Native free normalization

Write $z=(h,\phi )$ and $\zeta =(k,\psi )$. The field bundle is

The integrated pairing is

Define the action Hessian by

where

The gauge generator, gauge condition, and native gauge-fixed operator are

With four-dimensional trace reversal

retain

The principal differential part of $\mathcal{L}$ is ${\square }_{\bar{g}}\mathrm{id}$. Thus its Cauchy theory and causal inverses are those of a normally hyperbolic bundle operator BGP.

The contraction representative, when specified, obeys

Here

The factor of $\hslash$ is not included in $W$; a free contraction is $\hslash W$.

Write

It is real and symmetric in a real field trivialization. No positivity is assumed.

The complete auxiliary and ghost multiplet is retained. The kinematic lapse has zero functional derivatives with respect to those components. Hence they contribute no direct free internal contraction to the insertion. This observation does not eliminate ghost or auxiliary contributions to an interacting calculation.