Manuscript En05-A@1 · En05-A.tex · theorem · f076e703

Theorem: Conditional algebraic microcausality

A conditional continuum package with matched realizations › Observable algebras require their own interface · lines 1945–1975 · proof included or linked

En05-A@1#thm:microcausality

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Assume C1–C3 and C6. For bounded open regions $O_{1}, O_{2}$ whose closures are spacelike separated at positive margin,

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

Their admitted non-selective operations also obey operational no-signalling.

Proof. Uniform C2, the positive margin, and shrinking block footprints exclude scheduled dependency paths between the two regions at sufficiently small scales. By C6(i), corresponding discrete generators $a_{ε}, b_{ε}$ commute.

Let their comparison images converge to $a, b$ . Asymptotic multiplicativity gives

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

Finite polynomials in each set of generators commute with finite polynomials in the other. Norm continuity of multiplication extends commutation to the generated $C^{*}$ -algebras. Operational no-signalling is Corollary cor:contNS.

Displayed formulas in this passage

unlabeled · line 1949
$[A (O_{1}), A (O_{2})] = {0} .$
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$[a, b] = ε \to 0 lim j_{ε} (a_{ε} b_{ε} - b_{ε} a_{ε}) = 0.$

Theorem: Conditional algebraic microcausality

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