Charge, canonical photon field and observer units

Use the oriented, time-oriented Minkowski model, length coordinates, metric, and SI/natural dictionary of Quantum field quantities and compatible SI normalizations, with $\eta =\mathrm{diag}(-1,1,1,1)$ and $x^0=ct$. Use the gamma matrices, Dirac adjoint, and effective Clifford metric of Charged gauge data and SI-natural conventions: ${\gamma }^0=\beta$, $\{{\gamma }^a,{\gamma }^b\}=-2{\eta }^{ab}{I}_4$, and $\bar{\psi }={\psi }^{†}\beta$. Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) for the mathematical QFT gauge-data supplier; the unit conversions below do not themselves spend a choice principle.

Identify the SI charge and mass symbols $q,m$ in Charged gauge data and SI-natural conventions with $q_{\mathrm{S}\mathrm{I}},m_{\mathrm{S}\mathrm{I}}$ here. Its natural coupling $g$ is $q_{\mathrm{n}}$ below, and its inverse-length mass $\mu$ is the same $m_{\mathrm{S}\mathrm{I}}c\mathrm{/}\mathrm{\hslash }$.

Let $q_{\mathrm{S}\mathrm{I}}\in \mathbb{R}$ be a selected charge parameter in coulombs and $m_{\mathrm{S}\mathrm{I}}>0$ a selected rest-mass parameter in kilograms. They are inputs of the chosen model; neither is derived from Maxwell's equations. Set $\mu =m_{\mathrm{S}\mathrm{I}}c\mathrm{/}\mathrm{\hslash }$ in ${\mathrm{m}}^{-1}$ and $q_{\mathrm{n}}=q_{\mathrm{S}\mathrm{I}}\sqrt{{\mu }_{0}c\mathrm{/}\mathrm{\hslash }}$, which is dimensionless under the framework units of Quantum field quantities and compatible SI normalizations. For an electron use $q_{\mathrm{S}\mathrm{I}}=-e_{\mathrm{S}\mathrm{I}}$ and $e_{\mathrm{n}}=e_{\mathrm{S}\mathrm{I}}\sqrt{{\mu }_{0}c\mathrm{/}\mathrm{\hslash }}>0$, so $q_{\mathrm{n}}=-e_{\mathrm{n}}$. Use ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ with the measured current-SI ${\mu }_0$ of Quantum field quantities and compatible SI normalizations. These definitions therefore do not assign an exact numerical value to $q_{\mathrm{n}}$, $e_{\mathrm{n}}$, or ${\epsilon }_0$.

In a named inertial chart, let $A_{\mathrm{S}\mathrm{I}}^{\mu }=(\phi \mathrm{/}c,\mathbf{A})$ be the electromagnetic potential when one is supplied, with components in $\mathrm{T}\mathrm{m}$ under Framework-local SI units, the vacuum constants and the exact/measured status of their values and The electromagnetic two-form, its observer decomposition and the four-current. Define the canonical potential and field-strength components by

$a^{\mu }:=\frac{A_{\mathrm{S}\mathrm{I}}^{\mu }}{\sqrt{{\mu }_0\mathrm{\hslash }c}},f^{\mu \nu }:=\frac{F_{\mathrm{S}\mathrm{I}}^{\mu \nu }}{\sqrt{{\mu }_0\mathrm{\hslash }c}}\mathrm{.}$

They have engineering dimensions ${\mathrm{m}}^{-1}$ and ${\mathrm{m}}^{-2}$, respectively. With the metric-lowered potential $A_{\mathrm{S}\mathrm{I},a}={\eta }_{ab}{A}_{\mathrm{S}\mathrm{I}}^b$, the potential component convention is $A_{\mathrm{S}\mathrm{I},0}=-\phi \mathrm{/}c$ and $A_{\mathrm{S}\mathrm{I},i}=A_i$. The covariant derivative in the selected QED model is

$D_a:={\mathrm{\partial }}_a-i{q}_{\mathrm{n}}{a}_a={\mathrm{\partial }}_a-\frac{i{q}_{\mathrm{S}\mathrm{I}}}{\mathrm{\hslash }}{A}_{\mathrm{S}\mathrm{I},a},$

since $q_{\mathrm{n}}{a}_a=q_{\mathrm{S}\mathrm{I}}{A}_{\mathrm{S}\mathrm{I},a}\mathrm{/}\mathrm{\hslash }$. Both terms in $D_a$ have unit ${\mathrm{m}}^{-1}$. If the SI gauge function $\chi$ has unit $\mathrm{W}\mathrm{b}=\mathrm{T}{\mathrm{m}}^2$, then $q_{\mathrm{S}\mathrm{I}}\chi \mathrm{/}\mathrm{\hslash }$ is dimensionless; the corresponding natural gauge function $\lambda =\chi \mathrm{/}\sqrt{{\mu }_0\mathrm{\hslash }c}$ is dimensionless and $q_{\mathrm{n}}\lambda =q_{\mathrm{S}\mathrm{I}}\chi \mathrm{/}\mathrm{\hslash }$. These conversions apply to smooth local gauge transformations; they assert no boundary condition, asymptotic charge sector, or global gauge-fixing result.

If a separately specified finite regulator introduces Hamiltonian eigenvalues, they have unit joule; this definition introduces no regulator. The field components and units refer to the selected inertial chart, not to a new observer congruence or a cross-framework identification.