Manuscript En06-A@1 · En06-A.tex · theorem · e04c9892

Theorem: Conditional algebraic microcausality

Observable algebras: what commutation actually establishes › The required algebraic interface · lines 1970–1988 · proof included or linked

En06-A@1#thm:algmicro

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Under C6 $^{obs}$ ,

[A (O_{1}), A (O_{2})] = {0}

for bounded regions whose closures are positively spacelike separated.

Proof. For generating limits $a, b$ , choose discrete representatives $a_{n}, b_{n}$ . They commute for all large $n$ . Cross-product preservation gives

[a, b] = n lim j_{n} (a_{n} b_{n} - b_{n} a_{n}) = 0.

Finite polynomials in the two generator families commute. Norm closure extends the result to their generated $C^{*}$ -algebras.

Displayed formulas in this passage

unlabeled · line 1973
$[A (O_{1}), A (O_{2})] = {0}$
unlabeled · line 1983
$[a, b] = n lim j_{n} (a_{n} b_{n} - b_{n} a_{n}) = 0.$

Nearby in this section

hypothesis: Hypothesis: C6 obs : Localized observable limit
section: The required algebraic interface