Re: Nonlocality and Possible Worlds\-\-\--T. Bigaj (2006)

To Michael Revzen <mrevzen@phys.ualberta.ca>

Dear Michael,

Thank you for bringing Bigaj's book to my attention. I promised to give
you my comments after I got hold of it. As you indicated, Bigaj's book
seems largely devoted to analyzing my various arguments that the
non-locality claim \[that theories that reproduce certain predictions of
quantum, and that embrace the idea that choices of experiments can be
treated, effectively, as localized "free choices", must allow some sort
of faster-than-light transfer of information\] can be strengthened by
using the framework of counterfactual reasoning \[as contrasted to
hidden-variable frameworks that effectively assume the existence of an
underlying "reality" that determines, for each of the (four) alternative
possible combinations of measurements between which the experimenters
are considered able to freely choose, which outcome appears.\] The
non-existence of an underlying 'determining reality' would nullify the
Bell-type (hidden-variable) proofs of the need for some sort of
faster-than-light transfer of information. But counterfactual-based
proofs can overcome that limitation.

This present reply is based on an examination only of Bigaj's book: I
have not gone back to look over my original papers, or those of their
critics, and re-warmed up the relevant brain circuits.

One point worth emphasizing right off is that Bigaj's analysis finds
fault with the main arguments that have claimed to have found faults in
my arguments\-\--most specifically with the criticism in the paper and
book of Redhead, and in the paper of Clifton, Butterfield, and Redhead,
and those of Shimony and Stein, and of Mermin

A second main point is this: Bigaj's account, like the accounts of some
critics, tend to make it appear that my development of several different
proofs was in response to the recognition that there were mistakes or
errors in some of these proofs. That is not the case (aside from one
"minor oversight" \-\--see Bigaj p.165). My different proofs are based
on similar but technically *differing assumptions*. I believe that in
all my proofs the stated conclusions follow from the stated assumptions.
The question at issue is the status of the assumptions. I believe that
the assumptions in all of my proofs are appropriate and pertinent to the
issue in question, which is the logical need\-\--in the defined classes
of theories that entail the validity of the predictions of QT, and that
accept the concept of the "free choice of experiments"\-\--for
*information* about the experimental setting chosen one space-time
region R to be present also in a second space-time region R', which is
space-like separated from the first.

My 1971 original counterfactual-based proof used\-\--in place of a
reality/hidden-variable assumption that *some pre-existing hidden
realities determine*, for each possible choice of experiment, what the
outcomes will be if that experiment is performed, rather\-\--the
assumption \[Cf. Bigaj p.109\] that for each of the two alternative
possible experiments that *could have been performed* in region 1, some
particular outcome *would have appeared there if that experiment had
been performed there*, and that for each of the two alternative possible
measurements that *could have been performed* in region 2, some
particular outcome *would have appeared there if that experiment had
been performed there.* These *assumptions*, plus the assumption of the
validity of certain orthodox predictions of QT, are incompatible with
the "locality" property that no information about the free choice of the
experiment performed in a region can be present in a region all points
of which are space-like separated from all points of the first region,
and hence that that any outcome that appears in a space-time region R if
one experiment is freely chosen and performed in a faraway region R'
that is spacelike separated from R "would have appeared" also if the
choice of measurement performed in the other (faraway) region R', had
gone the other way.

Redhead, in his 1987 book, claims (See Bigaj p.110) that my assumptions
imply "determinism", and hence that the assumption of "determinism" is
not successfully evaded.

Bigaj examines Redhead's argument in support of that conclusion, and
argues that it is not conclusive, because "it is based on some sort of
semantic ambiguity regarding the proper interpretation of counterfactual
conditionals.". My own argument is simply that if some proposed
theoretical structure is such that the choices of the experimental
conditions can be treated as free variables\-\--which essentially means
that the choice of experiment could go \[or could have gone\] either
way, and no matter which way the choice goes \[went\], some particular
outcome will \[would have\] appear\[ed\]\-\--then my theorem shows that
this theory will need to allow information about the choice of the
measurement performed in one region R to be present in the other region.
R', whether or not the theory assumes some substructure that determines
outcomes in terms of the elements of this substructure. My proof never
mentions, nor involves in any way, such a substructure. Hence it is a
stronger result: I do not accept the claim that one can deduce the
existence of a deterministic hidden-variable reality/substructure from
my "much-too-weak-for-deriving-that-conclusion" assumptions. Bigaj's
analysis supports my position on this issue.

Thus this "first" proof is, in my opinion, perfectly sound, and it is
stronger than the hidden-variable-based theorems because it covers
theories that contain/specify no structure that determines the
"outcomes" of the alternative possible measurements before\-\--or
independently of whether\-\--they actually appear/occur.

I take as a counter-example to the claim that "determinism is implicit"
simply quantum theory itself, in the case of the Stern-Gerlach
apparatus, or of a photon going through a bi-refractive crystal: two
primary quantum systems. One or the other of two properly positioned
counters will fire, but there is nothing in the theory that
predetermines which one will fire, even in the case of the one
experiment that is actually performed. My proof thus encompasses quantum
theory itself, embellished not by any deterministic substructure, but
only by the idea that the "free choices of experiment"\-\--which Bohr
endorses\-\--can be embedded in the conceptual structure of "possible
worlds". Classical physics embraces the idea of "possible worlds", so it
is no radical step to allow such a general concept also in quantum
theory, and, indeed, it is hard to contemplate what a "free choice of
the experiment" would mean in a conceptual framework devoid of the idea
of possibilities that will not be realized/actualized.

Bigaj (p. 111) separates the (counterfactual) Definite Response idea DR
\[that an experiment that could have been performed would have had some
particular outcome if it had been had been performed\] from the
(locality) Matching Condition MC that in all allowed "possible worlds"
the outcome appearing/occurring in either region is independent of the
"local free choice of measurement" made in the other region. He says
"There is no question that we should accept the Definite Response
principle," but he invokes a Lewis-type analysis to conclude that what
the outcome "would have been" is indeterminate. In a formula (3.1),
which he writes down, but which does not incorporate the locality
condition MC, this indeterminateness could be a problem because there
are symbols for outcomes in one region that specify which experiment was
performed in the faraway region, and these differing symbols could
(without MC) represent different numbers, in which case no contradiction
would appear. Thus MC is certainly needed in order to get a
contradiction!

But, as Bigaj emphasizes, the locality (MC) condition is certainly not
entailed by general logical principles. Indeed, that is precisely what
we want. When we impose MC\-\--in association with the other seemingly
much more secure assumptions\-\--we get a contradiction. Thus the fact
that MC does not follow from general logical considerations is a virtue,
not a defect or deficiency, of my proof. I need MC to be a logically
unsupported link in a logical chain that leads to a contradiction. For I
want to argue that it must fail. So the fact that MC is not logically
supported is a virtue, not a flaw in my proof, as.Bigaj's wording seems
to suggest. The MC condition flows from the assumption of
no-FTL-transfer-of-information, which is the assumption that I claim to
be incompatible with the seemingly more secure assumptions.

Bigaj analyses Redhead's argument that my locality condition (MC)
entails determinism,

and finds it defective. Paraphrased, Bigaj's argument is this. Consider,
in a quantum context, a situation in which experiment B1 is performed at
time Tb in region R, and a radio-active decay Aa is detected and
recorded in R' at time Ta\<Tb. Can we claim that the assertion "If
instead (of B1) B2, then Aa would (still) occur at Ta", without
appealing to some notion of determinism? Within QT, the occurrence of Aa
at Ta happens without there being any appeal to\-\--or explanation in
terms of\-\--any deterministic substructure. Yet within QT there is a
principle that no "signal" can be transmitted faster than light: the
information about which choice B1 or B2 is made in R' at time Tb cannot
become known to an observer in R before point traveling at the speed of
light from region R'at Tb can get to region R. This property motivates
the *conjecture* that the outcome Aa at Ta cannot depend upon whether
the experimenter in R' performs B1 or B2. This conjecture does not arise
from any presumed deterministic substructure. The occurrence of Aa in R
at Ta did not arise from any deterministic substructure represented in
the theory, and the conjectured non-dependence of this
outcome/occurrence Aa upon whether the (later) free choice in R' is B1
or B2 expresses the intuition that because QT forbids "signals" to be
transmitted faster-than-light there is probably no transfer of any
information faster than the speed of light (or backward in time). These
arguments do not entail or depend upon any presumption of the existence,
within the contemplated theoretical structure, of any presumed
deterministic substructure\-\--there is only the idea that whatever
appeared (and was recorded) at the earlier time in region L under the
actual condition B1 "would have appeared" in L also if, instead (of B1),
B2 had been performed. This "would have" expresses not "determinism" but
rather the {trial) conjecture that the information about the choice made
later in R did not exist in L when the outcome there was recorded.

Redhead's claim that an assumption of "determinism" is implicit in
"would have".

is not valid in the situation under discussion..

The above argument is my elaboration upon Bigaj's argument. that
Redhead's

claim \[that determinism is a hidden element in the formulation of my
"no-faster-than-light-transfer-of-information" conjecture\] lacks
adequate support.. Of course, this no-faster-than-light conjecture is
only that: it is a conjecture that is formulated, not as something that
is meant ultimately to be believed, but rather as an idea that is
ultimately to be rejected as false, because it is incompatible with
"Predictions of QT + Free Choice".

Redhead's argument was incorporated into the 1990 paper of Clifton,
Butterfield, and Readhead, and the flaw in this argument has fatally
infected, I believe, all of the claims of these authors to the effect
that my arguments are somehow flawed.

Bigaj and Readhead refer to the "broken square problem", which arises if
one rejects the no-faster-than-light-transfer-of-information conjecture,
and the associated condition that the outcome in each region be
independent of which experiment is performed in the other region. But my
argument proceeds by accepting (provisionally) that conjecture, and
showing that this leads to a contradiction. So my line of argument
encounters no "broken square problem". The "broken square" is, rather,
the *solution* to the problem: once MC is rejected there is no longer a
contradiction, because the square is broken.

\[Mermin's objection was to the notion that I was trying to prove that
"actual outcomes" in one region must depend upon the faraway choice. I
was aiming to prove less: merely that information about the choice of
experiment in one region R must be present in a region R' that is
space-like separated from R.\]

Bigaj considered also my argument based on the GHZ three-particle
experiments, and also my argument based upon the experiment proposed by
Hardy.

In my argument pertaining to the QHZ experiment I used a technical
assumption:

"Elimination of Eliminated Conditions" (EEC). This assumption asserts if
a statement of the following form is true:

"If, in place of any conflicting aspect of the previously imposed
condition C, one substitutes the overriding condition C1, then if in
place of any conflicting aspect of the previously imposed condition C2
one substitutes the overriding condition C3, then if in place of any
conflicting aspect of the previously imposed condition C4 one
substitutes the overriding condition C5, then P(o) is true"

then the corresponding assertion of the following (shorter) form is also
true:

.

"If, in place of any conflicting aspect of the previously imposed
condition C, one imposed the overriding condition C5, then P(o) is
true".

provided o is a value that is specified by an aspect of condition C that
is unaffected by the

imposition of condition C1, and P(o) is a value that is specified by
eigenvalues made definite by condition C3 that is unaffected by imposing
the overriding condition C5.

This EEC principle is technical rather than physical, It is logically
plausible because it asserts that we can drop intermediate conditions
that leave the essential quantities intact, and that are subsequently
overridden.

Bigaj does not dispute that the QHZ argument proof goes through, given
this technical assumption, but tries and fails to reconcile EEC with
Lewis's rules for determining truth values for statements containing
counterfactual conditionals.

He considers a toy universe adapted to the CHZ situation. Because the
GHZ experiment has three dichotomic measurements each with two possible
outcomes, the toy universe will have 64 possible worlds, There are four
GHZ predictions, each which excludes 4 of the original 64, leaving 48.
Can we construct a set of rules of (Lewis-theoretic) "closeness" such
that the assumption EEC holds?

The answer is 'Yes'. The construction is simple In closeness space put
all the worlds

{(Y1,X2,X3, x2x3=+1) ,(X1,X2,X3, x2x3=+1), (X1,Y2,Y3,y2y3= -1),

(Y1,Y2,Y3, y2y3= - 1)} at one point and all the worlds with the +1
values and -1 values interchanged at a different point. \[One could
expand each "point" to a cluster of four "close-together" states that is
faraway from the other cluster. \] Then Lewis rules ensure that EEC
holds. Thus the plausible-sounding EEC is compatible with Lewis's rules.

That (Y1,X2,X3, x2x3=+1) is close to (X1,X2,X3, x2xd3=+1) follows from
the sameness of x2x3. That (X1,Y2,Y3,y2y3= -1) is close to (Y1,Y2,Y3,
y2y3= - 1)

follows from the sameness of y2y3. That (X1,X2,X3, x2x3=+1) is close to
(X1,Y2,Y3,y2y3= -1) follows from the sameness of x1, which follows from
GHZ1 and GHZ2 \[Bigaj p.145\]. The closeness of these worlds to each
other entails the needed equation (4.8) \[Bigaj p.152\]. So the assumed
EEC seems perfectly in accord with Lewis's rules.

The assumption EEC is, however, a technical assumption, and as such is
perhaps not 100% satisfactory. Thus I prefer to rely upon the
Hardy-experiment-based proof, which is also much simpler

The Hardy-based proof.

Bigaj notes that my most recent proof of based upon the quantum
theoretical predictions in the experimental conditions described by
Hardy. The original formulation assumes a rule of inference called LOC2,
which Bigaj describes as "the most hotly debated topics in all
subsequent polemics,,," Since this assumption enters in the middle of
the proof of contradiction that is supposed to show that LOC2 is false,
the debate can get tangled by discussions of the justification of an
assumption that is "false". To make the logical argument more
transparent, it is helpful to follow a slightly different exposition.
The first part of the proof is then again simply the derivation of
Bigaj's (4.23) \[p. 165\] which is the line *before* the introduction of
LOC2. This first conclusion says that, under the condition that L2 is
performed in region L, the following statement SR is true:

SR: If.R2 is performed and gives outcome +1 then if (instead of R2) R1
is performed the outcome in R will be R1-.

Now suppose SR were true [also]{.underline} under the condition that L1
(instead of L2) were performed in L and that the outcome there was L1-.
Under that condition we could combine H1 (p.161), which says that under
these very conditions the outcome in R must be R2+., with SR and hence
conclude that if (instead of R2) R1 is performed in R the outcome in R
must be R1-

But H4 (p.161) is exactly the negation of this assertion. Thus we can
conclude that our trial assumption that SR is true also under the
condition that L1 (instead of L2) is performed in L, and the outcome L1-
appears there, must be false.

Now we have a simple question: Can a theory make SR true if L2 is
performed in region L but false if L1 is performed in L, without
allowing some presence in region R of the information about whether L1
or L2 was chosen in region L

Or we have the theorem:

Any theory that agrees the predictions of QT in the Hardy case,
satisfies LOC 1, \[p. 165\] and allows the choices of the experiments to
be treated as free variables, must make SR true if L2 is performed in
region l but false if L1 is performed in L \[I use the fact that QT
predicts that if L1 is performed in L then the outcome L1- sometimes
(roughly half the time) occurs.\].

The condition LOC1 \[Bigaj p. 165\] is simply

If L2 and R2 are performed and L2+ occurs, then if (instead of R2) R1 is
performed

the outcome in L will still be L2+..

LOC1 is the"uncontroversial" locality condition, and it is only this one
single case that enters into this argument! We put in a LOC1 condition
of no FTL action from R to L, and get out a conclusion that requires an
apparent FTL action from L to R.

I believe this formulation to be so clear and simple as to forestall or
answer the questions

discussed by Bigaj..

Bigaj does emphasize his conclusion \[p.175}that the Shimony and Stein's
criticisms are

not valid. I think the best way to see this is to try to apply them to
the simple proof described above. which is similar to the one that I
described to you during your visit, and that is given at the end of my
book "Mindful Universe".

Shimony and Stein argue that in this argument the meaning of the
counterfactual has become entangled with the condition of nonlocality in
such a way as to prevent any clean conclusions about nonlocality to be
drawn. But in their analysis the two directions of the possible FTL
transfer of info are not cleanly separated. What is actually impossible
is to excluded simultaneously a FTL transfer in both directions. My
proof precludes a FTL transfer from R to L in order to deduce under that
condition the need for a FTL transfer of info from L to R. The exclusion
of FTL transfer from R to L can be made particularly understandable by
taking a frame where R is later than L and arguing that the outcome
recorded in L cannot depend upon a choice in R that has not yet been
made, This is expressed by LOC1. The meaning of the counterfactual is
merely this condition that what has already occurred and been recorded
cannot depend upon a free choice that has not yet been made. Given that
no-FTL-from-R to L, we easily conclude by simple logic that SR is valid
if L2 is performed in L but false if L1 is performed in L. But how could
this be achieved in a theoretical structure that does not allow the info
about the choice made in L to be present in R?

I hope this clarifies the nonlocality situation, and provides a very
brief outline of an adequate response to Bigaj's detailed and useful
book. Thanks again for alerting me to its existence.

Best regards,

Henry