From: naturman@uctvax.uct.ac.za
Newsgroups: sci.math
Subject: Re: Bayes' theorem and QM
Date: 19 Dec 92 14:52:07 +0200
Keywords: quantum mechanics, probability
Organization: University of Cape Town

> The second statement is unwarranted, in my opinion. Bell's Theorem
> shows that there is no deterministic completion of QM that uses
> classical probability theory. However, Bell's Theorem does *not* say
> that there are no deterministic completions of quantum mechanics, if
> one is willing to give up classical probability theory.


The following types of models of a Bell Inequality type experiment are known to
be equivalent in the sense that if a model of one of these types exists then so
do models of the other types:

1) Any model in which the Bell Inequality holds.
2) A "traditional" hidden variable theory.
3) A "weak" hidden variable theory of the type investigated by Kochen and
   Specher with exception that the Kochen-Specker style functional
   correspondence is allowed to fail on a set of measure zero.
4) Any model in which the four main observables in the BI type experiments
   have a well-defined joint distribution
5) Any model in which the mean counterfactual values of the main observables
   is the same as the QM expectation value.
6) Any stochastic factorizable model

and several other types of models which I can't remember.

Since there are deterministic models which are non-equivalent to these models,
one cannot conclude that deterministic models are ruled out by the Bell
Inequality. In fact, a computer program algorithm modelling QM provides a
a very deterministic model of QM although it is unlikely to be a physically
relevant model. Research is being done into finding a plausible pseudorandom
recursive mechanism behind QM. (I have not seen any results yet though.)

Beware of the fact that many books and popular articles on QM claim that the
Bell Inequality must hold in any realist interpretation of QM. This is false,
the supposed proofs they use tacitly assume a model of type (5) or (6) above.

If you are talking about ONTOLOGICAL determinism then the current
evidence is that QM must be deterministic otherwise one would get violations
of locality (which I am not willing to accept barring a major revision of
physics.)


> 
> As I have mentioned several times before, there *is* a
> hidden-variables theory for quantum mechanics developed by Pitowsky
> and Gudder. However, the probabilities associated with the hidden
> variables are non-classical; in particular, the measurable sets do not
> form a sigma algebra.
> 
> Daryl McCullough

The debate over whether QM uses "classical" probability would be aided if
someone actually bothered to define what they mean by "classical" or
"non-classical" The failure of traditional hidden variable theories shows
that QM is non-classical in the sense that it works very differently to
the Kinetic Theory of Gases (the original motivation for hidden variables).

BUT QM probabilty is still classical in the sense that for each observable and
for each state (pure or mixed) we have a well defined ordinary very classical 
probability measure on the Borel sets which is interpreted as the probability
that the observable takes a value in any particular Borel set.

In my opinion and my experience with attempting to argue with physicists on
sci.physics, most physicists think that QM is different to other probabilistic
theories for the simple reason that they are not familiar with the weird things
that can happen in ordinary probability theory.

For a good book on weird examples in ordinary probability try get hold of
the book "Counterexamples in Probability and Statistics" (I can't remember the
authors).

The branch of math known as Quantum Probabilty deals with the type of
probability thingies that arise in QM. It uses "generalized probabilities"
which are not defined on Boolean sigma algebras but on orthomodular lattices.
It is very convenient for dealing with QM problems but it is still classical
since we can translate between traditional probability and quantum probability.
The reason for its usefulness seems to be that it is a more natural discription
of probabilities defined on states which are not distinct states of knowledge
but which in a sense overlap. (Maybe I am not expressing myself well but to
get a feel for what I mean, compare QM states that can be superposed or have
non-zero transition probabilities between them, with states in classical
mechanics that are in a sense all distinct and non-overlapping.)


Colin Naturman

PS As you probably know, one can have an ordinary sigma algebra style hidden 
variable theory for QM if we allow violations of locality, but again most
people would reject such an idea. 
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