Manuscript VII-C@1 · VII-C.tex · section · 8b2ade29

The mixed covariance and its reflected branch

Cauchy covariances and the full tensor audit › The mixed covariance and its reflected branch · lines 692–736

VII-C@1#eq:K

The component needed for the obstruction is

K (x, y) := H_{w h}^{0}_{00} (x, y) .

By (eq:CH),

K (x, y) = E_{0}^{x} \partial_{t} Δ_{0}^{+} (x - y) .

For $r > 0$ ,

E_{0} (e^{- i r t}) = - \frac{i t}{2 r} e^{- i r t} + \frac{i sin ( r t )}{2 r ^{2}} .

Consequently,

K (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4 r} + \frac{i}{8 r ^{2}}) e^{- i r (t - s)} - \frac{i}{8 r ^{2}} e^{i r (t + s)}] .

The terms $r^{- 1}$ and $r^{- 2}$ are locally integrable at $p = 0$ . At large momentum this is an oscillatory distribution with symbol amplitudes.

The last term in (eq:K) is the Cauchy-reflected branch. Its first time frequency is opposite to that of the direct Hadamard branch, while its second time frequency is unchanged. This is the branch responsible for the obstruction.

Reality of $E_{0} C$ and Hermiticity of the fixed covariance imply

H_{h w 00}^{0} (y, x) = K (x, y) .

Displayed formulas in this passage

VII-C@1#eq:K · line 710
$K (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4 r} + \frac{i}{8 r ^{2}}) e^{- i r (t - s)} - \frac{i}{8 r ^{2}} e^{i r (t + s)}] .$
unlabeled · line 695
$K (x, y) := H_{w h}^{0}_{00} (x, y) .$
unlabeled · line 699
$K (x, y) = E_{0}^{x} \partial_{t} Δ_{0}^{+} (x - y) .$
unlabeled · line 703
$E_{0} (e^{- i r t}) = - \frac{i t}{2 r} e^{- i r t} + \frac{i sin ( r t )}{2 r ^{2}} .$
unlabeled · line 733
$H_{h w 00}^{0} (y, x) = K (x, y) .$