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Further Evidence for Near-Tsirelson Bell-CHSH Violations in Quantum Field Theory via Haar Wavelets

David Dudal, Ken Vandermeersch

TL;DR

The work reframes near-Tsirelson Bell-CHSH violations in the vacuum of a free massless 1+1D spinor QFT as a spectral problem for a family of symmetric matrices built from Haar-wavelet integrals. Step 1 demonstrates an exact wavelet-based construction that yields explicit test functions achieving correlations close to the bound, with the key link being the maximal eigenvalue of $A(N,K)$ approaching $\pi$ in the large-resolution limit for $K=1$, giving a numerical value $\approx 3.1105202$ for the asymptotic maximum. Step 2 then shows how to replace Haar wavelets by smooth bumpified versions via Planck-taper windows, preserving the necessary inner-product relations to arbitrary precision and producing $C^{\infty}$ test functions compatible with Algebraic QFT. The general case $K\ge1$ is discussed through block Toeplitz and matrix-valued Fourier-series analysis, with numerical evidence suggesting the maximal eigenvalue tends to $\pi$ as the translate count grows, supporting the conjecture that near-maximal Bell-CHSH violations exist in this QFT setting. Together, these results offer a more efficient route to certify near-Tsirelson violations in QFT and illuminate how wavelet-based constructions can yield explicit, smooth test functions in the vacuum sector.

Abstract

This paper investigates a recent construction using bumpified Haar wavelets to demonstrate explicit violations of the Bell-Clauser-Horne-Shimony-Holt inequality within the vacuum state in quantum field theory. The construction was tested for massless spinor fields in $(1+1)$-dimensional Minkowski spacetime and is claimed to achieve violations arbitrarily close to an upper bound known as Tsirelson's bound. We show that this claim can be reduced to a mathematical conjecture involving the maximal eigenvalue of a sequence of symmetric matrices composed of integrals of Haar wavelet products. More precisely, the asymptotic eigenvalue of this sequence should approach $π$. We present a formal argument using a subclass of wavelets, allowing us to reach $3.11052$. Although a complete proof remains elusive, we present further compelling numerical evidence to support it.

Further Evidence for Near-Tsirelson Bell-CHSH Violations in Quantum Field Theory via Haar Wavelets

TL;DR

The work reframes near-Tsirelson Bell-CHSH violations in the vacuum of a free massless 1+1D spinor QFT as a spectral problem for a family of symmetric matrices built from Haar-wavelet integrals. Step 1 demonstrates an exact wavelet-based construction that yields explicit test functions achieving correlations close to the bound, with the key link being the maximal eigenvalue of approaching in the large-resolution limit for , giving a numerical value for the asymptotic maximum. Step 2 then shows how to replace Haar wavelets by smooth bumpified versions via Planck-taper windows, preserving the necessary inner-product relations to arbitrary precision and producing test functions compatible with Algebraic QFT. The general case is discussed through block Toeplitz and matrix-valued Fourier-series analysis, with numerical evidence suggesting the maximal eigenvalue tends to as the translate count grows, supporting the conjecture that near-maximal Bell-CHSH violations exist in this QFT setting. Together, these results offer a more efficient route to certify near-Tsirelson violations in QFT and illuminate how wavelet-based constructions can yield explicit, smooth test functions in the vacuum sector.

Abstract

This paper investigates a recent construction using bumpified Haar wavelets to demonstrate explicit violations of the Bell-Clauser-Horne-Shimony-Holt inequality within the vacuum state in quantum field theory. The construction was tested for massless spinor fields in -dimensional Minkowski spacetime and is claimed to achieve violations arbitrarily close to an upper bound known as Tsirelson's bound. We show that this claim can be reduced to a mathematical conjecture involving the maximal eigenvalue of a sequence of symmetric matrices composed of integrals of Haar wavelet products. More precisely, the asymptotic eigenvalue of this sequence should approach . We present a formal argument using a subclass of wavelets, allowing us to reach . Although a complete proof remains elusive, we present further compelling numerical evidence to support it.

Paper Structure

This paper contains 14 sections, 17 theorems, 123 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Summary of previous results
  3. An exact solution in terms of wavelets (Step 1)
  4. Haar wavelets
  5. Problem statement
  6. Numerical solution
  7. Exploiting the (anti)symmetry relations
  8. The matrix A
  9. The special case $K = 1$
  10. The series $\sum_{n=1}^\infty \iota_n$.
  11. Outlook: The general case K ≥ 1
  12. Bumpification (Step 2)
  13. Bumpified Haar wavelets
  14. The final test functions

Key Result

Lemma 3.2

For all $n,k \in \mathbb Z$ we have

Figures (11)

  • Figure 1: The mother Haar wavelet $\psi = \psi_{0,0}.$
  • Figure 2: The a.e. equality $\psi_{n,k}(-x) = - \psi_{n,-k-1}(x)$, visually.
  • Figure 3: The graph of the function $J$ from the proof of Lemma \ref{['prop:A']}.
  • Figure 4: Test function components for $\eta=0.99$ where the resolution was set to $N_0 = -10$, $N_1 = 120$, $K = 5$. Dudal2023
  • Figure 5: Numerical evidence that the test function components found in Dudal2023 satisfy $f_2 = -cf_1$.
  • ...and 6 more figures

Theorems & Definitions (38)

  • Definition 3.1: Haar wavelets
  • Lemma 3.2
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Conjecture A
  • Remark 3.6
  • Lemma 3.7
  • Remark 3.8: Minimal eigenvalue
  • Conjecture B
  • ...and 28 more