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Causally disjoint free Weyl local algebras commute

Statement

Assume Countable Choice and Dependent Choice. Let $(M, g, N, P, E_{+}, E_{-})$ be a Green-data background of Green-hyperbolic free scalar interface, and let $W_{M} (O)$ be the algebraic local Weyl subalgebra assigned to an open set $O \subseteq M$ by Algebraic free Weyl net on Green backgrounds. If $O_{1} \subseteq O_{2}$ , then $W_{M} (O_{1}) \subseteq W_{M} (O_{2})$ . If $O_{1}$ and $O_{2}$ are causally disjoint, meaning that no point of either set is causally related in either direction to a point of the other, then every element of $W_{M} (O_{1})$ commutes with every element of $W_{M} (O_{2})$ .

Statement references

Green-hyperbolic free scalar interface

Algebraic free Weyl net on Green backgrounds

Causal propagator induces the free scalar symplectic quotient

Facts & Assumptions

Given: Countable Choice, Dependent Choice, a Green-data background on $M$ , and open sets $O_{1}, O_{2} \subseteq M$ .

[A1]

Assume ${A C}_{ω}$ and Dependent Choice as stated in The Axiom of Countable Choice ( ${A C}_{ω}$ ) and The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain for the Green-data and causal-curve suppliers.

[F1]

Green-hyperbolic free scalar interface: for compact real test $g$ , $E g = E_{-} g - E_{+} g$ is smooth and $supp (E g) \subseteq J^{-} (supp g) \cup J^{+} (supp g)$ .

[F2]

Causal propagator induces the free scalar symplectic quotient: $σ_{M} ([f], [g]) = \int_{M} f (E g) μ_{g}$ is the symplectic form on the test-function quotient.

[F3]

Algebraic free Weyl net on Green backgrounds: $W_{M} (O)$ is generated as a unital algebra closed under $*$ by $W_{M} ([f])$ for real compact tests supported in $O$ , and $W_{M} (v) W_{M} (w) = e^{- i σ_{M} (v, w) / 2} W_{M} (v + w)$ .

[F4]

Causal curves, chronological and causal reachability, and inextendibility: $J^{\pm} (K)$ consists of points causally reachable from or to $K$ ; causal disjointness of the two regions makes their test supports disjoint from the causal future and past of each other.

Proof

1.1F3givenalgebra

If $O_{1} \subseteq O_{2}$ , every real compact test supported in $O_{1}$ is also supported in $O_{2}$ . Thus the generating set for $W_{M} (O_{1})$ is included in the generating set for $W_{M} (O_{2})$ , so the generated unital $*$ -subalgebra is included.

2.1A1F1F2F3F4givenalgebra∎

Let $f, g$ be real compact tests supported in $O_{1}, O_{2}$ , respectively. By causal disjointness and [F1], $supp f \cap supp (E g) = \emptyset$ . Therefore [F2] gives $σ_{M} ([f], [g]) = \int_{M} f (E g) μ_{g} = 0$ . The Weyl relation [F3] now gives $W_{M} ([f]) W_{M} ([g]) = W_{M} ([f] + [g]) = W_{M} ([g]) W_{M} ([f]),$ because the same argument gives $σ_{M} ([g], [f]) = 0$ . The adjoints of these generators are $W_{M} (- [f])$ and $W_{M} (- [g])$ , whose test functions have the same supports, so they commute across the two regions as well. Finite sums and products of mutually commuting cross-generators still commute, by distributivity and associativity; these sums and products are exactly the generated algebraic local subalgebras. Hence every pair of elements of the two local algebras commutes.

Remarks

This is an algebraic locality result conditional on the Green-data interface. It constructs no Hilbert-space representation, state or C*-completion. A represented local algebra inherits the same commutation relation under any *-representation where that representation is defined.

Causally disjoint free Weyl local algebras commute

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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