arXiv:2406.20033 · Physical review. D/Physical review. D.

Numerical approach to the Bell-Clauser-Horne-Shimony-Holt inequality in quantum field theory

Philipe De Fabritiis, M. S. Guimarães, I. Roditi, Silvio Paolo sorella

Published Sep 9, 2024• 9 citations

Abstract

The Bell-CHSH (Clauser-Horne-Shimony-Holt) inequality in the vacuum state of a relativistic scalar quantum field is analyzed. Using Weyl operators built with smeared fields localized in the Rindler wedges, the Bell-CHSH inequality is expressed in terms of the Lorentz invariant inner products of test functions. A numerical framework for these inner products is devised. Causality is also explicitly checked by a numerical evaluation of the Pauli-Jordan function. Violations of the Bell-CHSH inequality are reported for different values of the particle mass parameter. Published by the American Physical Society 2024

DOI: 10.1103/physrevd.110.065006

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arXiv:2406.20033 · Physical review. D/Physical review. D.

Numerical approach to the Bell-Clauser-Horne-Shimony-Holt inequality in quantum field theory

Philipe De Fabritiis, M. S. Guimarães, I. Roditi, Silvio Paolo sorella

Published Sep 9, 2024• 9 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevd.110.065006

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\mathcal{H}

and

{\mathcal{H}}_{AB}

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

AB

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.

(A,B)

refer to Alice and Bob, respectively. A set of bounded Hermitian operators is introduced by making use of the Weyl operators. A CHSH-type correlator is constructed and evaluated in the Fock vacuum by means of canonical quantization. Although the observed violation of the CHSH inequality turns out to be rather small as compared to Tsirelson's bound of quantum mechanics, the model can be employed for the study of Bell's inequalities in the more physical case of gauge theories such as the Higgs models, for which local Becchi-Rouet-Stora-Tyutin (BRST) invariant operators describing both the massive gauge boson as well as the Higgs particle have been devised. These operators can be naturally exponentiated, leading to BRST invariant type of Weyl operators useful to analyze Bell's inequalities within an invariant BRST environment.

1+1

Minkowski spacetime is scrutinized.

T

. By treating the Feynman propagator as a distribution and by introducing a suitable localizing set of compact support smooth test functions, we work out the path integral setup for the Bell-CHSH inequality, recovering the same results of the canonical quantization.

1+1

)-dimensional Minkowski space-time. Alice's and Bob's test functions are explicitly constructed, first by employing Haar wavelets which are then bumpified into proper test functions via a smoothening procedure relying on the Planck-taper window function. Relativistic causality is implemented by requiring the support of Alice's and Bob's test functions to be located in the left and right Rindler wedges, respectively. Violations of the Bell-CHSH inequality as close as desired to Tsirelson's bound are reported. We briefly comment on the extra portal, compared to earlier works; this opens the opportunity to scrutinize Bell-CHSH inequalities with generic, interacting quantum field theories.

sign(\varphi(f))

, where

\varphi(f)

stands for the smeared scalar field, it turns out that the Bell-CHSH correlator can be evaluated in closed analytic form in terms of the imaginary error function. Being the superposition of two coherent states, cat states allow for the existence of interference terms which give rise to a violation of the Bell-CHSH inequality. As such, the present setup can be considered as an explicit realization of the results obtained by Summers-Werner.