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

TL;DR

This paper explores violations of Bell inequalities in quantum field theory using Haar wavelets, providing numerical evidence and reducing the problem to a conjecture about eigenvalues approaching pi.

Contribution

It introduces a new wavelet-based construction for Bell inequality violations and links the problem to a conjecture on eigenvalues, advancing understanding in quantum field theory.

Findings

Violations close to Tsirelson's bound are demonstrated.

Eigenvalue conjecture related to Haar wavelet integrals is proposed.

Numerical evidence supports the conjecture, reaching 3.11052.

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 $\pi$. 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.