Manuscript XIII-B@1 · XIII-B.tex · section · b8a5c0f5

The sheet and the inherited coefficient

Prerequisite sheet and convention audit › The sheet and the inherited coefficient · lines 388–440

XIII-B@1#eq:O, XIII-B@1#eq:slots

Use

M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2}, m \geq 0,

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

The fixed smooth ramp $σ_{ε}$ is zero for $t \leq 0$ and one for $t \geq ε$ , with $0 < ε < T$ . The switch $g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1])$ is external. All configurations are ambient smooth multiplets; spatial decay is not required.

The operators are

P r = r - \frac{1}{2} η tr_{η} r, D^{ν} r = \partial_{μ} (P r)^{μν}, (K v)_{μν} = \partial_{μ} v_{ν} + \partial_{ν} v_{μ}, R = K /2.

Parentheses in indexed expressions include $\frac{1}{2}$ . The complete free shell is

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

The auxiliary field remains independent off shell.

The named composite is

O = Φ^{(2)}_{D} + A_{1} [f, g] (r_{g} [ϕ]; ϕ), Φ^{(2)}_{D} = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D},

with

r_{D} = - u_{D}, r_{g} [ϕ] = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]) .

It is an even, ghost-number-zero homogeneous cubic polynomial: its sectors have degrees $(2, 1)$ and $(0, 3)$ in $(γ, ϕ)$ . Its compact polarized kernel has the covector-sum-zero bound $Z_{3}$ proved in XII-D and XII-E XIID,XIIE. Its metric derivatives include the admitted insertion's metric slots. Nonlocality is not removed by suppressing those derivatives.

Displayed formulas in this passage

XIII-B@1#eq:O · line 421
$O = Φ^{(2)}_{D} + A_{1} [f, g] (r_{g} [ϕ]; ϕ), Φ^{(2)}_{D} = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D},$
XIII-B@1#eq:slots · line 429
$r_{D} = - u_{D}, r_{g} [ϕ] = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ]) .$
unlabeled · line 391
$M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2}, m \geq 0,$
unlabeled · line 397
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3} .$
unlabeled · line 406
$P r = r - \frac{1}{2} η tr_{η} r, D^{ν} r = \partial_{μ} (P r)^{μν}, (K v)_{μν} = \partial_{μ} v_{ν} + \partial_{ν} v_{μ}, R = K /2.$
unlabeled · line 414
$C_{sol} = {□ γ = 0, P_{m} ϕ = 0, □ c = □ c ˉ = 0, b = D γ} .$