Frozen conventions and the operator split

Let $M={\mathbb{R}}^4$ with $\eta =\mathrm{diag}(1,-1,-1,-1)$, $\square ={{\partial }_t}^2-{\Delta }_{\mathbf{x}}$, and $P_m=\square +m^2$, $m\ge 0$. Fix $T>0$, $M_T=[0,T]\times {\mathbb{R}}^3$, and ${M_T}^{\circ }=(0,T)\times {\mathbb{R}}^3$. The ramp ${\sigma }_{\epsilon }$ is smooth, has fixed width $\epsilon >0$, and vanishes for $t\le 0$. The nonzero switch $g$ belongs to the frozen admissible class of XI-G; in particular its compact support lies in ${M_T}^{\circ }$. Additional plateau hypotheses used by XI-G are part of that input, not inferred from compact support alone.

There is one metric field and no second classical metric fluctuation. The species, parities, and ghost numbers are

The symbols $f,g,{\sigma }_{\epsilon },\eta$ are external data, not field generators. Both $\hslash$ and $\kappa$ are even, ghost-number-zero formal parameters.

Trace reversal, divergence, and the symmetrized gradient are

Parentheses include the factor $\frac{1}{2}$. With $A=2G_{\mathrm{lin}}$ and formal adjoint $D^{\ast }$, the frozen identities are

They fix the normalization ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E_0}^{\mathrm{R}}$.