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

Switching before variation

Sign, density, switch, and regulator audit › Switching before variation · lines 2196–2210

For any smooth symmetric coefficient $A^{μν}$ and any smooth switch $s$ ,

\partial_{μ} (s A^{μν} \partial_{ν} ψ) = s \partial_{μ} (A^{μν} \partial_{ν} ψ) + (\partial_{μ} s) A^{μν} \partial_{ν} ψ .

Multiplying an unswitched equation by $s$ is therefore not the Euler equation of the switched density. This applies twice here: to the interaction switch $g$ in (eq:V1g)–(eq:V2g), and to the regulator $σ_{ε}$ in (eq:V)–(eq:W). The frozen prescriptions are the divergence-form ones.

Displayed formulas in this passage

unlabeled · line 2200
$\partial_{μ} (s A^{μν} \partial_{ν} ψ) = s \partial_{μ} (A^{μν} \partial_{ν} ψ) + (\partial_{μ} s) A^{μν} \partial_{ν} ψ .$