Manuscript XIV-E@1 · XIV-E.tex · section · e89fd2a5

Operators and complete retained derivatives

E1: exact calculation of the complete precomposition sum › Operators and complete retained derivatives · lines 543–639

XIV-E@1#eq:operators, XIV-E@1#eq:dotdensity, XIV-E@1#eq:sixbranches

For a smooth metric tensor $r$ , put

a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r,

b_{2} [G] = \frac{1}{8} (tr_{η} G)^{2} - \frac{1}{4} G_{ρ σ} G^{ρ σ},

a_{2}^{μν} [G] = G^{μ}_{ρ} G^{ρ ν} - \frac{1}{2} (tr_{η} G) G^{μν} + b_{2} [G] η^{μν} .

Define

X^{μ} [r] U [r] v V_{1, g} [r] ψ p [r; ψ] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α μ} [r]), = \partial_{α} (σ_{ε} a_{1}^{α β} [r] \partial_{β} v), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ψ) + m^{2} g b_{1} [r] ψ, = - E_{m}^{R} (V_{1, g} [r] ψ) .

Let $G$ be the residual metric direction, $h$ the contracted metric direction, and $ψ$ the contracted scalar direction. These are smooth directions of the classical functional, not values of singular line distributions. Write

X_{G} = X [G], X_{h} = X [h], r_{h} = r_{D}^{'} [G] h = P E_{0}^{R} (J_{D}^{'} [G] h) .

The source $J_{D}$ is the unchanged source of XII-C:

J_{D} [γ] = EL_{γ} (σ_{ε} ℓ_{3} [γ]), ℓ_{3} [γ] = - 2 [ρ^{3}] (- det (η + ρ γ) R [η + ρ γ]) .

The unintegrated curvature representative, and hence every coefficient of $J_{D}^{'} [G] h$ , is fixed as in XII-C. No source coefficient is set to zero by the following notation XIIC,XIIE.

The density derivative is

b ˙_{2} [G; h] a_{2} ˙^{μν} [G; h] = \frac{1}{4} (tr_{η} G) (tr_{η} h) - \frac{1}{2} G_{ρ σ} h^{ρ σ}, = h^{μ}_{ρ} G^{ρ ν} + G^{μ}_{ρ} h^{ρ ν} - \frac{1}{2} ((tr_{η} h) G^{μν} + (tr_{η} G) h^{μν}) + b ˙_{2} [G; h] η^{μν} .

The copied mixed branch derivatives are

M_{4} [G; h, ψ] M_{5} [G; h, ψ] M_{6} [G; h, ψ] M_{7} [G; h, ψ] M_{8} [G; h, ψ] M_{9} [G; h, ψ] = \int f (X_{h}^{ν} \partial_{ν} X_{G}^{μ} + X_{G}^{ν} \partial_{ν} X_{h}^{μ}) \partial_{μ} ψ d^{4} x, = \int f E_{0}^{R} (U [h] X_{G}^{μ} + U [G] X_{h}^{μ}) \partial_{μ} ψ d^{4} x, = \int f E_{0}^{R} (\partial_{α} (σ_{ε} a_{2} ˙^{α μ} [G; h])) \partial_{μ} ψ d^{4} x, = \frac{1}{2} \int f (X_{h}^{μ} X_{G}^{ν} + X_{G}^{μ} X_{h}^{ν}) \partial_{μ} \partial_{ν} ψ d^{4} x, = - \int f X^{μ} [r_{h}] \partial_{μ} ψ d^{4} x, = \int f p [r_{h}; ψ] d^{4} x .

They include all metric-slot choices, source derivatives, and both response legs. Set

M_{ret} [G; h, ψ] = j = 4 \sum^{9} M_{j} [G; h, ψ] .

Displayed formulas in this passage

XIV-E@1#eq:operators · line 563
$X^{μ} [r] U [r] v V_{1, g} [r] ψ p [r; ψ] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α μ} [r]), = \partial_{α} (σ_{ε} a_{1}^{α β} [r] \partial_{β} v), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ψ) + m^{2} g b_{1} [r] ψ, = - E_{m}^{R} (V_{1, g} [r] ψ) .$
XIV-E@1#eq:dotdensity · line 597
$b ˙_{2} [G; h] a_{2} ˙^{μν} [G; h] = \frac{1}{4} (tr_{η} G) (tr_{η} h) - \frac{1}{2} G_{ρ σ} h^{ρ σ}, = h^{μ}_{ρ} G^{ρ ν} + G^{μ}_{ρ} h^{ρ ν} - \frac{1}{2} ((tr_{η} h) G^{μν} + (tr_{η} G) h^{μν}) + b ˙_{2} [G; h] η^{μν} .$
XIV-E@1#eq:sixbranches · line 612
$M_{4} [G; h, ψ] M_{5} [G; h, ψ] M_{6} [G; h, ψ] M_{7} [G; h, ψ] M_{8} [G; h, ψ] M_{9} [G; h, ψ] = \int f (X_{h}^{ν} \partial_{ν} X_{G}^{μ} + X_{G}^{ν} \partial_{ν} X_{h}^{μ}) \partial_{μ} ψ d^{4} x, = \int f E_{0}^{R} (U [h] X_{G}^{μ} + U [G] X_{h}^{μ}) \partial_{μ} ψ d^{4} x, = \int f E_{0}^{R} (\partial_{α} (σ_{ε} a_{2} ˙^{α μ} [G; h])) \partial_{μ} ψ d^{4} x, = \frac{1}{2} \int f (X_{h}^{μ} X_{G}^{ν} + X_{G}^{μ} X_{h}^{ν}) \partial_{μ} \partial_{ν} ψ d^{4} x, = - \int f X^{μ} [r_{h}] \partial_{μ} ψ d^{4} x, = \int f p [r_{h}; ψ] d^{4} x .$
unlabeled · line 546
$a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r,$
unlabeled · line 552
$b_{2} [G] = \frac{1}{8} (tr_{η} G)^{2} - \frac{1}{4} G_{ρ σ} G^{ρ σ},$
unlabeled · line 556
$a_{2}^{μν} [G] = G^{μ}_{ρ} G^{ρ ν} - \frac{1}{2} (tr_{η} G) G^{μν} + b_{2} [G] η^{μν} .$
unlabeled · line 579
$X_{G} = X [G], X_{h} = X [h], r_{h} = r_{D}^{'} [G] h = P E_{0}^{R} (J_{D}^{'} [G] h) .$
unlabeled · line 584
$J_{D} [γ] = EL_{γ} (σ_{ε} ℓ_{3} [γ]), ℓ_{3} [γ] = - 2 [ρ^{3}] (- det (η + ρ γ) R [η + ρ γ]) .$
unlabeled · line 636
$M_{ret} [G; h, ψ] = j = 4 \sum^{9} M_{j} [G; h, ψ] .$