Manuscript XIII-D@1 · XIII-D.tex · section · bab69024

D0: the route and the unchanged sheet

D0: the route and the unchanged sheet · lines 447–516

XIII-D@1#eq:shell, XIII-D@1#eq:bracket, XIII-D@1#eq:UCP, XIII-D@1#eq:full-remainder

Route A is frozen; Route D is its obstruction alternative.

Use

M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2},

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

The backgrounds $g$ and $σ_{ε}$ are fixed; neither is a spurion. The independent fields are $(γ, ϕ, c, c ˉ, b)$ , with no antifields. The complete shell is

C_{sol} = {□ γ = 0, P_{m} ϕ = 0, □ c = □ c ˉ = 0, b = D γ} .

Fields are ambient smooth configurations. No spatial decay is required. Grassmann-valued identities are interpreted coefficientwise.

Fix $f, g, m, ε, T, W$ before selecting correction maps, with

W \neq = \emptyset, W \subset P_{g} \cap {ε < t < T}, P_{g} = int {g = 1},

and

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

Its bracket is the ordinary vector-field bracket

[ξ, ζ]^{μ} = ξ^{ν} \partial_{ν} ζ^{μ} - ζ^{ν} \partial_{ν} ξ^{μ} .

This is not the free ghost solution space.

Impose the uniform hypothesis

(UCP_{W}) g = 1 near J^{-} (supp f) \cap J^{+} (W) \cap M_{T} .

Then the copied remainder, including the scalar-sourced slot, is

Δ_{ξ} O_{C_{sol}} = B_{ξ}_{C_{sol}} .

The domain of the cancellation target is the complete shell. $C_{W}$ is only a subsequent evaluation set.

The order is classical, $ℏ^{0}$ , at the displayed $κ^{2}$ coefficient. One additional even correction is allowed. No insertion counted by $λ$ is added, and no coefficient of a quantum retarded product is constructed.

Displayed formulas in this passage

XIII-D@1#eq:shell · line 468
$C_{sol} = {□ γ = 0, P_{m} ϕ = 0, □ c = □ c ˉ = 0, b = D γ} .$
XIII-D@1#eq:bracket · line 488
$[ξ, ζ]^{μ} = ξ^{ν} \partial_{ν} ζ^{μ} - ζ^{ν} \partial_{ν} ξ^{μ} .$
XIII-D@1#eq:UCP · line 497
$(UCP_{W}) g = 1 near J^{-} (supp f) \cap J^{+} (W) \cap M_{T} .$
XIII-D@1#eq:full-remainder · line 504
$Δ_{ξ} O_{C_{sol}} = B_{ξ}_{C_{sol}} .$
unlabeled · line 450
$Route A is frozen; Route D is its obstruction alternative.$
unlabeled · line 455
$M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2},$
unlabeled · line 461
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, 0 < ε < T .$
unlabeled · line 478
$W \neq = \emptyset, W \subset P_{g} \cap {ε < t < T}, P_{g} = int {g = 1},$
unlabeled · line 484
$g_{W} = C_{c}^{\infty} (W; T M) .$