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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 reference^{6.2} 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:** Polarized light **Up:** Relativity plus
quantum mechanics **Previous:** Locality and quantum mechanics
**Contents**

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