EPR-Bohr-Bell and Nonlocality.

John Bell[1] wrote an interesting remark about Bohr's rebuttal
to the argument of Einstein, Podolsky, and Rosen[2]. Commenting
on the statement that was the core of Bohr's argument[3], Bell
wrote:

"Indeed I have very little idea of what this means. I do not
understand in what sense the word `mechanical' is used, in characterizing
the disturbances that Bohr does not contemplate, as distinct from those 
he does. I do not know what the  italicized passage means---
`an influence on the very conditions...' . Could it mean just that
different experiments on the first system give different kinds of
information about the second? But this was one of the main points of EPR,
who observed that one could learn *either* the position *or* the momentum
of the second system..... Is Bohr just rejecting the premise---
`no action at a distance'---rather than refuting the argument?"

I think Bohr's meaning is clear, once the whole situation is laid out
clearly, and that he is rationally refuting the EPR argument by
distinguishing between two kinds of influences, the usual `mechanical'
ones and another more subtle one, and claiming that although
mechanical influences cannot act at a distance the other one can.
The EPR argument is thus refuted  because it is based on a 
`no action at a distance' premise that is stronger than what 
quantum theory allows. 

The EPR argument is a tight one that, given its no-action-at-a-distance
premise, proves that quantum theory would be incomplete.
 
Rosenfeld[4] has described the consternation that the EPR argument
produced in Copenhagen. The EPR argument is very solid, and Bohr
did not immediately see how to refute it. In the end he saw that he had
to deny the no-action-at-a-distance premise, and was faced with the
task of distinguishing between the `mechanical' influences that were required
by the physical demands of theory of relativity to act no faster than
light and another kind influence that was required by quantum theory
to act over spacelike intervals.

To make this all perfectly clear one needs to look closely at the
EPR argument.

The EPR paper was entitled "Can Quantum Mechanical Description Of 
Physical Reality Be Considered Complete? 

The immediate  problem that EPR had to face was that the Copenhagen
philosophy rejected all talk about  `physical reality'. It claimed  that
science, properly understood, recognized that the job of science  is to
make testable predictions about connections between observations:
metaphysical speculations about `physical reality' are outside science. 

EPR cracked that line of defense by proposing a sufficient condition
for the existence of an element of physical  reality that the 
orthodox Copenhagen physicist found hard to deny:

"If, without in any way disturbing a  system, one can predict with
certainty the value of a physical quantity, then there exists an
element of physical reality corresponding to this physical quantity."

They then found a quantum system consisting of two far apart subsystems 
such that if an experimenter measured property 1 of the first subsystem 
then he could predict with certainty the value of property 1' of the
second subsystem, and if, alternatively, he measured property 2 of
the first subsystem then he could predict with certainty the
value of the property 2' of the second system.

Thus the experimenter can predict with certainty the value of *either* 
property 1' *or* property  2' of the faraway system. Assuming "no action at a 
distance" one can conclude from the `EPR criterion of physical reality'
stated above that the value of *either* property 1' *or* property 2' 
can be an element of physical reality, and the choice of which one of the two 
properies  of the faraway system is real lies in the hands of the experimenter
situated here.  

In their key penultimate paragraph EPR note that the quantum theorist
could say that EITHER property 1' OR property  2' is an element of reality,
BUT NOT BOTH. Their  answer is this"

"This makes the reality of P and Q [read 1' and 2'] depend on the 
process of measurement carried out on the first system, which does not
disturb the second system in any way. No reasonable definition of physical
reality could be expected to permit this."

In short, if 1' OR 2' is an element of reality, and the choice of
which one is real is determined by an action that cannot affect
faraway realities, then BOTH 1' and 2' must be real. But everyone
agrees that quantum theory would then be incomplete: it cannot
accommodate both values simultaneously.

This is the EPR argument.

Bohr denies this final step in the EPR argument: he recognizes
that according to quantum theory what the experimenter does here
affects the available information about the faraway system: it affects
the predictions that can be made about the faraway system. 

Thus from the Copenhagen perspective what one chooses to do here
does affect the faraway reality in this situation.

But this subtle action at a distance in no way contradicts
the physical demands made by the theory of realivity. Those demands
are that no "controllable signal" of "message" can be transmitted
faster than light. That sort of influence is carried by matter/energy,
and can rightly be called `mechanical'. But quantum theory is
basically about `available information', and the available information
about a system can be affected by a faraway observation.

Bohr's argument is abstract, but the whole situation is made
manifest and concrete by the von Neuman-Tomonaga-Schwinger
objective formulation of quantum theory, where an observation
in  one region is associated with an action S-->PSP,
where S is  the state (density matrix) before the observation, 
and P is a projection  operator that acts on the directly observered
system. If P acts on one part of an entangled system then the
reduction S-->PSP instantly affects the other part, where
"instantly" is defined by some Tomonaga-Schwinger spacelike
surface.  Since the Bohr reply to EPR recognizes the need for
these faster than light influence in quantum theory, it seems to me
more sensible to incorporate them explicity into a concrete objective 
formulation, than to engage, as is often done, in heroic attempts to 
try to deny their existence.

The contibution of Bell to this discussion is that he ruled out Einstein's
idea of local realism, and thus buttressed Bohr's solution:

"Of course, there is in a case like that just considered no question
of a mechanical disturbance of the system under investigation during 
the last critical stage of the measuring procedure. But even at this  
stage there is essentially the question of *an influence on the very 
conditions which define the possible types of predictions regarding the  
future behaviour of the system.* Since these conditions constitute an inherent
element of the description of any phenomenon to which the term `physical
reality' can be properly attached, we see that the argumentation of the 
mentioned authors does not justify their conclusion that quantum-mechanical
description is essentially  incomplete."

This is the core of Bohr's argument that Bell was connenting upon.

It is clear that Bohr is saying that although the  measurement has no
`mechanical' effect on the faraway system, it does have some sort of 
influence. He is saying that by disturbing the conditions that
control the possible predictions (available information) one is, from 
the quantum point of view, disturbing "reality". Thus the EPR criterion
of physical reality can be accepted, since it is hard to deny, and the
consequent claim that 1' OR 2' could become an element of reality, depending
or what the experimenter does. can also be accepted. But because
the available information about  the (faraway) system is disturbed
by the (nearby) measurement one cannot, according to quantum thinking,
justify the final step in the EPR argument.