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- TL;DR Summary
- The notion of locality in (quantum) physics should be clearly defined

Unfortunately one of the threads about entanglement and Bell tests has again been closed prematurely. It has not been clarified what "locality" means.

In the physics community, not involved in philosophical arguments about foundations of QT, it's clearly defined as the property of a relativistic theory that obeys the causality principle of relativistic spacetime, which implies that there cannot be any causal relation between space-like separated events. As far as I know that's also the meaning Bell gives always to this notion in his work on Bell inequalities valid in local (in this meaning!) realistic (which means that all observables always take determined values).

In relativistic QFT this locality principle is implemented by construction through the demand that local observables must commute at space-like separated arguments, i.e., (using the signature ##(+---)## for the Minkowski form)

$$[\hat{A}(x),\hat{B}(y)]_{-}=0 \quad \text{for} \quad (x-y)^2<0,$$

if ##\hat{A}(x)## and ##\hat{B}(x)## are representing local observables (like, e.g., charge-current densities, energy-momentum tensors/intensities, etc.). Since the energy density ##\mathcal{H}(x)## is an observable, this indeed implies that there cannot be any nonlocal (inter)actions between distant parts of a quantum system.

Of course this does not rule out entanglement between far distant parts of quantum systems. In the mostly discussed case of entangled photon pairs you can of course have entangled photon states with the corresponding Bell-inequality violating correlations between the outcomes of measurements on the single photons in the pair at far distant places. All Bell tests are in accordance with standard local QED, which shows that standard relativsitic local QFT, which is local by construction, is able to describe the observed violations of Bell's inequality. That's no contradiction to Bell's derivation of these inequalities, but in addition he has assumed "realism", i.e., that all observables always take determined values, independent of the state of the system. This is, of course, not the case in relativistic local QFTs, which obeys the Heisenberg uncertainty principle, as any QT, which implies that incompatible observables usually do not take determined values, i.e., there are always states, in which one of these observables is determined (or in the case of observables with continues spectra very sharply determined) and the other then is necessarily indetermined (or pretty unsharply determined). The conclusion thus is that Nature behaves according to a local but non-realistic theory, namely relativistic local QFT.

In the physics community, not involved in philosophical arguments about foundations of QT, it's clearly defined as the property of a relativistic theory that obeys the causality principle of relativistic spacetime, which implies that there cannot be any causal relation between space-like separated events. As far as I know that's also the meaning Bell gives always to this notion in his work on Bell inequalities valid in local (in this meaning!) realistic (which means that all observables always take determined values).

In relativistic QFT this locality principle is implemented by construction through the demand that local observables must commute at space-like separated arguments, i.e., (using the signature ##(+---)## for the Minkowski form)

$$[\hat{A}(x),\hat{B}(y)]_{-}=0 \quad \text{for} \quad (x-y)^2<0,$$

if ##\hat{A}(x)## and ##\hat{B}(x)## are representing local observables (like, e.g., charge-current densities, energy-momentum tensors/intensities, etc.). Since the energy density ##\mathcal{H}(x)## is an observable, this indeed implies that there cannot be any nonlocal (inter)actions between distant parts of a quantum system.

Of course this does not rule out entanglement between far distant parts of quantum systems. In the mostly discussed case of entangled photon pairs you can of course have entangled photon states with the corresponding Bell-inequality violating correlations between the outcomes of measurements on the single photons in the pair at far distant places. All Bell tests are in accordance with standard local QED, which shows that standard relativsitic local QFT, which is local by construction, is able to describe the observed violations of Bell's inequality. That's no contradiction to Bell's derivation of these inequalities, but in addition he has assumed "realism", i.e., that all observables always take determined values, independent of the state of the system. This is, of course, not the case in relativistic local QFTs, which obeys the Heisenberg uncertainty principle, as any QT, which implies that incompatible observables usually do not take determined values, i.e., there are always states, in which one of these observables is determined (or in the case of observables with continues spectra very sharply determined) and the other then is necessarily indetermined (or pretty unsharply determined). The conclusion thus is that Nature behaves according to a local but non-realistic theory, namely relativistic local QFT.