Manuscript XI-D@1 · XI-D.tex · section · 994a5c1b

Geometry, regulator, and fields

Prerequisite and convention sheet › Geometry, regulator, and fields · lines 327–379

Retain the supplied conventions:

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

η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.

All coefficient fields are smooth on a neighbourhood of the closed slab. Parentheses on indices include the factor $1/2$ .

Fix $0 < ε < T$ and

σ_{ε} (t) = χ (t / ε), χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.

Fix

g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), P_{g} = int {g = 1} \neq = \emptyset,

and an open relatively compact $U$ with

U \subset P_{g} \cap {ε < t < T}, g_{U} = C_{c}^{\infty} (U; T M) .

The observation test is a real $f \in C_{c}^{\infty} (M_{T}^{\circ})$ . The width is never sent to zero.

For a symmetric tensor $r$ define

P r = r - \frac{1}{2} η tr_{η} r, D^{μ} r = \partial_{ν} (P r)^{ν μ},

(K v)_{μν} = \partial_{μ} v_{ν} + \partial_{ν} v_{μ}, R = \frac{1}{2} K .

The independent multiplet is $(γ, ϕ, c, c ˉ, b)$ with parities $(0, 0, 1, 1, 0)$ and ghost numbers $(0, 0, 1, - 1, 0)$ .

Displayed formulas in this passage

unlabeled · line 330
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, Σ = {t = 0},$
unlabeled · line 335
$η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.$
unlabeled · line 344
$σ_{ε} (t) = χ (t / ε), χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.$
unlabeled · line 355
$g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), P_{g} = int {g = 1} \neq = \emptyset,$
unlabeled · line 360
$U \subset P_{g} \cap {ε < t < T}, g_{U} = C_{c}^{\infty} (U; T M) .$
unlabeled · line 368
$P r = r - \frac{1}{2} η tr_{η} r, D^{μ} r = \partial_{ν} (P r)^{ν μ},$
unlabeled · line 372
$(K v)_{μν} = \partial_{μ} v_{ν} + \partial_{ν} v_{μ}, R = \frac{1}{2} K .$