Project — The β(L) decay law for vacuum Bell-CHSH violation (OQP #12)

Project: The β(L) decay law for vacuum Bell-CHSH violation

Target: IQOQI Open Quantum Problem #12 (Verch) — long-range-bell-decay. The spine of the three-project OQP program.

This is the rare case where a named open problem sits directly on machinery the group has already published. The proved endpoint (Summers–Werner, $L\to 0$) and the group’s finite-$L$ Weyl-operator results are the two ends of the same curve; this project fills in the middle and pins the decay rate.

1. Thesis statement

Establish the functional form and decay rate of the maximal vacuum CHSH violation $\beta (L)$ between two spacelike-separated regions, as a two-sided estimate: a lower bound $\beta (L)\ge 2\sqrt{1+c(L{)}^2}$ from explicit Weyl-operator constructions, and an upper bound $\beta (L)-2\le F(L)$ from nuclearity / split-property estimates. This resolves the quantitative half of Verch’s problem and ties the decay class (exponential vs. power-law) to the mass gap.

2. Why it fits the group (assets already in hand)

• Core method exists — Phys. Rev. D 108, 085026 (2023): Weyl operators + Tomita–Takesaki, with Bob’s observables generated from Alice’s by modular conjugation $J$. This is the engine.
• Compact-region results — Eur. Phys. J. C 85 (2025): Bell-CHSH in causal diamonds, the right geometry for finite separation $L$.
• Optimization toolkit — Phys. Rev. D 112 (2025): the characterized class of bounded Hermitian dichotomic operators that optimize the violation; needed to make $\beta (L)$ a genuine supremum.
• Numerics — Phys. Rev. D 110 (2024): systematic evaluation across mass, dimension, test function.
• Proved endpoint — Summers–Werner: $\beta =2\sqrt{2}$ at region tangency ($L\to 0$), from type III₁ + Reeh–Schlieder.

3. Phased plan (~12 months)

Phase 0 — Precise formulation (2–3 wks)

Define $\beta (L)$ as a supremum over admissible dichotomic pairs in two diamonds at separation $L$. Catalog status: PROVED at $L\to 0$, OPEN for $L>0$. Deliverable: formal problem-statement note → the seeded question long-range-bell-decay.

Phase 1 — Lower bound: the violation persists (4–6 wks)

Use the modular-conjugation construction (PRD 108) and the optimal operator class (PRD 112) to compute ${\beta }_{\text{lower}}(L)$ for two diamonds, extending Eur. Phys. J. C 85 to explicit $L$-dependence. Extract the leading law: massive scalar $\to e^{-mL}$; massless $\to L^{-p}$, with $c(L)$ set by the smeared Pauli–Jordan correlator. Validate with the PRD 110 numerics. Deliverable: explicit lower bound — standalone paper #1, ships regardless of Phase 2. People: the two PhD students (analytics + numerics), silvio-paolo-sorella, itzhak-roditi.

Phase 2 — Upper bound: the hard, novel step (8–12 wks)

Bound the supremum over all observables. Route: Tsirelson’s bound expresses the maximal CHSH value between $\mathcal{A}({\mathcal{O}}_A)$ and $\mathcal{A}({\mathcal{O}}_B)$ through a maximal-correlation / operator-norm quantity; control it with Buchholz–Wichmann–Yngvason nuclearity and the split property. For massive free fields the split property gives exponential clustering $\Rightarrow \beta (L)-2\le C{e}^{-mL}$, matching the Phase-1 class. Deliverable: rigorous upper bound (CMP/LMP-grade if it closes). This is where Rainer Verch (or a Fewster-style AQFT collaborator) is decisive.

Phase 3 — Synthesis & extensions (4–6 wks)

Combine into the $\beta (L)$ decay-law theorem (or sharp two-sided estimate). Extend to: massless (power-law tied to failure of the split property — a clean physical dichotomy), higher spin (reuse the massless-spinor Haar-wavelet machinery, PRD 108 L081701; cf. higher-spin-bell-inequalities), and the Rindler/thermal setting. Deliverable: headline paper answering OQP #12.

4. Go / no-go decision points

• After Phase 1: the lower-bound paper ships unconditionally → guaranteed progress on OQP #12 even if Phase 2 stalls.
• After Phase 2: if nuclearity proves intractable, fall back to “sharp lower bound + strong numerics + conjectured matching upper bound.” Still a citable advance on the named problem.

5. Risks

Risk	Likelihood	Mitigation
Nuclearity/split machinery too heavy	Medium-high	Partner with Verch; Phase 1 is a complete standalone result
Massless case resists	Medium	Scope to massive first; massless as a follow-up
sup-over-all-observables not reducible to a tractable quantity	Medium	Use the PRD 112 operator class to argue near-optimality of the Weyl family, converting the gap into a controlled estimate

6. Targets & people

PRL (headline) + PRD or CMP (long companion). Lead: Marcelo + silvio-paolo-sorella; the two PhD students; external Rainer Verch (decisive for Phase 2), david-dudal optional.

7. Connection to existing wiki questions

• long-range-bell-decay — the open problem this project executes
• higher-spin-bell-inequalities — folded in at Phase 3
• impossible-measurements-qft — whether the optimal observables are operationally realizable

8. Sequencing within the OQP program

The spine. bell-in-type-iii-project (companion, OQP #1/26/32) rides in this project’s slipstream — same Weyl/modular toolkit and people, near-free marginal cost. embezzlement-capacity-project (OQP #48) runs in parallel on a separate track. The one external dependency worth securing early is Verch’s involvement for Phase 2.

9. Suggested next actions

1. Email Verch to warm up the AQFT side before the lower-bound work finishes.
2. Assign Phase 1 analytics to the two PhD students; reuse the PRD 110 numerical code.
3. Draft the Phase 0 formulation note from the PRD 108 / Eur. Phys. J. C 85 results.

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Project plan committed 2026-06-19. Companion to bell-in-type-iii-project; parallel to embezzlement-capacity-project. Part of the bell-inequalities-qft line.