From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Hidden variable theories
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Sun, 13 Sep 92 06:05:38 GMT

>Paul Budnik writes:
>In article <1992Sep12.194540.16125@galois.mit.edu>, jbaez@riesz.mit.edu 
(John C. Baez) writes:
>Having redefined causality are physicists now in the process of redefining
>influence? Of course its true that you can only give a mathematically
>precise definition of these terms from within a mathematically
>precise model. However the model that is appropriate for discusing experimental
>tests of Bell's inequality is classical. After all the purpose of the
>experiment is to test the correctness of quantum mechanics and it is only
>macroscopic observations (that classical theory deals well with)
>that are relevant to the experiment. Within this model I believe
>these terms have well defined and accepted meanings.

Which for some reason you are loath to tell us....

I think you could define causality in a manner suitable for classical mechanics
in such a way that quantum mechanics violated it, if you wished to do so.  (I
would still like to see a precise definition, though.)  The definition I gave 
in terms of the "diamond property," however, applies equally well to
classical and quantum theory, both classical and quantum field theory
turn out to be causal under this definition, and frankly it's a pretty
darn simple and nice definition.  Perhaps my mathematical jargonizing
made it sound more mysterious than it need be - I was trying to be
precise, but maybe I should be heuristic instead.  Let me give it one
more time, just so everyone out there gets it, and then call it quits.
It is perfectly clear to me that we are not arguing about facts, only
hopes, fears, and definitions, so this will be my last post on this. 
Really, what follows is not so much for Paul as for folks who were
wondering what I meant by the sort of causality I called the "diamond
property."  

Okay, we are in Minkowski space, which I'll just call "spacetime."  I
will call open sets of spacetime "regions".  We say that a region A is
in the "causal shadow" of a region B if every timelike geodesic
passing through A hits B.  One should think of B as being in the past
(or future) of A, and all I am saying is that anything moving slower than 
the speed of light that passes through A has to have gone through B.  
If this condition is met, and nothing moves faster than light, we
would naturally expect that everything going on in A will be
determined - to the extent that it's determined at all! - by what went
on in B.  So we say a theory is "causal" or satisfies the "diamond
property" if the state of affairs in A is determined by the state of
affairs in B whenever A is in the causal shadow of B.  

Here I am being deliberately sloppy: one has to have a mathematical
formalism to make precise the notion of a "state of affairs in A".
In classical mechanics this would be a measure on a phase space, while
in quantum theory it would be a density matrix on a Hilbert space or,
in a more sophisticated approach, a (normalized) positive linear
functional on a C*-algebra.  (See the Haag-Kastler stuff.)  

In any event, knowing the "state of affairs in A" in a classical or 
quantum theory tells you how to calculate the probability that any
experiment that you can perform *just inside A* will yield any
outcome.  For example, A might be "this room, from now until ten
minutes from now".  Knowing the state of affairs in A would tell me
everything there is to know about how likely it is, for example, that
I will measure the spin of some photon to be +1 in this room 5 minutes
from now.  

So, quantum field theory as we know it satisfies the diamond property. 
(It's called that, by the way, because if you draw the causal shadow
of a given region in a 2d spacetime diagram it looks like a diamond.)  
If Sam sends me a photon from many lightyears away, and I measure it
5 minutes from now, nothing he does "right now" can affect the
probability that if I measure X about the photon that I get the answer
Y.  I put "right now" in quotes because this is ambiguous in
relativity; what I really mean is "outside my past lightcone".  

Now it is true in quantum field theory that if Sam repeatedly makes a 
pair of photons, sends one on to me for experimentation and plays around with 
the other himself, and later on Sam and I compare notes, we can get results 
that aren't consistent with classical mechanics and causality.  To be precise, 
they can violate Bell's inequality.  Of course, this is no big surprise,
really, since classical mechanics is wrong!  :-)  We do NOT - at least,
nobody ever has - get results that violate the diamond property.

>> 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.  

>So you *have* redefined influence! 

No, it would only count as REdefining it if you or someone else had
already defined it!  In fact, when people started thinking about
causality in relativity they rather quickly arrived at this notion.  
Haag and Kastler merely stated in C*-algebraic language something that
had been floating around for quite some time.  

>If I am correct about Bell's inequality
>not being violated these new definitions will be in the dustbin of history
>before you know it.

If.

>Always keep in mind that the rich and elegant
>mathematics that you are working with is different from physical reality.
>You do not not know what physical reality is `really like' nor does anyone 
>else.

I don't recall saying I knew what physical reality is "really like" -
if I did, I take it back.  :-)

>There is a philosophical disagreement that underlys our disagreement
>about specific questions. There are certain aspects of physical phenomena
>that no current theory can explain in a way consistent with conventional
>notions of determinism, mechanistic models etc. Most physicists
>have concluded that therefore there must be no explanations in those
>terms and they are free to redefine fundamental concepts related to these
>classes of models in a way that best suits their current theories. I think
>you are jumping the gun. You certainly haven't considered all possible
>models or even all plausible models.

Naturally.  Life is short.  I work on the models I do because I
consider them the most likely.  If I'm wrong, so be it.  It's a
calculated risk.  

>What we know to be true are experimental results in a macroscopic universe.
>In those terms Bell's inequality presents us with a result that seems
>most improbable. We cannot assume that observations made at one site
>are independent of manipulations made at a distant site. This is true
>even if the observations and manipulations are separated by billions
>of light years. Yet there is not a shred of experimental evidence
>to suggest that this affect occurs. The theoretical basis for
>it is the instantaneous collapse postulate, which was simply asserted
>to be true, possibly because anything else would have implied that quantum
>mechanics is an incomplete theory. 

I *certainly* don't believe in "instantaneous collapse of the wave
function," and have inveighed against it repeatedly on the net.  Now
is not the time for another round of *that*.  But briefly, A) I don't
believe in "collapse of the wavefunction" and B) "instantaneous"
collapse does not make sense in special relativity, since simultaneity
of distant events is observer-dependent.  

But one futile argument at time, eh?

>Isn't it possible that you are taking your models too seriously?

If they turn out to be wrong, no doubt I'll kick myself and agree with
you, being the emotional sort of guy I am.  

>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 such a subtly phrased bet?  Why don't you come out and bet that there will
be substantial experimental evidence that Bell's equality HOLDS
whenever the events are spacelike separated?  

I would be glad to bet $50 that the preponderance of tests of Bell's inequality 
up to year X (you name it) will show that it is VIOLATED.  Name a year
by which you think they will have done the experiment right, so that
the events are really spacelike separated.

To repeat, I'm tired out and will only continue to post on one aspect
of this business - the bet.  Enough talk, time to lay down your bets, folks!


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