Potential representations, gauge transformations and the Lorenz and Coulomb conditions

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.