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

Conventions, defining kernels, and exact adjoints

Conventions, defining kernels, and exact adjoints · lines 1617–1706

Em02-A@6#eq:seed-charts, Em02-A@6#eq:P47, Em02-A@6#eq:P5a, Em02-A@6#eq:P5z, Em02-A@6#eq:P6, Em02-A@6#eq:P8, Em02-A@6#eq:P9d, Em02-A@6#eq:P9m

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Use the determinant-one independent charts

Γ_{47} (b, s, t, y) Γ_{5 a} (a, s, t, y) Γ_{5 z} (z, s, t, y) Γ_{resp} (a, s, y) = (b - t, b, b - s, y), = (a, a + s, a - t, y), = (z - t, z + s, z, y), = (a, a + s, a, y) .

Let

I_{RR} (H) = \int H (x, s, t, y) d^{4} x d R (s) d R (t) d^{4} y, I_{resp} (H) = \int H (a, s, y) d^{4} a d R (s) d^{4} y .

The base $x$ is $b, a, z$ for the three two-line charts. Write $Ψ_{e} = Ψ \circ Γ_{e}$ and

J_{α, β} Ψ (a, s, y) = (\partial_{z}^{α} \partial_{a}^{β} Ψ) (a, a + s, a, y) .

Ambient derivatives precede restriction.

Put

B_{μ} = \partial_{b^{μ}} + \partial_{s^{μ}} + \partial_{t^{μ}}, h_{47} = σ (b - t) σ (b - s), f_{ρ} = \partial_{ρ} f .

The exact adjoints are

P_{47}^{I, J} [Ψ] = P_{5 a}^{I, J} [Ψ] = P_{5 z}^{I, J} [Ψ] = P_{6}^{I, J} [Ψ] = P_{8}^{I, J} [Ψ] = P_{9 d}^{I, J} [Ψ] = P_{9 m}^{I, J} [Ψ] = - A^{λ μ, J} A^{θ ν, I} \partial_{s^{ν}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}] - A^{λ ρ, J} A^{θ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f_{ρ} (b) B_{μ} Ψ_{47}] + A^{λ ρ, J} A^{θ μ, I} \partial_{s^{ρ}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}], A^{λ θ, I} A^{ρ μ, J} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (a + s) σ (a) σ (a - t) \partial_{s^{μ}} Ψ_{5 a}], A^{λ θ, J} A^{ρ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (z + s) σ (z) σ (z - t) \partial_{s^{μ}} Ψ_{5 z}], - D^{λ μ, I, J} \partial_{s^{λ}} [f (a + s) σ (a) \partial_{s^{μ}} Ψ_{resp}], α, β, L, M \sum q_{α β}^{M, I, J} (a) A^{κ μ}_{L} P^{L}_{M} \frac{s _{κ}}{2} J_{α, β} [f (b) \partial_{b^{μ}} Ψ], α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} A^{μν}_{L} J_{α, β} [g (b) \partial_{b^{μ}} F (b) \partial_{b^{ν}} Ψ], - m^{2} α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} b_{L} J_{α, β} [g (b) F (b) Ψ] .

All source sums satisfy $∣ α ∣ + ∣ β ∣ \leq 2$ .

Displayed formulas in this passage

Em02-A@6#eq:seed-charts · line 1620
$Γ_{47} (b, s, t, y) Γ_{5 a} (a, s, t, y) Γ_{5 z} (z, s, t, y) Γ_{resp} (a, s, y) = (b - t, b, b - s, y), = (a, a + s, a - t, y), = (z - t, z + s, z, y), = (a, a + s, a, y) .$
Em02-A@6#eq:P47 · line 1650
$P_{47}^{I, J} [Ψ] = P_{5 a}^{I, J} [Ψ] = P_{5 z}^{I, J} [Ψ] = P_{6}^{I, J} [Ψ] = P_{8}^{I, J} [Ψ] = P_{9 d}^{I, J} [Ψ] = P_{9 m}^{I, J} [Ψ] = - A^{λ μ, J} A^{θ ν, I} \partial_{s^{ν}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}] - A^{λ ρ, J} A^{θ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f_{ρ} (b) B_{μ} Ψ_{47}] + A^{λ ρ, J} A^{θ μ, I} \partial_{s^{ρ}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}], A^{λ θ, I} A^{ρ μ, J} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (a + s) σ (a) σ (a - t) \partial_{s^{μ}} Ψ_{5 a}], A^{λ θ, J} A^{ρ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (z + s) σ (z) σ (z - t) \partial_{s^{μ}} Ψ_{5 z}], - D^{λ μ, I, J} \partial_{s^{λ}} [f (a + s) σ (a) \partial_{s^{μ}} Ψ_{resp}], α, β, L, M \sum q_{α β}^{M, I, J} (a) A^{κ μ}_{L} P^{L}_{M} \frac{s _{κ}}{2} J_{α, β} [f (b) \partial_{b^{μ}} Ψ], α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} A^{μν}_{L} J_{α, β} [g (b) \partial_{b^{μ}} F (b) \partial_{b^{ν}} Ψ], - m^{2} α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} b_{L} J_{α, β} [g (b) F (b) Ψ] .$
Em02-A@6#eq:P5a · line 1650
$P_{47}^{I, J} [Ψ] = P_{5 a}^{I, J} [Ψ] = P_{5 z}^{I, J} [Ψ] = P_{6}^{I, J} [Ψ] = P_{8}^{I, J} [Ψ] = P_{9 d}^{I, J} [Ψ] = P_{9 m}^{I, J} [Ψ] = - A^{λ μ, J} A^{θ ν, I} \partial_{s^{ν}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}] - A^{λ ρ, J} A^{θ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f_{ρ} (b) B_{μ} Ψ_{47}] + A^{λ ρ, J} A^{θ μ, I} \partial_{s^{ρ}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}], A^{λ θ, I} A^{ρ μ, J} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (a + s) σ (a) σ (a - t) \partial_{s^{μ}} Ψ_{5 a}], A^{λ θ, J} A^{ρ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (z + s) σ (z) σ (z - t) \partial_{s^{μ}} Ψ_{5 z}], - D^{λ μ, I, J} \partial_{s^{λ}} [f (a + s) σ (a) \partial_{s^{μ}} Ψ_{resp}], α, β, L, M \sum q_{α β}^{M, I, J} (a) A^{κ μ}_{L} P^{L}_{M} \frac{s _{κ}}{2} J_{α, β} [f (b) \partial_{b^{μ}} Ψ], α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} A^{μν}_{L} J_{α, β} [g (b) \partial_{b^{μ}} F (b) \partial_{b^{ν}} Ψ], - m^{2} α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} b_{L} J_{α, β} [g (b) F (b) Ψ] .$
Em02-A@6#eq:P5z · line 1650
$P_{47}^{I, J} [Ψ] = P_{5 a}^{I, J} [Ψ] = P_{5 z}^{I, J} [Ψ] = P_{6}^{I, J} [Ψ] = P_{8}^{I, J} [Ψ] = P_{9 d}^{I, J} [Ψ] = P_{9 m}^{I, J} [Ψ] = - A^{λ μ, J} A^{θ ν, I} \partial_{s^{ν}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}] - A^{λ ρ, J} A^{θ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f_{ρ} (b) B_{μ} Ψ_{47}] + A^{λ ρ, J} A^{θ μ, I} \partial_{s^{ρ}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}], A^{λ θ, I} A^{ρ μ, J} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (a + s) σ (a) σ (a - t) \partial_{s^{μ}} Ψ_{5 a}], A^{λ θ, J} A^{ρ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (z + s) σ (z) σ (z - t) \partial_{s^{μ}} Ψ_{5 z}], - D^{λ μ, I, J} \partial_{s^{λ}} [f (a + s) σ (a) \partial_{s^{μ}} Ψ_{resp}], α, β, L, M \sum q_{α β}^{M, I, J} (a) A^{κ μ}_{L} P^{L}_{M} \frac{s _{κ}}{2} J_{α, β} [f (b) \partial_{b^{μ}} Ψ], α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} A^{μν}_{L} J_{α, β} [g (b) \partial_{b^{μ}} F (b) \partial_{b^{ν}} Ψ], - m^{2} α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} b_{L} J_{α, β} [g (b) F (b) Ψ] .$
Em02-A@6#eq:P6 · line 1650
$P_{47}^{I, J} [Ψ] = P_{5 a}^{I, J} [Ψ] = P_{5 z}^{I, J} [Ψ] = P_{6}^{I, J} [Ψ] = P_{8}^{I, J} [Ψ] = P_{9 d}^{I, J} [Ψ] = P_{9 m}^{I, J} [Ψ] = - A^{λ μ, J} A^{θ ν, I} \partial_{s^{ν}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}] - A^{λ ρ, J} A^{θ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f_{ρ} (b) B_{μ} Ψ_{47}] + A^{λ ρ, J} A^{θ μ, I} \partial_{s^{ρ}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}], A^{λ θ, I} A^{ρ μ, J} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (a + s) σ (a) σ (a - t) \partial_{s^{μ}} Ψ_{5 a}], A^{λ θ, J} A^{ρ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (z + s) σ (z) σ (z - t) \partial_{s^{μ}} Ψ_{5 z}], - D^{λ μ, I, J} \partial_{s^{λ}} [f (a + s) σ (a) \partial_{s^{μ}} Ψ_{resp}], α, β, L, M \sum q_{α β}^{M, I, J} (a) A^{κ μ}_{L} P^{L}_{M} \frac{s _{κ}}{2} J_{α, β} [f (b) \partial_{b^{μ}} Ψ], α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} A^{μν}_{L} J_{α, β} [g (b) \partial_{b^{μ}} F (b) \partial_{b^{ν}} Ψ], - m^{2} α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} b_{L} J_{α, β} [g (b) F (b) Ψ] .$
Em02-A@6#eq:P8 · line 1650
$P_{47}^{I, J} [Ψ] = P_{5 a}^{I, J} [Ψ] = P_{5 z}^{I, J} [Ψ] = P_{6}^{I, J} [Ψ] = P_{8}^{I, J} [Ψ] = P_{9 d}^{I, J} [Ψ] = P_{9 m}^{I, J} [Ψ] = - A^{λ μ, J} A^{θ ν, I} \partial_{s^{ν}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}] - A^{λ ρ, J} A^{θ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f_{ρ} (b) B_{μ} Ψ_{47}] + A^{λ ρ, J} A^{θ μ, I} \partial_{s^{ρ}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}], A^{λ θ, I} A^{ρ μ, J} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (a + s) σ (a) σ (a - t) \partial_{s^{μ}} Ψ_{5 a}], A^{λ θ, J} A^{ρ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (z + s) σ (z) σ (z - t) \partial_{s^{μ}} Ψ_{5 z}], - D^{λ μ, I, J} \partial_{s^{λ}} [f (a + s) σ (a) \partial_{s^{μ}} Ψ_{resp}], α, β, L, M \sum q_{α β}^{M, I, J} (a) A^{κ μ}_{L} P^{L}_{M} \frac{s _{κ}}{2} J_{α, β} [f (b) \partial_{b^{μ}} Ψ], α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} A^{μν}_{L} J_{α, β} [g (b) \partial_{b^{μ}} F (b) \partial_{b^{ν}} Ψ], - m^{2} α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} b_{L} J_{α, β} [g (b) F (b) Ψ] .$
Em02-A@6#eq:P9d · line 1650
$P_{47}^{I, J} [Ψ] = P_{5 a}^{I, J} [Ψ] = P_{5 z}^{I, J} [Ψ] = P_{6}^{I, J} [Ψ] = P_{8}^{I, J} [Ψ] = P_{9 d}^{I, J} [Ψ] = P_{9 m}^{I, J} [Ψ] = - A^{λ μ, J} A^{θ ν, I} \partial_{s^{ν}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}] - A^{λ ρ, J} A^{θ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f_{ρ} (b) B_{μ} Ψ_{47}] + A^{λ ρ, J} A^{θ μ, I} \partial_{s^{ρ}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}], A^{λ θ, I} A^{ρ μ, J} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (a + s) σ (a) σ (a - t) \partial_{s^{μ}} Ψ_{5 a}], A^{λ θ, J} A^{ρ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (z + s) σ (z) σ (z - t) \partial_{s^{μ}} Ψ_{5 z}], - D^{λ μ, I, J} \partial_{s^{λ}} [f (a + s) σ (a) \partial_{s^{μ}} Ψ_{resp}], α, β, L, M \sum q_{α β}^{M, I, J} (a) A^{κ μ}_{L} P^{L}_{M} \frac{s _{κ}}{2} J_{α, β} [f (b) \partial_{b^{μ}} Ψ], α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} A^{μν}_{L} J_{α, β} [g (b) \partial_{b^{μ}} F (b) \partial_{b^{ν}} Ψ], - m^{2} α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} b_{L} J_{α, β} [g (b) F (b) Ψ] .$
Em02-A@6#eq:P9m · line 1650
$P_{47}^{I, J} [Ψ] = P_{5 a}^{I, J} [Ψ] = P_{5 z}^{I, J} [Ψ] = P_{6}^{I, J} [Ψ] = P_{8}^{I, J} [Ψ] = P_{9 d}^{I, J} [Ψ] = P_{9 m}^{I, J} [Ψ] = - A^{λ μ, J} A^{θ ν, I} \partial_{s^{ν}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}] - A^{λ ρ, J} A^{θ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f_{ρ} (b) B_{μ} Ψ_{47}] + A^{λ ρ, J} A^{θ μ, I} \partial_{s^{ρ}} \partial_{s^{λ}} \partial_{t^{θ}} [h_{47} f (b) B_{μ} Ψ_{47}], A^{λ θ, I} A^{ρ μ, J} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (a + s) σ (a) σ (a - t) \partial_{s^{μ}} Ψ_{5 a}], A^{λ θ, J} A^{ρ μ, I} \partial_{s^{λ}} \partial_{t^{θ}} \partial_{t^{ρ}} [f (z + s) σ (z) σ (z - t) \partial_{s^{μ}} Ψ_{5 z}], - D^{λ μ, I, J} \partial_{s^{λ}} [f (a + s) σ (a) \partial_{s^{μ}} Ψ_{resp}], α, β, L, M \sum q_{α β}^{M, I, J} (a) A^{κ μ}_{L} P^{L}_{M} \frac{s _{κ}}{2} J_{α, β} [f (b) \partial_{b^{μ}} Ψ], α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} A^{μν}_{L} J_{α, β} [g (b) \partial_{b^{μ}} F (b) \partial_{b^{ν}} Ψ], - m^{2} α, β, L, M \sum q_{α β}^{M, I, J} (a) P^{L}_{M} b_{L} J_{α, β} [g (b) F (b) Ψ] .$
unlabeled · line 1628
$I_{RR} (H) = \int H (x, s, t, y) d^{4} x d R (s) d R (t) d^{4} y, I_{resp} (H) = \int H (a, s, y) d^{4} a d R (s) d^{4} y .$
unlabeled · line 1635
$J_{α, β} Ψ (a, s, y) = (\partial_{z}^{α} \partial_{a}^{β} Ψ) (a, a + s, a, y) .$
unlabeled · line 1643
$B_{μ} = \partial_{b^{μ}} + \partial_{s^{μ}} + \partial_{t^{μ}}, h_{47} = σ (b - t) σ (b - s), f_{ρ} = \partial_{ρ} f .$

Conventions, defining kernels, and exact adjoints

Displayed formulas in this passage

Nearby in this section