Manuscript IX-E-2@1 · IX-E-2.tex · proposition · e70d9077

Proposition: Explicit nonzero control

A nonzero observation-spacelike control · lines 1044–1107 · proof included or linked

IX-E-2@1#eq:control-tests, IX-E-2@1#eq:control-pairing

There are nonnegative compactly supported tests $f, g$ whose observation supports are strictly spacelike separated and for which $C_{f, g} \neq = 0$ . The calculation alone does not certify double ordering of their actual functional supports.

Proof. Use the bumps (eq:bumps), set $η = 1/20$ , and choose

f (t, x) g (s, y) = β ((t - 1) / η) χ (x / η), = β ((s - 1) / η) χ ((y - e_{1}) / η) .

On their product support,

∣ t - s ∣ \leq \frac{1}{10}, \frac{9}{10} \leq R \leq \frac{11}{10}, t + s \geq \frac{19}{10} .

Thus the whole product support lies strictly inside

∣ t - s ∣ < R < t + s .

No cone distribution meets it.

Choose the unrestricted smooth off-shell configuration

ϕ (t, x) = t, h_{00} (s, y) = α > 0, h_{μν} = 0 for the other components .

Then (eq:commutator) and (eq:imag-regions) give

C_{f, g} [h, ϕ] = - \frac{i α}{8 π} \int_{M_{+}^{2}} \frac{f ( x ) g ( y )}{∣ x - y ∣} d^{4} x d^{4} y \neq = 0.

The integral is finite and strictly positive before multiplication by the nonzero prefactor: the denominator is bounded away from zero and both bumps are nonnegative and nonzero. More explicitly, writing $I$ for the integral,

\frac{10}{11} (\int f) (\int g) \leq I \leq \frac{10}{9} (\int f) (\int g) .

This is an exact pairing with explicit tests, not numerical sampling.

Displayed formulas in this passage

IX-E-2@1#eq:control-tests · line 1053
$f (t, x) g (s, y) = β ((t - 1) / η) χ (x / η), = β ((s - 1) / η) χ ((y - e_{1}) / η) .$
IX-E-2@1#eq:control-pairing · line 1084
$C_{f, g} [h, ϕ] = - \frac{i α}{8 π} \int_{M_{+}^{2}} \frac{f ( x ) g ( y )}{∣ x - y ∣} d^{4} x d^{4} y \neq = 0.$
unlabeled · line 1062
$∣ t - s ∣ \leq \frac{1}{10}, \frac{9}{10} \leq R \leq \frac{11}{10}, t + s \geq \frac{19}{10} .$
unlabeled · line 1070
$∣ t - s ∣ < R < t + s .$
unlabeled · line 1076
$ϕ (t, x) = t, h_{00} (s, y) = α > 0, h_{μν} = 0 for the other components .$
unlabeled · line 1098
$\frac{10}{11} (\int f) (\int g) \leq I \leq \frac{10}{9} (\int f) (\int g) .$

Proposition: Explicit nonzero control

Displayed formulas in this passage

Nearby in this section