In quantum mechanics we cant get rid of probability, but can we conserve locality by using new mathematics and new assumptions?
Usually, the notion of locality is used to distinguish between classical (hidden variable) theories which might mimic the quantum-mechanical behaviour observed in nature. Local hidden-variable theories can be excluded experimentally by Bell inequality tests. Although it is difficult to close all the "loop holes" in an experiment, it is widely believed that no local hidden-variable theory is responsible for the quantum-mechanical behaviour observed in this world. Non-local hidden variable theories cannot be excluded, but believing in them is as good as believing that this world is a big computer that keeps track about non-local correlations --- nobody can prove or disprove it, but such knowledge is useless.
Quantum mechanics might seem to be non-local, if you judge it from the classical point of view. But from the point of view of quantum mechanics, it is as local as it can get, meaning that one can always introduce particles carrying the interaction, continuously, from point to point in space and with time. The EPR paradox is not contradictory from the point of view of quantum mechanics. The authors did not realize that knowledge (or information) is local. So, the fact that someone far away knows something about particle B, does not change a thing about particle A, as long as that (physical) knowledge about B remains far away and cannot be used to make a prediction about A.
Dear Vitaly thank you for your participation,
as you know from Bell's experiment the measurement process of one particle may affect the state of the other even if the particles are far away from each other this contradicts locality if we looked to the problem from our four dimensions.
The state of particle A is described by the density matrix rho_A of that particle. Since the particles are far away from each other, the density matrix rho_A is the most complete description one can give, locally, to that particle. It is easy to convince yourself that rho_A does not change if particle B is measured by somebody far away. The authors of the EPR paper were confused, because they treat information as global (i.e. if something becomes known, then it becomes known to everybody and everywhere). However, information is physical, i.e. has to be stored in some form in a physical system. The description of a quantum-mechanical system depends on the information available to the observer. If you describe the EPR pair locally (at particle A), then there is no contradiction. Similarly, if you describe it globally (assuming that A and B are not that far away and communication/transport of information can take place between A and B), then again no contradiction arises. However, if you mix the two ways, i.e. you imagine that your knowledge is global, but the description is local, then of course you run into a contradiction.
The best way to understand these things is to read the EPR paper. The authors are very careful in analyzing the things and it is easy to see at which point they contradict themselves. As I remember, it is somewhere at the bottom of page 2 or the top of page 3. Good luck!
Dear professor thanks for your effort. The problem of nonlocality in quantum mechanics is due to instantaneous transfer of data between the entangled particles. Iagree with you that desity of states can be assumed local but that's solve nothing of the nonlocality paradox. Bell's theorem proved the nonlocality of quantum mechanics. so if we confined in our four dimentional space it is impossible to get a localized quantum theory.but how about assuming extra dimensions this may solve the problem or not I think this subject can be put under discussion if you are interested
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