From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Hidden variable theories, was: Uncertainty Princi
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Sat, 12 Sep 92 19:45:40 GMT

paul@mtnmath.UUCP (Paul Budnik) writes:
>In article <1992Sep12.020632.9597@galois.mit.edu>, jbaez@riesz.mit.edu (John C. Baez) writes:

>> Your latest post confirms that we have no argument over facts, just
>>massive disagreements about vocabulary, which reassures me that I can
>> let the matter rest.

>Good, but I think it will be worthwhile to confirm this by trying to state
>the facts clearly.

>	1. Bell's inequality is a mathematical result (it does not involve
>	quantum mechanics) that any local theory must satisfy.

As I stated before, tossing around words like "local" and "causal" in
this context without giving a mathematically precise definition can
lead to endless argument.  I gave 3 examples of definitions of "local"
a while ago and I would agree with the above if you are using
definition number 3.

I'd prefer, actually, to say it's a result that any *classical* theory 
must satisfy.  Bell's inequality relies crucially on the assumption that 
states are described by probability measures on a measure space, the measure
space for the entire (two-part) system being a direct product of
measure spaces for each of the two subsystems.   It is violated when
instead states are described by density matrices on Hilbert spaces,
the Hilbert space for the entire system being a tensor product of
Hilbert spaces for each of the two subsystems.  The difference lies
solely in the fact that in the classical case, any pure state is a
tensor product of pure states, while in quantum theory this fails. 
This allows for correlations of a sort that are impossible in
classical mechanics.

>	2. Quantum mechanics predicts the equality is violated. It is possible
>	to test this experimentally.

Fine.

>	3. In an experimental tests of Bell's inequality manipulations of a local
>	polarizer must instantaneously influence results obtained by a distant
>	observer or Bell's inequality cannot be violated.

Nope.  That'd be true in a classical theory, but not in a quantum
theory.  At least that's how I talk about it.  Since you have not
defined "instantaneously influence" we can argue all day without ever
getting anywhere.  There is a precise sense in which nothing done in
any region can influence what's going on in any spacelike separated
region.  This precise sense is what has been formalized in axiomatic
quantum field theory as "causality", or the "diamond property" (see my
previous post and the references).  When I talk about causality and
say there are no instantaneous influences in quantum field theory,
this is the sense I mean.  

By the way, for those of you who are wondering why the heck I'm
dragging quantum field theory into this mess, it's not out of some
love of needless erudition.  It's just because when you try to combine
special relativity and quantum theory, which is what we're doing here,
you get quantum field theory.  To talk about this stuff in a mathematically
precise manner, you need to use the language of quantum field theory. 

>	4. This influence is not detectable at the time it occurs but is
>	detectable at a later time by comparing a series of polarizer angles
>	and detections observed at both sites.

Fine.  I don't call something an "influence" if it cannot be detected
at the time it "occurs", but we can agree to disagree on this one.  

>The arguments about this center on item 3 with item 4 being used to dismiss
>its significance. Experimentally detectable influence, by standard definitions,
>involves the transfer of information and, if it occurs instantaneously,
>violates either causality or Lorentz invariance.

I never said the violation of Bell's inequality was insignificant.  I
think it's very interesting.  Since it does not violate causality or
Lorentz invariance *in the sense of mathematically rigorous quantum
field theory* I don't think it's anything "bad".  In other words, it's
weird and wonderful but it does not indicate any inconsistency in the
currently accepted approach to quantum field theory.

>Do you disagree with item 3? 

See above.

> See any book which explains the various axiomatic approaches to quantum
> field theory. ...  There
> are, in fact, two distinct notions of causality in quantum field
> theory, "microscopic causality," which is called "Local commutativity" on
> p. 597 of the book cited above, and another sort of causality
> sometimes called the "diamond property," also discussed on the same
> page.

>>Why is there any need to introduce new definitions of a term that has
>>a well established meaning? 

To quantum field theorists, who spend all day thinking about
relativity and quantum theory and their relationships, the definitions I
gave ARE the "well established" definitions of causality, and your
definition, which by the way you have never given (certainly not
with any mathematical precision) - is the "new" one.  I have no
problem with multiple definitions of the same word so long as they are
precise and explicitly stated.  It's only when one tosses around words
like "causality" and "influence" claiming that they are well
understood but not providing precise definitions that things get
confusing.  

>Of course you may need to introduce new
>concepts that are variations or related to the traditional term, but
>why redefine it in two different ways. Doesn't this make things
>needlessly confusing and lead to pointless arguments?

"Philosophy is the abuse of language specially designed for that
purpose."  

>I will be happy to bet you $50 that by the year 2005 there will be
>substantial experimental evidence that there is a space-like separation
>between correlated events that would otherwise violate Bell's
>inequality. 

Now I am really confused.  Are you saying that you think Bell's
inequality is violated or is not?



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