Manuscript X-D@3 · X-D-v3.tex · section · b7774529

Coordinates, bare tests, and completed tests

Frozen tests and regulator-independent Euler coefficients › Coordinates, bare tests, and completed tests · lines 399–478

X-D@3#eq:coordinate-equations, X-D@3#eq:bare, X-D@3#eq:Euler, X-D@3#eq:completed

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The coordinate coefficients have zero Cauchy position and velocity at $Σ$ , and satisfy

□ X_{1, ε}^{ρ} □ X_{2, ε}^{ρ} = - \partial_{μ} (σ_{ε} a^{μ ρ}), = - \partial_{μ} [σ_{ε} (A^{μ ρ} + a^{μν} \partial_{ν} X_{1, ε}^{ρ})] .

These are the coefficient equations of the divergence-form coordinate problem with density

A_{ε} [q] = η^{- 1} + σ_{ε} (- q q^{- 1} - η^{- 1}) .

They agree with retarded propagation of the copied coordinate sources, locally by finite propagation.

Set

B_{ε}^{μ} = X_{1, ε}^{ν} \partial_{ν} X_{1, ε}^{μ} - X_{2, ε}^{μ} .

The bare tests are

c_{1, ε} c_{2, ε} = \partial_{μ} (f X_{1, ε}^{μ}), = - \partial_{μ} (f B_{ε}^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f X_{1, ε}^{μ} X_{1, ε}^{ν}) .

They are actual off-shell tests supported in $K$ .

The positive Euler coefficients are

L_{1, g} u = \partial_{μ} (g a^{μν} \partial_{ν} u) + g m^{2} b u, L_{2, g} u = \partial_{μ} (g A^{μν} \partial_{ν} u) + g m^{2} B u .

They contain $g, h, k$ , never $σ_{ε}$ . The notation $L_{ε}, M_{ε}$ in the limit plan therefore reduces to the fixed operators $L_{1, g}, L_{2, g}$ .

The completed tests are

d_{1, ε} d_{2, ε} = c_{1, ε} - L_{1, g} E_{m}^{A} f, = c_{2, ε} - L_{1, g} E_{m}^{A} c_{1, ε} - L_{2, g} E_{m}^{A} f + L_{1, g} E_{m}^{A} L_{1, g} E_{m}^{A} f .

Equivalently,

e_{ε} = f + κ d_{1, ε} + κ^{2} d_{2, ε} = R_{q} (f + κ c_{1, ε} + κ^{2} c_{2, ε}), R_{q} = (1 + (κ L_{1, g} + κ^{2} L_{2, g}) E_{m}^{A})^{- 1} (mod κ^{3}) .

This is X-C's test-level recursion, not a new map.

The free-field output is

ϕ (c) = [F_{c}], F_{c} (u) = \int c u d^{4} x,

in the scalar algebra $A_{0} = A_{off} / J_{0}$ , where $ϕ (P_{m} v) = 0$ for compact smooth $v$ . The off-shell warning remains: $R_{1} (V; F_{P_{m} v}) = ϕ (- L v)$ need not vanish. Intermediate tests are not replaced by unspecified on-shell representatives.

Displayed formulas in this passage

X-D@3#eq:coordinate-equations · line 403
$□ X_{1, ε}^{ρ} □ X_{2, ε}^{ρ} = - \partial_{μ} (σ_{ε} a^{μ ρ}), = - \partial_{μ} [σ_{ε} (A^{μ ρ} + a^{μν} \partial_{ν} X_{1, ε}^{ρ})] .$
X-D@3#eq:bare · line 428
$c_{1, ε} c_{2, ε} = \partial_{μ} (f X_{1, ε}^{μ}), = - \partial_{μ} (f B_{ε}^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f X_{1, ε}^{μ} X_{1, ε}^{ν}) .$
X-D@3#eq:Euler · line 439
$L_{1, g} u = \partial_{μ} (g a^{μν} \partial_{ν} u) + g m^{2} b u, L_{2, g} u = \partial_{μ} (g A^{μν} \partial_{ν} u) + g m^{2} B u .$
X-D@3#eq:completed · line 450
$d_{1, ε} d_{2, ε} = c_{1, ε} - L_{1, g} E_{m}^{A} f, = c_{2, ε} - L_{1, g} E_{m}^{A} c_{1, ε} - L_{2, g} E_{m}^{A} f + L_{1, g} E_{m}^{A} L_{1, g} E_{m}^{A} f .$
unlabeled · line 415
$A_{ε} [q] = η^{- 1} + σ_{ε} (- q q^{- 1} - η^{- 1}) .$
unlabeled · line 423
$B_{ε}^{μ} = X_{1, ε}^{ν} \partial_{ν} X_{1, ε}^{μ} - X_{2, ε}^{μ} .$
unlabeled · line 458
$e_{ε} = f + κ d_{1, ε} + κ^{2} d_{2, ε} = R_{q} (f + κ c_{1, ε} + κ^{2} c_{2, ε}), R_{q} = (1 + (κ L_{1, g} + κ^{2} L_{2, g}) E_{m}^{A})^{- 1} (mod κ^{3}) .$
unlabeled · line 467
$ϕ (c) = [F_{c}], F_{c} (u) = \int c u d^{4} x,$

Nearby in this section

lemma: Lemma: Reduction to the bare tests