Manuscript XI-B@1 · XI-B.tex · section · 063b989e

The independent auxiliary block

Causal Green operators and their normalization › The independent auxiliary block · lines 686–711

XI-B@1#eq:Dstar, XI-B@1#eq:Laux

Let $D^{*}$ be the formal adjoint of $D$ . With flat index identifications,

(D^{*} v)_{μν} = - \partial_{(μ} v_{ν)} + \frac{1}{2} η_{μν} \partial_{ρ} v^{ρ} .

Write $A = 2 G_{lin}$ . The even metric–auxiliary operator is

L_{γ b} = (A 2 D 2 D^{*} - 2 id) .

Its source equation is

A γ + 2 D^{*} b = J, 2 D γ - 2 b = j .

Displayed formulas in this passage

XI-B@1#eq:Dstar · line 690
$(D^{*} v)_{μν} = - \partial_{(μ} v_{ν)} + \frac{1}{2} η_{μν} \partial_{ρ} v^{ρ} .$
XI-B@1#eq:Laux · line 697
$L_{γ b} = (A 2 D 2 D^{*} - 2 id) .$
unlabeled · line 707
$A γ + 2 D^{*} b = J, 2 D γ - 2 b = j .$