Theorem: Charged Friedrichs vacuum current

For Assumptions ass:inputs and ass:ren, the current of Definition def:current is

In particular,

The formula is valid for every $0<\alpha <1$ and every $0\le \beta <1$, including image-boundary parameters.

Proof. Use a longitudinal spacelike split:

Equivalently on the Euclidean section, $\Delta \tau =0$ and

This split requires no convention for a Euclidean time component of the current.

At fixed $s\ne 0$, differentiate (eq:GE) in the angular variables before setting their separation to zero. Absolute convergence holds even after multiplication by $n+\beta$. The angular derivative of the direct flat term is zero at $\phi ={\phi }^{\prime }$. Hence

By Lemma lem:signed,

This proves (eq:main). Analyticity of the regular remainder makes the result the coincidence limit, independently of the chosen split.

The Euclidean kernel is even in $\Delta \tau$ and $\Delta z$, so its regular remainder has vanishing first difference derivatives in those variables at coincidence. It is also symmetric in $r,r^{\prime }$ at fixed angular difference, so the radial difference derivative vanishes. Continuation gives the stated Lorentzian components.