Lemma: Regular-part identities and ambiguities

Under Assumption ass:ren:

1. The Wick-square ambiguity $a{m}^2+bR$, the usual stress ambiguities, and local background-gauge-curvature terms vanish on the regular part. The possible current ambiguity proportional to ${\nabla }^{\nu }{F}_{\nu \mu }$ also vanishes.

2. Local conservation corrections and the regular conformal trace anomaly vanish.

3. For $F_h=[h]$,

   $${\nabla }^{\mu }{T}_{\mu \nu }=0,T^{\mu }{}_{\mu }=\frac{1}{2}(6\xi -1){\square }_g{F}_h,$$
   $$T_{\mu \nu }(\xi )-T_{\mu \nu }(1/6)=\left(\xi -\frac{1}{6}\right)(g_{\mu \nu }{\square }_g-{\nabla }_{\mu }{\nabla }_{\nu })F_h.$$

   For the complex field $F_h=2F$.

4. The current defined by (eq:current-prescription) is real and conserved.

These statements do not classify completion-supported distributions or contact terms in higher products.

Proof. All admissible local geometric ambiguity and anomaly expressions vanish when $m=0$, the Riemann tensor is zero, and the external connection curvature is zero. Holonomy is not a local curvature invariant.

In a parallel Cartesian chart put

Symmetry, the bisolution equations, and Synge's rule give

Thus

Taking a divergence, trace, and coupling difference proves (iii). Hermiticity makes the current real. Applying a divergence to its coincidence expression gives $\mathrm{i}e[({\square }_X-{\square }_{X^{\prime }})H]=0$.