Manuscript IX-E-2@1 · IX-E-2.tex · section · e70d9077

Sharp insertion, locked covariance, and free product

Explicit inherited inputs and their limits › Sharp insertion, locked covariance, and free product · lines 316–392

IX-E-2@1#eq:sharp-insertion, IX-E-2@1#eq:locked-kernel, IX-E-2@1#eq:ordered-coefficients, IX-E-2@1#eq:free-reversal

Work in the flat half-space

M_{+} = [0, \infty) \times R^{3}, Σ = {0} \times R^{3}, M_{+}^{\circ} = (0, \infty) \times R^{3} .

The configuration space is the unrestricted smooth off-shell scalar and metric configuration space inherited from IX-A. Neither an on-shell quotient nor a gauge-fixed quotient is used.

The sharp insertion is

L = E_{0} C, B_{f} [h, ϕ] = - \int_{M_{+}} f (x) (L h)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x .

For $x = (t, x)$ , $y = (s, y)$ , the locked mixed covariance supplied from VII-C and restated on the VII-D convention sheet is

K (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4∣ p ∣} + \frac{i}{8∣ p ∣ ^{2}}) e^{- i ∣ p ∣ (t - s)} - \frac{i}{8∣ p ∣ ^{2}} e^{i ∣ p ∣ (t + s)}] .

No graviton positivity is assumed. The inherited statements $H_{w w} = 0$ and the characteristic-cone failure of $B_{f}$ remain unchanged.

The declared free product $⋆_{c}$ and its first-stage coefficients are inputs:

B_{f} ⋆_{c} W_{g} W_{g} ⋆_{c} B_{f} = B_{f} W_{g} + ℏ Q_{f, g}, = B_{f} W_{g} + ℏ Q^{R}_{g, f}, Q_{f, g} Q^{R}_{g, f} = - 2 ⟨ f (x) g (y) K (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩, = - 2 ⟨ g (y) f (x) K (x, y), h_{00} (y) \partial_{t} ϕ (x) ⟩ .

These are the inherited exact first-stage formulas in the truncated coefficient space. Their reversal identity is

⋆_{c} (d)^{*} = ⋆_{c} (d^{†}) .

The released degree-two and degree-four descent, the word through $ℏ^{3}$ , the evaluations of $ker p_{L}, ker p_{R}, C_{desc}, C_{glue}$ , and F9a fixed-support continuity are not reopened. F9a is not promoted to hypocontinuity.

Full support of $K$ , where inherited, is not used to determine its imaginary part. A real distribution may have full support.

Displayed formulas in this passage

IX-E-2@1#eq:sharp-insertion · line 331
$L = E_{0} C, B_{f} [h, ϕ] = - \int_{M_{+}} f (x) (L h)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x .$
IX-E-2@1#eq:locked-kernel · line 342
$K (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4∣ p ∣} + \frac{i}{8∣ p ∣ ^{2}}) e^{- i ∣ p ∣ (t - s)} - \frac{i}{8∣ p ∣ ^{2}} e^{i ∣ p ∣ (t + s)}] .$
IX-E-2@1#eq:ordered-coefficients · line 362
$B_{f} ⋆_{c} W_{g} W_{g} ⋆_{c} B_{f} = B_{f} W_{g} + ℏ Q_{f, g}, = B_{f} W_{g} + ℏ Q^{R}_{g, f}, Q_{f, g} Q^{R}_{g, f} = - 2 ⟨ f (x) g (y) K (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩, = - 2 ⟨ g (y) f (x) K (x, y), h_{00} (y) \partial_{t} ϕ (x) ⟩ .$
IX-E-2@1#eq:free-reversal · line 379
$⋆_{c} (d)^{*} = ⋆_{c} (d^{†}) .$
unlabeled · line 319
$M_{+} = [0, \infty) \times R^{3}, Σ = {0} \times R^{3}, M_{+}^{\circ} = (0, \infty) \times R^{3} .$