Manuscript IX-C@1 · IX-C.tex · section · 03ce8e6b

The actual-functional carrier

Inherited data and the separation from V › The actual-functional carrier · lines 303–366

IX-C@1#eq:BW, IX-C@1#eq:admission

Let

M = R^{1 + 3}, M_{+} = {t > 0}, X = M_{+}^{3}, η = diag (- 1, 1, 1, 1) .

Write $x = (t, x)$ , $y = (s, y)$ , and $x^{'} = (t^{'}, x^{'})$ . Set

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

The operator $E_{0}$ is the zero-value, zero-first-time-derivative Cauchy inverse on ${t = 0}$ .

Configurations $(h, ϕ)$ are smooth and off shell. The selected functionals are

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

IX-A defines $P_{Σ}$ as a space of actual finite polynomial functionals with compact symmetric distributional kernels. Its admission criterion is

(D_{η} F)_{h}^{'} (ψ) \in S_{1} = M_{1} + Q_{Σ} for every smooth ψ and finite direction list η .

Here

q_{a} = L^{*} a = C^{*} (θ G_{A}^{0} a), Q_{Σ} = {q_{a} : a \in C_{c}^{\infty} (M_{+}; T^{*} M)},

and $M_{1}$ consists of compact linear metric kernels with no nonzero nonspacelike wavefront covectors.

The topology $τ_{Σ}$ records the strong compact-distribution coefficients and every intrinsic boundary-trace coordinate

b ((D_{η} F)_{h}^{'} (ψ)) .

We inherit the carrier, intrinsic trace, classical closure, directional closure, and Hausdorff topology from IX-A IXA (def:Padm, lem:charge, thm:closure, def:topology). The truncated coefficient module is

V_{Σ}^{[3]} = P_{Σ} \otimes_{C} C [ℏ] / (ℏ^{4}),

not a constructed quantum algebra IXA (eq:module).

Displayed formulas in this passage

IX-C@1#eq:BW · line 323
$B_{f} [h, ϕ] = - \int f (x) (L h)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, W_{g} [h] = \int g (y) h_{00} (y)^{2} d^{4} y .$
IX-C@1#eq:admission · line 333
$(D_{η} F)_{h}^{'} (ψ) \in S_{1} = M_{1} + Q_{Σ} for every smooth ψ and finite direction list η .$
unlabeled · line 306
$M = R^{1 + 3}, M_{+} = {t > 0}, X = M_{+}^{3}, η = diag (- 1, 1, 1, 1) .$
unlabeled · line 312
$□ = - \partial_{t}^{2} + Δ_{x}, (C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}), L = E_{0} C .$
unlabeled · line 342
$q_{a} = L^{*} a = C^{*} (θ G_{A}^{0} a), Q_{Σ} = {q_{a} : a \in C_{c}^{\infty} (M_{+}; T^{*} M)},$
unlabeled · line 351
$b ((D_{η} F)_{h}^{'} (ψ)) .$
unlabeled · line 361
$V_{Σ}^{[3]} = P_{Σ} \otimes_{C} C [ℏ] / (ℏ^{4}),$