Manuscript IX-A@1 · IX-A.tex · section · 55aa6f6d

Linear coordinate functionals

The sharp adjoint and an intrinsic boundary trace › Linear coordinate functionals · lines 598–622

IX-A@1#eq:Jcoord, IX-A@1#eq:Cstar

For a compact smooth covector test $a$ supported in $M_{+}$ , define

J (a) [h] = \int a_{μ} (L h)^{μ} d^{4} x .

Let $G_{A}^{0}$ be the advanced inverse of $□_{η}$ , and set

b = G_{A}^{0} a .

The superscript distinguishes this massless inverse from any massive scalar inverse appearing in a later interacting construction.

The formal adjoint of $C$ is

(C^{*} b)^{α β} = - \partial^{(α} b^{β)} + \frac{1}{2} η^{α β} \partial^{μ} b_{μ} .

Displayed formulas in this passage

IX-A@1#eq:Jcoord · line 601
$J (a) [h] = \int a_{μ} (L h)^{μ} d^{4} x .$
IX-A@1#eq:Cstar · line 615
$(C^{*} b)^{α β} = - \partial^{(α} b^{β)} + \frac{1}{2} η^{α β} \partial^{μ} b_{μ} .$
unlabeled · line 608
$b = G_{A}^{0} a .$