Manuscript VI@1 · VI.tex · section · b5b60afb

Conventions

Action, signs, background, and coefficient domain › Conventions · lines 326–375

VI@1#eq:DK, VI@1#eq:L, VI@1#eq:pure-gauge-operator

Let

M = R^{1 + 3}, η = diag (- 1, 1, 1, 1), □_{η} = - \partial_{t}^{2} + Δ, P = □_{η} - m^{2}, m > 0.

Fix a finite time $T > 0$ , an initial surface $t = 0$ , and the slab

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

All fields are smooth on a neighborhood of the slab. Compact test functions used for observations and interactions have support in its interior.

The metric is denoted by $γ$ ; the letter $g$ denotes only an interaction switch.

For a symmetric tensor $a$ , write

a ˉ_{μν} (K ξ)_{μν} = a_{μν} - \frac{1}{2} η_{μν} tr_{η} a, = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, D (a)^{ν} X_{ξ} = \partial_{μ} a ˉ^{μν}, = ξ^{μ} \partial_{μ} .

Then

D (K ξ)^{μ} = □_{η} ξ^{μ} .

The scalar first-variation operator is

L_{a} = - a^{μν} \partial_{μ} \partial_{ν} - D (a)^{ν} \partial_{ν} .

In particular,

L_{K ξ} = - [P, X_{ξ}] .

These identities follow by expanding the derivatives.

Displayed formulas in this passage

unlabeled · line 348
$a ˉ_{μν} (K ξ)_{μν} = a_{μν} - \frac{1}{2} η_{μν} tr_{η} a, = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, D (a)^{ν} X_{ξ} = \partial_{μ} a ˉ^{μν}, = ξ^{μ} \partial_{μ} .$
VI@1#eq:DK · line 359
$D (K ξ)^{μ} = □_{η} ξ^{μ} .$
VI@1#eq:L · line 364
$L_{a} = - a^{μν} \partial_{μ} \partial_{ν} - D (a)^{ν} \partial_{ν} .$
VI@1#eq:pure-gauge-operator · line 370
$L_{K ξ} = - [P, X_{ξ}] .$
unlabeled · line 329
$M = R^{1 + 3}, η = diag (- 1, 1, 1, 1), □_{η} = - \partial_{t}^{2} + Δ, P = □_{η} - m^{2}, m > 0.$
unlabeled · line 337
$M_{T} = [0, T] \times R^{3} .$