Manuscript X-I-ST@1 · X-I-ST.tex · proposition · 9745fd40

Proposition: Precise zeroth-order comparison

A finite reference action through third order › The tadpole correction to the free-limit assertion · lines 951–979 · proof included or linked

X-I-ST@1#prop:tadpole, X-I-ST@1#eq:tadpole

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At order zero,

I_{0} = I_{free} + ⟨ r, A h ⟩ mod divergences .

The free Hessian and the field differential $s_{0}$ are unchanged for all fixed $h, k$ . The full zeroth-order BV Hamiltonian equals the free one in the centered sector $h = k = 0$ . For a general off-shell $h$ , it instead retains the affine source $A h$ in the metric antifield equation.

Proof. The quadratic gravitational coefficient is $\frac{1}{2} ⟨ h + r, A (h + r) ⟩$ . Subtracting its background-only term gives

\frac{1}{2} ⟨ r, A r ⟩ + ⟨ r, A h ⟩ .

The scalar and gauge-fixing coefficients at zero are those of (eq:Ifree). The extra term is linear in $r$ , so it changes no Hessian and no field rule. Its bracket with $r^{*}$ is $A h$ .

Displayed formulas in this passage

X-I-ST@1#eq:tadpole · line 954
$I_{0} = I_{free} + ⟨ r, A h ⟩ mod divergences .$
unlabeled · line 971
$\frac{1}{2} ⟨ r, A r ⟩ + ⟨ r, A h ⟩ .$

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section: The tadpole correction to the free-limit assertion