From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Subject: Re: Why Wave Functions aren't Physical (Was: Re: QM non-causal?)
Organization: Purdue University Statistics Department
Date: Mon, 14 Dec 1992 18:51:12 GMT

In article <1992Dec13.204233.10835@galois.mit.edu> jbaez@riesz.mit.edu (John C. Baez) writes:
>In article <1992Dec11.220139.14066@newshost.lanl.gov> jlg@cochiti.lanl.gov (J. Giles) writes:
>>In article <1992Dec10.192451.9924@lmpsbbs.comm.mot.com>, bhv@areaplg2.corp.mot.com (Bronis Vidugiris) writes:
>>|> [...]
>>|> P2:  Mathematically, if some measurable phenomenon C affects the
>>|> probabilities of both A and B, one can mathematically state that

>>|> P(A) = intergal of Pa|c() * delta P(C)
>>|> P(B) = intergal of pb|c() * delta P(C)

>>|> where Pa|c is the 'conditional' probability of A given C
>>|> (I belive this is known as Baynes thereom)

>>There are several problems with this.  One is that the formal mathematical
>>properties of probability are not necessarily applicable to sub-QM physics.
>>The second is that the definition of integration is not clear when the
>>underlying functions are not continuous - or, at least, the number of
>>discontinuities are infinite (and what proof have you that they are
>>continuous?).  There are many other assumptions buried in the math.  
>>The idea that conventional mathematics is necessarily applicable to 
>>these physical domains is pure speculation.  Even the definition of 
>>*multiplication* is a human invention which we have no reason - beyond

Bayes' Theorem is quite applicable for objects which have probabilities.
However,

>The relationship between probability theory and quantum mechanics is far
>better understood that 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. 

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,
can easily fail to be non-negative, nor need the "density" even be
Legesgue integrable.  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.

				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.)  

I am quite well acquainted with Prof. Segal, but I was not aware that
he ever was a statistician.  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.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@snap.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
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