From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Bayes' theorem and QM
Organization: ORA Corporation
Date: Thu, 24 Dec 1992 10:14:52 GMT

bhoughto@sedona.intel.com (Blair P. Houghton) writes:

>How would a "non-classical probability theory" look?  What is
>the limitation of "classical probability theory," and just
>what is a "classical probability theory?"

Classical and quantum probability are described in the book "Quantum
Probability" by Stanley Gudder. There is nothing especially "quantum"
about quantum probability, except for the fact that it can be applied
to quantum mechanics.

Briefly, the differences between classical and nonclassical
probability theory is mostly about the issue of nonmeasurable sets.
(Probability theory being a special case of measure theory).

Nonmeasurable sets:

Take the simple case of the good old Lebesgue measure for
three-dimensional space. In this case measure is essentially
the same as volume. 

Suppose that we know that a point particle is somewhere inside the
unit cube centered at the origin (that is, its x coordinate is between
+ 1/2 and - 1/2, and so is its y coordinate, and so is its z
coordinate). Suppose further that the particle is equally likely to be
anywhere in that region. Then we can answer a question like: What is
the probability that the point is in region A (where A is a subset of
the unit cube)? The probability is simply the three-dimensional volume
of region A.

It would be nice if the volume of a region of space were always
well-defined, but it is not. A demonstration due to Banach and Tarski
showed that it is possible (mathematically, rather than physically
possible) to decompose a sphere into a finite number of pieces and
then recombine them by rotations and translations to get two complete
spheres. It is easy to see that these pieces cannot possibly have
volumes, since that would lead to the conclusion that the volume of
two spheres equals the volume of one sphere.

So one cannot always assume that *every* set has a measure (or volume,
or probability).

Sigma algebras:

Of course, these pieces are very weird, and are not regions one is
typically interested in. Therefore, a measure theory does not have to
assign measures to every set, but only to the sets of interest. The
assumption behind classical measure theory is that the measurable sets
form a sigma algebra. All this means is that we have some collection
of basis sets for which measures are given, and all other measurable
sets are formed from these by combinations of the operations of:

   set difference,
   set intersection (countable intersections),
   set union (countable unions)

These are the operations of a sigma algebra. These operations, though
they won't necessarily give you all sets, gives a nice collection of
sets, which have a nice, intuitive interpretation in the case of
probability measures: Every set can be thought of as a proposition;
for instance we can associate a set A with the proposition "The
particle is somewhere in set A". The fact that the measurable sets
form a sigma algebra means that any logical combination of such
statements is given a probability of being true.

Nonclassical measure theory:

Nonclassical measure theory simply drops the assumption that the
measurable sets form a sigma algebra. It allows for the possibility
that proposition A may have a well-defined probability, and proposition
B may also have a well-defined probability, while the proposition A and
B may be given no probability whatsoever.

The applicability to quantum mechanics should be obvious: wave
functions allow us to calculate the probability that the particle can
be found in region A, and they allow us to calculate the probability
that its momentum is in region B, but they don't allow us to calculate
the probability that the particle is in region A *and* its momentum is
in region B. Quantum mechanics does not give a unique answer to such
questions.

So nonclassical probability theory differs from classical probability
theory in that it may fail to give probabilities for logical
combinations of meaningful propositions.

Daryl McCullough
ORA Corp.
Ithaca, NY


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