Manuscript XII-F@1 · XII-F.tex · section · 54cb3169

F1: the sheet and the four-row ledger

F1: the sheet and the four-row ledger · lines 465–542

XII-F@1#eq:first-completion, XII-F@1#eq:roots

Use

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

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

The inherited ramp $σ_{ε}$ is smooth, zero for $t \leq 0$ , and one for $t \geq ε$ , with $0 < ε < T$ . The switch and parameter space are

g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), U \subset P_{g} \cap {ε < t < T}, P_{g} = int {g = 1}, g_{U} = C_{c}^{\infty} (U; T M) .

Fields are ambient smooth configurations; no decay is required. All tensor pairings use the flat Lorentz identifications. Parentheses include $\frac{1}{2}$ .

For a symmetric covariant tensor $r$ ,

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}, (D^{*} v)_{μν} = - \partial_{(μ} v_{ν)} + \frac{1}{2} η_{μν} \partial_{ρ} v^{ρ} .

Define

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

The inherited first completion is

X_{1}^{μ} [r] V_{1, g} [r] ϕ p_{1} [r; ϕ] A_{1} [f, g] (r; ϕ) = - E_{0}^{R} \partial_{λ} (σ_{ε} a_{1}^{λ μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ϕ) + m^{2} g b_{1} [r] ϕ, = - E_{m}^{R} V_{1, g} [r] ϕ, = \int f (p_{1} [r; ϕ] - X_{1} [r] \cdot \partial ϕ) d^{4} x .

In particular, $\partial g$ remains inside the scalar divergence.

The root notation is fixed by

A_{1} [f, g] (r; ϕ) = \int α_{f, g} [ϕ] \cdot r d^{4} z, β_{f, g} [ϕ] = - E_{0}^{A} (P α_{f, g} [ϕ]) .

These are XII-D's roots, not new choices. The raw advanced root can extend below $t = 0$ ; multiplication by the compact parameter in $U$ makes all pairings used here compact without a new boundary prescription.

For a contravariant density $v$ ,

(L_{ξ}^{dens} v)^{μν} = ξ^{ρ} \partial_{ρ} v^{μν} - v^{ρ ν} \partial_{ρ} ξ^{μ} - v^{μ ρ} \partial_{ρ} ξ^{ν} + (\partial_{ρ} ξ^{ρ}) v^{μν} .

The Lie derivative of $β$ is the ordinary covariant tensor Lie derivative.

Displayed formulas in this passage

XII-F@1#eq:first-completion · line 506
$X_{1}^{μ} [r] V_{1, g} [r] ϕ p_{1} [r; ϕ] A_{1} [f, g] (r; ϕ) = - E_{0}^{R} \partial_{λ} (σ_{ε} a_{1}^{λ μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ϕ) + m^{2} g b_{1} [r] ϕ, = - E_{m}^{R} V_{1, g} [r] ϕ, = \int f (p_{1} [r; ϕ] - X_{1} [r] \cdot \partial ϕ) d^{4} x .$
XII-F@1#eq:roots · line 521
$A_{1} [f, g] (r; ϕ) = \int α_{f, g} [ϕ] \cdot r d^{4} z, β_{f, g} [ϕ] = - E_{0}^{A} (P α_{f, g} [ϕ]) .$
unlabeled · line 469
$M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2},$
unlabeled · line 473
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3} .$
unlabeled · line 478
$g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), U \subset P_{g} \cap {ε < t < T}, P_{g} = int {g = 1}, g_{U} = C_{c}^{\infty} (U; T M) .$
unlabeled · line 489
$P r = r - \frac{1}{2} η tr_{η} r, D^{ν} r = \partial_{μ} (P r)^{μν},$
unlabeled · line 493
$(K v)_{μν} = \partial_{μ} v_{ν} + \partial_{ν} v_{μ}, (D^{*} v)_{μν} = - \partial_{(μ} v_{ν)} + \frac{1}{2} η_{μν} \partial_{ρ} v^{ρ} .$
unlabeled · line 500
$a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r .$
unlabeled · line 533
$(L_{ξ}^{dens} v)^{μν} = ξ^{ρ} \partial_{ρ} v^{μν} - v^{ρ ν} \partial_{ρ} ξ^{μ} - v^{μ ρ} \partial_{ρ} ξ^{ν} + (\partial_{ρ} ξ^{ρ}) v^{μν} .$

F1: the sheet and the four-row ledger

Displayed formulas in this passage

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