Manuscript X-A@1 · X-A.tex · lemma · dc7f5c0a

Lemma: Divergence form of density pullback

The varied action (Route II) · lines 1707–1723 · proof included or linked

X-A@1#lem:Z

For every smooth scalar density $r$ ,

F_{dens}^{*} r - r = \partial_{μ} Z_{F}^{μ} [r], Z_{F}^{μ} [r] = κ ξ^{μ} r + κ^{2} (ζ^{μ} r + \frac{1}{2} ξ^{μ} D_{ξ} r) + O (κ^{3}),

with $Z_{F} [r]$ supported in $supp ξ \cup supp ζ$ .

Proof. From (eq:densitypull), $D_{ξ} r = \partial_{μ} (ξ^{μ} r)$ , $D_{ζ} r = \partial_{μ} (ζ^{μ} r)$ , and $D_{ξ}^{2} r = \partial_{μ} (ξ^{μ} \partial_{ν} (ξ^{ν} r)) = \partial_{μ} (ξ^{μ} D_{ξ} r)$ .

Displayed formulas in this passage

unlabeled · line 1710
$F_{dens}^{*} r - r = \partial_{μ} Z_{F}^{μ} [r], Z_{F}^{μ} [r] = κ ξ^{μ} r + κ^{2} (ζ^{μ} r + \frac{1}{2} ξ^{μ} D_{ξ} r) + O (κ^{3}),$

Nearby in this section

section: The varied action (Route II)
proposition: Proposition: Exact varied-action identity with all switch terms
section: The varied action (Route II)
proposition: Proposition: First-order display for a general compact gauge parameter
corollary: Corollary: Vanishing on the frozen group; the Noether identity
section: The varied action (Route II)