Manuscript XVI-J@1 · XVI-J.tex · section · 6c1f306d

Geometry and fiber pairing

Conventions, inherited inputs, and the selected sector › Geometry and fiber pairing · lines 225–260

XVI-J@1#eq:fiberpair

Work on ambient Minkowski space $M = R^{4}$ , with

η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ.

The symmetric-tensor fiber is

V = Sym^{2} (R^{4})^{*} .

Its fixed nondegenerate bilinear pairing is

b (A, C) = A_{μν} C^{μν} = tr (η A η C) .

All raised indices use $η$ ; all pairings are bilinear, without complex conjugation. Parentheses on indices denote normalized symmetrization.

Write

P A = A - \frac{1}{2} η tr (η A), P^{2} = id, K = - □ P .

The map $P$ is self-adjoint for $b$ .

The scalar test is denoted by $φ$ . The relative-coordinate map will be denoted by $r$ , never by $φ$ .

Displayed formulas in this passage

XVI-J@1#eq:fiberpair · line 238
$b (A, C) = A_{μν} C^{μν} = tr (η A η C) .$
unlabeled · line 228
$η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ.$
unlabeled · line 234
$V = Sym^{2} (R^{4})^{*} .$
unlabeled · line 248
$P A = A - \frac{1}{2} η tr (η A), P^{2} = id, K = - □ P .$