Manuscript IX-D-ST@1 · IX-D-ST.tex · section · b1dc631a

Inherited free sector and locked comparison kernel

Inherited free sector and locked comparison kernel · lines 423–504

IX-D-ST@1#eq:generators, IX-D-ST@1#eq:K, IX-D-ST@1#eq:firststages, IX-D-ST@1#eq:trees

Configurations are $Ψ = (h, ϕ)$ . Set

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 .

Pairings are complex linear in their tests.

Use $x = (t, x)$ , $y = (s, y)$ , $x^{'} = (t^{'}, x^{'})$ , and

τ = t - s, u = t + s, z = x - y, R = ∣ z ∣.

The locked mixed comparison kernel 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)}] .

It is a chart comparison formula, not a covariance derived from the conical host.

The first stages are locked as

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

The minus sign and the factor $2$ are fixed.

For later support bookkeeping write

w = L h, S_{μν} (x, x^{'}) = \partial_{x^{μ}} \partial_{x^{' ν}} Δ_{m}^{+} (x - x^{'}),

T_{f, f^{'}} = ⟨ f (x) f^{'} (x^{'}) S_{μν} (x, x^{'}), w^{μ} (x) w^{ν} (x^{'}) ⟩ .

Define

G_{12} = Q_{f, g} B_{f^{'}}, G_{23} = B_{f} Q_{g, f^{'}}^{R}, G_{13} = T_{f, f^{'}} W_{g} .

The inherited two-edge values have residual forms

G_{12, 23} G_{12, 13} G_{13, 23} = 2 ⟨ f (x) g (y) f^{'} (x^{'}) K (x, y) K (x^{'}, y), \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = 2 ⟨ f (x) g (y) f^{'} (x^{'}) K (x, y) S_{0 ν} (x, x^{'}), h_{00} (y) w^{ν} (x^{'}) ⟩, = 2 ⟨ f (x) g (y) f^{'} (x^{'}) S_{μ 0} (x, x^{'}) K (x^{'}, y), w^{μ} (x) h_{00} (y) ⟩ .

Displayed formulas in this passage

IX-D-ST@1#eq:generators · line 427
$B_{f} [h, ϕ] = - \int f (x) (L h)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, W_{g} [h] = \int g (y) h_{00} (y)^{2} d^{4} y .$
IX-D-ST@1#eq:K · line 443
$K (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4∣ p ∣} + \frac{i}{8∣ p ∣ ^{2}}) e^{- i ∣ p ∣ (t - s)} - \frac{i}{8∣ p ∣ ^{2}} e^{i ∣ p ∣ (t + s)}] .$
IX-D-ST@1#eq:firststages · line 459
$B_{f} ⋆_{c} W_{g} W_{g} ⋆_{c} B_{f} = B_{f} W_{g} + ℏ Q_{f, g}, = W_{g} B_{f} + ℏ Q^{R}_{g, f}, Q_{f, g} Q^{R}_{g, f} = - 2 ⟨ f g K, \partial_{t} ϕ (x) h_{00} (y) ⟩, = - 2 ⟨ f g K, \partial_{t} ϕ (x) h_{00} (y) ⟩ .$
IX-D-ST@1#eq:trees · line 489
$G_{12, 23} G_{12, 13} G_{13, 23} = 2 ⟨ f (x) g (y) f^{'} (x^{'}) K (x, y) K (x^{'}, y), \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = 2 ⟨ f (x) g (y) f^{'} (x^{'}) K (x, y) S_{0 ν} (x, x^{'}), h_{00} (y) w^{ν} (x^{'}) ⟩, = 2 ⟨ f (x) g (y) f^{'} (x^{'}) S_{μ 0} (x, x^{'}) K (x^{'}, y), w^{μ} (x) h_{00} (y) ⟩ .$
unlabeled · line 438
$τ = t - s, u = t + s, z = x - y, R = ∣ z ∣.$
unlabeled · line 471
$w = L h, S_{μν} (x, x^{'}) = \partial_{x^{μ}} \partial_{x^{' ν}} Δ_{m}^{+} (x - x^{'}),$
unlabeled · line 477
$T_{f, f^{'}} = ⟨ f (x) f^{'} (x^{'}) S_{μν} (x, x^{'}), w^{μ} (x) w^{ν} (x^{'}) ⟩ .$
unlabeled · line 483
$G_{12} = Q_{f, g} B_{f^{'}}, G_{23} = B_{f} Q_{g, f^{'}}^{R}, G_{13} = T_{f, f^{'}} W_{g} .$