Manuscript XIV-D@1 · XIV-D.tex · section · 1a7f173d

Every branch derivative displayed

D1: the complete coefficient before joint composition › Every branch derivative displayed · lines 704–794

XIV-D@1#eq:dotcoeff, XIV-D@1#eq:allmixed, XIV-D@1#eq:allscalar

Write

X_{γ} = X [γ], X_{h} = X [h], p_{γ} = p [γ; ψ], p_{h} = p [h; ψ],

and

U [r] q = \partial_{α} (σ_{ε} a_{1}^{α β} [r] \partial_{β} q) .

Set

b ˙_{2} [γ; h] a_{2} ˙^{μν} [γ; h] V ˙_{2} [γ; h] ψ r_{h} = \frac{1}{4} (tr_{η} γ) (tr_{η} h) - \frac{1}{2} γ_{ρ σ} h^{ρ σ}, = h^{μ}_{ρ} γ^{ρ ν} + γ^{μ}_{ρ} h^{ρ ν} - \frac{1}{2} ((tr_{η} h) γ^{μν} + (tr_{η} γ) h^{μν}) + b ˙_{2} [γ; h] η^{μν}, = \partial_{μ} (g a_{2} ˙^{μν} [γ; h] \partial_{ν} ψ) + m^{2} g b ˙_{2} [γ; h] ψ, = r_{D}^{'} [γ] h = 2 D (γ, h) .

Let $\int_{f}$ denote integration against $f$ over the observation variable. The mixed branch derivatives $M_{j} [γ; h, ψ] = (B_{j})_{γ ϕ}^{''} [h, ψ]$ are

M_{1} M_{2} M_{3} M_{4} M_{5} M_{6} M_{7} M_{8} M_{9} M_{10} = - \int_{f} E_{m}^{R} (V_{1, g} [h] p_{γ} + V_{1, g} [γ] p_{h}), = - \int_{f} E_{m}^{R} (V ˙_{2} [γ; h] ψ), = - \int_{f} (X_{h}^{μ} \partial_{μ} p_{γ} + X_{γ}^{μ} \partial_{μ} p_{h}), = \int_{f} (X_{h}^{ν} \partial_{ν} X_{γ}^{μ} + X_{γ}^{ν} \partial_{ν} X_{h}^{μ}) \partial_{μ} ψ, = \int_{f} E_{0}^{R} (U [h] X_{γ}^{μ} + U [γ] X_{h}^{μ}) \partial_{μ} ψ, = \int_{f} E_{0}^{R} (\partial_{α} (σ_{ε} a_{2} ˙^{α μ} [γ; h])) \partial_{μ} ψ, = \frac{1}{2} \int_{f} (X_{h}^{μ} X_{γ}^{ν} + X_{γ}^{μ} X_{h}^{ν}) \partial_{μ} \partial_{ν} ψ, = - \int_{f} X^{μ} [r_{h}] \partial_{μ} ψ, = \int_{f} p [r_{h}; ψ], = 0, M_{11} = 0.

Thus

O^{''}_{γ ϕ} = j = 1 \sum^{11} M_{j} .

For the scalar–scalar block, put $N_{j} = (B_{j})_{ϕϕ}^{''}$ . Its complete ledger is

N_{j} = 0, j = 1, \dots, 9,

N_{10} [ψ_{1}, ψ_{2}] = - 2 \int_{f} (N_{11} [ψ_{1}, ψ_{2}] = 2 \int_{f} (X^{μ} [G (ψ_{1}, ψ_{2})] \partial_{μ} ϕ + X^{μ} [G (ϕ, ψ_{1})] \partial_{μ} ψ_{2} + X^{μ} [G (ϕ, ψ_{2})] \partial_{μ} ψ_{1}), p [G (ψ_{1}, ψ_{2}); ϕ] + p [G (ϕ, ψ_{1}); ψ_{2}] + p [G (ϕ, ψ_{2}); ψ_{1}]) .

Displayed formulas in this passage

XIV-D@1#eq:dotcoeff · line 716
$b ˙_{2} [γ; h] a_{2} ˙^{μν} [γ; h] V ˙_{2} [γ; h] ψ r_{h} = \frac{1}{4} (tr_{η} γ) (tr_{η} h) - \frac{1}{2} γ_{ρ σ} h^{ρ σ}, = h^{μ}_{ρ} γ^{ρ ν} + γ^{μ}_{ρ} h^{ρ ν} - \frac{1}{2} ((tr_{η} h) γ^{μν} + (tr_{η} γ) h^{μν}) + b ˙_{2} [γ; h] η^{μν}, = \partial_{μ} (g a_{2} ˙^{μν} [γ; h] \partial_{ν} ψ) + m^{2} g b ˙_{2} [γ; h] ψ, = r_{D}^{'} [γ] h = 2 D (γ, h) .$
XIV-D@1#eq:allmixed · line 737
$M_{1} M_{2} M_{3} M_{4} M_{5} M_{6} M_{7} M_{8} M_{9} M_{10} = - \int_{f} E_{m}^{R} (V_{1, g} [h] p_{γ} + V_{1, g} [γ] p_{h}), = - \int_{f} E_{m}^{R} (V ˙_{2} [γ; h] ψ), = - \int_{f} (X_{h}^{μ} \partial_{μ} p_{γ} + X_{γ}^{μ} \partial_{μ} p_{h}), = \int_{f} (X_{h}^{ν} \partial_{ν} X_{γ}^{μ} + X_{γ}^{ν} \partial_{ν} X_{h}^{μ}) \partial_{μ} ψ, = \int_{f} E_{0}^{R} (U [h] X_{γ}^{μ} + U [γ] X_{h}^{μ}) \partial_{μ} ψ, = \int_{f} E_{0}^{R} (\partial_{α} (σ_{ε} a_{2} ˙^{α μ} [γ; h])) \partial_{μ} ψ, = \frac{1}{2} \int_{f} (X_{h}^{μ} X_{γ}^{ν} + X_{γ}^{μ} X_{h}^{ν}) \partial_{μ} \partial_{ν} ψ, = - \int_{f} X^{μ} [r_{h}] \partial_{μ} ψ, = \int_{f} p [r_{h}; ψ], = 0, M_{11} = 0.$
XIV-D@1#eq:allscalar · line 779
$N_{10} [ψ_{1}, ψ_{2}] = - 2 \int_{f} (N_{11} [ψ_{1}, ψ_{2}] = 2 \int_{f} (X^{μ} [G (ψ_{1}, ψ_{2})] \partial_{μ} ϕ + X^{μ} [G (ϕ, ψ_{1})] \partial_{μ} ψ_{2} + X^{μ} [G (ϕ, ψ_{2})] \partial_{μ} ψ_{1}), p [G (ψ_{1}, ψ_{2}); ϕ] + p [G (ϕ, ψ_{1}); ψ_{2}] + p [G (ϕ, ψ_{2}); ψ_{1}]) .$
unlabeled · line 707
$X_{γ} = X [γ], X_{h} = X [h], p_{γ} = p [γ; ψ], p_{h} = p [h; ψ],$
unlabeled · line 712
$U [r] q = \partial_{α} (σ_{ε} a_{1}^{α β} [r] \partial_{β} q) .$
unlabeled · line 770
$O^{''}_{γ ϕ} = j = 1 \sum^{11} M_{j} .$
unlabeled · line 776
$N_{j} = 0, j = 1, \dots, 9,$

Nearby in this section

lemma: Lemma: Completeness of the branch ledger