Classical normalization and the switch-divergence tensor

These are the X-E normalization and dressing identifications XE (Classical value and the dressing formula). Off the scalar shell the extra source $g(P_{m}u){\partial }^{\mu }u$ must also be retained.

The same tensor $({\partial }_{\nu }g){T_0}^{\nu \mu }[u]$ appears in the necessary invariant-jet condition. Nevertheless the gauge-fixed response always exists for the compact smooth source. Existence of $k_1[u]$ does not establish a solution of the invariant coupled equations.

As a global zero-past field identity, $w[k_1[u]]=0$ implies that tensor vanishes, by applying ${\square }^2$. Vanishing of one observation pairing or vanishing only on $K$ does not imply global source conservation.