Manuscript Em02-A@6 · Em02-A-v6.tex · section · cb2f0a9a

Conventions, defining kernels, and exact adjoints

Conventions, defining kernels, and exact adjoints · lines 1505–1617

Em02-A@6#eq:K5a, Em02-A@6#eq:K5z, Em02-A@6#eq:K47, Em02-A@6#eq:K6, Em02-A@6#eq:response-definition, Em02-A@6#eq:K8, Em02-A@6#eq:K9d, Em02-A@6#eq:K9m

Back to manuscript

Fix arbitrary constant arrays $A^{λ μ, I}$ , $A^{κ μ}_{L}$ , $D^{λ μ, I, J}$ , $P^{L}_{M}$ , $b_{L}$ ; $m \geq 0$ ; smooth $f, g, F, σ$ ; and smooth source coefficients $q_{α β}^{L, I, J}$ for $∣ α ∣ + ∣ β ∣ \leq 2$ . These choices define a mathematical class. They do not instantiate the inherited family.

Put

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

Define

K_{5}^{[a], I, J} K_{5}^{[z], I, J} K_{47}^{I, J} K_{6}^{I, J} = \partial_{b^{μ}} [f (b) σ (a) \partial_{a^{λ}} R (b - a) A^{λ θ, I} \partial_{a^{θ}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ (z) \partial_{z^{λ}} R (b - z) A^{λ θ, J} \partial_{z^{θ}} 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 (a) D^{λ μ, I, J} \partial_{a^{λ}} R (b - a) δ^{(4)} (a - z)] .

In $K_{6}$ , the displayed $a$ derivative acts on $R$ only.

The source

Q^{L, I, J} (v; a, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a)

exists in the independent linear coordinates $(v, v - z, v - a)$ . Its action is

⟨ Q, Ψ ⟩ = α, β \sum \int q_{α β} (v) (\partial_{z}^{α} \partial_{a}^{β} Ψ) (v, v, v) d^{4} v .

The source signs cancel under endpoint transposition.

For $S = R$ or $B$ , use the precise response definition

\int S (b - v) Q (v; a, z) d^{4} v := α, β \sum (- 1)^{∣ α ∣ + ∣ β ∣} \partial_{z}^{α} \partial_{a}^{β} [q_{α β} (a) S (b - a) δ^{(4)} (a - z)] .

The bracket is a seed in the independent coordinates $(a, b - a, z - a)$ .

Set

T^{L, I, J} K_{8}^{I, J} K_{9 d}^{I, J} K_{9 m}^{I, J} = 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}], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} .

All rows are extended independently of $y$ .

Displayed formulas in this passage

Em02-A@6#eq:K5a · line 1525
$K_{5}^{[a], I, J} K_{5}^{[z], I, J} K_{47}^{I, J} K_{6}^{I, J} = \partial_{b^{μ}} [f (b) σ (a) \partial_{a^{λ}} R (b - a) A^{λ θ, I} \partial_{a^{θ}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ (z) \partial_{z^{λ}} R (b - z) A^{λ θ, J} \partial_{z^{θ}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ (a) D^{λ μ, I, J} \partial_{a^{λ}} R (b - a) δ^{(4)} (a - z)] .$
Em02-A@6#eq:K5z · line 1525
$K_{5}^{[a], I, J} K_{5}^{[z], I, J} K_{47}^{I, J} K_{6}^{I, J} = \partial_{b^{μ}} [f (b) σ (a) \partial_{a^{λ}} R (b - a) A^{λ θ, I} \partial_{a^{θ}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ (z) \partial_{z^{λ}} R (b - z) A^{λ θ, J} \partial_{z^{θ}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ (a) D^{λ μ, I, J} \partial_{a^{λ}} R (b - a) δ^{(4)} (a - z)] .$
Em02-A@6#eq:K47 · line 1525
$K_{5}^{[a], I, J} K_{5}^{[z], I, J} K_{47}^{I, J} K_{6}^{I, J} = \partial_{b^{μ}} [f (b) σ (a) \partial_{a^{λ}} R (b - a) A^{λ θ, I} \partial_{a^{θ}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ (z) \partial_{z^{λ}} R (b - z) A^{λ θ, J} \partial_{z^{θ}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ (a) D^{λ μ, I, J} \partial_{a^{λ}} R (b - a) δ^{(4)} (a - z)] .$
Em02-A@6#eq:K6 · line 1525
$K_{5}^{[a], I, J} K_{5}^{[z], I, J} K_{47}^{I, J} K_{6}^{I, J} = \partial_{b^{μ}} [f (b) σ (a) \partial_{a^{λ}} R (b - a) A^{λ θ, I} \partial_{a^{θ}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ (z) \partial_{z^{λ}} R (b - z) A^{λ θ, J} \partial_{z^{θ}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ (a) D^{λ μ, I, J} \partial_{a^{λ}} R (b - a) δ^{(4)} (a - z)] .$
Em02-A@6#eq:response-definition · line 1582
$\int S (b - v) Q (v; a, z) d^{4} v := α, β \sum (- 1)^{∣ α ∣ + ∣ β ∣} \partial_{z}^{α} \partial_{a}^{β} [q_{α β} (a) S (b - a) δ^{(4)} (a - z)] .$
Em02-A@6#eq:K8 · line 1594
$T^{L, I, J} K_{8}^{I, J} K_{9 d}^{I, J} K_{9 m}^{I, J} = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v, = - \partial_{b^{μ}} [f (b) A^{κ μ}_{L} P^{L}_{M} \partial_{b^{κ}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J}], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} .$
Em02-A@6#eq:K9d · line 1594
$T^{L, I, J} K_{8}^{I, J} K_{9 d}^{I, J} K_{9 m}^{I, J} = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v, = - \partial_{b^{μ}} [f (b) A^{κ μ}_{L} P^{L}_{M} \partial_{b^{κ}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J}], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} .$
Em02-A@6#eq:K9m · line 1594
$T^{L, I, J} K_{8}^{I, J} K_{9 d}^{I, J} K_{9 m}^{I, J} = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v, = - \partial_{b^{μ}} [f (b) A^{κ μ}_{L} P^{L}_{M} \partial_{b^{κ}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J}], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} .$
unlabeled · line 1518
$L^{μ, I} (s, a) = σ (a) A^{λ μ, I} \partial_{a^{λ}} R (s - a) .$
unlabeled · line 1562
$Q^{L, I, J} (v; a, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a)$
unlabeled · line 1572
$⟨ Q, Ψ ⟩ = α, β \sum \int q_{α β} (v) (\partial_{z}^{α} \partial_{a}^{β} Ψ) (v, v, v) d^{4} v .$

Conventions, defining kernels, and exact adjoints

Displayed formulas in this passage

Nearby in this section