Is Bell's inequality Lorentz invariant?

[jk22]

I browsed the net and found :

https://arxiv.org/abs/quant-ph/0408127

It is said the value of Bell's operator depends on the speed, so how can it be Lorentz invariant ?

 

[atyy]

Assuming classical relativistic spacetime (which is assumed in both classical and quantum relativistic theory), the violation of a Bell inequality is not compatible with classical relativistic causality. Classical relativistic causality does have a "special" speed, which is commonly called the speed of light.

Note that a Bell inequality is a statement about classical relativistic causality, and does not depend on relativistic quantum theory. Relativistic quantum theory does violate Bell inequalities, showing that relativistic quantum theory is not compatible with classical relativistic causality. The "Bell operator" or "Bell observable" mentioned in the paper is a part of relativistic quantum theory, and is not needed in the derivation of the Bell inequality. The Bell operator or Bell observable is used to show that relativistic quantum theory violates a Bell inequality in any reference frame.

In relativistic quantum theory, neither operators (including the Bell operator or Bell observable) nor states need to be separately Lorentz invariant, only the combined use of operators and states to predict the probabilities of measurement outcomes needs to be Lorentz invariant.

 

[Tendex]

Assuming classical relativistic spacetime (which is assumed in both classical and quantum relativistic theory), the violation of a Bell inequality is not compatible with classical relativistic causality. Classical relativistic causality does have a "special" speed, which is commonly called the speed of light.

Note that a Bell inequality is a statement about classical relativistic causality, and does not depend on relativistic quantum theory. Relativistic quantum theory does violate Bell inequalities, showing that relativistic quantum theory is not compatible with classical relativistic causality. The "Bell operator" or "Bell observable" mentioned in the paper is a part of relativistic quantum theory, and is not needed in the derivation of the Bell inequality. The Bell operator or Bell observable is used to show that relativistic quantum theory violates a Bell inequality in any reference frame.

In relativistic quantum theory, neither operators (including the Bell operator or Bell observable) nor states need to be separately Lorentz invariant, only the combined use of operators and states to predict the probabilities of measurement outcomes needs to be Lorentz invariant.

That "combined-only" requirement is referred to sometimes as Lorentz(Poincare) covariance in relativistic quantum theory. This follows the requisite of local gauge invariance, from local quantum fields, which is not explicitly enforced in the case of classical SR in Minkowski spacetime, that is determined only by its spacetime symmetries rather than also by the "internal' ones of quantum relativistic gauge theory.