Near-Tsirelson Bell-CHSH Violations in Quantum Field Theory via Carleman and Hankel Operators

Abstract

We study Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) violations in the vacuum state of free spinor fields in $(1+1)$-dimensional Minkowski spacetime. We construct explicit smooth compactly supported test functions with spacelike separated supports whose Bell-CHSH correlators converge to Tsirelson's bound $2\sqrt2$. In the massless case, after passage to the time-zero slice and a natural symmetry reduction, the problem reduces to the quadratic form of the Carleman operator on $L^2([0,\infty))$. Near-maximal Bell violation is then governed by the spectral edge $\pi$, and explicit near-extremizers are obtained from compactly supported cutoffs of the generalized eigenfunction $x^{-1/2}$. This also explains the appearance of the constant $\pi$ in earlier wavelet-based formulations. In the massive case, the same reduction leads to a Hankel operator with kernel $mK_1(m(x+y))$, where $K_1$ denotes the modified Bessel function of the second kind of order $1$, and exponentially damped variants of the massless test functions again yield Bell-CHSH values converging to $2\sqrt2$. Therefore, we establish a direct link between Bell-CHSH violations for free $(1+1)$-dimensional spinor fields and the spectral theory of Carleman and Hankel operators on the half-line.