Manuscript Psy02-A@1 · Psy02-A.tex · proposition · d92ce402

Proposition: Ordered correlations need not be a joint law

Static concentration and normalized fluctuations › Ordered expectations for noncommuting families · lines 808–829 · proof included or linked

Psy02-A@1#prop:not-joint

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The ordered expression in (eq:ordered) need not be the characteristic function of a classical joint probability law.

Proof. Take residual vacuum, zero targets, $v_{1} = e$ , $v_{2} = i e$ , with $\norm e = 1$ . Then

f (t, s) = exp (- \frac{ϵ}{2} (t^{2} + s^{2}) - i ϵ t s) .

A joint characteristic function must satisfy $f (- t, - s) = f (t, s)$ . The displayed function violates this identity whenever $sin (ϵ t s) \neq = 0$ .

Displayed formulas in this passage

unlabeled · line 818
$f (t, s) = exp (- \frac{ϵ}{2} (t^{2} + s^{2}) - i ϵ t s) .$

Nearby in this section

section: Ordered expectations for noncommuting families
theorem: Theorem: Ordered Weyl identity and bound
section: Ordered expectations for noncommuting families