Manuscript XI-F@2 · XI-F-v2.tex · lemma · f4899be0

Lemma: Order- κ variation of the de Donder term

F2: the gauge-fixing companion › The de Donder variation from the fermion · lines 924–955 · proof included or linked

XI-F@2#lem:gf, XI-F@2#eq:Egfvar, XI-F@2#eq:divgf

Let $s$ be a smooth symmetric tensor and shift $γ \mapsto γ + κ s$ with $(b, c, c ˉ)$ held fixed. Then

s_{0} Ψ [γ + κ s] = s_{0} Ψ [γ] + 2 κ \int_{M_{T}} b_{μ} D^{μ} s d^{4} x,

the auxiliary equation becomes $b = D (γ + κ s)$ , and after elimination

S_{gf} [γ + κ s] = S_{gf} [γ] + 2 κ \int D γ \cdot D s + κ^{2} \int D s \cdot D s, E_{gf} [γ + κ s] - E_{gf} [γ] = 2 κ D^{*} D s .

The ghost term $i c ˉ_{μ} □ c^{μ}$ is $γ$ -independent: it has no external metric leg at this order. Moreover

\partial_{μ} (2 D^{*} D s)^{μν} = - □ D^{ν} s .

Proof. Read off from (eq:fermion): only $2 b \cdot D γ$ depends on $γ$ , linearly. Elimination of $b$ and expansion of the square give (eq:Egfvar), with $E_{gf} = 2 D^{*} D γ$ as in XI-A. For (eq:divgf), with $v = D s$ , $\partial_{μ} (D^{*} v)^{μν} = - \frac{1}{2} □ v^{ν} - \frac{1}{2} \partial^{ν} \partial \cdot v + \frac{1}{2} \partial^{ν} \partial \cdot v = - \frac{1}{2} □ v^{ν}$ . This is XI-A's separation of the gauge-fixing breaking $- □ D$ from the invariant Noether identity, now at order $κ$ in $s$ .

Displayed formulas in this passage

XI-F@2#eq:Egfvar · line 933
$S_{gf} [γ + κ s] = S_{gf} [γ] + 2 κ \int D γ \cdot D s + κ^{2} \int D s \cdot D s, E_{gf} [γ + κ s] - E_{gf} [γ] = 2 κ D^{*} D s .$
XI-F@2#eq:divgf · line 941
$\partial_{μ} (2 D^{*} D s)^{μν} = - □ D^{ν} s .$
unlabeled · line 928
$s_{0} Ψ [γ + κ s] = s_{0} Ψ [γ] + 2 κ \int_{M_{T}} b_{μ} D^{μ} s d^{4} x,$

Lemma: Order- κ variation of the de Donder term

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