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The free Weyl algebra obeys time slice

Statement

Assume Countable Choice and Dependent Choice. Let $ι : U \to M$ be a Cauchy morphism of Green-data backgrounds: it is an orientation-preserving, time-orientation-preserving isometric embedding with causally convex open image, compatible scalar operators, and image containing a Cauchy hypersurface of $M$ . Assume the Green maps and Cauchy uniqueness in Green-hyperbolic free scalar interface. Then the induced algebraic Weyl map $α_{ι} : W (U) \to W (M)$ is surjective. If the induced symplectic quotient map $S_{ι} : V (U) \to V (M)$ is injective, then $α_{ι}$ is an isomorphism.

Statement references

Green-hyperbolic free scalar interface

Algebraic free Weyl net on Green backgrounds

Green morphisms induce algebraic Weyl maps

Compact test classes localize to a Cauchy neighborhood

Facts & Assumptions

Given: Countable Choice, Dependent Choice, a Cauchy morphism $ι : U \to M$ with its Green-data interfaces, and the induced quotient map $S_{ι}$ .

[A1]

Assume ${A C}_{ω}$ and Dependent Choice, with the exact principles of 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.

[F1]

Algebraic free Weyl net on Green backgrounds: $W (M)$ has formal basis $W_{M} (v)$ indexed by $v \in V (M)$ , and its elements are finite linear combinations of those basis symbols.

[F2]

Compact test classes localize to a Cauchy neighborhood: every target class $[f] \in V (M)$ has a representative $[ι_{*} g]$ with $g \in D_{g} (U)$ .

[F3]

Green morphisms induce algebraic Weyl maps: $α_{ι} (W_{U} (v)) = W_{M} (S_{ι} v)$ is a unital $*$ -homomorphism; $S_{ι}$ is symplectic.

Proof

1.1A1F1F2F3givenalgebra

Let $W_{M} ([f])$ be any basis generator of $W (M)$ . By [F2] there is $g \in D_{g} (U)$ with $[f] = [ι_{*} g]$ . Hence $W_{M} ([f]) = W_{M} (S_{ι} [g]) = α_{ι} (W_{U} ([g]))$ by [F3]. Every element of $W (M)$ is a finite linear combination of basis generators by [F1], so every element has a preimage under $α_{ι}$ . This proves surjectivity.

2.1A1F1F3step 1.1givenalgebra∎

Suppose $S_{ι}$ is injective. If $\sum_{j = 1}^{n} c_{j} W_{U} (v_{j})$ maps to zero, first combine any repeated $v_{j}$ so the remaining $v_{j}$ are distinct. Their images $S_{ι} v_{j}$ are then distinct, and $α_{ι}$ sends the sum to $\sum_{j = 1}^{n} c_{j} W_{M} (S_{ι} v_{j})$ . The latter can be zero only if every $c_{j} = 0$ , since the $W_{M} (w)$ are a formal vector-space basis by [F1]. Thus $α_{ι}$ is injective. Together with step 1.1 it is bijective; the inverse of a bijective algebra homomorphism preserves products, the unit and involution, so it is an isomorphism.

Remarks

The Weyl algebra here is the explicit algebraic $*$ -algebra of Algebraic free Weyl net on Green backgrounds. No $C^{*}$ -norm or completion is asserted. Reeh–Schlieder and local factor type remain nonlogical context because their source proof requires an edge-of-the-wedge uniqueness input not supplied by this pair.

The free Weyl algebra obeys time slice

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources