Potentials and gauge

Let $c>0$, let $k\ge 2$ be an integer, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be an open set that is star-shaped with respect to a point $a\in \mathrm{\Omega }$ (Star-shaped open subsets of Euclidean space), and let $I\subseteq \mathbb{R}$ be an interval with at least two elements (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). Joint regularity on $\mathrm{\Omega }\times I$ is read in the restriction convention of Joint regularity of the radial vector and scalar potentials on a star-shaped domain: a map on $\mathrm{\Omega }\times I$ is jointly $C^m$ when it is the restriction of a $C^m$ map on some open neighbourhood of $\mathrm{\Omega }\times I$ in ${\mathbb{R}}^4$. For open $I$ this is the ordinary open-domain class of $C^k$ Euclidean maps and diffeomorphisms and agrees with the field class of The electric and magnetic fields as fixed-time spatial vector fields; for non-open $I$ it is the convention that makes the statements below meaningful, and no time jet is asserted beyond the restricted values.

Let $E,B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ be jointly $C^k$ and satisfy the two source-free Maxwell equations of The microscopic vacuum Maxwell equations: smooth and weak branches pointwise on $\mathrm{\Omega }\times I$:

$\mathrm{div}B=0,\mathrm{curl}E=-{\mathrm{\partial }}_{t}B,$

with the divergence and curl of Divergence and curl of a $C^1$ vector field and the partial derivative ${\mathrm{\partial }}_t$ at fixed position of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ (at included endpoints of a non-open $I$, as relative derivatives in the restriction convention). Then there are a scalar potential $\phi :\mathrm{\Omega }\times I\to \mathbb{R}$ that is jointly $C^{k-1}$ and a vector potential $A:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ that is jointly $C^k$, both in the restriction convention, with

$B=\mathrm{curl}A,E=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A\text{on }\mathrm{\Omega }\times I,$

so that $(\phi ,A)$ is a potential pair whose field pair is $(E,B)$, in the vocabulary of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression. If $E$ and $B$ are jointly $C^{\mathrm{\infty }}$, then $\phi$ and $A$ may be chosen jointly $C^{\mathrm{\infty }}$. Conversely, if $\phi$ and $A$ are jointly $C^{\mathrm{\infty }}$ on $\mathrm{\Omega }\times I$ in the restriction convention and $E:=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A$, $B:=\mathrm{curl}A$ are their field pair, then $\mathrm{div}B=0$ and $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$ on $\mathrm{\Omega }\times I$.

The theorem asserts existence on the stated star-shaped domain only: it claims no global potential on a general open set, no boundary, decay or growth condition on the potentials, and no uniqueness. Potentials are auxiliary objects of the framework and are not claimed to be observable.

• The star-shaped hypothesis is substantive. On a general open $\mathrm{\Omega }$ the two source-free equations need not admit a global potential pair; the divergence-free radial witness of [F1] and its scalar companion both use the radial segments from the centre, and the companion examples page records the obstruction on a punctured region.
• No uniqueness is claimed. The witnesses of [F1] are single specimens. Gauge transformations preserving their fields and the domain hypotheses for a converse are analyzed later on this page; this existence theorem does not classify all representations of one field pair.

Let $c>0$, let $U\cong I\times \mathrm{\Omega }$ be the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields with its restricted inertial chart, let fields and units be those of The electric and magnetic fields as fixed-time spatial vector fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let the scalar-vector potential vocabulary on that region be that of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression: a potential pair $(\phi ,A)$ of class $C^k$ on $\mathrm{\Omega }\times I$ consists of $\phi :\mathrm{\Omega }\times I\to \mathbb{R}$ and $A:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ of class $C^k$ jointly in the sense of $C^k$ Euclidean maps and diffeomorphisms, and for a non-open time interval $I$ the joint classes in this item are read in the open-neighbourhood restriction convention of Joint regularity of the radial vector and scalar potentials on a star-shaped domain (for open $I$ this is the ordinary class of $C^k$ Euclidean maps and diffeomorphisms); for $k\ge 1$ the field pair is

$E(\phi ,A):=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A,B(\phi ,A):=\mathrm{curl}A,$

where $\mathrm{\nabla }$ is the gradient, $\mathrm{div}$ and $\mathrm{curl}$ are those of Divergence and curl of a $C^1$ vector field, and ${\mathrm{\partial }}_t$ is the partial derivative at fixed position of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$. For $k\ge 1$ the Lorenz divergence of the pair is $G(\phi ,A):=\mathrm{div}A+c^{-2}{\mathrm{\partial }}_t\phi$. The definitions in this item are stated on the laboratory region; the framework's fields are those of The electric and magnetic fields as fixed-time spatial vector fields. No field equation is required to define the vocabulary; the existence theorem recalled in (a) has the additional hypotheses stated there.

(a) [Representations] Let $k\ge 2$ be an integer and let $(E,B)$ be a field pair of class $C^{k-1}$ on $\mathrm{\Omega }\times I$, meaning $E,B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ of class $C^{k-1}$ jointly as in The electric and magnetic fields as fixed-time spatial vector fields. A potential representation of class $C^k$ of $(E,B)$ is a potential pair $(\phi ,A)$ of class $C^k$ on $\mathrm{\Omega }\times I$ whose field pair is $(E,B)$, that is, $E=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A$ and $B=\mathrm{curl}A$ on $\mathrm{\Omega }\times I$. More generally, a potential pair of class $C^1$ represents a specified field pair whenever these two identities hold, even if the fields have greater regularity than the defining derivatives alone guarantee. This definition asserts neither existence nor uniqueness. When $\mathrm{\Omega }$ is open and star-shaped, $I$ is an interval with at least two elements, and the jointly $C^r$ field data, $r\ge 2$, satisfy $\mathrm{div}B=0$ and $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$ pointwise on $\mathrm{\Omega }\times I$, Smooth scalar and vector potentials on a star-shaped space-time region supplies $\phi \in C^{r-1}$ and $A\in C^r$ in the stated restriction convention, and smooth potentials for smooth data satisfying the same equations; it does not assert a $C^r$ scalar potential for merely $C^r$ data. Equality of field pairs does not by itself assert gauge relatedness on an arbitrary region; a converse needs its own domain and regularity hypotheses.

(b) [Gauge functions and gauge equivalence] A gauge function of class $C^m$, $m\ge 2$, on $\mathrm{\Omega }\times I$ is a function $\chi :\mathrm{\Omega }\times I\to \mathbb{R}$ of class $C^m$ jointly; its gauge transformation sends a potential pair $(\phi ,A)$ of class $C^m$ to

$T_{\chi }(\phi ,A):=(\phi -{\mathrm{\partial }}_t\chi ,A+\mathrm{\nabla }\chi ),$

a potential pair of class $C^{m-1}$. Two potential pairs are gauge related when one is the gauge transform of the other by some gauge function. On pairs and generators of class $C^{\mathrm{\infty }}$, $T_0$ is the identity and $T_{{\chi }_1}(T_{{\chi }_2}(\phi ,A))=T_{{\chi }_1+{\chi }_2}(\phi ,A)$ for all $C^{\mathrm{\infty }}$ generators ${\chi }_1,{\chi }_2$; hence gauge relatedness is an equivalence relation on the $C^{\mathrm{\infty }}$ potential pairs, the inverse of $T_{\chi }$ being $T_{-\chi }$. For a fixed finite class the transform drops one order, and relatedness is used only through the transform it records.

(c) [Conditions] A potential pair is in Lorenz gauge on $\mathrm{\Omega }\times I$ when $G(\phi ,A)=\mathrm{div}A+c^{-2}{\mathrm{\partial }}_t\phi =0$ throughout $\mathrm{\Omega }\times I$, and in Coulomb gauge in the chosen spatial chart when $\mathrm{div}A=0$ throughout $\mathrm{\Omega }\times I$. These are properties of a pair in the stated conventions. The existence of a gauge function achieving either condition, with or without prescribed initial or boundary data, is a separate PDE conclusion with its own hypotheses and is not asserted by this definition.

(d) [Physical status] The physical content of a representation is its field pair $(E,B)$ of The electric and magnetic fields as fixed-time spatial vector fields; the values of $\phi$, of $A$ and of any gauge function are convention dependent and are auxiliary, and gauge related representations have the same field pair. The scalar potential $\phi$ carries the volt, the vector potential $A$ carries the tesla metre, and a gauge function carries V s $=$ T m${}^2$, so its time derivative and gradient have the units of $\phi$ and $A$, respectively, consistently with Framework-local SI units, the vacuum constants and the exact/measured status of their values. No value of $\phi$ or $A$ is by itself an observable or evidence for a field.

• The composition identity is forced by linearity. Componentwise, ${\mathrm{\partial }}_t({\chi }_1+{\chi }_2)={\mathrm{\partial }}_t{\chi }_1+{\mathrm{\partial }}_t{\chi }_2$ and $\mathrm{\nabla }({\chi }_1+{\chi }_2)=\mathrm{\nabla }{\chi }_1+\mathrm{\nabla }{\chi }_2$, because partial derivatives are one-variable derivatives (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$) and differentiation is additive (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$); substituting into $T_{{\chi }_1}(T_{{\chi }_2}(\phi ,A))$ gives $T_{{\chi }_1+{\chi }_2}(\phi ,A)$. Reflexivity is $T_0=\mathrm{i}\mathrm{d}$, and symmetry is the inverse $T_{-\chi }$, with $\chi$ and $-\chi$ of the same class. No choice principle is used by the composition computation.
• What is deliberately absent. This is a vocabulary and bookkeeping definition: it draws no conclusion that a pair can be put in either gauge, no statement about prescribed boundary or initial data, and no Maxwell equation. The two gauge conditions are recorded as properties, not as achievable normal forms.
• Signs and spelling. The signs $E=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A$, $B=\mathrm{curl}A$ and $\phi -{\mathrm{\partial }}_t\chi$ are the conventions of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression; a different sign convention would produce a different but equivalent calculus. Lorenz here refers to Ludvig Lorenz; the condition is the coordinate form of ${\mathrm{\partial }}_{\mu }{A}^{\mu }=0$ under these conventions and is not a consequence of any field equation.

Let $c>0$, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be open, let $I\subseteq \mathbb{R}$ be an interval with at least two elements (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), let $k\ge 2$ be an integer, let $(\phi ,A)$ be a potential pair of class $C^k$ jointly on $\mathrm{\Omega }\times I$ and let $\chi$ be a gauge function of class $C^k$ jointly on $\mathrm{\Omega }\times I$, all in the conventions of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression and Potential representations, gauge transformations and the Lorenz and Coulomb conditions: $E(\phi ,A)=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A$, $B(\phi ,A)=\mathrm{curl}A$ and $T_{\chi }(\phi ,A)=(\phi -{\mathrm{\partial }}_t\chi ,A+\mathrm{\nabla }\chi )$; for a non-open $I$ the joint $C^k$ classes are read in the open-neighbourhood restriction convention of Joint regularity of the radial vector and scalar potentials on a star-shaped domain (for open $I$ this is the ordinary class of $C^k$ Euclidean maps and diffeomorphisms).

Then $T_{\chi }(\phi ,A)$ is a potential pair of class $C^{k-1}$ jointly on $\mathrm{\Omega }\times I$ and its field pair is the field pair of $(\phi ,A)$:

$E(T_{\chi }(\phi ,A))=E(\phi ,A),B(T_{\chi }(\phi ,A))=B(\phi ,A)\text{on }\mathrm{\Omega }\times I,$

the common values being maps $\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ of class $C^{k-1}$ jointly, that is, fields of the framework in the sense of The electric and magnetic fields as fixed-time spatial vector fields. In particular gauge related potential representations of one field pair have equal field pairs: the electric and magnetic fields are gauge invariants of the framework, and no value of $\phi$, of $A$ or of $\chi$ is asserted to be observable.

The proof uses only the defining formulas, the vanishing of $\mathrm{curl}\mathrm{\nabla }\chi$ for the $C^2$ function $\chi$ and the equality of the time-space mixed partials ${\mathrm{\partial }}_t{\mathrm{\partial }}_i\chi ={\mathrm{\partial }}_i{\mathrm{\partial }}_t\chi$; no Maxwell equation, no hypothesis on $\mathrm{\Omega }$ beyond openness, and no choice principle is used.

• Why $k\ge 2$ and not $k\ge 1$. The scalar identity uses $\mathrm{curl}\mathrm{\nabla }\chi =0$, which needs $\chi$ to be $C^2$, and the electric identity uses the symmetry of the second mixed partial of $\chi$. For a merely $C^1$ generator the two field expressions are still formally equal, but the intermediate quantities $\mathrm{curl}\mathrm{\nabla }\chi$ and the mixed partials are then not defined.
• No converse. The theorem says gauge transformations do not change the fields. Which potential pairs represent the same field pair, and which gauge conditions are reachable, are separate questions treated elsewhere on this page.

Let $c>0$, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be open, let $I\subseteq \mathbb{R}$ be an interval with at least two elements (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and let $(\phi ,A)$ be a potential pair of class $C^2$ jointly on $\mathrm{\Omega }\times I$ in the restriction convention of Joint regularity of the radial vector and scalar potentials on a star-shaped domain (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression): for open $I$ this is the ordinary open-domain class of $C^k$ Euclidean maps and diffeomorphisms, and for non-open $I$ the pair is the restriction of a $C^2$ pair on an open neighbourhood of $\mathrm{\Omega }\times I$, with relative derivatives at included endpoints and no time jet beyond the restricted values. Write $(E,B)$ for the field pair of $(\phi ,A)$ (The electric and magnetic fields as fixed-time spatial vector fields) and

$G:=\mathrm{div}A+c^{-2}{\mathrm{\partial }}_t\phi ,W:=c^{-2}{\mathrm{\partial }}_t^2-\mathrm{\Delta }$

for the Lorenz divergence and the wave expression of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, with $\mathrm{\Delta }$ the Laplacian of The Laplacian of a $C^2$ function and of a $C^2$ vector field. Let $\rho :\mathrm{\Omega }\times I\to \mathbb{R}$ and $J:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ be continuous source fields of the framework (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current).

(a) [Exact identities] Independently of any field equation,

$\mathrm{div}E=W\phi -{\mathrm{\partial }}_{t}G,\mathrm{curl}B-c^{-2}{\mathrm{\partial }}_{t}E=WA+\mathrm{\nabla }G\text{on }\mathrm{\Omega }\times I\mathrm{.}$

Consequently the sourced Maxwell equations of The microscopic vacuum Maxwell equations: smooth and weak branches hold for $(E,B,\rho ,J)$ if and only if

$W\phi -{\mathrm{\partial }}_{t}G=\frac{\rho }{{\epsilon }_0},WA+\mathrm{\nabla }G={\mu }_{0}J\text{on }\mathrm{\Omega }\times I,$

with ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ the constant of that postulate.

(b) [Gauge shift] If $\chi :\mathrm{\Omega }\times I\to \mathbb{R}$ is a gauge function of class $C^2$ jointly on $\mathrm{\Omega }\times I$ (Potential representations, gauge transformations and the Lorenz and Coulomb conditions), then $G(T_{\chi }(\phi ,A))=G-W\chi$ on $\mathrm{\Omega }\times I$ and the transformed pair has the same field pair $(E,B)$ (Gauge invariance of the electric and magnetic fields); hence $T_{\chi }(\phi ,A)$ is in Lorenz gauge if and only if $W\chi =G$ on $\mathrm{\Omega }\times I$.

(c) [Decoupling in Lorenz gauge] Suppose in addition that the given pair is in Lorenz gauge, $G=0$ on $\mathrm{\Omega }\times I$. Then its field pair solves the sourced Maxwell system of The microscopic vacuum Maxwell equations: smooth and weak branches if and only if the pair satisfies the decoupled sourced wave equations

$W\phi =\rho \mathrm{/}{\epsilon }_0,WA={\mu }_{0}J\text{on }\mathrm{\Omega }\times I\mathrm{.}$

The two source-free equations $\mathrm{div}B=0$ and $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$ hold automatically for every potential pair of class $C^2$. For an arbitrary given potential pair, the conjunction of Lorenz gauge with these two wave equations implies the sourced Maxwell system, but Maxwell's equations alone do not impose $G=0$ on that representative: their exact equivalent conditions remain the residual equations of (a), whose right-hand sides retain ${\mathrm{\partial }}_{t}G$ and $\mathrm{\nabla }G$.

(d) [Residual generators] If the $C^2$ gauge function of (b) satisfies $W\chi =0$ on $\mathrm{\Omega }\times I$, then the transform of a pair in Lorenz gauge is again in Lorenz gauge and has the same field pair. When $\mathrm{\Omega }$ is nonempty the Lorenz condition does not determine the pair: the generator $\chi (x,t)=x_1$, read in numerical SI components with gauge units as in Potential representations, gauge transformations and the Lorenz and Coulomb conditions, has $W\chi =0$ and changes $A$ by the nonzero constant vector $(1,0,0)$.

(e) [Scope] The identities and equivalences are properties of the given $C^2$ representation. No gauge function with $W\chi =G$, and no generator meeting prescribed initial or boundary values, is asserted to exist; that is a separate PDE conclusion with its own hypotheses, treated by the gauge-reachability theorem later on this page. No boundary, decay or initial condition on the pair is imposed, and no empirical claim or observability of $\phi$ or $A$ is included. A pair of class $C^1$ defines $E$, $B$ and $G$ but need not possess the second derivatives appearing in $W\phi$, $WA$, $\mathrm{\Delta }$ or ${\mathrm{\partial }}_{t}G$, so $C^2$ is the exact standing regularity here.