Manuscript XVI-I@1 · XVI-I.tex · section · a7046a8d

Geometry and component conventions

Classical kernel, domains, and exact matching › Geometry and component conventions · lines 287–338

Let

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

and set

M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3} .

Fix

0 < ε < T, m \geq 0, f \in C_{c}^{\infty} (M_{T}^{\circ}; R), g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]) .

The fixed preparation ramp is

σ_{ε} (t) = χ (t / ε), χ (s) = 0 (s \leq 0), χ (s) = 1 (s \geq 1),

with $χ$ real and smooth.

The metric perturbation is a covariant symmetric tensor. Indices are raised only with $η$ . We use a fixed basis of the ten-dimensional symmetric-tensor fiber and its dual, with component indices $I, J, K, L$ . Thus summation over field components counts independent symmetric components once. It does not introduce a second factor for off-diagonal spacetime components.

The fixed Minkowski fiber pairing identifies the tensor bundle and its dual. Trace reversal is

P h = h - \frac{1}{2} η tr_{η} h, P^{2} = id .

Its two-leg kernel matrix, with the fixed fiber identifications understood, is denoted by $P^{I K}$ . All component contractions below use that convention.

The volume density is $d^{4} x$ . Distributional pairings are bilinear, without conjugation.

Displayed formulas in this passage

unlabeled · line 290
$η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2},$
unlabeled · line 298
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3} .$
unlabeled · line 304
$0 < ε < T, m \geq 0, f \in C_{c}^{\infty} (M_{T}^{\circ}; R), g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]) .$
unlabeled · line 310
$σ_{ε} (t) = χ (t / ε), χ (s) = 0 (s \leq 0), χ (s) = 1 (s \geq 1),$
unlabeled · line 327
$P h = h - \frac{1}{2} η tr_{η} h, P^{2} = id .$