Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

Gauge algebra and formal transformations

Convention sheet and frozen data › Gauge algebra and formal transformations · lines 428–475

X-A-conv@1#eq:gaugealg, X-A-conv@1#eq:formalgroup, X-A-conv@1#eq:infaction, X-A-conv@1#eq:hprime, X-A-conv@1#eq:kprime

For a vector field $ξ$ , set

(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, X_{ξ} = ξ^{μ} \partial_{μ} .

The frozen gauge algebra is

g_{U} = C_{c}^{\infty} (U; T M) .

Each member vanishes on a neighborhood of the edge of $U$ , hence vanishes there to every order. It also vanishes near both temporal boundaries.

The corresponding group is generated by flows of members of $g_{U}$ . At second formal order we use

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

This parametrization is closed under formal composition because $g_{U}$ is closed under Lie brackets.

For the single-field metric (eq:baremetric), the infinitesimal action is

δ_{ξ} h = K ξ + κ L_{ξ} h, δ_{ξ} ϕ = κ X_{ξ} ϕ .

For (eq:jetmetric), finite coefficient extraction gives

h^{'} k^{'} = h + K ξ, = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η .

The nonlinear term in (eq:infaction) and the second coefficient in (eq:kprime) are retained.

Displayed formulas in this passage

X-A-conv@1#eq:gaugealg · line 438
$g_{U} = C_{c}^{\infty} (U; T M) .$
X-A-conv@1#eq:formalgroup · line 448
$F_{κ}^{*} = exp (κ L_{ξ} + κ^{2} L_{ζ}) (mod κ^{3}), ξ, ζ \in g_{U} .$
X-A-conv@1#eq:infaction · line 460
$δ_{ξ} h = K ξ + κ L_{ξ} h, δ_{ξ} ϕ = κ X_{ξ} ϕ .$
X-A-conv@1#eq:hprime · line 467
$h^{'} k^{'} = h + K ξ, = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η .$
X-A-conv@1#eq:kprime · line 467
$h^{'} k^{'} = h + K ξ, = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ}^{2} η .$
unlabeled · line 431
$(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, X_{ξ} = ξ^{μ} \partial_{μ} .$