Compact gauge variation is boundary-free for conserved source

Assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), inherited through the componentwise fundamental-variation step of The fundamental lemma of the calculus of variations for continuous coefficients and through Distributional differentiation is continuous and commutes. Let $U,N,A,j,F,S$ and the constants $c,{\mu }_0,{\epsilon }_0$ be as in clauses (a)-(b) of Smooth finite-region Maxwell action, compact variations and the fixed-endpoint particle functional: $U$ bounded open with $\bar{U}$ compact in the laboratory chart image $cI\times \mathrm{\Omega }$, $N$ an open neighbourhood of $\bar{U}$, $A$ and $j$ smooth on $N$, $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$, $j=(c\rho ,J)$ with the source fields of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, and $S[A]=c^{-1}{\int }_U[-(4{\mu }_0{)}^{-1}{F}_{\mu \nu }{F}^{\mu \nu }+j^{\nu }{A}_{\nu }]d\lambda$. For $\chi \in C_c^{\mathrm{\infty }}(U;\mathbb{R})$ write ${\mathrm{\partial }}^{\mathrm{♯}}\chi$ for the vector field with contravariant components $({\mathrm{\partial }}^{\mathrm{♯}}\chi {)}^{\mu }:={\mathrm{\partial }}^{\mu }\chi ={\eta }^{\mu \nu }{\mathrm{\partial }}_{\nu }\chi$ (Test function space d of an open set, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$). It is a compactly supported smooth field variation. Thus $A^{\mu }\mapsto A^{\mu }+{\mathrm{\partial }}^{\mu }\chi$ lowers to $A_{\nu }\mapsto A_{\nu }+{\mathrm{\partial }}_{\nu }\chi$; $A+{\mathrm{\partial }}^{\mathrm{♯}}\chi$ denotes this vector addition.

(a) [Exact gauge difference, no boundary term] For every such $\chi$ one has $F_{\mu \nu }(A+{\mathrm{\partial }}^{\mathrm{♯}}\chi )=F_{\mu \nu }(A)$ on $U$, because ${\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }\chi -{\mathrm{\partial }}_{\nu }{\mathrm{\partial }}_{\mu }\chi =0$ for the smooth $\chi$ (Clairaut--Schwarz theorem for continuous second partial derivatives); consequently the field term of $S$ is unchanged and

$S[A+{\mathrm{\partial }}^{\mathrm{♯}}\chi ]-S[A]=c^{-1}{\int }_U{j}^{\mu }{\mathrm{\partial }}_{\mu }\chi d\lambda =-c^{-1}{\int }_U({\mathrm{\partial }}_{\mu }{j}^{\mu })\chi d\lambda ,$

the second equality being the distributional integration-by-parts identity for the locally integrable fields $j^{\mu }$ and the smooth compactly supported $\chi$ (Distributional derivative, Regular distribution from a locally integrable function, Distributional differentiation is continuous and commutes); no boundary term arises precisely because $\chi$ is compactly supported in $U$.

(b) [Equivalence for smooth sources] If ${\mathrm{\partial }}_{\mu }{j}^{\mu }=0$ on $U$ (the source is conserved), then $S[A+{\mathrm{\partial }}^{\mathrm{♯}}\chi ]=S[A]$ for every $\chi \in C_c^{\mathrm{\infty }}(U;\mathbb{R})$. Conversely, if $S[A+{\mathrm{\partial }}^{\mathrm{♯}}\chi ]=S[A]$ for every such $\chi$, then ${\mathrm{\partial }}_{\mu }{j}^{\mu }=0$ pointwise on $U$: the function ${\mathrm{\partial }}_{\mu }{j}^{\mu }$ is continuous on the open $U$ and pairs to zero against every compactly supported smooth $\chi$, so The fundamental lemma of the calculus of variations for continuous coefficients applies. Hence for smooth prescribed sources the invariance of the field action under all compact gauge variations is equivalent to the conservation of the four-current.

(c) [Maxwell-compatible sources] Suppose in addition that the field pair $(E,B)$ represented by $A$ through the potential dictionary of clause (c) of The electromagnetic two-form, its observer decomposition and the four-current and the source fields $(\rho ,J)$ satisfy the smooth branch of the vacuum Maxwell equations on $U$ (equivalently, by clause (a) of Tensor Maxwell equations equal the four laboratory Maxwell equations, the tensor system ${\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }$ together with the cyclic identity). Then clause (a) of Charge continuity as a compatibility condition of the vacuum Maxwell equations gives ${\mathrm{\partial }}_t\rho +\mathrm{div}J=0$ on $U$ (Divergence and curl of a $C^1$ vector field), which in the chart reads ${\mathrm{\partial }}_{\mu }{j}^{\mu }={\mathrm{\partial }}_0(c\rho )+\mathrm{div}J=c^{-1}{\mathrm{\partial }}_t(c\rho )+\mathrm{div}J=0$; hence $S[A+{\mathrm{\partial }}^{\mathrm{♯}}\chi ]=S[A]$ for every compact gauge variation.

(d) [Distributional reading] If $j^{\mu }$ is a distribution on $U$ (Distribution), the gauge-difference pairing $c^{-1}⟨j^{\mu },{\mathrm{\partial }}_{\mu }\chi ⟩$ is defined for every test function $\chi \in C_c^{\mathrm{\infty }}(U;\mathbb{R})$ and equals $-c^{-1}⟨{\mathrm{\partial }}_{\mu }{j}^{\mu },\chi ⟩$ by the signed definition of the distributional derivative; hence the pairing vanishes for every compact gauge variation if and only if ${\mathrm{\partial }}_{\mu }{j}^{\mu }=0$ as a distribution. This is a statement about the pairing only: the smooth $j$ of the definition is retained whenever the functional $S$ itself is used, and no products of distributions are formed.

Scope: the item computes the gauge variation of the field action and its compatibility with source conservation; it does not derive the field equations from gauge invariance, does not assert that conservation of a source implies stationarity, does not treat global or topological obstructions, and uses no Noether machinery.

The gauge-shift computation is Tong §5.5.3 (printed pp. 125-126), which computes $S\to S-\frac{1}{c}\int \chi {\mathrm{\partial }}_{\mu }{J}^{\mu }$ and reads invariance as ${\mathrm{\partial }}_{\mu }{J}^{\mu }=0$; Landau–Lifshitz §§28-29 and Binney §§3.6-3.7 supply the four-current, continuity and interaction-term context. The exact boundary-free formulation here uses the framework's own compact-support hypothesis and the published distribution interface; the converse direction replaces the sources' informal 'arbitrary gauge direction at a point' step by the fundamental variation lemma, and no Noether formalism is adopted.