Manuscript XV-F-patch-ii@1 · XV-F-patch-ii.tex · section · 9674ac67

The six operational entries, enumerated

Patch 2: the six-entry presentation package (Q2) › The six operational entries, enumerated · lines 593–664

XV-F-patch-ii@1#eq:retained, XV-F-patch-ii@1#eq:L, XV-F-patch-ii@1#eq:K5a, XV-F-patch-ii@1#eq:K5z, XV-F-patch-ii@1#eq:K47, XV-F-patch-ii@1#eq:K6, XV-F-patch-ii@1#eq:Dfrom, XV-F-patch-ii@1#eq:Q, XV-F-patch-ii@1#eq:T, XV-F-patch-ii@1#eq:K8, XV-F-patch-ii@1#eq:K9, XV-F-patch-ii@1#eq:cross

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The retained kernel and the supported sector are II2, Jreference

K_{ret} = K_{47} + K_{5} + K_{6} + K_{8} + K_{9}, K_{5} = K_{5}^{[a]} + K_{5}^{[z]}, K_{supp} = K_{6} + K_{8} + K_{9} .

The coordinate-response line is XIV-H's HeqL

L^{μ, I} (s, a) = σ_{ε} (a) A^{λ μ, I} \partial_{a^{λ}} R (s - a) .

The displays, copied for enumeration (copying is not instantiation; see sec:patch3), are

K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],

from XIV-H Hkernel and XIV-I II2, where XIV-I defines

a_{2} ˙^{α μ} [G; h] = D^{α μ, I, J} h_{I} G_{J},

“the constant bilinear fiber map determined by” this identity; and from XIV-I's input statement copied from XIV-H Iresponse, with the order bound of XIV-H Hq,

Q^{L, I, J} (v; a, z) T^{L, I, J} (b; a, z) K_{8}^{I, J} (a, b, z) K_{9}^{I, J} (a, b, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q^{L, I, J}_{α β} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a), = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v, = - \partial_{b^{μ}} [f (b) A^{κ μ}_{L} P^{L}_{M} \partial_{b^{κ}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z), F = E^{A}_{m} f .

(In (eq:K8) the inner tensor index is written $κ$ to keep it distinct from the multi-index $α$ of (eq:Q); the source writes $α$ for both.) The cross-line operator of the patch is recalled for Lemma lem:index:

C_{cross} = π^{*} (H^{'}_{γ γ} \otimes H^{'}_{ϕϕ} - H_{γ γ} \otimes H_{ϕϕ}), π (a, b, z, y) = (y - a, y - b) .

Displayed formulas in this passage

XV-F-patch-ii@1#eq:retained · line 597
$K_{ret} = K_{47} + K_{5} + K_{6} + K_{8} + K_{9}, K_{5} = K_{5}^{[a]} + K_{5}^{[z]}, K_{supp} = K_{6} + K_{8} + K_{9} .$
XV-F-patch-ii@1#eq:L · line 603
$L^{μ, I} (s, a) = σ_{ε} (a) A^{λ μ, I} \partial_{a^{λ}} R (s - a) .$
XV-F-patch-ii@1#eq:K5a · line 610
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],$
XV-F-patch-ii@1#eq:K5z · line 610
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],$
XV-F-patch-ii@1#eq:K47 · line 610
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],$
XV-F-patch-ii@1#eq:K6 · line 610
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],$
XV-F-patch-ii@1#eq:Dfrom · line 630
$a_{2} ˙^{α μ} [G; h] = D^{α μ, I, J} h_{I} G_{J},$
XV-F-patch-ii@1#eq:Q · line 637
$Q^{L, I, J} (v; a, z) T^{L, I, J} (b; a, z) K_{8}^{I, J} (a, b, z) K_{9}^{I, J} (a, b, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q^{L, I, J}_{α β} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a), = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v, = - \partial_{b^{μ}} [f (b) A^{κ μ}_{L} P^{L}_{M} \partial_{b^{κ}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z), F = E^{A}_{m} f .$
XV-F-patch-ii@1#eq:T · line 637
$Q^{L, I, J} (v; a, z) T^{L, I, J} (b; a, z) K_{8}^{I, J} (a, b, z) K_{9}^{I, J} (a, b, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q^{L, I, J}_{α β} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a), = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v, = - \partial_{b^{μ}} [f (b) A^{κ μ}_{L} P^{L}_{M} \partial_{b^{κ}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z), F = E^{A}_{m} f .$
XV-F-patch-ii@1#eq:K8 · line 637
$Q^{L, I, J} (v; a, z) T^{L, I, J} (b; a, z) K_{8}^{I, J} (a, b, z) K_{9}^{I, J} (a, b, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q^{L, I, J}_{α β} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a), = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v, = - \partial_{b^{μ}} [f (b) A^{κ μ}_{L} P^{L}_{M} \partial_{b^{κ}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z), F = E^{A}_{m} f .$
XV-F-patch-ii@1#eq:K9 · line 637
$Q^{L, I, J} (v; a, z) T^{L, I, J} (b; a, z) K_{8}^{I, J} (a, b, z) K_{9}^{I, J} (a, b, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q^{L, I, J}_{α β} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a), = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v, = - \partial_{b^{μ}} [f (b) A^{κ μ}_{L} P^{L}_{M} \partial_{b^{κ}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z), F = E^{A}_{m} f .$
XV-F-patch-ii@1#eq:cross · line 658
$C_{cross} = π^{*} (H^{'}_{γ γ} \otimes H^{'}_{ϕϕ} - H_{γ γ} \otimes H_{ϕϕ}), π (a, b, z, y) = (y - a, y - b) .$

Nearby in this section

proposition: Proposition: Recalled from XV-F-patch