Manuscript XX-E@1 · XX-E.tex · section · cf7a8f40

Restoring the identical smeared target

The identical freeze and a necessary smearing correction › Restoring the identical smeared target · lines 451–478

XX-E@1#eq:b, XX-E@1#eq:O

Henceforth $f$ is a fixed XX-A label-density test. Write

L_{f} (z) = \int f ℓ z d t d^{3} x .

Let $b (z, w)$ be the symmetric polarization of $q$ :

b (z, w) = b_{loc} (z, w) - \calRz \cdot \nabla ℓ w - \calRw \cdot \nabla ℓ z .

Define

B_{f} (z, w) = \int f b (z, w) d t d^{3} x .

The full classical functional is

O_{f} = O_{0, f} + s L_{f} (z) + \frac{s ^{2}}{2} (L_{f} (ζ) + B_{f} (z, z)) (mod s^{3}) .

Displayed formulas in this passage

XX-E@1#eq:b · line 459
$b (z, w) = b_{loc} (z, w) - \calRz \cdot \nabla ℓ w - \calRw \cdot \nabla ℓ z .$
XX-E@1#eq:O · line 470
$O_{f} = O_{0, f} + s L_{f} (z) + \frac{s ^{2}}{2} (L_{f} (ζ) + B_{f} (z, z)) (mod s^{3}) .$
unlabeled · line 455
$L_{f} (z) = \int f ℓ z d t d^{3} x .$
unlabeled · line 466
$B_{f} (z, w) = \int f b (z, w) d t d^{3} x .$