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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[19]. 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 is `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 is meant 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. Measuring the position of one
particle puts 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 the experiment is setup,
the measurement will fall within some range of possible values that
could be large. Measuring one particle's position with high
accuracy gives the other particles position to a similar accuracy.
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 a pair of measurements
are made 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. An experimental
manipulation, like changing the angle of a polarizing filter, can
be involved in a measurement. Bell proved that experimental
manipulations can influence a distant detection instantaneously if
quantum mechanics is correct. 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 reference^{8.2} but it
does not matter because the causal affect goes in a direction that
is indeterminate.

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