Statement
A group-specific spin-off of Theme 4 of Goto et al., which flags the Tsirelson problem — whether the set of quantum correlations defined by tensor-product measurements equals that defined by commuting-operator measurements — as the sharp infinite-dimensional subtlety behind nonlocal games (resolved negatively at the operator-algebra level by ). Question: does this distinction affect the group’s Bell-CHSH results in QFT, where Alice’s and Bob’s observables live in commuting wedge/diamond von Neumann algebras that do not tensor-factorize?
Why It Matters
The group’s Bell-CHSH-in-QFT program is built on Weyl operators in commuting spacelike algebras with modular-conjugation-related observables — i.e. it lives natively on the commuting-operator side of the Tsirelson dichotomy, not the tensor-product side that finite-dimensional Bell intuition assumes. Whether the Tsirelson gap constrains the achievable violation (or its decay) is the algebraic crux of the type-III₁ Bell chapter (OQP #1/26/32) and bears on the β(L) decay law (OQP #12).
What We Know
- Local QFT algebras satisfy microcausality (spacelike-commuting), and the vacuum is cyclic/separating (Reeh–Schlieder) — the setting is commuting-operator, and generically does not admit a Hilbert-space tensor factorization across the cut.
- Summers–Werner: the vacuum maximally violates Bell-CHSH () on suitable wedge algebras — a commuting-algebra statement.
- The group achieves numerical values – (bumpified Haar wavelets) approaching Tsirelson; these are commuting-algebra constructions.
- Tsirelson equality holds in finite dimensions; the gap is an infinite-dimensional/operator-algebraic phenomenon.
Possible Approaches
- Make precise whether any tensor-factorized model could reproduce the vacuum wedge-algebra CHSH value, or whether the commuting-operator value is strictly a III₁ artifact.
- Test whether the commuting-vs-tensor distinction changes the scaling of the maximal violation with spacelike separation — feeding long-range-bell-decay.
- Connect to the split property / nuclearity (which does supply an intermediate type-I factor and a tensor structure) as the bridge between the two sides.