Selected classical field topology and optional circle normalization

Let $(M,g,N,o)$ be the smooth, Hausdorff, second-countable, oriented and time-oriented Lorentzian four-manifold of Einstein–Maxwell primitive quantities and SI conventions, with signature $(-,+,+,+)$ and no boundary. Write ${\mathrm{\Omega }}^k(M)$ for the smooth real sections of ${\mathrm{\Lambda }}^k{T}^{\ast }M$. The classical electromagnetic field type is a smooth real two-form $F\in {\mathrm{\Omega }}^2(M)$. In length-coordinate charts its components have SI unit tesla; its integral over an oriented smooth two-surface has unit weber. The exterior derivative is denoted $d$. A field is closed when $dF=0$. Closedness is a field condition, not the same condition as having a global potential.

For an open set $U\subseteq M$, a local potential for $F$ on $U$ is a smooth real one-form $A_U\in {\mathrm{\Omega }}^1(U)$ satisfying $d{A}_U=F{\mathrm{\mid }}_U$. It is a global potential when $U=M$. In a length-coordinate chart the components of $A_U$ have unit tesla metre. This classical real-field description chooses $F$ itself and requires no bundle or global $A$.

An optional circle-normalized connection sector adds the mathematical data of a smooth right principal $U(1)$ bundle $\pi :P\to M$ and a connection, using the local-potential and circle-data conventions of Local potentials and smooth circle-bundle connection data. Here $U(1)=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ with multiplication. Concretely, supply a countable open cover $\{U_i\}$ and smooth transition maps $g_{ij}:U_i\cap U_j\to U(1)$ satisfying $g_{ii}=1$ and $g_{ij}{g}_{jk}=g_{ik}$. On overlap pieces with smooth real lifts $g_{ij}=e^{i{f}_{ij}}$, the normalized local connection one-forms $a_i$ satisfy $a_j=a_i+d{f}_{ij}$; in local fibre angles the convention is ${\theta }_j={\theta }_i-f_{ij}$, so $d{\theta }_i+a_i$ patches. The local curvature forms $d{a}_i$ agree. Fix an independent constant ${\kappa }_0>0$ with unit weber, and set $A_i={\kappa }_0{a}_i$ and $F{\mathrm{\mid }}_{U_i}={\kappa }_{0}d{a}_i$. Thus $a_i$ has component unit ${\mathrm{m}}^{-1}$ in length coordinates, while $A_i$ has component unit tesla metre. Choosing $P$ and ${\kappa }_0$ is extra model data; no claim is made that every smooth closed real $F$ is the curvature of such a fixed-normalization connection.

The selected topological sector is the fixed pair consisting of the bundle $P$ and the normalization ${\kappa }_0$; its connection is data on that fixed bundle. Changing $P$ or ${\kappa }_0$ selects a different sector.

In a trivialization, a gauge transformation is a single-valued smooth map $h:U\to U(1)$; its local real lifts may differ by integer multiples of $2\pi$. A connection on a trivial bundle is gauge trivial when one such global map takes its normalized connection form to zero. For a closed piecewise-$C^1$ loop $\gamma$ lying in a trivialization, horizontal transport is defined by $\dot{\theta }=-a(\dot{\gamma })$; the endpoint fibre rotation is its holonomy. When $a$ is one global one-form along the loop this is $\exp (-i{\int }_{\gamma }a)$. These definitions introduce no quantum state or quantum phase. The constant ${\kappa }_0$ is not identified with Planck's constant or a unit of electric charge.