Proposition: Well-definedness, gauge invariance, and conservation

Definition def:current gives a real, smooth, gauge-invariant one-form on $M$. In the class of Assumption ass:ren, no nonzero local background ambiguity survives on this regular flat host. Moreover,

as an identity of smooth functions, and hence of distributions on $M$.

Proof. Smoothness and independence of the direction of coincidence follow from Proposition prop:kernel and (eq:localHadamard). Hermiticity proves reality. Using covariant derivatives in both endpoints makes the coincidence expression gauge invariant.

The standard charged-current ambiguity includes a multiple of ${\nabla }^{\nu }{F}_{\nu \mu }$, which vanishes here BalakumarWinstanley,KleinZahn. More directly, a locally covariant, state-independent background covector on this host can be evaluated in a flat parallel chart. The curvature jets vanish, and a flat connection is locally gauge equivalent to zero. There is no nonzero Lorentz-invariant covector built from these local data. Thus the allowed local ambiguity is zero. This reasoning does not classify nonlocal additions or distributions on the removed worldsheet.

Finally, in a parallel Cartesian chart, differentiation of a coincidence limit acts as $\partial +{\partial }^{\prime }$. Therefore

The mixed derivatives cancel, and the two remaining terms vanish by the bisolution equations. The calculation is covariant under the local identifications used above.