Spacetime and configurations

Let

Coordinates are denoted $x=(t,\mathbf{x})$, and

All indices are raised and lowered with $\eta$ unless explicitly stated otherwise. Parentheses include the factor $1/2$.

The configuration space is

with its compact-open smooth Fréchet topology. Its elements are $(h,\varphi )$. Fields are defined on all of $M$, so convolution requires no extension choice at the slab boundary.

For coefficient-jet calculations we use

with elements $(h,k,\varphi )$. These are smooth off-shell configuration spaces. No equation-of-motion, gauge, or gauge-fixed quotient is taken.

The metric is formal:

or, when evaluating a coefficient jet,

There is no factor $1/2$ in $k$. No numerical smallness, convergence, or global invertibility is asserted.

We fix a massless minimally coupled scalar. The coupling $\kappa$ has the sign and normalization specified by the action in Section sec:action.