Manuscript XIV-B@1 · XIV-B.tex · lemma · 53202eaa

Lemma: Zero-past uniqueness

B2: the free-shell ghost boundary · lines 736–749 · proof included or linked

XIV-B@1#lem:uniq

Let $u$ be a smooth $Λ$ -valued section on a neighbourhood of $M_{T}$ with $□ u = 0$ and $supp u \subset {t \geq t_{1}}$ for some $t_{1} > 0$ . Then $u = 0$ on $M_{T}$ .

Proof. Work coefficientwise in $Λ$ . Choose $t_{0} \in (0, t_{1})$ ; $u$ vanishes on a neighbourhood of ${t = t_{0}}$ , so $u$ and $\partial_{t} u$ vanish there. Uniqueness for the smooth homogeneous Cauchy problem on $[t_{0}, T] \times R^{3}$ gives $u = 0$ there BGP (Ch. 3); for $t \leq t_{0}$ , $u = 0$ by hypothesis.

Nearby in this section

theorem: Theorem: Free-shell ghost sector of
corollary: Corollary: No interacting ghost differential on
remark: Remark: Tests versus configurations
section: B2: the free-shell ghost boundary