From: baez@guitar.ucr.edu (john baez)
Newsgroups: sci.physics
Subject: QM and probability theory
Date: 14 Dec 92 20:53:17 GMT
Organization: University of California, Riverside

I wrote:
>The relationship between probability theory and quantum mechanics is far
>better understood than Giles seems to imply.  Quantum mechanics can be
>thought of as a generalization of classical probability theory, and it's
>easy to check in many cases which theorems of classical probability
>theory extend to the quantum case.

Herman Rubin writes:
>If definitely is NOT that well understood.  Quite a few people in the
>40s and 50s attempted to do this, and no one obtained a satisfactory
>solution.  The problem is that the static wave function is not determined
>by the corresponding probability distribution.  Now formally it is
>determined, apart from trivial trasformations, by the "joint distribution"
>in phase space, and in some cases, this is surprisingly simple.  But this
>joint distribution, relatively easily calculated from the wave function,

Ah, the dangers of the passive.  How about: *I* understand the relationship
between probability theory and quantum mechanics far better than Giles seems
to think anyone does.  :-)  (See what vanity can be concealed by a trick
of grammar.)

The work you describe attempts to REDUCE quantum mechanics to classical
probability theory and I agree that this is hopeless.  While the 
"joint distribution in phase space," usually called the Wigner distribution in
this context, is a very interesting thing, it wasn't the sort of thing I was
talking about. What I was talking about is a treatment of quantum theory 
using noncommutative algebras in such a way as to reveal it as a natural 
GENERALIZATION of probability theory.  The theory of C*-algebras and von 
Neumann algebras and their associated noncommutative integration theory reduces, 
in the commutative case, to the theory of locally compact Hausdorff spaces
(topology) and measures on them.  Many theorems about the latter have
been generalized to the noncommutative context and are thus applicable
to quantum theory.  But, of course, not all, because strange things can happen 
in quantum theory that can't happen classically.  A few examples:

1)  Pure states need not be dispersion-free.
2)  The restriction of a pure state to a sub-C*-algebra need not be pure.
3)  The entropy of the restriction of a state can be greater than the entropy 
    of the state (this is closely related to 2).
4)  A pure state on a tensor product need not be a tensor product of pure
    states (this is why Bell's inequality is violated; see my paper "Bell's
    inequality for C*-algebras" Lett. Math. Phys. 13 (1987), 135-136).

Now this stuff may not sound like probability theory, but if one realizes
that "state" is just a generalization of "measure" and "pure state" is
a generalization of "Dirac delta measure," one sees that it is all a generalization
of probability theory.

Herman Rubin writes:
>And while an infinitesimal quantum transition looks
>a lot like an infinitesimal Wiener process movement with a purely imaginary
>variance, that is enough to move from meaure theory to something quite
>different; there have been numerous attempts to relate the Feynman integral
>to the Wiener integral, none really successful.  About the best which has
>been done here is to show that the solution satisfies the Heisenberg
>equation, but this does not help in finding a principle which will work
>when that much regularity is not present.

I guess you mean Schrodinger equation, not Heisenberg equation.  Again,
this stuff isn't what I was referring to, although it's interesting.  (For people
interested: the "joint distribution on phase space" Rubin refers to above
is often called the Wigner distribution, and a nice exposition appears
in Gerald Folland's "Harmonic Analysis in Phase Space".  The relationship
between Wiener measure and Schrodinger's equation *has* been made precise and
is applicable to situations where the potential is about as nasty as it
would ever be in real physics problems; here a nice reference is Barry Simon's
book on functional integration.)

I wrote:
>Irving Segal (who was a statistician
>before he went into mathematical physics) developed the formalism of
>C*-algebras and noncommutative integration theory in the 40's and 50's
>for exactly this reason (among others).  Bayes' theorem holds, in
>particular.  (But Vidugiris has not stated Bayes' theorem... maybe he
>really DID mean Baynes' thereom!  :-)  Bayes' theorem is that the
>probability of F given E is the probability of E given F times the
>probability of F divided by the probability of E.)  

Herman Rubin writes:
>I am quite well acquainted with Prof. Segal, but I was not aware that
>he ever was a statistician. 

I'm quite acquainted with him too, having written a book with him, and I assure
you that it's true.  :-)  His first paper was on statistics, in fact.

>The algebras you mention were developed 
>as an abstract characterization of bounded operators on Hilbert space,
>similar to the earlier characterization of commutative Banach algebras
>as algebras of functions.
 
That was part of the point, but the generalization of probability theory
was also very much on Segal's mind.

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