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

Proposition: Classical values and the retained coefficient -2

F3: three response kernels, support, and admission › Matter normalization and classical values · lines 1630–1674 · proof included or linked

X-F-ST@1#prop:classical, X-F-ST@1#eq:classical, X-F-ST@1#eq:minus2

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For smooth inputs $u, r$ , the normal-symbol values of the three trees are

T_{mat} [u] T_{mix} [h; u, r] T_{self} [u, r] = ⟨ d_{1} [- 2 P_{tr} E_{0}^{R} (g T_{0} [u])], u ⟩, = ⟨ d_{1} [- Δ_{gr}^{R} (σ Q^{(1)} [h] r)], u ⟩, = ⟨ d_{1} [- \frac{1}{2} Δ_{gr}^{R} (σ G_{2} [r, r])], u ⟩ .

Together they are the completed affine insertion $d_{1} [k_{q} [h; r, u]]$ , counted once.

If $P_{m} u = 0$ , the matter coordinate response is

w^{μ} [- 2 P_{tr} E_{0}^{R} (g T_{0} [u])] = - 2 (E_{0}^{R})^{2} ((\partial_{ν} g) T_{0}^{ν μ} [u]) .

Proof. Pair the kernels (eq:treekernels) with smooth fields and transpose their indicated differential operators. This gives the three sources in (eq:response). The coefficient $- 2$ follows from $Δ_{gr}^{R} (\frac{1}{2} g T_{0}) = - 2 P_{tr} E_{0}^{R} (g T_{0})$ .

The matter response vanishes before $supp g$ ; there $σ = 1$ on every contributing coordinate source. Since $a_{1} [l] = - P_{tr} l$ ,

w [l] = E_{0}^{R} C [l], C^{μ} [- 2 P_{tr} E_{0}^{R} (g T_{0} [u])] = - 2 E_{0}^{R} \partial_{ν} (g T_{0}^{ν μ} [u]) .

Use $\partial_{ν} T_{0}^{ν μ} [u] = 0$ on the free scalar shell.

Displayed formulas in this passage

X-F-ST@1#eq:classical · line 1634
$T_{mat} [u] T_{mix} [h; u, r] T_{self} [u, r] = ⟨ d_{1} [- 2 P_{tr} E_{0}^{R} (g T_{0} [u])], u ⟩, = ⟨ d_{1} [- Δ_{gr}^{R} (σ Q^{(1)} [h] r)], u ⟩, = ⟨ d_{1} [- \frac{1}{2} Δ_{gr}^{R} (σ G_{2} [r, r])], u ⟩ .$
X-F-ST@1#eq:minus2 · line 1649
$w^{μ} [- 2 P_{tr} E_{0}^{R} (g T_{0} [u])] = - 2 (E_{0}^{R})^{2} ((\partial_{ν} g) T_{0}^{ν μ} [u]) .$
unlabeled · line 1667
$w [l] = E_{0}^{R} C [l], C^{μ} [- 2 P_{tr} E_{0}^{R} (g T_{0} [u])] = - 2 E_{0}^{R} \partial_{ν} (g T_{0}^{ν μ} [u]) .$

Nearby in this section

section: Matter normalization and classical values