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Bosonization, vertex operators, and maximal violation of the Bell-Clauser-Horne-Shimony-Holt inequality in wedge regions

J. G. A. Caribé, M. S. Guimarães, I. Roditi, Silvio Paolo sorella

Published Aug 15, 2026• 0 citations
Publisher

Abstract

It is pointed out that the vertex operators of a chiral boson in 1 + 1 dimensions provide an explicit realization of dichotomic, bounded, Hermitian operators that saturate the Tsirelson bound of the Bell-Clauser-Horne-Shimony-Holt inequality in the vacuum state.
DOI
10.3929/ethz-c-000807488

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The violation of the Bell-Clauser-Horne-Shimony-Holt inequality in a relativistic scalar quantum field theory is analyzed by means of a set of bounded Hermitian operators constructed out of the unitary Weyl operators. These operators allow for both analytic and numerical approaches. While the former relies on the modular theory of Tomita-Takesaki, the latter is devised through an explicit construction of the test functions needed for the localization of the aforementioned operators. The case of causal tangent diamonds in 1+11+11+1 Minkowski spacetime is scrutinized.
The Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality in the vacuum state of a relativistic scalar quantum field is revisited by making use of the Hilbert space $\mathcal{H}\ensuremath{\bigotimes}{\mathcal{H}}_{AB}$, where H\mathcal{H}H and HAB{\mathcal{H}}_{AB}HAB​ stand, respectively, for the Hilbert space of the scalar field and of a generic bipartite quantum mechanical system. The construction of Hermitian, field-dependent, dichotomic operators is devised as well as the Bell-CHSH inequality. Working out the ABABAB part of the inequality, the resulting Bell-CHSH correlation function for the quantum field naturally emerges from unitary Weyl operators. Furthermore, introducing a Jaynes-Cummings type Hamiltonian accounting for the interaction between the scalar field and a pair of qubits, the quantum corrections to the Bell-CHSH inequality in the vacuum state of the scalar field are evaluated till the second order in perturbation theory.
We discuss the relationship between the Bogoliubov transformations, squeezed states, entanglement, and maximum violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. In particular, we point out that the construction of the four bounded operators entering the Bell-CHSH inequality can be worked out in a simple and general way, covering a large variety of models, ranging from quantum mechanism to relativistic quantum field theories. Various examples are employed to illustrate the above mentioned framework. We start by considering a pair of entangled spin-one particles and a squeezed oscillator in quantum mechanics, moving then to the relativistic complex quantum scalar field and to the analysis of the vacuum state in Minkowski space-time in terms of the left and right Rindler modes. In the latter case, the Bell-CHSH inequality turns out to be parametrized by the Unruh temperature.
Bell-type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson's bound constitutes a global upper limit for quantum violations of the Clauser-Horne-Shimony-Holt (CHSH) and the Clauser-Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell-type inequalities by solving the eigenvalue problem for the operator associated with each Bell-type inequality. Thereby, we use the min-max principle to calculate the norm of these self-adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell-type inequality is maximally violated.
The Bell-CHSH inequality provides a powerful probe of quantum correlations in relativistic quantum field theory. We study its violation in the vacuum of the free massless Majorana field in 1+1 dimensions using modular wedge localization. By introducing a rapidity variable, we obtain a particularly simple representation of the one-particle Hilbert space as a direct sum of two scalar components, in which the spinor mixing present in previous formulations is eliminated. This allows us to give an explicit algorithm for constructing four bounded, Hermitian, dichotomic operators whose Bell-CHSH expectation value can be made arbitrarily close to the Tsirelson bound ( 2 2 ). We derive the resulting Bell-CHSH functional in terms of the modular spectral weight and provide a semi-analytic family of test functions that realizes this near-saturation explicitly.