From: paul@mtnmath.UUCP (Paul Budnik)
Newsgroups: sci.physics
Subject: Re: Defining local, causal etc., was: Hidden variable theories
Summary: I think we have arrived at the crux of the matter.
Date: 15 Sep 92 21:17:38 GMT
Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070

In article <1992Sep14.195331.3379@galois.mit.edu>, jbaez@riesz.mit.edu (John C. Baez) writes:
> paul@mtnmath.UUCP (Paul Budnik) writes:
> 
> >Any definition of these properties based on probabilistic models,
> >including the diamond property you described, *cannot* be mathematically
> >rigorous because there is no rigorous mathematical definition of a
> >random sequence. There is reason to believe that no such definition is
> >possible.
> 
> Hmm, this is a big tangent, but are you aware of Chaitin's definition
> of algorithmically random sequences?  To my mind it's a perfectly fine
> mathematically rigorous definition of a random sequence.  Check out
> his book.  

There was a brief thread on this several months ago. At that time I
posted a sketch of a proof that any sequence that satisfied Chaitin's
definition of random (as posted to the net) must encode an infinite
number of theorems of elementary number theory that are not decidable
by any finite axiom system. I cross posted to sci.logic to give a wider
range of people a chance to take pot shots. This is the problem with
defining random sequence. Any property that is not satisfied by a
recursively enumerable set has a higher degree of unsolvability and thus
seems to define a set even more structured in some sense that recursively
enumerable sets.

> >A theory is local iff the future state at a given point
> >is determined by the previous state in a local region surrounding this point.
> >In a continuous theory the size of this region must go to zero as
> >the time between states goes to zero. 
> 
> Good, at last!  Now if you use the usual definition of "state" in
> quantum theory, that is, a unit vector in a Hilbert space, or (better)
> density matrix on a Hilbert space, or (better) normalized positive
> linear functional on a C*-algebra, quantum field theories ARE required 
> to be causal according to the definition you offer!  This is, in fact,
> almost exactly the diamond property (which also specifies how big a
> local region you need).  

I believe we have arrived at the heart of the matter. If so the journey, well
tedious at times, was well worth it at least for me. What you say is true.
However in quantum mechanics predictions are determined by two factors:

	1. The state vector as you describe.

	2. The conservation laws.

The nonlocal, in the classical sense, effects of quantum mechanics all result
from the conservation laws. The state vector evolves in a completely local way.
The issue under test in Bell's inequality is whether the state vector as
defined in quantum mechanics completely characterizes the state, in the
classical sense, of some region of space-time. If this is the case, then
the experimentally observable affects of the conservation laws 
appear as if by magic. My metaphor for this is that there is a cosmic
accountant that makes sure all the observed results conform to the
conservation laws within the limits of the uncertainty principle.

I do not believe in this `magic'. I believe there is additional physical
state information that `enforces' the conservation laws. I believe this
information propagates in a local fashion. I also believe it obeys nonlinear
laws and often exhibits chaotic behavior. This accounts for the seeming
randomness of quantum effects. It is also why these results seem so
inexplicable in mechanistic terms to physicists who are use to thinking only
in terms of linear or approximately linear models.

It seems to me that you are doing precisely what I warned you about.
You assume that the state vector in quantum mechanics is a complete
description of the state in the classical sense. That is you
assume quantum mechanics is a complete theory. This 
implys that Bell's inequality is violated. It seems to me wrong
and dangerous to describe experimental tests of Bell's inequality using
language that implies that the issue is settled.

> I know that now you and Daryl and others will pounce on me, saying
> that I am being seduced by mathematical formalism, and that I am
> ignoring how observations "collapse the wave function" or do
> something, at any rate, different than unitary time evolution.  I will
> not respond to such pouncing since this is an argument that I have
> already engaged in to my heart's content.

The problem I see is simple and direct. You assume your
formalism is a complete description of reality when that is 
the question that is being tested experimentally.

>...
> >Well it is not surprising that classical mechanics is wrong, it will be
> >extraordinary if Bell's inequality is violated. 
> 
> I think it will be extraordinary if it isn't, but I think we can agree
> that it will be interesting either way.

Yes.

> ...
> >But you believe in a model that has the same experimentally detectable
> >consequences. What are your objective reasons for preferring this model
> >to one in which collapse is an objective Lorentz invariant process?
> 
> Is there a model of the latter sort which makes the verified
> predictions of standard quantum field theory?

Yes. Take the standard axioms of quantum mechanics and replace the collapse
postulate with the following explicitly incomplete axiom:

   Quantum collapse is engendered by experimental conditions designed to
   observe the classical parameters of a system. It is an objective event
   that occurs for a variety of reasons that are not fully understood or
   part of current theory. It has a local Lorentz invariant structure and
   will thus occur spontaneously before the point that an experiment would
   result in violating locality.

> Does it make any
> verified predictions not made by standard quantum field theory? 

No.

> If the answer to the first question is "yes" and the second is "no", we
> will just have to wait for them to do better tests of Bell's inequality.

The experimenters always have the last word in these matters. But it does
seem to me that physicists have prematurely abandonded some extremely
important principles of science without justification. Why would anyone
prefer the `cosmic accountant' approach to enforcing the conservation laws
to a local mechanistic theory. The only reasons that I have come up with
are:

	1. They wanted to develop a complete theory.

	2. They could not conceive of how a mechanistic theory could account
	   for the observed experimental results. Thus they concluded such a
	   theory is not possible. Brilliant people are susceptible to the
	   sin of hubris.

	3. I think Bohr and his ideas on `complementarity' was a projection
	   onto physics of what applys correctly to far more complex fields
	   such as human psychology. When asked what was complementary to truth
	   Bohr replied clarity. I think this is dead wrong for mathematics
	   or physics. Results in these fields should be both clear and true.
	   In contrast, books on pop-psychology are often clear and far from
	   the truth. The psychology of Jung, which is hardly clear, may begin to
	   capture some of the rich complexity of human truth.

>...
> >> >I will be happy to bet you $50 that by the year 2005 there will be
> >> >substantial experimental evidence that that the correlated events
> >> >observed in tests of Bell's do not have a space-like separation.
>...
> >Why do you object to my wording?
> 
> Because I don't want to bet on the competence of experimenters.  I
> don't want to lose the bet just because by 2005 nobody has gotten
> around to doing a decent test of Bell's inequality that ensures
> spacelike separation.  I want to have a bet such that, WHEN they
> finally get around to doing the experiment RIGHT, I win if Bell's
> inequality is violated, and you win if it holds.   

I thought by my wording I would loose the bet if a proper experiment
had not been done by 2005. Is the following wording acceptable?

When an experiment has been done that we both agree is an effective
test of Bell's inequality, the bet will be decided on the outcome of
the experiment. If it confirms that the inequality is violated you win,
if it confirms that there is not a space-like separation between correlated
manipulations and observations I win. There is no time limit. I would expect
at least a 2 year delay between the time the experiment is published and the
time we are ready to agree that it is reasonably conclusive.

Paul Budnik
Newsgroups: sci.physics