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current: $\rho$ is compactly supported and of class $C^{\mathrm{\infty }}$, hence continuous, and for every bounded Lebesgue-measurable $V$ with compact closure inside $\mathrm{\Omega }$ and containing $\mathrm{supp}f$, the charge is $Q(V,t)=q\int f=q$; the same value holds for the retained compact Jordan/Riemann branch at every time $t$, so the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches applies to this source.

(b) [Point source] Let $I\subseteq \mathbb{R}$ be an open interval and $z:I\to {\mathbb{R}}^3$ a $C^1$ trajectory whose graph map $t\mapsto (z(t),t)$ is proper into $\mathrm{\Omega }\times I$ (Point-support spacetime densities: distributions, continuity equation and endpoint fluxes; properness is automatic when $\mathrm{\Omega }\times I$ is the whole ambient domain or when $I=\mathbb{R}$). For the same charge $q$ as in (a), the pairings $⟨\rho ,\psi ⟩:=q{\int }_I\psi (z(t),t)dt$ and $⟨J,\psi ⟩:=q{\int }_I{z}^{\mathrm{\prime }}(t)\psi (z(t),t)dt$, for $\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega }\times I)$, are the point-source distributions of Point-support spacetime densities: distributions, continuity equation and endpoint fluxes: they are distributions (Distribution, Test function space d of an open set) satisfying ${\mathrm{\partial }}_t\rho +\mathrm{div}J=0$ in the distributional sense. At each fixed time $t$ this model assigns the charge $q\cdot {\mathbf{1}}_{z(t)\in V}$ to a Jordan-measurable $V\subseteq \mathrm{\Omega }$, and its instantaneous spatial density is the evaluation functional $q{\delta }_{z(t)}$ (Dirac delta and its derivatives), $\phi \mapsto q\phi (z(t))$; a source field of the smooth branch instead gives the integrals $\phi \mapsto {\int }_{\mathrm{\Omega }}\rho (y,t)\phi (y)dy$ of its continuous density.

(c) [The two classes differ] Fix $t\in I$, put $z_0:=z(t)\in \mathrm{\Omega }$ and suppose $q\mathrm{\ne }0$. Retain Countable Choice for the regular-distribution embedding and the Lebesgue-null singleton result cited below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Then no locally integrable $\rho \in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ represents the point charge: there is no such $\rho$ whose regular distribution (Regular distribution from a locally integrable function) satisfies $⟨\rho ,\phi ⟩=q\phi (z_0)$ for every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. Indeed, if $\rho$ did, then on the open set $\mathrm{\Omega }\setminus \{z_0\}$ every test vanishes at $z_0$, so the regular distribution of the restriction of $\rho$ is zero, and injectivity gives $\rho =0$ almost everywhere there (Measure-null sets and almost-everywhere statements relative to a measure); since $\{z_0\}$ is Lebesgue-null (Every at most countable subset of ${\mathbb{R}}^n$ is Lebesgue null; in particular ${\lambda }_1(\mathbb{Q})=0$) and a union of two null sets is null (Finite and countable subadditivity of measures), $\rho =0$ almost everywhere on $\mathrm{\Omega }$. The regular distribution depends only on the almost-everywhere class, so $⟨\rho ,{\phi }_0⟩=0$ for a cutoff ${\phi }_0\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with ${\phi }_0=1$ near $z_0$ (Test function cutoffs and euclidean localization), while the assumed identity gives $⟨\rho ,{\phi }_0⟩=q{\phi }_0(z_0)=q\mathrm{\ne }0$: a contradiction. Hence the point source is not a source field of the smooth branch of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current (whose densities are continuous functions), and the smooth-branch equations are not applied to it pointwise; the point source enters the framework only through its distributional pairings.

(d) [Conclusion] The two source classes carry the same parameter $q$ and the same total charge $q$ (the smooth density integrates to $q$; the point model assigns charge $q$ to regions containing the particle). For $q\mathrm{\ne }0$ their spatial distributions are distinct and the point source has no locally integrable representative. For $q=0$ both constructions give the zero charge and zero current distributions, represented by the smooth zero functions. The nonzero point source is used only through its pairings; neither branch of The microscopic vacuum Maxwell equations: smooth and weak branches supplies a singular-source Maxwell adoption. This example asserts no field value at $z(t)$ and contains no measurement or empirical claim.