Manuscript III@2 · III.tex · theorem · 82dad3e5

Theorem: Nontrivial compact coupled quotient observable

Compact coupled Ward observables › An explicit nonzero compact pair · lines 2179–2242 · proof included or linked

III@2#thm:compact-pair, III@2#eq:positive-observable

The pair $(J, c)$ satisfies (eq:Ward) and defines a nonzero functional on $S_{C}^{red}$ . In particular,

(h, φ) = (0, \frac{Φ}{A _{⋆}}) \in S_{C}

and

O_{J, c} (0, \frac{Φ}{A _{⋆}}) = \frac{2}{A _{⋆}} (\int_{τ_{1}}^{τ_{2}} a (τ) csc^{2} (m τ) d τ) (\int_{R^{3}} b (x) d^{3} x) > 0.

Moreover,

c \in / P C_{c}^{\infty} (M) .

Proof. On the coefficient support,

\partial_{i} Φ = 0, \partial^{0} Φ = A_{⋆} m sin (m τ) .

For $ν = 0$ , the Ward expression is

- 2 a^{'} b + \frac{2 a ^{'} b}{A _{⋆} m sin ( m τ )} A_{⋆} m sin (m τ) = 0.

For spatial $ν$ , both terms vanish. The identity also holds outside the compact support.

The test solution has $h = 0$ and a free scalar with compact data. Its value is

\int_{M} c \frac{Φ}{A _{⋆}} d^{4} x = \frac{2}{A _{⋆} m} (\int a^{'} (τ) cot (m τ) d τ) (\int b d^{3} x) = \frac{2}{A _{⋆}} (\int a (τ) csc^{2} (m τ) d τ) (\int b d^{3} x) .

Integration by parts has no endpoint term. Both factors are positive.

If $c = P ψ$ for compact smooth $ψ$ , then

\int_{M} c \frac{Φ}{A _{⋆}} d^{4} x = \int_{M} ψ P (\frac{Φ}{A _{⋆}}) d^{4} x = 0,

contradicting positivity.

Displayed formulas in this passage

III@2#eq:positive-observable · line 2188
$O_{J, c} (0, \frac{Φ}{A _{⋆}}) = \frac{2}{A _{⋆}} (\int_{τ_{1}}^{τ_{2}} a (τ) csc^{2} (m τ) d τ) (\int_{R^{3}} b (x) d^{3} x) > 0.$
unlabeled · line 2184
$(h, φ) = (0, \frac{Φ}{A _{⋆}}) \in S_{C}$
unlabeled · line 2200
$c \in / P C_{c}^{\infty} (M) .$
unlabeled · line 2207
$\partial_{i} Φ = 0, \partial^{0} Φ = A_{⋆} m sin (m τ) .$
unlabeled · line 2212
$- 2 a^{'} b + \frac{2 a ^{'} b}{A _{⋆} m sin ( m τ )} A_{⋆} m sin (m τ) = 0.$
unlabeled · line 2222
$\int_{M} c \frac{Φ}{A _{⋆}} d^{4} x = \frac{2}{A _{⋆} m} (\int a^{'} (τ) cot (m τ) d τ) (\int b d^{3} x) = \frac{2}{A _{⋆}} (\int a (τ) csc^{2} (m τ) d τ) (\int b d^{3} x) .$
unlabeled · line 2237
$\int_{M} c \frac{Φ}{A _{⋆}} d^{4} x = \int_{M} ψ P (\frac{Φ}{A _{⋆}}) d^{4} x = 0,$