Manuscript VII-I@1 · VII-I.tex · section · 907e96ab

Restricted gauge transformations

One classical observable and its actual Cauchy data › Restricted gauge transformations · lines 466–503

Fix a scalar-action switch $g$ , a nonempty plateau $U$ , and

g \in C_{c}^{\infty} (M_{+}^{\circ}), U \subset int {g = 1}, g_{U} = C_{c}^{\infty} (U; T M) .

The group is generated by flows of these vector fields. Its second-order formal pullbacks are

F_{κ}^{*} = exp (κ L_{ξ} + κ^{2} L_{ζ}) (mod κ^{3}), ξ, ζ \in g_{U} .

They are the identity near $Σ$ .

For the unswitched coordinate equation,

X_{F^{*} q} = X_{q} \circ F

by naturality and Cauchy uniqueness. Therefore

O_{f} [F^{*} q, F^{*} ϕ] = O_{f} [q, ϕ] (mod κ^{3}) .

For the switched scalar solution with fixed free input, the additional plateau intertwiner is the theorem of VII-A CheungVIIA. It is not unrestricted fixed-switch covariance.

At the background coordinate map, preserving fixed values and fixed coordinate-time derivatives requires

ξ ∣_{Σ} = 0, \partial_{t} ξ ∣_{Σ} = 0

at first order. Vanishing only of $ξ ∣_{Σ}$ is insufficient. Support in $U$ supplies a stronger sufficient condition.

Displayed formulas in this passage

unlabeled · line 469
$g \in C_{c}^{\infty} (M_{+}^{\circ}), U \subset int {g = 1}, g_{U} = C_{c}^{\infty} (U; T M) .$
unlabeled · line 476
$F_{κ}^{*} = exp (κ L_{ξ} + κ^{2} L_{ζ}) (mod κ^{3}), ξ, ζ \in g_{U} .$
unlabeled · line 484
$X_{F^{*} q} = X_{q} \circ F$
unlabeled · line 488
$O_{f} [F^{*} q, F^{*} ϕ] = O_{f} [q, ϕ] (mod κ^{3}) .$
unlabeled · line 497
$ξ ∣_{Σ} = 0, \partial_{t} ξ ∣_{Σ} = 0$