Manuscript IX-D@3 · IX-D-v3.tex · section · 3ad4586e

Geometry and actual functionals

Release conventions and inherited inputs › Geometry and actual functionals · lines 299–346

IX-D@3#eq:L, IX-D@3#eq:generators

Let

M = R^{1 + 3}, M_{+} = {t > 0}, Σ = {t = 0}, X = M_{+}^{3},

with

η = diag (- 1, 1, 1, 1), □ = - \partial_{t}^{2} + Δ_{x} .

Set

(C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}), L = E_{0} C, w = L h .

The operator $E_{0}$ is the zero-value, zero-first-time-derivative Cauchy inverse at $Σ$ . Configurations $ψ = (h, ϕ)$ and configuration directions are smooth and off shell.

For

\Tspace = C_{c}^{\infty} (M_{+}; C),

the generators are

B_{f} = - \int f (x) w^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, W_{g} = \int g (y) h_{00} (y)^{2} d^{4} y .

Complex tests define complex-valued actual functionals on real smooth configurations.

The mixed covariance $K (x, y)$ and scalar covariance $Δ_{m}^{+} (x - x^{'})$ are fixed flat free kernels. They are not configuration-dependent metric propagators. The separate notation $K ξ$ below denotes a metric gauge variation, not the mixed kernel $K$ .

Displayed formulas in this passage

IX-D@3#eq:L · line 314
$(C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}), L = E_{0} C, w = L h .$
IX-D@3#eq:generators · line 332
$B_{f} = - \int f (x) w^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, W_{g} = \int g (y) h_{00} (y)^{2} d^{4} y .$
unlabeled · line 302
$M = R^{1 + 3}, M_{+} = {t > 0}, Σ = {t = 0}, X = M_{+}^{3},$
unlabeled · line 309
$η = diag (- 1, 1, 1, 1), □ = - \partial_{t}^{2} + Δ_{x} .$
unlabeled · line 328
$\Tspace = C_{c}^{\infty} (M_{+}; C),$

Geometry and actual functionals

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