Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

One regulator, fixed before all products

Convention sheet and frozen data › One regulator, fixed before all products · lines 476–523

X-A-conv@1#eq:epsrange, X-A-conv@1#eq:regulator, X-A-conv@1#eq:auxmetric

Define the real even compact mollifier

ϱ (v) = ⎩ ⎨ ⎧ Z^{- 1} exp (- \frac{1}{1 - ∣ v ∣ _{E}^{2}}), 0, ∣ v ∣_{E} < 1, ∣ v ∣_{E} \geq 1, Z = \int_{R^{4}} exp (- \frac{1}{1 - ∣ v ∣ _{E}^{2}}) 1_{{∣ v ∣_{E} < 1}} d^{4} v .

Freeze one number

0 < ε \leq 1/8.

All statements concern this fixed number. Define

ϱ_{ε} (v) = ε^{- 4} ϱ (v / ε), S_{ε} a = ϱ_{ε} * a, T_{g, ε} a = S_{ε} (g a),

componentwise in the displayed background coordinates.

This is a spacetime convolution, not a Lorentz-invariant or diffeomorphism-natural regulator. It regularizes the metric arguments of the coordinate functional. It does not regulate the action, the scalar field argument, or the Cauchy inverse itself.

For a metric jet define the auxiliary metric

q_{g, ε} = η + κ H + κ^{2} J, H = T_{g, ε} h, J = T_{g, ε} k .

The action uses $q$ , not $q_{g, ε}$ . This distinction is part of the prescription.

Displayed formulas in this passage

X-A-conv@1#eq:epsrange · line 492
$0 < ε \leq 1/8.$
X-A-conv@1#eq:regulator · line 497
$ϱ_{ε} (v) = ε^{- 4} ϱ (v / ε), S_{ε} a = ϱ_{ε} * a, T_{g, ε} a = S_{ε} (g a),$
X-A-conv@1#eq:auxmetric · line 513
$q_{g, ε} = η + κ H + κ^{2} J, H = T_{g, ε} h, J = T_{g, ε} k .$
unlabeled · line 479
$ϱ (v) = ⎩ ⎨ ⎧ Z^{- 1} exp (- \frac{1}{1 - ∣ v ∣ _{E}^{2}}), 0, ∣ v ∣_{E} < 1, ∣ v ∣_{E} \geq 1, Z = \int_{R^{4}} exp (- \frac{1}{1 - ∣ v ∣ _{E}^{2}}) 1_{{∣ v ∣_{E} < 1}} d^{4} v .$