Manuscript VII-A-ST@1 · VII-A-ST.tex · section · c39a67a3

Switching before variation

Switching before variation · lines 517–593

VII-A-ST@1#eq:action, VII-A-ST@1#eq:Euler, VII-A-ST@1#eq:QT, VII-A-ST@1#eq:DW, VII-A-ST@1#eq:VW, VII-A-ST@1#eq:switch-derivatives

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The Lie bracket preserves $g_{U}$ , and the truncated Baker–Campbell–Hausdorff formula preserves the class (eq:formal-group). Including $ζ$ is necessary for this convenient composition-stable parametrization. Every admitted actual flow is the identity outside a compact subset of $U$ , preserves $g$ , and is the identity near $Σ_{0}$ .

Define the switched coefficients

A_{g} [γ] = (1 - g) A_{η} + g A_{γ}, ρ_{g} [γ] = (1 - g) + g ρ_{γ} .

Our scalar action and Euler operator are

S_{g}^{sc} [γ, ϕ] D_{g} [γ] ϕ = - \frac{1}{2} \int (A_{g} [γ]^{μν} \partial_{μ} ϕ \partial_{ν} ϕ + m^{2} ρ_{g} [γ] ϕ^{2}) d^{4} x, = \partial_{μ} (A_{g} [γ]^{μν} \partial_{ν} ϕ) - m^{2} ρ_{g} [γ] ϕ .

The action is variationally defined with compact variations; its numerical value need not be finite on every smooth input. Its Euler output represents a scalar density relative to $d^{4} x$ .

For a symmetric tensor $H$ , put

Q_{H}^{(m)} u T_{h}^{(m)} u = \partial_{μ} (A^{μν} (H) \partial_{ν} u) - m^{2} a (H) u, = \partial_{μ} (B^{μν} (h) \partial_{ν} u) - m^{2} b (h) u .

Then

D_{g} [γ] = P_{m} + κ V_{g} [h] + κ^{2} W_{g} [h, k] (mod κ^{3}),

where

V_{g} [h] W_{g} [h, k] u = Q_{g h}^{(m)}, = \partial_{μ} (g (A^{μν} (k) + B^{μν} (h)) \partial_{ν} u) - m^{2} g (a (k) + b (h)) u .

In particular,

V_{g} [h] u W_{g} [h, k] u = g Q_{h}^{(m)} u + (\partial_{μ} g) A^{μν} (h) \partial_{ν} u, = g (Q_{k}^{(m)} + T_{h}^{(m)}) u + (\partial_{μ} g) (A^{μν} (k) + B^{μν} (h)) \partial_{ν} u .

The quadratic metric vertex has one factor $g$ , not $g^{2}$ . The iterated first-order response below contains two switch insertions for a different reason.

Displayed formulas in this passage

VII-A-ST@1#eq:action · line 534
$S_{g}^{sc} [γ, ϕ] D_{g} [γ] ϕ = - \frac{1}{2} \int (A_{g} [γ]^{μν} \partial_{μ} ϕ \partial_{ν} ϕ + m^{2} ρ_{g} [γ] ϕ^{2}) d^{4} x, = \partial_{μ} (A_{g} [γ]^{μν} \partial_{ν} ϕ) - m^{2} ρ_{g} [γ] ϕ .$
VII-A-ST@1#eq:Euler · line 534
$S_{g}^{sc} [γ, ϕ] D_{g} [γ] ϕ = - \frac{1}{2} \int (A_{g} [γ]^{μν} \partial_{μ} ϕ \partial_{ν} ϕ + m^{2} ρ_{g} [γ] ϕ^{2}) d^{4} x, = \partial_{μ} (A_{g} [γ]^{μν} \partial_{ν} ϕ) - m^{2} ρ_{g} [γ] ϕ .$
VII-A-ST@1#eq:QT · line 555
$Q_{H}^{(m)} u T_{h}^{(m)} u = \partial_{μ} (A^{μν} (H) \partial_{ν} u) - m^{2} a (H) u, = \partial_{μ} (B^{μν} (h) \partial_{ν} u) - m^{2} b (h) u .$
VII-A-ST@1#eq:DW · line 564
$D_{g} [γ] = P_{m} + κ V_{g} [h] + κ^{2} W_{g} [h, k] (mod κ^{3}),$
VII-A-ST@1#eq:VW · line 570
$V_{g} [h] W_{g} [h, k] u = Q_{g h}^{(m)}, = \partial_{μ} (g (A^{μν} (k) + B^{μν} (h)) \partial_{ν} u) - m^{2} g (a (k) + b (h)) u .$
VII-A-ST@1#eq:switch-derivatives · line 579
$V_{g} [h] u W_{g} [h, k] u = g Q_{h}^{(m)} u + (\partial_{μ} g) A^{μν} (h) \partial_{ν} u, = g (Q_{k}^{(m)} + T_{h}^{(m)}) u + (\partial_{μ} g) (A^{μν} (k) + B^{μν} (h)) \partial_{ν} u .$
unlabeled · line 528
$A_{g} [γ] = (1 - g) A_{η} + g A_{γ}, ρ_{g} [γ] = (1 - g) + g ρ_{γ} .$

Switching before variation

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