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Modular Theory and the Bell-CHSH inequality in relativistic scalar Quantum Field Theory

J. G. A. Caribé, M. S. Guimaraes, I. Roditi, S. P. Sorella

Abstract

The Tomita-Takesaki modular theory is employed to discuss the Bell-CHSH inequality in wedge regions. By using the Bisognano-Wichmann results, the construction of a set of wedge localized vectors in the one-particle Hilbert space of a relativistic massive scalar field in $1+1$ dimensions is devised to establish whether violations of the Bell-CHSH inequality might occur for different choices of Bell's operators. In particular, the construction of the wedge localized vectors employed in the seminal work by Summers-Werner is scrutinized and applied to Weyl and other operators. We also outline a possible path towards the saturation of Tsirelson's bound.

Modular Theory and the Bell-CHSH inequality in relativistic scalar Quantum Field Theory

Abstract

The Tomita-Takesaki modular theory is employed to discuss the Bell-CHSH inequality in wedge regions. By using the Bisognano-Wichmann results, the construction of a set of wedge localized vectors in the one-particle Hilbert space of a relativistic massive scalar field in dimensions is devised to establish whether violations of the Bell-CHSH inequality might occur for different choices of Bell's operators. In particular, the construction of the wedge localized vectors employed in the seminal work by Summers-Werner is scrutinized and applied to Weyl and other operators. We also outline a possible path towards the saturation of Tsirelson's bound.

Paper Structure

This paper contains 14 sections, 141 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Construction of the vectors $\{ \psi \} \in L^2(d\mu_p, H_m)$ by employing the modular operators $(\delta,j)$
  3. Background
  4. The Bisognano-Wichmann results and the construction of modular localized vectors
  5. Applications to the Bell-CHSH inequality
  6. Construction of the Summers-Werner vectors in rapidity space
  7. Applications
  8. Weyl operators revisited
  9. Hunting Tsirelson's bound
  10. Conclusion
  11. Canonical quantization
  12. The real massive scalar field in $(1+1)$ Minkowski spacetime
  13. Saturation of the Tsirelson bound for free Fermi fields
  14. The violation of the Bell-CHSH inequality

Figures (2)

  • Figure 1: Behavior of the Bell-CHSH correlator $\langle 0 |{\cal C} |0\rangle$ from equation \ref{['Bccc']} as function of the free parameters $\eta$ and $\eta'$. The orange surface above the blue one corresponds to the regions where the violation takes place.
  • Figure 2: Behavior of the Bell-CHSH correlator $\langle 0|{\cal C} |0\rangle$ as function of the parameter $\eta'$, for $\eta= -0.395$. One sees that the size of the violation is of about $\approx 2.3$