**Comments on Shimony's "An Analysis of Stapp's 'A bell-type theorem
without hidden variables'"**

Henry P. Stapp

Lawrence Berkeley National Laboratory

University of California

Berkeley, California 94720.

The hidden-variable theorems of Bell and followers depend upon an
assumption, namely the hidden-variable assumption, that conflicts with
the precepts of quantum philosophy. Hence from an orthodox quantum
perspective those theorems entail no faster-than-light transfer of
information. They merely reinforce the ban on hidden variables. The need
for some sort of faster-than-light information transfer can be shown by
using counterfactuals instead of hidden variables. Shimony's criticism
of that argument fails to take into account the distinction between
no-faster-than-light connection in one direction and that same condition
in both directions. The argument can be cleanly formulated within the
framework of a fixed past, open future interpretation of quantum theory,
which neatly accommodates the critical assumptions that the
experimenters are free to choose which experiments they will perform.
The assumptions are compatible with the Tomonaga-Schwinger formulation
of quantum field theory, and hence with orthodox quantum precepts, and
with the relativistic requirement that no prediction pertaining to an
outcome in one region can depend upon a free choice made in a region
spacelike-separated from the first.

Bell's theorem, Nonlocality, Hidden variables, Counterfactuals

Asher Peres has had a long-standing interest in the subject matter of
this article, and I have benefited for numerous communications with him
on this topic. The form of this paper is in part a consequence of his
insightful demands for mathematical rigor combined with conceptual
clarity in the approach to fundamental issues.

Professor Shimony's article\[1\] is an extremely helpful contribution to
the subject. It summarizes in a lucid way the large areas of agreement
between us, and provides a back-to-basics proof of the two propositions
that are the main technical results of my paper\[2\]. Shimony's long and
detailed derivation of those two basic propositions should lay
completely to rest all but one of the objections that were raised
against my more compact 1997 proof\[3\]. I shall examine presently that
remaining objection, but first will emphasize some key points of
agreement mentioned by Shimony.

Shimony identifies the motivation of my work, namely the fact that the
theorems of J.S. Bell\[4\] and his followers\[5\] rest explicitly or
implicitly on the local-hidden-variable assumption that the values of
the pertinent observables exist whether they are measured or not. That
assumption conflicts with orthodox quantum philosophy, and that fact
undermines the idea that some sort of faster-than-light transfer of
information is implied by the conjunction of Bell's theorem and the
assumed validity of the predictions of quantum theory. The more likely
conclusion, from the orthodox perspective, is a failure of the
hidden-variable assumption. The orthodox interpretation of Bell's
theorem is not that faster-than-light transfer of information exists. It
is rather that the hidden-variable assumption is false. Shimony notes
that a proof not requiring a hidden-variable assumption of the need in
quantum theory for faster-than-light information transfer "would be a
profound scientific and philosophical achievement."

Shimony questions the sufficiency of my reasons for supplementing my
1997 proof with the 2004 version\[2\]. He examines, consequently, not my
new proof but rather the explicitly counterfactual approach that I
proposed in a published reply to his earlier comments. That approach
differs fundamentally from the one used in my 2004 paper, but his proof
of the validity of the two propositions covers both formulations.

The proof constructed and criticized by Shimony lies within the general
framework of counterfactual reasoning, whereas my 2004 proof, although
retaining some of the trappings and language of counterfactual
argumentation, is based on a substantially different foundation. The
combination of my assumptions of "free choices" and of "no
backward-in-time influence" amounts to the assumption that theories
covered by my new work *are to be compatible with* the idea of "*fixed
past, open future*". This conceptualization circumvents, at the
foundational level, the need for counterfactuals. It accords with the
notion of an advancing "now" in which events occur that "fix and settle"
first the free choice made by any agent about which experiment he will
perform, and later the outcome of that freely chosen experiment. The
future is "open" in the sense that the choices in regions R and L of
which experiments are to be performed in those regions are required to
be treatable, within the theory, as free choices that are made by the
agents when the moment "now" arrives. The subsequent "choice of the
outcome of the freely chosen experiment" is likewise required to be
treatable, within the class of theories to which the propositions apply,
as undetermined until the advancing moment "now" arrives, at which time
the outcome also becomes "fixed and settled". These latter choices are
termed "nature's choices" and are required to conform to the statistical
rules of quantum theory. Treating the theory in this way is supposed to
be one adequate way of expressing the content of the theory, although
perhaps not the only possible way.

This switch from an approach formulated in the framework of
"counterfactuals" to one formulated in the framework of "fixed past,
open future" has no significant effect on the proofs of the two
propositions. But it brings the concepts being used into closer accord
with those of orthodox quantum thinking. Although philosophers contend
that counterfactual concepts pervade science, and are needed for
science, the significance of results based on the use of counterfactuals
remains somewhat shakey in the minds of most quantum physicists. But the
idea that the events already observed in the past by somebody can be
treated as if they are fixed and settled, and that our future choices
can be treated as if they free, agrees with the way that physicists deal
with their theories, with their theoretical practices, and with their
lives in general.

Shimony's objection to my argument begins with the assertion "But SR is
not an assertion about actually occurring events. It is a counterfactual
conditional ". This statement alone activates the intuitive distrust of
scientists in arguments based on counterfactuals. I shall deal presently
with Shimony's specific objection, raised within the framework of the
counterfactual formulation. But first I shall describe the application
of the two propositions from the "fixed past, open future" point of view
that is more congenial with the normal thinking of physicists.

Why does Shimony claim that the validity of these two propositions lacks
scientific significance?

This wording is not exactly the way that Shimony put it. But scientific
significance is the basic issue. The theorems of Bell and his followers
are ultimately of value because they rule out certain possible models or
theories of nature. The pertinent questions are thus: Does the joint
validity of the two propositions rule out some models or theories of
nature that are not ruled out by Bell's theorems? And does the joint
validity of these two propositions rule out all of the local-hidden
variable theories that are eliminated by Bell's theorem? If the joint
validity of these two propositions does indeed rule out all of the
hidden-variable theories covered by Bell's theorem, and others besides,
then these propositions are jointly stronger than Bell's Theorem, both
because their consequences are stronger---they rule out more
theories\-\--and also because their assumptions are weaker. In this
connection it is important to notice that it is not *nature* that is
required to conform to the assumptions. It is rather that a *theory*
must, in order for these propositions to be applicable to that theory,
be such that the choice made by the experimenter in the later region R
*can be treated* as a free variable, effectively undetermined until the
moment of the decision, and that whatever outcome has already been
observed in the earlier region L *can be considered* to remain
undisturbed by the subsequent events. The premises of the two
propositions are thus *conditions on the class of theories* to which
these propositions apply.

To see how this works, suppose you are trying to construct a local
theory that agrees with the predictions of Quantum Theory. Then what has
been proved is that if this theory is merely such that (1) the
experimenter's choices can be considered "free" (i.e., without any
*relevant* causal roots), and (2) what is observed to happen in region L
can be considered to be fixed and settled independently of whether R1 or
R2 will later be freely chosen and performed by the experimenter in
region R, and (3) the predictions of quantum theory for the Hardy
experiments are valid, then the theory must, for these experiments,
satisfy the following two properties:

I.  If L2 is performed in L, then if R2 were to be performed and were to
    give outcome + then if R1 were to be performed, the outcome would
    always be \--.

> II If L1 is performed in L, then if R2 were to be performed and were
> to give outcome + then if R1 were to be performed the outcome would
> sometimes be +.

I have eliminated here the counterfactual terminology that was employed
in my counterfactual-based 1997 paper, and that was retained in my 2004
paper for the sake of historical continuity. I have adopted here the
language appropriate to the assumptions of my 2004 paper, which, as
emphasized above, are concordant with the idea of fixed past open
future. The two propositions pertain to the *structure of a theory* in
which the free variables are the open choices to be made by the
experimenters as to which experiments will be performed, and the two
propositions are assertions pertaining to relationships that then follow
from the combination of the assumption of the validity of the relevant
predictions of quantum theory, together with the idea that the outcomes
that have already been observed by some human witness in one region can
be treated as fixed and settled, completely unalterable by subsequent
free choices made in a region space-like separated from the first.

Although something akin to hidden variables might be *entailed* by these
propositions, any such structure is here a *consequence* of our fixed
past, open future assumptions, together with the assumptions of the
predictions of QM. These *consequences* are *not* hidden-variable
*assumptions*.

These two propositions, taken together, entail the presence in region R
of information about the free choice made in L between L1 and L2:

no theory that satisfies these to propositions can be "local" in the
sense that it is logically compatible with an exclusion of all
faster-than-light transfers of information.

No *local-hidden-variable theory* can satisfy both of these properties:
In such a theory the observable properties are fixed and definite
whether they are measured of not, and they do not depend upon which
experiment is chosen and performed far away. That combination of
conditions is not compatible with the validity of these two
propositions.

No Bell-type hidden-variable *assumption* has entered into the proof of
the two properties. These two propositions are consequences simply of
the assumptions that the theory is compatible with the theoretical
concept of "fixed past, open future", in conjunction with the validity
of predictions of quantum theory for this Hardy-type experiment.

Relativistic quantum field theory (RQFT) is compatible with the premises
of the propositions. This is shown by the works of Tomonaga\[6\] and
Schwinger\[7\] (T&S), where an advancing surface "now" is a
parameterized space-like surface σ(τ) such that for τ' ≤ τ the surface
σ(τ') lies nowhere later than σ(τ), but is somewhere earlier. In the T&S
formulation there exists a fixed history of the evolution of the state
vector Ψ(σ(τ)) for all σ(τ') up until the present time "Now". In that
formulation there is also, in association with the fixing of any
outcome, a change of the state vector ψ(σ(τ)) that produces an
instantaneous transfer of information along the space-like surfaces
σ(τ). But in spite of the existence within the T&S formulation of RQFT
of this instantaneous information transfer along space-like surfaces,
all the predictions of the theory about outcomes of measurements conform
to the requirement of relativity theory that no such prediction
pertaining to an experiment performed in one space-time region can
depend upon which experiment is chosen and performed in a second
space-time region that is situated space-like relative to the first.

The fact that the T&S formulation of RQFT does involve faster-than-light
information transfers does not by itself entail that the existence of
such transfers is an intrinsic feature of RQFT itself. There are other
formulations that focus directly on connections between observables, and
in which no trace of faster-than-light information transfer is evident.
However, application of the two propositions requires merely that theory
under examination be "compatible with" the concepts of "fixed past, open
future", in the sense that, without altering the content of the theory,
the choice between R1 and R2 *can be treated* as free, and the outcome
of the earlier observation in L *can be treated* as fixed and settled
prior to the fixing of the later choice in R The validity of this
assumption is entailed by the T&S formulation of RQFT, and hence RQFT
itself is *covered* by arguments based on the propositions..

The general conclusion is that *no theory* that can be treated in
accordance with the idea of fixed past, open future, and that accords
with the quantum predictions for the Hardy experiments, can be
reconciled with a locality requirement that bans all faster-than-light
transfer of information.

To illustrate the argument let us consider a complex model. Suppose in
region R there is a genie who receives the particle, and extracts
information from it, which he then combines with some random numbers,
and with the information about which experiment, R1 or R2, is being
performed in R, and then issues the output information in accordance
with some unstated rules, which, however, lead to results concordant
with the predictions of quantum theory. Without making any further
assumption about what the rules are, beyond the assertion that the free
choice between R1 and R2 made in R cannot disturb what has already
become fixed and settled in L, we know that if the experimenter in L
chooses to perform L2 then the genie's rules, whatever they are, must
entail a special connection between the outcomes that he would issue in
the two alternative cases R1 or R2.: If, in some instance, the agent in
R were to choose R2 and the genie were to choose + then if in that
instance the agent were to choose R1, the genie must definitely issues
the outcome -- . On the other hand, that same rule cannot always be
obeyed if the experimenter in L chooses L1. But this difference means
that information about whether L1 or L2 was chosen in L must be present
in R: the genie located in R cannot issue outcomes that depend upon the
choice between L1 and L2 made by the experimenter in L if no information
about that free choice in L is available in R.

If no information about the free choice between L1 and L2 can get to R,
then this genie-model is ruled out. But it is not ruled out by Bell's
theorem: Bell's hidden-variable assumptions are stronger than those of
the genie-model. Indeed, if one simply replaces the genie with "a
localized process" then the assumptions of that model are compatible not
only with quantum philosophy, but with relativistic quantum field theory
itself. Thus my results yield conclusions that are not entailed by the
theorems of Bell and his followers. They express in the form of a pair
of specific propositions an important feature of all theories that seek
to go beyond the mere expression of correlations between observations:
they must reject either the idea that our choices can be treated as
free, or the notion that there is no faster-than-light (including
backward in time) transfer of information, if they are to give the
quantum predictions for the Hardy-type experiments

We now turn to the two key questions:

1.  Does Shimony's argument reveal any flaw in this
    fixed-past-open-future argument?

2.  Does Shimony's argument reveal any flaw in the corresponding
    counterfactual-based argument?

I shall argue that the answer to both questions is No!

Shimony asserts that "The error in Stapp's argument is his claim that SR
is a statement about region R alone", But what I actually said, as he
correctly recorded, was that "the truth or falsity of SR is defined by
conditions on the truth or falsity of statements describing possible
events located in region R". This difference in wording is significant.
My argument, given above, is based on *my* wording: I displayed two
propositions that both follow from the stated assumptions, but that
are\-\--because of the fact that their truth or falsity is defined by
conditions on the truth or falsity of statements describing events
located in R\--jointly incompatible with a ban on transfer of
information to R of the choice between L1 and L2 made by the
experimenter in region L. Shimony treats the entire statement SR, which
involves counterfactuals, as a unit that incorporates, *within itself*,
my key assumption that what happened in L was fixed and settled before
the decision between R1 and R2 was made, whereas I take this key
assumption to be a restriction on the class of theories within which the
pair of propositions is proved to be true.

This latter approach of taking the stated assumptions to be conditions
on the class of theories in which the two propositions are jointly true
is a direct and completely legitimate way to proceed. Incorporating the
key assumption of no-backward-in-time influence into the meaning of a
counterfactual statement is less satisfactory for two reasons. In the
first place the mere use of statements about events that in principle
can never happen, because some contrary thing has been asserted to have
definitely happened tends by itself to render the argument less than
ideally rock solid in the minds of physicists. On the other hand,
speaking directly about properties of a class of theories that satisfy
certain specified conditions that are themselves in line with quantum
philosophy, and are actually satisfied by relativistic quantum field
theory, is a far more transparent approach that is less likely to
enshroud subtle difficulties. The second reason is the closely connected
fact that the scrambling the key causality assumption into the meaning
of the words that express contrary-to-fact assertions opens the door to
possible confusion.

The essential point here is that one must be careful not to introduce
any assumption that injects implicitly into the theory the transfer of
information from L to R that the joint validity of the two propositions
reveals to be present. Shimony's criticism possesses a certain initial
aura of credibility due to the fact that introducing *any* causal
connection between events in R and in L *harbors the danger* of
injecting implicitly some hidden assumption of the very influence from L
to R that the argument eventually reveals. If a hidden assumption of an
influence from R to L is smuggled into the assumptions then the fact
that such a connection eventually emerges would lack significance.. On
the other hand, if no such assumption is smuggled in, and the conclusion
that there must be transfer of information from L to R follows logically
from completely legitimate assumptions, including, in an essential way,
the pertinent predictions of quantum theory, then the conclusion
pertaining to the theories in question must be deemed to be logically
valid.

It is well-know that quantum theory is completely compatible with the
absence of faster-than-light influences in one direction, provided such
influences are allowed in other directions. The question at issue is
whether one can simultaneously forbid faster-than-light influences in
all directions. Hence if we wish to prove the need for faster-than-light
influence in *some* direction then we can legitimately proceed by
*excluding* faster-than-light action in one direction, say right to
left, and then showing that this restriction entails, when combined with
the assumption of the validity of pertinent predictions of quantum
theory, the need for faster-than-light transfer of information in the
other direction, namely from left to right. This is the completely
legitimate line of argument that I employ.

The first part of this legitimate argument is implemented by my
assumption that the earlier observed outcome in L is fixed and settled,
independently of what the later free choice in R will be. This
assumption of not a hidden assumption of the existence of an action from
left to right. It is the completely legitimate-in-this-context demand
that there be no action from right to left. This assumption, by itself,
does not entail any influence from left to right. Only when combined
with the predictions of quantum theory does it lead to the conclusion
that there must be information transfer from left to right. Thus the
requirement of no action from right to left, whether regarded as a
condition on the class of covered theories, or as part of the meaning of
SR, is completely legitimate, in the context of this proof. But
Shimony's analysis does not distinguish this
completely-legitimate-in-this-context assumption of no action from right
to left from what would be a completely illegitimate assumption of
action from left to right.

The logical structure of the proof\-\--with the two very different
statuses of (1) the input assumption of no action from right to left and
(2) the resulting output conclusion of a necessary transfer of
information from left to right \-\-- is revealed far more clearly and
directly in the fixed-past-open-future formulation of the conditions for
applicability of two propositions than in an approach that mixes
counterfactual concepts into the meanings of the words appearing in the
proofs. If that latter approach is used, then it is necessary in
principle to unpack the counterfactual statements in order to clearly
distinguish between legitimate inputs and possible illegitimate ones.
Shimony's counterfactual-based analysis fails make this crucial
distinction. In lieu of making this distinction within the
counterfactual approach, the alternative and simpler way to verify the
validity of the basic claim is to work directly from the assumptions of
my 2004 paper, in the way described above, and thereby circumvent the
complexities introduced by the avoidable use of counterfactuals.

ACKNOWLEDGEMENTS

I thank J. Finkelstein for very useful comments on an earlier much
shorter draft of this paper. This work was supported in part by the
Director, Office of Science, Office of High Energy and Nuclear Physics
of the U.S. Department of Energy under contract No. DE-AC03-76SF00098.

REFERENCES

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    hidden variables'", *Foundations of Physics* ??

2.  H. P. Stapp, "A Bell-type theorem without hidden variables",
    *American Journal of Physics* **72**, 30-33 (2004).

> 3\. H. P. Stapp, "Nonlocal character of quantum theory", *American
> Journal of Physics* **65**, 300-304 (1997).
>
> 4\. J. S. Bell, "On the Einstein-Podolsky-Rosen paradox", *Physics*
> **1**, 195-200 (1964).

5.  J.F. Clauser and A. Shimony, "Bell's theorem: Experimental tests and
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> 1881-1927 (1978).

6\. S. Tomonaga, "On a relativistically invariant formulation of the
quantum theory of wave fields," *Prog. Theor. Phys*. **1**, 27-42
(1946).

7\. J. Schwinger, "The theory of quantized fields I," *Phys. Rev*.
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