Manuscript XI-E@1 · XI-E.tex · section · 46e55514

Copied sheet and frozen conventions

Copied sheet and frozen conventions · lines 321–435

XI-E@1#eq:free, XI-E@1#eq:shell, XI-E@1#eq:L, XI-E@1#eq:B, XI-E@1#eq:rg, XI-E@1#eq:T

Use

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.

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

Fix $0 < ε < T$ and the ramp

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

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

The observation test is real $f \in C_{c}^{\infty} (M_{T}^{\circ})$ . An additional observation-plateau hypothesis will be stated only where it is needed.

For a symmetric tensor $r$ put

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 .

Then $P^{2} = id$ and $D R = \frac{1}{2} □$ .

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

G_{gr}^{R} = - P E_{0}^{R}, s_{0} γ = R c, s_{0} ϕ = s_{0} c = s_{0} b = 0, s_{0} c ˉ = ib .

Here $E_{0}^{R}$ is the zero-initial-data wave inverse on the slab. Its advanced counterpart has zero final data. The scalar equation inverse is $E_{m}^{R}$ ; its action-Hessian inverse is $- E_{m}^{R}$ .

Free configurations satisfy

- □ P γ = 0, b = D γ, P_{m} ϕ = 0, □ c = □ c ˉ = 0.

Ghost Cauchy data are unrestricted.

Define

a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν},

L_{ε} r = X_{1} [r] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α \cdot} [r]) = E_{0}^{R} (σ_{ε} D r - σ_{ε}^{'} a_{1}^{0 \cdot} [r]) .

The inherited symbols are

B_{f} [γ, ϕ] r_{g} [ϕ] T (f, g) [ϕ] = - \int f X_{1} [γ]^{μ} \partial_{μ} ϕ d^{4} x, = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]), = \int f (L_{ε} r_{g} [ϕ])^{μ} \partial_{μ} ϕ d^{4} x,

where

T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} (\partial_{α} ϕ \partial^{α} ϕ - m^{2} ϕ^{2}) .

XI-C's physical one-graviton response of $B_{f}$ is $- T$ , at coupling degree $κ^{2}$ XIC.

The factor in (eq:free) is $- 1$ . Neither $- 2 P E_{0}^{R}$ nor $- 4 P E_{0}^{R}$ is used. VII-C's Hadamard matrix, sharp word, and reflected-residue calculation are not inputs.

Displayed formulas in this passage

XI-E@1#eq:free · line 377
$G_{gr}^{R} = - P E_{0}^{R}, s_{0} γ = R c, s_{0} ϕ = s_{0} c = s_{0} b = 0, s_{0} c ˉ = ib .$
XI-E@1#eq:shell · line 389
$- □ P γ = 0, b = D γ, P_{m} ϕ = 0, □ c = □ c ˉ = 0.$
XI-E@1#eq:L · line 401
$L_{ε} r = X_{1} [r] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α \cdot} [r]) = E_{0}^{R} (σ_{ε} D r - σ_{ε}^{'} a_{1}^{0 \cdot} [r]) .$
XI-E@1#eq:B · line 408
$B_{f} [γ, ϕ] r_{g} [ϕ] T (f, g) [ϕ] = - \int f X_{1} [γ]^{μ} \partial_{μ} ϕ d^{4} x, = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]), = \int f (L_{ε} r_{g} [ϕ])^{μ} \partial_{μ} ϕ d^{4} x,$
XI-E@1#eq:rg · line 408
$B_{f} [γ, ϕ] r_{g} [ϕ] T (f, g) [ϕ] = - \int f X_{1} [γ]^{μ} \partial_{μ} ϕ d^{4} x, = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]), = \int f (L_{ε} r_{g} [ϕ])^{μ} \partial_{μ} ϕ d^{4} x,$
XI-E@1#eq:T · line 408
$B_{f} [γ, ϕ] r_{g} [ϕ] T (f, g) [ϕ] = - \int f X_{1} [γ]^{μ} \partial_{μ} ϕ d^{4} x, = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]), = \int f (L_{ε} r_{g} [ϕ])^{μ} \partial_{μ} ϕ d^{4} x,$
unlabeled · line 325
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, Σ = {t = 0},$
unlabeled · line 330
$η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.$
unlabeled · line 339
$σ_{ε} (t) = χ (t / ε), χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.$
unlabeled · line 350
$g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), P_{g} = int {g = 1} \neq = \emptyset,$
unlabeled · line 355
$U \subset P_{g} \cap {ε < t < T}, g_{U} = C_{c}^{\infty} (U; T M) .$
unlabeled · line 364
$P r = r - \frac{1}{2} η tr_{η} r, D^{μ} r = \partial_{ν} (P r)^{ν μ},$
unlabeled · line 368
$(K v)_{μν} = \partial_{μ} v_{ν} + \partial_{ν} v_{μ}, R = \frac{1}{2} K .$
unlabeled · line 397
$a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν},$
unlabeled · line 421
$T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} (\partial_{α} ϕ \partial^{α} ϕ - m^{2} ϕ^{2}) .$