Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · section · 8ca47088

The fixed positive-width regulator

The fully specified convolution contract › The fixed positive-width regulator · lines 413–446

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

Let

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

Freeze one width

0 < ε \leq \frac{1}{8}, ϱ_{ε} (v) = ε^{- 4} ϱ (v / ε), S_{ε} a = ϱ_{ε} * a, T_{g, ε} a = S_{ε} (g a) .

Convolution is componentwise in these completion coordinates.

The auxiliary metric is

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

The regulator acts on the metric arguments of the coordinate functional. It is not inserted into the action, the scalar argument, or the Cauchy inverse. It is neither Lorentz invariant nor diffeomorphism natural.

Displayed formulas in this passage

X-A-conv-ST@1#eq:regulator · line 426
$0 < ε \leq \frac{1}{8}, ϱ_{ε} (v) = ε^{- 4} ϱ (v / ε), S_{ε} a = ϱ_{ε} * a, T_{g, ε} a = S_{ε} (g a) .$
X-A-conv-ST@1#eq:auxmetric · line 437
$q = η + κ H + κ^{2} J, H = T_{g, ε} h, J = T_{g, ε} k .$
unlabeled · line 416
$ϱ (v) = ⎩ ⎨ ⎧ Z^{- 1} exp (- \frac{1}{1 - ∣ v ∣ _{E}^{2}}), 0, ∣ v ∣_{E} < 1, ∣ v ∣_{E} \geq 1, Z = \int_{∣ v ∣_{E} < 1} exp (- \frac{1}{1 - ∣ v ∣ _{E}^{2}}) d^{4} v .$