Charged gauge data and SI-natural conventions

Use the fixed Minkowski coordinates $x=(x^0,\mathbf{x})=(ct,x^1,x^2,x^3)$, with $\eta =\mathrm{diag}(-1,1,1,1)$, on an open coordinate domain $U\subseteq {\mathbb{R}}^4$ with its standard topology. Each $x^a$ has unit metre. For a smooth scalar component $u:U\to \mathbb{R}$ or $u:U\to \mathbb{C}$, define ${\mathrm{\partial }}_{a}u:=\mathrm{\partial }u\mathrm{/}\mathrm{\partial }{x}^a$; thus ${\mathrm{\partial }}_0=c^{-1}{\mathrm{\partial }}_t$ at fixed $\mathbf{x}$, ${\mathrm{\partial }}_j=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^j$ for $j=1,2,3$, and each ${\mathrm{\partial }}_a$ has unit ${\mathrm{m}}^{-1}$. Raise derivative indices with ${\mathrm{\partial }}^a:={\eta }^{ab}{\mathrm{\partial }}_b$. The fields in this definition are smooth maps $\varphi \in C^{\mathrm{\infty }}(U;\mathbb{R}),\mathbf{A}\in C^{\mathrm{\infty }}(U;{\mathbb{R}}^3),\psi \in C^{\mathrm{\infty }}(U;{\mathbb{C}}^4),\chi \in C^{\mathrm{\infty }}(U;\mathbb{R})\mathrm{.}$ Define the lower-potential component maps $A_a:U\to \mathbb{R}$ by $A_0=-\varphi \mathrm{/}c$ and $A_j={\mathbf{A}}_j$; collectively $(A_a)$ is the real covector potential in this chart. Then $F_{ab}:={\mathrm{\partial }}_a{A}_b-{\mathrm{\partial }}_b{A}_a$ has smooth real component maps $F_{ab}:U\to \mathbb{R}$. The spinor is an ordinary commuting ${\mathbb{C}}^4$-valued map, not a tuple of Grassmann generators. Its Dirac adjoint is the smooth row-valued map $\bar{\psi }:={\psi }^{†}\beta :U\to ({\mathbb{C}}^4{)}^{\ast }$, using the fixed matrices and adjoint of Dirac Clifford matrices and energy projections, so $\{{\gamma }^a,{\gamma }^b\}=-2{\eta }^{ab}{I}_4$. In fixed SI units, $\varphi$ is in volts, each $A_a$ is in $\mathrm{T}\mathrm{m}$, $\psi$ is assigned field dimension ${\mathrm{m}}^{-3\mathrm{/}2}$, and $\chi$ is in webers; these unit labels accompany the maps and do not identify $q$ with a measured charge. For periodic spatial domains, the same component and unit conventions apply to smooth periodic coordinate lifts, with the quotient domain specified by the item using them.

For the SI-normalization bookkeeping, let $c>0$ (m s${}^{-1}$), $\mathrm{\hslash }>0$ (J s), ${\mu }_0>0$ (N A${}^{-2}$), a real charge parameter $q$ (C), and a mass parameter $m>0$ (kg) be symbolic inputs; put ${ϵ}_0=({\mu }_0{c}^2{)}^{-1}$. In the current SI, $c$ and $h$ are defining constants while ${\mu }_0$ is experimentally determined; the formula below keeps ${\mu }_0$ symbolic and assigns it no numerical value. Use the lower-potential convention $A_a=(-\varphi \mathrm{/}c,\mathbf{A}),$ where $A_a$ has units T m, $\varphi$ is in volts, and $\mathbf{A}$ is in T m. Define $S_0=\sqrt{{\mu }_0\mathrm{\hslash }c},a_a=A_a\mathrm{/}{S}_0,f_{ab}={\mathrm{\partial }}_a{a}_b-{\mathrm{\partial }}_b{a}_a,$ $\mu =mc\mathrm{/}\mathrm{\hslash },g=q\sqrt{{\mu }_{0}c\mathrm{/}\mathrm{\hslash }}=q\mathrm{/}\sqrt{{ϵ}_0\mathrm{\hslash }c},D_a={\mathrm{\partial }}_a-ig{a}_a\mathrm{.}$ Here $D_a$ acts on $\psi \in C^{\mathrm{\infty }}(U;{\mathbb{C}}^4)$ by $(D_a\psi )(x)={\mathrm{\partial }}_a\psi (x)-ig{a}_a(x)\psi (x)$, again a smooth map $U\to {\mathbb{C}}^4$. Thus $a_a$ and $\mu$ have units m${}^{-1}$, $f_{ab}$ has units m${}^{-2}$, and $g$ is dimensionless. For $\psi$ with units m${}^{-3\mathrm{/}2}$, both $-\frac{1}{4}{f}_{ab}{f}^{ab}$ and $\bar{\psi }(i{\gamma }^a{D}_a-\mu )\psi$ have units m${}^{-4}$; their integral over $d^{4}x$ is dimensionless and represents the action divided by $\mathrm{\hslash }$ in this normalization.

For the real smooth gauge-parameter map $\chi :U\to \mathbb{R}$ in webers, put $\alpha :=\chi \mathrm{/}{S}_0\in C^{\mathrm{\infty }}(U;\mathbb{R})$ (dimensionless) and use $A_a^{\mathrm{\prime }}=A_a+{\mathrm{\partial }}_a\chi ,a_a^{\mathrm{\prime }}=a_a+{\mathrm{\partial }}_a\alpha ,{\psi }^{\mathrm{\prime }}=e^{ig\alpha }\psi \mathrm{.}$ The induced covariant derivative is $D_a^{\mathrm{\prime }}={\mathrm{\partial }}_a-ig{a}_a^{\mathrm{\prime }}$. These are formal mathematical conventions for the later conditional deductions. They do not adopt a measured value of ${\mu }_0$, identify $q$ with an electron charge, impose charge quantization, or assert that interacting continuum fields exist.