Manuscript X-F-ST@1 · X-F-ST.tex · section · e71f03ef

The retained action and the invariant-jet boundary

The retained action and the invariant-jet boundary · lines 515–548

X-F-ST@1#eq:action

This section fixes the inherited action and records the identities needed to prevent an equation-space misidentification. It does not reopen X-E-ST's action choice XEST,XE.

Use the curvature convention

R^{ρ}_{σ μν} = \partial_{μ} Γ^{ρ}_{ν σ} - \partial_{ν} Γ^{ρ}_{μ σ} + Γ^{ρ}_{μ λ} Γ^{λ}_{ν σ} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ σ}, R_{σ ν} = R^{ρ}_{σ ρ ν} .

The local action is

S_{gr} [q] S_{sc, g} [q, ϕ] L_{q} [q, ϕ] = - \frac{1}{2 κ ^{2}} \int J_{q} R [q] d^{4} x, = \int ((1 - g) L_{η} [ϕ] + g L_{q} [q, ϕ]) d^{4} x, = \frac{1}{2} J_{q} (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) .

These denote local Lagrangians with compact variations; global finiteness of the action integral is not assumed.

Displayed formulas in this passage

X-F-ST@1#eq:action · line 533
$S_{gr} [q] S_{sc, g} [q, ϕ] L_{q} [q, ϕ] = - \frac{1}{2 κ ^{2}} \int J_{q} R [q] d^{4} x, = \int ((1 - g) L_{η} [ϕ] + g L_{q} [q, ϕ]) d^{4} x, = \frac{1}{2} J_{q} (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) .$
unlabeled · line 524
$R^{ρ}_{σ μν} = \partial_{μ} Γ^{ρ}_{ν σ} - \partial_{ν} Γ^{ρ}_{μ σ} + Γ^{ρ}_{μ λ} Γ^{λ}_{ν σ} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ σ}, R_{σ ν} = R^{ρ}_{σ ρ ν} .$

The retained action and the invariant-jet boundary

Displayed formulas in this passage

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