Manuscript X-A-ST@1 · X-A-ST.tex · proposition · d9ccbde1

Proposition: Varied switched action

Varied-action derivation: Route II · lines 1271–1303 · proof included or linked

X-A-ST@1#prop:varied, X-A-ST@1#eq:varied

The finite variational difference satisfies

S_{g}^{sc} [F^{*} q, F^{*} ϕ] - S_{g}^{sc} [q, ϕ] = \int_{M_{T_{*}}} (1 - g) (L_{η} [F^{*} ϕ] - L_{η} [ϕ]) d^{4} x - \int_{M_{T_{*}}} (\partial_{μ} g) Z_{F}^{μ} [L_{q} [ϕ]] d^{4} x (mod κ^{3}) .

Every non-divergence term contains $1 - g$ or $\partial g$ .

Proof. Naturality of the curved density gives

L_{g} [F^{*} q, F^{*} ϕ] - L_{g} [q, ϕ] = (1 - g) (L_{η} [F^{*} ϕ] - L_{η} [ϕ]) + g (F_{dens}^{*} L_{q} - L_{q}) .

Apply (eq:Z) and $g \partial_{μ} Z^{μ} = \partial_{μ} (g Z^{μ}) - (\partial_{μ} g) Z^{μ}$ . The compact divergence integrates to zero. The difference is finite even when the individual unvaried actions have no finite numerical value.

Displayed formulas in this passage

X-A-ST@1#eq:varied · line 1274
$S_{g}^{sc} [F^{*} q, F^{*} ϕ] - S_{g}^{sc} [q, ϕ] = \int_{M_{T_{*}}} (1 - g) (L_{η} [F^{*} ϕ] - L_{η} [ϕ]) d^{4} x - \int_{M_{T_{*}}} (\partial_{μ} g) Z_{F}^{μ} [L_{q} [ϕ]] d^{4} x (mod κ^{3}) .$
unlabeled · line 1292
$L_{g} [F^{*} q, F^{*} ϕ] - L_{g} [q, ϕ] = (1 - g) (L_{η} [F^{*} ϕ] - L_{η} [ϕ]) + g (F_{dens}^{*} L_{q} - L_{q}) .$

Proposition: Varied switched action

Displayed formulas in this passage

Nearby in this section