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February 8, 2001 \hfill LBNL-46870 \\
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{\large \bf Reply to ``On Stapp's `Nonlocal Character of Quantum Theory' ''}
\footnote{This work is supported in part by the Director, Office of Science,
Office of High Energy and Nuclear Physics, Division of High Energy Physics,
of the U.S. Department of Energy under Contract DE-AC03-76SF00098}
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Henry P. Stapp\\
{\em Lawrence Berkeley National Laboratory\\
University of California\\
Berkeley, California 94720}
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\begin{abstract}
The question raised by Shimony and Stein is examined and used
to explain in more detail a key point of my proof that any
theory that conforms to certain general ideas of orthodox
relativistic quantum field theory must permit transfers of
information over spacelike intervals. It is also explained
why this result is not a problem for relativistic quantum
theory, but, on the contrary, opens the door to a satisfactory
realistic relativistic quantum theory based on the ideas of
Tomonaga, Schwinger, and von Neumann.
\end{abstract}
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Shimony and Stein[1] have raised a question about an essential claim made in
my 1997 paper[2]. I begin by explaining the claim, and the question they
raised.
Lines 1 through 5 of my proof[2] show that under certain
explicitly stated conditions the statement
$$
L2\Rightarrow [(R2\wedge R2+)\rightarrow(R1\Box\rightarrow R1-)] \eqno (1)
$$
is true, while lines 6 through 14 of that proof show that
under these same conditions the statement
$$
L1\Rightarrow [(R2\wedge R2+)\rightarrow(R1\Box\rightarrow R1-)] \eqno (2)
$$
is false. Shimony and Stein arrive at the same conclusion---namely
that (1) is true and (2) is false---under similar
conditions. I then claim that this fact, that (1) is true and (2) is
false, entails that information must sometimes be transferred over
space-like intervals. Shimony and Stein question this claim, and suggest
that one must make a hidden-variable assumption, as was done in
Bell's theorems [3,4], in order to arrive at this strong conclusion.
This issue is important, because all the assumptions used my proof are
elements of orthodox quantum philosophy, and hence my claim, if valid,
means that the precepts of orthodox quantum philosophy entail that
information must sometimes be transferred over spacelike intervals. That
conclusion is far stronger than what is proved by Bell's theorem[3],
and its usual generalizations[4,5], and it seems to have profound
implications for development of relativistic quantum theory.
To provide an adequate foundation for the discussion I need to
explain the meanings of (1) and (2), the assumptions that go into my
proof that (1) is true and (2) is false, and the technical differences
between my assumptions and those of Shimony and Stein.
The conditions under which I prove that (1) is true and
(2) is false are these:\\
A. The choices made by the experimenters in each of the two
regions {\bf R} and {\bf L} about which experiment will be
performed in that region can be treated as {\it free choices},
or {\it free variables}.
B. There is at least one Lorentz frame of reference, call it LF,
such that if in that frame every point of the spacetime region {\bf L}
is earlier than every point in the spacetime region {\bf R} then for any
experiment freely chosen and performed in the earlier region {\bf L}
the outcome that appears to observers in that region can be taken to be
independent of which experiment will be freely chosen and performed
later in the region {\bf R}: the universe can be regarded as evolving
forward in time in LF and, in particular, there is no action of a
free choice made {\it later} in {\bf R} upon an outcome that has
{\it already appeared earlier} in {\bf L}.
This assumption is called LOC1.
C. No matter which experiments are freely chosen and performed,
the predictions of quantum theory will be satisfied.
These assumptions are, I believe, compatible with the precepts of
orthodox quantum thinking, and are, in a broad sense, entailed
by them.
Notice that the truth of certain very special contrary-to-fact assertions
is entailed by these assumptions. In particular, if the set of
possible worlds is limited by conditions A, B, and C then,\\
SF: For any possible world $W$, the following statement is true:\\
If the situation in $W$ is such that\\
1. The Hardy experimental conditions are satisfied,\\
2. Experiment L2 is freely chosen and performed in {\bf L},\\
3. Experiment R2 is freely chosen and performed in {\bf R}, and\\
4. The outcome $L2+$ appears in {\bf L},\\
then in any possible world $W'$ that is the same as world $W$
except for possible consequences of choosing and performing in
region {\bf R} the experiment R1, instead of the experiment
chosen and performed in {\bf R} in world $W$, the outcome
in the earlier region {\bf L} is $L2+$.
The result asserted by SF is immediately entailed by the stated
assumptions, actually just A and B, and it is expressed symbolically as
$$
(L2 \wedge R2 \wedge L2+)\Rightarrow (R1\Box \rightarrow L2+).\eqno(3)
$$
This statement asserts, in brief, that if the theoretical
conditions A, B, and C are satified then freely choosing
and performing R1, instead of R2, in the later region {\bf R},
leaves the earlier outcome in {\bf L} undisturbed.
Similarly, the symbolic statement (1) asserts that, under the
three conditions A, B, and C, if experiment L2 is freely chosen
and performed in the earlier region {\bf L} then a certain statement
SR is true, whereas statement (2) makes the same claim under
condition L1. Thus the conjunction of the facts that (1) is true
and (2) is false implies that the truth of statement SR depends
nontrivially on which of the two alternative possible experiments,
L1 or L2, is freely chosen and performed in the space-time
region {\bf L}.
The statement SR just mentioned is represented symbolically by
$$
[(R2\wedge R2+)\rightarrow(R1\Box\rightarrow R1-)] \eqno (4)
$$
It is an assertion about a possible world $W$, and it states \\
\noindent SR: If in the possible world $W$ the experimenter in
the space-time region {\bf R} freely chooses and
performs experiment R2 and gets the outcome $R2+$, then
in any possible world $W'$ that is the same as world $W$
except for the possible consequences of choosing in the
region {\bf R} the experiment R1, instead of
whatever was chosen in $W$ (namely R2), the outcome in
{\bf R} is $R1-$.
In reference [2] I justified each step in the proof that
statement (1) is true and statement (2) is false by using
the machinery of David Lewis's rules of reasoning with
counterfactual statements. The Lewis machinery is reasonable
and orthodox, but was created in the climate where the ideas
of deterministic classical physics prevailed, and in the end
it is merely a set of conventions designed to cope in a
deterministic setting with the idea that something other than
what actually happens `could have happened'. The conventions
are designed to mesh with our intuitions about the
proper use of contrary-to-fact statements, but there are
other contending rules, and the whole situation is somewhat
controversial. But as I have emphasized, and Shimony and
Stein have agreed, the quantum situation permits a more
direct approach, which avoids leaning on the basically
conventional features of the classical approach. Instead,
one can exploit the fact the concept of a `free choice' is
compatible with quantum theory, due to its basically
indeterministic character. This allows one to stay with
ordinary logic plus a natural specified meaning for the needed
counterfactual assertion.
In order to have a common ground for dealing with the
concerns of Shimony and Stein I shall, in this paper, adopt
this alternative approach, which is strictly in line with
quantum thinking, rather than relying on Lewis's classical
rules. However, apart from this technical change, I shall
adhere to the logical form of the argument that I used in
reference [2]. In particular, I shall retain the following
natural meaning of the statement
$(E\Box \rightarrow O)$:\\
$(E\Box \rightarrow O)$ is, by definition, true in a possible world
$W$ if and only if outcome $O$ occurs in any possible world
$W'$ that is the same as world $W$ except for the possible
consequences of freely choosing and performing experiment E
instead of the alternative experiment freely chosen and performed
in $W$. The set of possible worlds is limited by the
specified conditions A, B, and C.
To use this definition one must limit ``the possible
consequences of ...'' This is always done by using LOC1.
Since this definition is toothless without this condition LOC1,
or some such condition, and since LOC1 is used only in
connection with this definition, it is not unreasonable to
incorporate LOC1 into the definition of the counterfactual
statement. Shimony and Stein have done essentially that.
However, they did not do {\it exactly} that. My condition
LOC1 excludes from the effects of changing a free
choice {\it only} effects on outcomes that have already
appeared {\it earlier}, in the special Lorentz frame LF.
But Shimony and Stein exclude all effects that lie outside the
forward light-cone of the region in which the change in
the free choice occurs.
I use the weaker assumption LOC1 because the truth of LOC1 is certainly
compatible with the principles of relativistic quantum field
theory, and is in fact entailed by them, whereas the stronger form used
by Shimony and Stein is incompatible those principles. It is much
clearer to argue directly from assumptions that are true, in the sense
of being consequences of orthodox quantum theory, rather that making an
assumption that is incompatible with relativistic quantum theory.
The fact that LOC1 is entailed in orthodox relativistic quantum
field theory is proved by noting that the possibility of defining
{\it one} such frame LF follows from the Tomonaga-Schwinger [6,7] formulation
of relativistic quantum field theory, in which advancing space-like
surfaces are the analogs of the advancing constant-time surfaces
of the non-relativistic formulation of von Neumann [8]. Of course,
an infinitude of {\it alternative} possible choices for LF
can be found: {\it any} frame will do. But the required property
follows for {\it only one frame {\bf or} another}, not for any two
or more together.
With the stage thus set, I can turn to the central question
of whether the conjunction of the truth of (1) and the
falsity of (2) can be reconciled, as Shimony and Stein appear to
suggest, with the idea that {\it no} information about the choice made
in region {\bf L} can get to region {\bf R}, which is situated
spacelike relative to {\bf L}.
To see the apparent conflict one can consider the consequence
of the fact that (1) is true and (2) is false in the context
of the orthodox idea that ``nature chooses the outcome'' of
the experiment chosen by the experimenter. In this context
the consequence of the truth of (1) and falsity of (2) is that
SR asserts the existence of a definite theoretical connection
between the outcomes that nature delivers under the two alternative
possible conditions, and that this theoretically necessary
condition on what nature can do in {\bf R} depends
nontrivially on which experiment is freely chosen and
performed by the experimenter in $L$.
But how can {\it any} theoretical model---hidden-variable or not---
fulfill conditions on Nature's choices in region {\bf R}
that depend nontrivially on which free choice is
made in {\bf L} if no information about this choice made
in {\bf L} can be present in {\bf R}?
This apparent result, that any theoretical model
that conforms to the conditions A, B, and C
must accomodate transfers of information over a
spacelike intervals, does not conflict with the
requirements of the theory of relativity, {\it in
the context of quantum theory}: it conflicts only
with a certain prejudice generated by uncritically
extending to indeterministic quantum theory a feature of
its deterministic classical approximation. This prejudice
has, in fact, been the barrier that has blocked for many years
the creation of a satisfactory realistic formulation of
relativistic quantum theory. In a quantum context the
Lorentz requirements of relativity theory pertain exclusively
to relationships among observables, not to the reality that
lies behind the phenomena. Thus the obvious realistic,
relativistic quantum theory is just relativistic
quantum field theory with a preferred sequence of advancing
Tomonaga-Schwinger [6,7] spacelike surfaces defining the
successive instants ``now''.
This entails, of course, a
reversion to the pre-relativity Newtonian idea of an
absolute time, or something similar to it, at the underlying
ontological level. But the founders of quantum theory
strongly stressed the fact that this theory, as they
conceived it, was only about relationships between
observations, not about properties of the underlying reality.
The Tomonage-Schwinger theory maintains all the observable
requirements of the theory of relativity, no matter how the
preferred sequence of advancing spacelike surfaces
is chosen. Hence the only thing actually blocking acceptance
of this theory as the relativistic quantum theory of reality
is the prejudicial assumption that the reality itself, like
the connections between observations, can have no transfer of
information over spacelike intervals. But the fact that
this condition can be maintained in the {\it deterministic}
classical limit, where the entire history of the universe
is determined by the initial conditions, and can immediately
be laid out on a space-time background, with no free choices
allowed, does not entail that it can be maintained in the
full indeterministic theory with free choices allowed.
The analysis of the Hardy case supports the view that the
reality behind the indeterministic quantum rules cannot
maintain this constraint. That observation immediately
elevates John von Neumann's [8] formulation of quantum theory,
applied to Tomonaga-Schwinger relativistic quantum field
theory, to prime candidacy as the paradigm relativistic
quantum theory of reality.\\
Shimony and Stein allege that this apparent result---that
the information about whether L1 or L2 was freely
chosen and performed in region {\bf L} must be available in region
{\bf R} of---is incorrect. They base their argument on the assertion
that {\it the semantical truth conditions for the counterfactual in
question refer explicitly to the entire exterior of the extended future
light-cone of {\bf R}}.
That claim about the {\it entire} exterior is not exactly true
in my version of the proof. The statement SR combine with LOC1 says:\\
\noindent SR-LOC1: ``If in the possible world $W$ the experiment
R2 is freely chosen and performed in {\bf R} and the outcome
there is $R2+$ then if $W'$ is a possible world that is the
same as $W$ in {\bf L}, but in which R1 is freely chosen and
performed in {\bf R}, instead of R2, the outcome in {\bf R}
in world $W'$ is $R1-$.
In spite of the difference between the light-cone version
of the causality condition used by Shimony and Stein and
the condition LOC1 used by me, this combined statement SR-LOC1
exhibits the feature pointed to by Shimony and Stein: a reference
to the region {\bf L}, which lies outside the forward light cone
of the region {\bf R}. It is this implicit reference of SR to {\bf L}
that Shimony and Stein are concerned about. The question is whether
this reference to {\bf L} upsets my essential claim that the conjunction
of the truth of (1) and the falsity of (2) requires the information
about whether L1 or L2 is performed in {\bf L} to be present in
{\bf R}.
Let me begin my answer by explaining the question in more detail.
The statement SR involves the words ``instead
of''. We have a clear idea of what we mean here
by ''instead of''. In the real situation the experimenter
in {\bf R} makes the choice R2. But we have assumed that, just
at the moment of choosing, the other choice R1 could have popped
out instead of R2. But the central idea is that everything
{\it prior to that moment of choosing} is exactly what it is
in the actual world: there is just {\it one} evolving quantum world,
which could go either way at the moment of choice.
This condition of {\it sameness prior to the moment of choice}
is the condition that limits the changes permitted by the phrase
``except for the possible consequences of the change in
the free choice'': no possible consequence of a changed choice can lie
earlier than the moment of choice.
The point raised by Shimony and Stein, as applied to my
argument, is that this implicit reference to the (unchanged)
state of affairs (in {\bf L}) prior to the moment of the choice
between R1 and R2 is an essential element of the very idea
of ``instead of'' that appears in the statement SR. Hence
there is in SR an essential implicit reference to region {\bf L},
even though all the symbols explicitly appearing in SR pertain
to possible events in {\bf R}.
Their concern about this implicit reference to {\bf L} stems from
the fact that in my 1997 paper I based my argument---for the claim that the
conjunction of the truth of (1) and falsity of (2) entails a
violation of the idea that ``observable effects can propagate
only into the future (light-cone)''---on the fact that
``everything mentioned in SR is an observable phenomenon in region
{\bf R}.'' Their concern is that the essential implicit reference
of SR to the region {\bf L} might upset my argument.
This essential implicit reference of SR to {\bf L} does not
affect my argument. To understand why it does not, one must
note that the steps in a logical argument are like a series
of black boxes, each of which displays explicitly only certain
of the variables of the system. These explicitly displayed variables
are like inputs and outputs: certain connections between
these variables are exhibited, but the reasons why these connections
hold are not shown. However, all of the relevant effects pertaining to
the inner workings must be controlled by the displayed variables.
In the statement (1),
$$
L2\Rightarrow [(R2\wedge R2+)\rightarrow (R1\Box\rightarrow R1-)],
$$
the only displayed variables are $L2, R2, R2+, R1,$ and $R1-$.
The input conditions are $L2, R2, R2+,$ and $R1$, and the output is $R1-$.
The statement asserts that if the input variables $L2, R2, R2+,$ and $R1$
are put into a certain logical expression, the output must be $R1-$,
never $R1+$, But the falseness of (2) says that if the inputs are changed
only by changing L2 to L1, then the output is no longer restricted to $R1-$:
it is now allowed to be $R1+$. So changing the input variable from L2
to L1 has affected the output variable $R1+/R1-$. There can be all
sorts of dependence on all sorts of inner variables, but whatever these
dependences are they {\it must}, to the extent that they are relevant
to the output conclusion, be controlled by the input variables, if
the statement is indeed logically correct. So, in this case at hand,
changing the input variable L1/L2 affects nontrivially the output
variable $R1+/R1-$. But then the information about whether L1 or L2 is
chosen in {\bf L} must get to the region {\bf R} where the value of the
output variable $R1+/R1-$ is displayed.\\
{\bf Reply to Part II.}
In their part II Shimony and Stein say that they prefer
their covariant form of the locality condition ``because
of its relativistic invariance, which is {\it demanded} by
relativity theory and hence should be respected in an
investigation of the compatibility of quantum mechanics
and that theory."
First of all, I must emphasize that I am not arguing that
``quantum mechanics implies a nonlocality that is
inconsistent with the locality of relativity theory" as
Shimony and Stein assert in their abstract. My intent is
rather to provide support for a reconciliation of quantum
theory with relativity theory, a reconciliation that
de-mystifies the ``mysterious actions at a distance''.
The nonlocality that I claim to exhibit is completely
compatible with the locality properties of relativity theory,
which, in a quantum context, pertain only to features of
{\it our observations}, not to features of a putative
underlying reality. I adopted the weaker
locality condition, which involves a preferred set of
spacelike surfaces, in order to have a condition
that is {\it provably compatible} with relativistic quantum
field theory. One can prove this compatibility from an
examination of the Tomonaga-Schwinger formulation of
quantum theory, which is built on the fact that even in
a fully relativistic quantum field theory the quantum
state of the system is defined as the state associated
with a spacelike surface. In Schwinger's words:
``The problem of constructing a complete set of commuting
operators, that is, of simultaneously measureable physical
quantities, necessarily involves specific properties of
the fields. Nevertheless, as a general principle associated
with relativistic requirements, we must expect such mutually
commuting operators to be formed from field quantities at
physically independent space-time points, that is, points
which cannot be connected even by light signals. A continuous
set of such points form a spacelike surface, which is a
geometric concept independent of the coordinate system.
Therefore, a base vector system $\Psi(\zeta', \sigma)$
will be specified by a spacelike surface $\sigma$ and by
the eigenvalues $\zeta'$ of a complete set of commuting
operators constructed from field operators attached to
that surface. A change of representation will correspond,
in general, to the introduction of another set of commuting
operators on a different spacelike surface. ...
A description of the temporal development of a system
is evidently accomplished by stating the relationship
between the eigenvectors associated with different
spacelike surfaces, or, in other words, by exhibiting
the tranformation function (2.5).''
The outcome of the work of Tomonaga and of Schwinger
is a relativistic quantum theory that generalizes the
nonrelativistic theory by replacing the advancing sequence
of constant-time surfaces of the latter theory by
an advancing sequence of spacelike surfaces $\sigma$,
and the set of states $\Psi (\zeta', t)$ of the
nonrelativistic theory by a set of states
$\Psi (\zeta', \sigma)$. By imposing appropriate
boundary conditions one can ensure an evolution of the
state that leaves the past fixed but the future open,
in the sense that the measurements can be freely chosen,
and a von Neumann Process One applied at each measurement,
with the consequent change of the state occurring
on a spacelike surface $\sigma$ rather than on a surface
at constant time t. In the context of a quantum theory defined
over a space-time manifold defined by general relativity
it is natural to use as the preferred sequence of
surfaces $\sigma$ the constant-time surfaces of the
Robertson-Walker metric. In a special-relativity context
it is most natural to use the surfaces at constant times
in the frame in which the cosmic background radiation is
isotropic. But the key point, in the present context,
is that one can ensure no backward action for any
{\it one} sequence of advancing spacelike surfaces $\sigma$,
but not in general for two different sequences simultaneously.
I have defined my locality condition so that it is relativistic
in the sense that it is compatible with relativistic quantum
field theory, and in particular with the Tomonaga-Schwinger
formulation of relativistic quantum field theory.
Shimony and Stein note that I have ``abandoned the enterprise
of justifying LOC2." That is correct. In my 1997 paper I
tried to justify LOC2, and then prove that it was was false,
in a reductio ad absurdum strategy. Although that form of
argument is logically correct, it is needlessly complicated.
In my reply to Shimony and Stein I have exploited the fact
that they confirm that my claim that (1) is true and (2) is
false follows from my premises. Proving that fact was the main
focus of my 1997 paper. Given that basic result it is simpler
to argue straightforwardly from premises that are compatible
with relativistic quantum field theory, and that means
jettisoning both LOC2 and the covariant formulation of the
locality condition.
My argument was, and continues to be, based on the
fact that ``everything mentioned in SR is an observable in
{\bf R}". Shimony and Stein based their challenge on the
fact that the statement SR has a certain potential
{\it implicit} reference to region {\bf L} built into it.
In order to identify their concern I introduced in my reply
a statement, SR-LOC, that explicitly exhibits the reference
to region {\bf L} that Shimony and Stein are concerned
about. However, all that I use in my argument, or logically
need, is the fact that the explicit condition for the truth
of statement SR is that [for {\it any} (unnamed) free choice
made in {\bf L}] if the free choice in {\bf R} is $R2$ and
the outcome there is $R2+$, then in any conceivable allowed
world in which $R1$ is freely choosen instead of $R2$, the
outcome must be $R1-$. My input-output analysis makes the
following point: the fact that (1), the truth or falsity of
this statement SR is, for any fixed choice made by the
experimenter in {\bf L}, determined explicitly by whether or
not a certain conceivable event, R1-, must occur in {\bf R}
under conditions defined in {\bf R}, coupled with the
agreed-upon fact that (2), the truth of SR depends upon which
choice is made by the experimenter in {\bf L}, means that
whether or not this conceivable event in {\bf R} must occur
depends upon which choice is made by the experimenter in
{\bf L}. This dependence of whether or not a certain event
must occur in {\bf R} upon the free choice made in {\bf L} is
a necessary constraint on any theory or model that satisfies
the assumptions of my non-hidden-variable theorem.
It is of course true that in order for the condition
for the truth of SR to necessarily hold if $L2$ is freely
chosen in {\bf L}, and to necessarily not hold if $L1$ is
chosen in {\bf L}, there must be some logical linkage
between what can occur in {\bf R} and the free choice
made in {\bf L}. This linkage comes in via the various
conditions of the theorem. The no-backward-in-time-
influence condition specifies that the free choice made
later in {\bf R} can have no effect on which outcome
appeared already earlier in {\bf L}. This condition
together with the quantum predictions impose a set
of constraints that link the outcomes in {\bf R}
to the free choices made in {\bf L} in the specified
way: R1- must occur if $L2$ is freely chosen but need not
occur if $L1$ is chosen.
In the way that I have explicitly formulated things the
statement SR refers {\it only} to conceivable possible events
in {\bf R}. Then the conditions of the theorem impose
constraints that link the truth of SR to the free choice
made in {\bf L}, thus entailing that, under the conditions
of the theorem, the information about the free choice made
in region {\bf L} must get to region {\bf R}.
On the other hand one, could alter the logic slightly and
incorporate the locality condition immediately into the
definition of SR, so that the condition of doing $R1$
{\it instead of} $R2$ would already incorporate the condition
that no {\it outcome} in {\bf L} would be affected by this later
change of the free choice made in {\bf R}.
Shimony and Stein base their challenge on this
alternative way of organizing the proof. However,
this slight change in the order of introducing the
constraints of the theorem should not affect the
conclusion. The net effect is unchanged: the
truth of a statement about a conceivable possible event
in {\bf R} is logically linked by the assumptions of the
theorem to the free choice made in {\bf L}.
Of course, the very fact that within the framework supplied
by the assumptions of the theorem one can prove that the
truth of a certain statement about what can occur in
{\bf R} depends on the free choice made in {\bf L}
means that the constraints imposed by the assumptions of the
theorem {\it must} link these possible events. It is no
valid objection to the conclusion to point to the occurrence
of such a dependence. What is relevant is not the fact
that such connections are introduced by the premises
of the theorem, but rather the fact that in the context of
the Hardy experiment the combination of these premises makes
the linkage between these possible events so strong.
In short, I believe that by basing the proof directly in the
proven truth of Eq. (1) and falseness of Eq. (2), as suggested
by Shimony and Stein themselves, one evades their challenge, which
is rooted in the inherent complication associated with justifying the
false premise LOC2.
Shimony and Stein admit that ``the choice between L1 and L2
does makes a difference of some kind,'' but they suggest
that the proof does not demonstrate the need for the presence
in region {\bf R} of information about the choice made by the
experimenter in {\bf L}, because this dependence may be
simply a "brute fact" about the structure of certain sets
of possible worlds.
The argument of Shimony and Stein is perhaps not altogether
clear at this point, but they are evidently asking for a
proof of the claimed transfer of information within the
formal framework of possible worlds, which we both have used
to give precise meaning to the logical statements.
So let me supply that proof.
The statement in question is;
$$
SR: (R2\wedge R2+)\rightarrow [R1 \Box\rightarrow R1-]
$$
The meaning of this statement SR in terms of possible
worlds, as given in ref. 9, applied to the present case,
is this:
SR is true in world $W$ if and only if the conditions that
$R2$ is performed and that the outcome $R2+$ occurs in $W$
entail that for every world $W'$ such that
(1) R1 is performed in $W'$, and
(2) $W'$ coincides with $W$ for times earlier than
$\sigma$ (where $\sigma$ is an element of the preferred
advancing sequence of spacelike surfaces that separates
``earlier'' from ``later'', and {\bf L} and {\bf R}
are, respectivly, earlier and later than $\sigma$)
the outcome $R1-$ occurs in region {\bf R}.
The assumptions of the theorem have been shown to entail
that this claim that $R1-$ occurs in region {\bf R} is true
provided the set of possible worlds W is restricted to
those in which the choice made by the experimenter in
{\bf L} is L2, but is false if the set of possible worlds W
is restricted to those in which the free choice made by
the experimenter in {\bf L} is L1.
Thus the analysis in terms of possible worlds
asserts that the free choice in {\bf L} is correlated
(within a subset of pairs of worlds fixed by a condition
of no backward-in-time influence) with what can occur in
{\bf R}:
If L2 is selected the outcome R1- must occur in $W'$ but
if L1 is selected the outcome R1- need not occur in $W'$.
So within the theoretical structure created by the
assumptions of the theorem the information about
which experiment is chosen in {\bf L} in $W$ appears
in {\bf R} as the truth or falsity of the assertion that
$R1+$ cannot occur in the related world $W'$.
I say that this conclusion means that any theory that
conforms to the assumptions of the theorem must allow the
information about the choice made in {\bf L} to get to
{\bf R}. But choice of wording is a matter of definition
and terminology. What is important is the use to be made
of the technical result. In this connection one must remember
that the purpose in science of theorems of this kind is to
place conditions on allowed theories and models. My theorem
places a logical condition on a broad class of theories
that do not satisfy the assumptions of the theorems about
hidden-variable theories.
The theoretical result described above shows that theories
that meet certain weak conditions that are fully compatible
with ordinary relativistic quantum field theory, and which
express the general idea that choices made by experimenters
can be treated as free variables that do not affect
outcomes that have already been observered, must permit,
{\it within a logical framework built directly upon those
assumptions themselves,} conditions on whether or not
certain conceivable possible events can occur in the region
{\bf R} to depend upon which choice is made by experimenters
in the region {\bf L}, which is situated spacelike relative
to {\bf R}.
This result is essentially a development of the line of
argument instigated by Einstein, Podolsky, and Rosen, but
formulated now strictly within a relativistic quantum field
theory framework. Once the need for {\it some kind} of
spacelike transfer of information is recognized it becomes
reasonable to consider a theory in which {\it some one} of
the possible advancing sequences of Tomonaga-Schwinger
surfaces $\sigma$ defines the actual temporal evolution of
the universe. The ``mysterious actions at a distance'' are
immediately de-mystified by this theoretical move,
in conjunction with von Neumann's theory of measurement,
generalized by the replacement of the advancing sequence of
constant-time surfaces in von Neumann's nonrelativistic
formulation of quantum theory by {\it some particular}
advancing sequence of spacelike surfaces $\sigma$.
\noindent {\bf Part III. Postscript}
It is gratifying that all of the probing and discussion has
boiled our differences down to this simple, easily stated point.
We seem to be in essential agreement on almost everything. It is,
of course, manifestly obvious that there must be some structure that makes
$SR$ true under the condition that $L2$ is performed in {\bf L} but false
under the condition that $L1$ is performed in {\bf L}, or, equivalently,
that ensures, under certain fixed conditions in {\bf R}, that the
outcome $R1-$, must occur if $L2$ is performed in {\bf L}, but need not
occur if $L1$ is performed there. This means that the difference in the
free choice made in {\bf L} between $L2$ and $L1$ entails, within the
structure provided by the assumptions of the theorem, this difference in
whether or not $R1-$ must occur in {\bf R}.
One may wish to say that this fact that $SR$ is true if the free choice
in {\bf L} is $L2$ but is false if the free choice in {\bf L} is $L1$
means that the ``specific meaning'' of $SR$ is different in the two
cases. As Shimony and Stein assert, my statement that ``X must occur''
means ``X is true in every one of a class of possible worlds
that is picked out by a certain protocol.'' But the only difference
in the two protocols is a difference in the free choice made in
region {\bf L}, whereas the resulting change in what ''must occur''
is a conceivable event located in region {\bf R}. This is the transfer
to {\bf R} of information about the free choice made in {\bf L} that I am
talking about.
Shimony and Stein counter that this ``conceivable event located on {\bf R}''
is not well identified, because it is connected to the protocol,
which involves the free choice made in {\bf L}. But that is exactly the
point: the conceivable event in {\bf R} identified by this protocol
``must occur'' when the free choice in {\bf L} that enters into this
protocol is $L2$, but need not occur when that free choice is $L1$,
and this latter difference of the free choice made in {\bf L}
is the sole difference between the two protocols.
The bottom line, of course, is whether this theorem, whose assumptions
are, unlike those of the hidden-variable theorem, compatible with
the principles of quantum theory, rules out possible models of
quantum reality. I have given an example that is ruled out, namely
the model where nature chooses outcomes in each given region
on the basis of information available in that region, with information
about the free choices made in any region confined to the closed forward
light-cone from that region.
Thus the theorem has interesting non trivial consequences.
Shimony and Stein suggest that I have made some ``concession'',
which I, however, view as a simply a helpful observation.
Shimony and Stein assert that my reply contains a ``surprising
modification of his position in 1997.'' But a careful reading
of my statements shows that this is not the case. When I asserted
in this paper that ``I am not arguing that `quantum mechanics implies
a nonlocality that inconsistent with the locality of relativity
theory' '' I was emphasizing that in quantum mechanics the locality
of relativity theory pertains to {\it connections between observations,}
and the nonlocality asserted by my theorem does not conflict
with that locality.
Shimony and Stein do not see the connection between my main
objective, which is to ``de-mystify'' the ``mysterious actions
at a distance'', and my claimed strengthening of Bell's theorem.
The connection is this: The Tomonaga-Schwinger formulation of
quantum theory, with some specified advancing sequence of spacelike
surfaces would de-mystify the action at a distance, in the
sense that it would provide a perfectly well defined specification
of these actions. However, it has always seemed that the empirical
equivalence of all the elements of the infinite set of possible choices
of sequences renders all of them devoid of physical meaning, and
the notion that there really is some instantaneous action at a distance
a chimera, But if it is established, on the basis of reasonable
assumptions, that there must be faster-than-light transfers of information,
then it becomes much more sensible to accept the idea of a preferred
set of Tomonaga-Schwinger spacelike surfaces along which these
faster-than-light transfers act. Then the Tomonaga-Schwinger
formulation constitutes an already-worked-out theory of the
needed faster-than-light transfers, and it is a theory that is,
in spite of all the interactions at a distance, known to be
compatible with all the requirements of relativistic quantum field theory.
\noindent {\bf References}
1. Abner Shimony and Howard Stein, American Journal of Physics.
2. Henry P. Stapp, American Journal of Physics, {\bf 65}, 300-304 (1997).
3. John Bell, Physics {\bf 1}, 195 (1964).
4. John Bell, Proc. Int. School of Pysics `Enrico Fermi', course II,\\
New York, Academic, 171 (1971).
5. John F. Clauser and Abner Shimony, Rep. Prog. Phys. {\bf 41}, 1881 (1978)
6. Sin-itiro Tomonaga, Progress of Theoretical Physics, {\bf 1}, 27 (1946).
7. Julian Schwinger, Physical Review, {\bf 82}, 914 (1951).
8. John von Neumann, {\it Mathematical Foundations of Quantum Theory,}\\
Princeton Univ. Press, Princeton NJ, 1955.
9. H.P. Stapp, {\it Nonlocality, Counterfactuals, and Consistent
Histories}, Lawrence Berkeley National Laboratory Report
LBNL 43201: quant-ph 9905055
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