Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

Switching before variation

Sign, density, and switch audit › Switching before variation · lines 2084–2099

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

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

Therefore multiplying an unswitched equation by $g$ is not the same as multiplying the interaction Lagrangian density by $g$ before variation.

The main construction uses the latter prescription. This fixes both the switch derivatives and the second-order scalar operator.

Displayed formulas in this passage

unlabeled · line 2087
$\partial_{μ} (g A^{μν} \partial_{ν} ψ) = g \partial_{μ} (A^{μν} \partial_{ν} ψ) + (\partial_{μ} g) A^{μν} \partial_{ν} ψ .$