Manuscript XI-D@1 · XI-D.tex · section · 994a5c1b

The copied XI-C split

Prerequisite and convention sheet › The copied XI-C split · lines 417–476

XI-D@1#eq:L, XI-D@1#eq:B, XI-D@1#eq:rg, XI-D@1#eq:T

The coordinate operator is

L_{ε} r = X_{1} [r] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α \cdot} [r]) = E_{0}^{R} (σ_{ε} D r - σ_{ε}^{'} a_{1}^{0 \cdot} [r]),

where

a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν} .

The bare insertion and the classical tree are

B_{f} [γ, ϕ] r_{g} [ϕ] T (f, g) [ϕ] = - \int_{M_{T}} f X_{1} [γ]^{μ} \partial_{μ} ϕ d^{4} x, = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]), = \int_{M_{T}} f (L_{ε} r_{g} [ϕ])^{μ} \partial_{μ} ϕ d^{4} x,

with

T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} (\partial_{ρ} ϕ \partial^{ρ} ϕ - m^{2} ϕ^{2}) .

The physical one-graviton response of $B_{f}$ is $- T$ , at coupling degree $κ^{2}$ .

XI-C proves that these are compact distributional polynomial symbols satisfying its microcausal exclusion XIC. It does not construct their quantum admission. Its mixed insertion kernel is not smooth finite-rank. The noncharacteristic diagonal symbol has a nonzero $00$ component, with the stated numerator $- 1/2$ at $p = d t$ .

Above the ramp, for the named input

ϕ_{m} (t) = {sin (m t) / m, t, m > 0, m = 0,

the copied response is

X_{1}^{0} [r_{g} [ϕ_{m}]] = - \frac{1}{4} E_{0}^{R} E_{0}^{R} (\partial_{t} g) \neq \equiv 0.

XI-A's empty Einstein jet remains empty XIA. That source obstruction is not a quantum-admission obstruction.

Displayed formulas in this passage

XI-D@1#eq:L · line 420
$L_{ε} r = X_{1} [r] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α \cdot} [r]) = E_{0}^{R} (σ_{ε} D r - σ_{ε}^{'} a_{1}^{0 \cdot} [r]),$
XI-D@1#eq:B · line 432
$B_{f} [γ, ϕ] r_{g} [ϕ] T (f, g) [ϕ] = - \int_{M_{T}} f X_{1} [γ]^{μ} \partial_{μ} ϕ d^{4} x, = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]), = \int_{M_{T}} f (L_{ε} r_{g} [ϕ])^{μ} \partial_{μ} ϕ d^{4} x,$
XI-D@1#eq:rg · line 432
$B_{f} [γ, ϕ] r_{g} [ϕ] T (f, g) [ϕ] = - \int_{M_{T}} f X_{1} [γ]^{μ} \partial_{μ} ϕ d^{4} x, = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]), = \int_{M_{T}} f (L_{ε} r_{g} [ϕ])^{μ} \partial_{μ} ϕ d^{4} x,$
XI-D@1#eq:T · line 432
$B_{f} [γ, ϕ] r_{g} [ϕ] T (f, g) [ϕ] = - \int_{M_{T}} f X_{1} [γ]^{μ} \partial_{μ} ϕ d^{4} x, = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]), = \int_{M_{T}} f (L_{ε} r_{g} [ϕ])^{μ} \partial_{μ} ϕ d^{4} x,$
unlabeled · line 427
$a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν} .$
unlabeled · line 445
$T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} (\partial_{ρ} ϕ \partial^{ρ} ϕ - m^{2} ϕ^{2}) .$
unlabeled · line 462
$ϕ_{m} (t) = {sin (m t) / m, t, m > 0, m = 0,$
unlabeled · line 470
$X_{1}^{0} [r_{g} [ϕ_{m}]] = - \frac{1}{4} E_{0}^{R} E_{0}^{R} (\partial_{t} g) \neq \equiv 0.$