Manuscript Em06-A@1 · Em06-A-v1.tex · assumption · ee9923c6

Model specification: Fixed classical data

The classical object before the limit › Background, scalar equation, switch, and gauge domain · lines 158–195

Em06-A@1#spec:data

Fix

M = R^{1 + 3}, η = diag (1, - 1, - 1, - 1), Σ_{0} = {t = 0},

and a mass $m > 0$ . Set

□ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2} .

Fix $T > 0$ , a real switch

χ \in C_{c}^{\infty} ((0, T) \times R^{3}), 0 \leq χ \leq 1,

and an open set $U$ with compact closure in $(0, T) \times R^{3}$ , such that $χ = 1$ on a neighborhood of $U$ .

The admitted gauge parameters are exactly

g_{U} = C_{c}^{\infty} (U; T M) .

For $ξ \in g_{U}$ , define

(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, X_{ξ} = ξ^{μ} \partial_{μ} .

The metric coefficient $h$ is an arbitrary real smooth symmetric covariant tensor on $M$ . The scalar input $ϕ_{0}$ is a real smooth solution of $P_{m} ϕ_{0} = 0$ . The observation density is

f \in C_{c}^{\infty} ((0, T) \times R^{3}; R) .

All these data are independent of the classicality parameter introduced later.

Displayed formulas in this passage

unlabeled · line 161
$M = R^{1 + 3}, η = diag (1, - 1, - 1, - 1), Σ_{0} = {t = 0},$
unlabeled · line 167
$□ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2} .$
unlabeled · line 171
$χ \in C_{c}^{\infty} ((0, T) \times R^{3}), 0 \leq χ \leq 1,$
unlabeled · line 179
$g_{U} = C_{c}^{\infty} (U; T M) .$
unlabeled · line 183
$(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, X_{ξ} = ξ^{μ} \partial_{μ} .$
unlabeled · line 190
$f \in C_{c}^{\infty} ((0, T) \times R^{3}; R) .$

Nearby in this section

section: Background, scalar equation, switch, and gauge domain