From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Defining local, causal etc., was: Hidden variable theories
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Mon, 14 Sep 92 19:53:31 GMT

paul@mtnmath.UUCP (Paul Budnik) writes:
In article <1992Sep13.060538.19628@galois.mit.edu>, jbaez@riesz.mit.edu 
(John C. Baez) 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.  

I will not get into another argument about *this*, I'm tired out, and
I don't think it has anything to do with the point I was trying to
make, I just offer it in case you are interested in this subject and
didn't know of Chaitin's work.

>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 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.

>> 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.

>And no one has gotten results that violate Bell's inequality either, because
>no one has shown that the correlated events have a space-like separation.

Yes, that's why I said "in theory".

>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.

>Do you see any problem in applying the Oxford dictionary of influence
>to classical mechanics in the way I just described? Do you not think that
>in applying the word to quantum mechanics one should make the definition
>consistent with how it is used in classical mechanics if this is possible?

See above.  I believe the use of "causal" in QFT is the obvious
application of your definition to the case when a state is modelled as
it is in quantum theory.  In quantum field theory, the future state at a given
point is determined by the previous state in a local region surrounding 
this point.  I think your objection may really be an objection to how
I use the word "state" (see above for how I do).

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

>> >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.

>The problem with your wording is that I do not think any tests have shown
>that it is violated.

I agree.  Neither of us would win my bet today.  What's the problem?

>All the experimental evidence to date is that it holds.

Come on now, according to you there is simply no evidence either way,
since nobody has gotten things to be spacelike separated.

>Experimental results consistent with quantum mechanics are not the same
>thing as experimental results against Bell's inequality, unless you have
>more faith in existing formulations of quantum mechanics then you do in
>locality. 

I do, as a matter of fact, if we mean "locality" in one of your senses.
But I would just say that there is no good experimental evidence
either way.  Here I am simply trusting your word, by the way; I
haven't looked at the literature.

>This is a matter of judgement or religion and not a good basis
>for deciding a wager. You need to show the events are space-like separated
>by direct measurement, if you want to claim you have evidence against Bell's
>inequality. By your definition and my reasoning I could name the year 1992
>and would almost certainly win the bet.

>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.