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Realistic theories and randomness

Often Bell's result is presented as showing that quantum mechanics is not a realistic theory rather than showing that it is nonlocal. The focus is on the reference to hidden variables in Bell's proof. Eberhard developed a version of Bell's argument that did not involve hidden variables[13]. In turn some physicists objected to Eberhard's proof because he assumed "contrafactual definiteness". That is he assumed one could argue about all possible outcomes of an experiment including those that did not happen.

Arguments like those about hidden variables and contrafactual definiteness are philosophical. They have no clear resolution unlike problems that can be formulated mathematically. Such arguments are rare in the hard sciences. They occur here because of the claim in quantum mechanics that probabilities are fundamental or irreducible.

There is no mathematical model for irreducible probabilities. There is not even a mathematically definition of a random number sequence. There are sequences that are recursively random. Loosely speaking this means that no recursive process can do better than chance at guessing the next element in the sequence. The problem with recursively random sequences is that they are more complex than any recursive sequence. If somehow one could generate such a sequence one could use it to solve recursively unsolvable problems.

This suggests that a truly random sequence cannot exist. Any sequence that we would consider to be truly random must be recursively random. Otherwise there is some computer program that can guess with some degree of accuracy the elements in the sequence. Yet no recursive random sequence can be truly random. This presents a philosophical problem for the claim that quantum mechanics is truly random.

The randomness claimed for quantum mechanics has no foundation in mathematics and it appears to be impossible to construct such a foundation. This does not make it wrong but suggests there are problems in our existing conceptual framework. It also means that physicists when arguing about these issues are debating philosophy with no objective way of deciding the issue. My prejudice is with Einstein. I do not see a need to go beyond conventional logic or mathematics. I only see a problem with developing a better theory.

Understanding Bell's result can be difficult even though it is simple. It involves phenomena that can only exist in a theory like quantum mechanics in which probabilities are irreducible. At the macroscopic level of everyday experience the world seems to be causal. What we mean by an experimenter influencing a result is straight forward. In quantum mechanics this is not the case. An observation can be influenced by an experimenter but it usually also has a probabilistic component. Thus we can never tell how much the experimental manipulation contributed to the final observation.

Consider an experiment in which a pair of photons (particles of light) are emitted in a single event such as a particle decay. The conservation of momentum requires that the two photons be emitted in exactly opposite directions. Yet one cannot measure the position of a particle perfectly. By measuring the position of one particle we put constraints on the position of the other particle. Of course the same thing is true in classical physics. Information about each particle is implicit in the trajectory of the other particle. The difference is that quantum mechanics denies the existence of a particle trajectory independent of a series of position measurements. Depending on how we set up the experiment the measurement will fall within some range of possible values that could be large. Yet if we measure one particles position with high accuracy we know the other particles position to a similar accuracy without any measurement on it. Before that first measurement there was far more uncertainty in the second particles position. So if the particle does not have a classical trajectory does our first measurement actually influence the second measurement? It cannot do so in a direct causal way without violating relativity. But suppose we do a pair of measurements simultaneously on both particles. Could there be a correlation between those two measurements that implies non local influence without either measurement affecting the other in a way that would violate relativity?

Bell proved the answer to that question is yes. As part of a measurement we can make an experimental manipulation like changing the angle of a polarizing filter. What Bell proved is that such experimental settings can influence a distant detection instantaneously. This is only possible when there are a pair of experimental manipulations and distant detections. It is never possible to know which experimental manipulation affected which detection but it is possible to measure the influence that one of them had. The probabilistic element in the measurements is sufficient to mask any information about which measurement influenced which distant observation but the mathematics requires that one of the two measurements did influence the more distant observation. The order of the measurements can be different in different frames of reference6.2 but it does not matter because we can assume the causal effect goes in either direction.


Completed second draft of this book

PDF version of this book
next up previous contents
Next: Polarized light Up: Relativity plus quantum mechanics Previous: Locality and quantum mechanics   Contents


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