Manuscript XII-C@1 · XII-C.tex · section · dead1004

The selected formal action

C0: one choice and the lower-order consistency correction › The selected formal action · lines 478–526

XII-C@1#eq:selected-action, XII-C@1#eq:scalar

Let $ℓ_{3}$ be the precise cubic density fixed in Section sec:density. Define

S_{grav, D} = S_{EH} + κ \int_{M_{T}} (σ_{ε} - 1) ℓ_{3} [γ] d^{4} x, S_{EH} = - \frac{2}{κ ^{2}} \int_{M_{T}} - q R [q] d^{4} x .

The unchanged scalar action is

S_{g}^{sc} = \int_{M_{T}} ((1 - g) L_{η} [ϕ] + g L_{q} [ϕ]) d^{4} x, L_{q} [ϕ] = \frac{1}{2} - q (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) .

Write

S_{D} = S_{grav, D} + S_{g}^{sc}, S_{D, gf} = S_{D} + S_{gf}, S_{gf} = \int_{M_{T}} D_{ν} γ D^{ν} γ d^{4} x .

Actions mean formal bulk variational functionals, tested with compact interior variations. Numerical finiteness on all smooth configurations is not required. The unweighted linear Einstein–Hilbert divergence is treated as in XI-A. The selected cubic representative is not changed by a later integration by parts.

Equation (eq:selected-action) weights only the cubic coefficient. The quadratic coefficient and all quartic and higher coefficients remain unweighted. This fixes the higher-order completion rather than leaving it unspecified.

Displayed formulas in this passage

XII-C@1#eq:selected-action · line 482
$S_{grav, D} = S_{EH} + κ \int_{M_{T}} (σ_{ε} - 1) ℓ_{3} [γ] d^{4} x, S_{EH} = - \frac{2}{κ ^{2}} \int_{M_{T}} - q R [q] d^{4} x .$
XII-C@1#eq:scalar · line 494
$S_{g}^{sc} = \int_{M_{T}} ((1 - g) L_{η} [ϕ] + g L_{q} [ϕ]) d^{4} x, L_{q} [ϕ] = \frac{1}{2} - q (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) .$
unlabeled · line 507
$S_{D} = S_{grav, D} + S_{g}^{sc}, S_{D, gf} = S_{D} + S_{gf}, S_{gf} = \int_{M_{T}} D_{ν} γ D^{ν} γ d^{4} x .$

The selected formal action

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