Manuscript Em02-D@4 · Em02-D-v4.tex · section · fcf2fac7

Seed adjoints

Exact response and seed assembly › Seed adjoints · lines 691–767

Em02-D@4#eq:47-adjoint, Em02-D@4#eq:5a-adjoint, Em02-D@4#eq:5z-adjoint, Em02-D@4#eq:first-action

The substitutions for the integrated vertex are

M_{47} = M_{δ, 2} (a, s, s - t), M_{5 a} = M_{δ, 2} (a, s, - t),

M_{5 z} = M_{δ, 2} (c - t, s + t, t) .

Their exact actions are

⟨ K_{47}^{22}, M_{δ, 2} ⟩ ⟨ K_{5 a}^{22}, M_{δ, 2} ⟩ ⟨ K_{5 z}^{22}, M_{δ, 2} ⟩ = - \int d^{4} a d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}} (\partial_{s^{0}} + \partial_{t^{0}}) M_{47}, = \int d^{4} a d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}}^{2} [f (a + s) \partial_{s^{0}} M_{5 a}], = \int d^{4} c d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}}^{2} [f (c + s) \partial_{s^{0}} M_{5 z}] .

For $47$ , the outer transfer gives a minus sign and the two line transfers give positive combined sign. For each $5$ row,

⟨ \partial_{s^{0}} [f R^{'} (s) R^{''} (t)], M ⟩ = - ⟨ R^{'} (s) R^{''} (t), f \partial_{s^{0}} M ⟩ = ⟨ R (s) R (t), \partial_{s^{0}} \partial_{t^{0}}^{2} (f \partial_{s^{0}} M)⟩ .

These are identities before cone restriction.

The derivative conversions are

Seed	Substitution	Temporal derivatives on $M$
$47$	$M (a, s, s - t)$	$\partial_{s^{0}} = D + \partial_{r^{0}}$ , $\partial_{t^{0}} = - \partial_{r^{0}}$
$5 a$	$M (a, s, - t)$	$\partial_{s^{0}} = D$ , $\partial_{t^{0}} = - \partial_{r^{0}}$
$5 z$	$M (c - t, s + t, t)$	$\partial_{s^{0}} = D$ , $\partial_{t^{0}} = - \partial_{a^{0}} + D + \partial_{r^{0}}$

The first component contains only row $9 m$ :

⟨ K_{ret}^{11}, M_{δ, 1} ⟩ = - \int d^{4} a d R (S) [f (M_{δ, 1} + \partial_{r^{1}} M_{δ, 1})]_{r = 0} .

Multiplication by $- i$ gives the corresponding contribution to $t_{δ}^{ζ}$ .

Displayed formulas in this passage

Em02-D@4#eq:47-adjoint · line 702
$⟨ K_{47}^{22}, M_{δ, 2} ⟩ ⟨ K_{5 a}^{22}, M_{δ, 2} ⟩ ⟨ K_{5 z}^{22}, M_{δ, 2} ⟩ = - \int d^{4} a d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}} (\partial_{s^{0}} + \partial_{t^{0}}) M_{47}, = \int d^{4} a d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}}^{2} [f (a + s) \partial_{s^{0}} M_{5 a}], = \int d^{4} c d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}}^{2} [f (c + s) \partial_{s^{0}} M_{5 z}] .$
Em02-D@4#eq:5a-adjoint · line 702
$⟨ K_{47}^{22}, M_{δ, 2} ⟩ ⟨ K_{5 a}^{22}, M_{δ, 2} ⟩ ⟨ K_{5 z}^{22}, M_{δ, 2} ⟩ = - \int d^{4} a d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}} (\partial_{s^{0}} + \partial_{t^{0}}) M_{47}, = \int d^{4} a d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}}^{2} [f (a + s) \partial_{s^{0}} M_{5 a}], = \int d^{4} c d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}}^{2} [f (c + s) \partial_{s^{0}} M_{5 z}] .$
Em02-D@4#eq:5z-adjoint · line 702
$⟨ K_{47}^{22}, M_{δ, 2} ⟩ ⟨ K_{5 a}^{22}, M_{δ, 2} ⟩ ⟨ K_{5 z}^{22}, M_{δ, 2} ⟩ = - \int d^{4} a d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}} (\partial_{s^{0}} + \partial_{t^{0}}) M_{47}, = \int d^{4} a d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}}^{2} [f (a + s) \partial_{s^{0}} M_{5 a}], = \int d^{4} c d R (s) d R (t) \partial_{s^{0}} \partial_{t^{0}}^{2} [f (c + s) \partial_{s^{0}} M_{5 z}] .$
Em02-D@4#eq:first-action · line 757
$⟨ K_{ret}^{11}, M_{δ, 1} ⟩ = - \int d^{4} a d R (S) [f (M_{δ, 1} + \partial_{r^{1}} M_{δ, 1})]_{r = 0} .$
unlabeled · line 694
$M_{47} = M_{δ, 2} (a, s, s - t), M_{5 a} = M_{δ, 2} (a, s, - t),$
unlabeled · line 698
$M_{5 z} = M_{δ, 2} (c - t, s + t, t) .$
unlabeled · line 725
$⟨ \partial_{s^{0}} [f R^{'} (s) R^{''} (t)], M ⟩ = - ⟨ R^{'} (s) R^{''} (t), f \partial_{s^{0}} M ⟩ = ⟨ R (s) R (t), \partial_{s^{0}} \partial_{t^{0}}^{2} (f \partial_{s^{0}} M)⟩ .$