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Nov 25, 1996 \hfill LBNL-38803new \\
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{\large \bf Nonlocal Character of Quantum Theory}
\footnote{This work was supported by the Director, Office of Energy
Research, 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 Theoretical Physics Group\\
Lawrence Berkeley Laboratory\\
University of California\\
Berkeley, California 94720}
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\begin{abstract}
According to a common conception of causality, the truth of a statement that
refers only to phenomena confined to an earlier time cannot depend upon which
measurement an experimenter will freely choose to perform at a later time.
According to a common idea of the theory of relativity this causality condition
should be valid in all Lorentz frames. It is shown here that this concept of
relativistic causality is incompatible with some simple predictions of quantum
theory.
\end{abstract}
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\noindent {\bf 1. Introduction}
It widely believed by physicists that the work of Einstein, Podolsky, and
Rosen[1], John Bell[2], and others[3] has demonstrated a conflict between
quantum mechanics and the theory of relativity. This conflict is suppose to be
a logical incompatibility between some elementary predictions of quantum theory
and the idea that causal influences can propagate only into the forward light
cone.
This belief, though widespread, is far from universal. A principal cause of
dissent is this: the simplest demonstrations of the supposed conflict merely
prove a conflict between quantum mechanics and a property called "local
realism". As this phrase indicates, these proofs demonstrate no direct conflict
between quantum mechanics and relativity, for there is the added assumption of
"realism". This latter assumption can take various forms. It is essentially an
assumption of the existence of hidden variables of a kind banned by quantum
philosophy. Or perhaps it is an assumption of the existence of "elements of
reality" of the kind proposed by Einstein, Podolsky, and Rosen, but rejected by
Bohr. Alternatively, there might be an assumption akin to determinism, which
likewise is rejected by quantum philosophy.
The aim of this paper is to give a new proof of a logical conflict between some
prediction of quantum mechanics and the idea that no causal influence can
propagate faster than light. This new proof is perhaps simpler than ones I have
given before[4,5], and has some added technical virtues that I shall describe
later.
To speak clearly about causal influences one must identify some "causes". The
basic assumption in all Einstein-Podolsky-Rosen-Bell-type arguments is that the
choices to be made by experimenters concerning which measurement they will
eventually perform on some individual quantum system can be regarded as
unconstrained, or free: these choices are assumed to be sufficiently
uncontrolled, and independent of the quantum system under consideration, to be
treated as free variables, within the context of the analysis. Indeed, Bohr
repeatedly emphasized the freedom of experimenters to examine either one aspect
or another of an individual quantum system.
The word ``local'' refers here to a putative causality condition that claims
that no free choice can influence observable phenomena lying outside its
forward light cone: if a statement S refers only to choices between possible
experiments and their possible observable outcomes that are all localized in a
spacetime region lying earlier than some time T then the truth of this
statement S cannot depend upon which experiment is freely chosen and performed
in a spatially separated region at a time later than T. Statement S cannot be
true if one choice is made at the later time, but false if another choice is
made at the later time. The crucial locality assertion, suggested by the theory
of relativity, is that any Lorentz frame can be use to determine what it
earlier and what is later: the non-dependence of the earlier upon the later
should hold in all frames.
The argument to be used here depends on the idea that a choice to be made by an
experimenter at a later time can be regarded as a free variable, and that the
possible consequences of making different choices can be compared. Thus the
argument involves a certain weak form of counterfactual reasoning. This fact is
sometimes taken as a sufficient reason to discard wholesale all EPR-Bell-type
arguments. That tactic is not rational.
Imagine in classical mechanics an electron of unknown velocity entering an
apparatus having, say, various crossed electric and magnetic fields, so that
if the electron is observed to land at a certain point then one knows from the
theory what its incoming velocity was, and hence where it ``would have landed''
if the experimenter had used in the experiment a second apparatus, ``instead
of'' the first one.
This example illustrates the fact that theoretical assumptions often allow one
to say with certainty, on the basis of the outcome of a certain experiment,
what "would have happened" if an alternative possible apparatus had been used.
If logical arguments of this kind---that follow directly from certain
theoretical assumptions---lead to a contradiction, then one must, if rational,
accept the fact that at least one of the explicit or implicit theoretical
assumptions is incorrect.
Logicians have developed a symbolic way of expressing conditionals of this
kind, which abound in physics. It will be useful to adopt their notation.
The experiment to be examined here, from a theoretical perspective, involves
two space-like separated space-time regions, {\bf R} and {\bf L}. There is
initially one single physical system that is subjected in each region to a
bivalent free choice between just two alternative possible measurements. The
chosen measurement will then immediately be performed in that region.
Suppose $R1$ and $R2$ represent the two alternative possible measurements that
might be performed in the space-time region {\bf R}. The symbolic statement
$$(X\wedge R2)\Longrightarrow [R1 \Box \rightarrow Y]$$ asserts that the truth
of conditions $X$ and $R2$ entails that if measurement $R1$ were to be chosen,
{\it instead of $R2$}, then condition $Y$ {\it would hold}. The notation is
suppose to signify that the other free choice (i.e., the one in {\bf L}) is
left unchanged.
In the following proof the "locality" condition is asserted in three forms.
They all express the idea that the truth of a statement pertaining to
(macroscopic) phenomena confined to one region is independent of which
measurement will be freely chosen and performed in the other region at a later
time.
In the experiment to be considered here each of the possible measurements has
two possible outcomes. Let the two possible outcome of $R1$ be called $R1+$ and
$R1-$, etc..
Symbolically stated, the first locality condition is this: $$LOC1: (L2\wedge
R2\wedge L2+) \Longrightarrow [R1\Box \rightarrow (L2\wedge R1\wedge L2+)].$$
It asserts that if under the condition that the choices were $L2$ and $R2$ the
outcome in {\bf L} at some earlier time were $L2+$, then if the (later) choice
in {\bf R} were to be $R1$, instead of $R2$, but the free choice in {\bf L}
were to remain unchanged, then the outcome $L2+$ in {\bf L} would likewise
remain unchanged. This assumption is an expression of the theoretical idea that
the present facts are independent of which free choice will be made later.
The second locality condition is called $LOC2$. Suppose the measurement made in
{\bf L} is performed {\it before} the measurement performed in {\bf R} is
chosen. And suppose that under the condition that the measurement made in {\bf
L} is $L2$ one can prove (using $LOC1$) the truth of the following statement
SR:
$$ [(R2\wedge R2+) \Longrightarrow (R1\Box \rightarrow R1-)].$$
Notice that everthing mentioned in SR is an observable phenomena in region {\bf
R}. This statement has the same form as the one from classical mechanics that I
used as an illustration. The assumption in $LOC2$ is that statement SR has been
proved under the condition that $L2$ was performed in the far-away region {\bf
L} at an earlier time. Now special relativity comes in. According to the ideas
of special relativity it should not matter which frame is used to specify
``earlier'' and ``later''. So let's use a frame in which the choice to perform
$L2$ in region {\bf L} is made {\it later} than all the phenomena referred to
in SR. But a statement about what would happen in {\bf R}, under certain
condition defined there, that is known to be true under the condition that the
later free choice in {\bf L} is $L2$, cannot be rendered false by changing that
later free choice, provided the idea is upheld that observable effects can
propagate only into the future (light-cone). So $LOC2$ asserts this: ``If SR is
proved to be true under the condition that $L2$ is freely chosen in {\bf L}
then SR must be true also under the condition that $L1$ is freely chosen
there instead.''
The final locality condition, $LOC3$, asserts the commutability of
$L1-\Longrightarrow$ and $R1 \Box \rightarrow$. This result follows from the
fact that, according to the putative locality property, the truth of $L1-$ is
unaffected by changing the choice in {\bf R} from $R2$ to $R1$: $L1-$ is the
same condition whether asserted under condition $R2$ or $R1$.
These three locality conditions will be used in the context of an analysis
of a Hardy-type experiment.
\noindent {\bf 2. Hardy-type Experiment}
The important features of a Hardy-type experiment are shown in Figure 1,
along with the four pertinent predictions of quantum theory.
\begin{figure}[h]
\vspace{-2in}
\epsfxsize=6in
\epsfbox{fig1.ps}
\vspace{-2in}
\caption{The logical structure of the Hardy
experiment is represented, together with the four pertinent predictions of
quantum theory. The three solid paths between pairs of labelled points
represent predictions that have conditional probability equal to unity. The
dashed line represents a connection that has conditional probabilty equal to
50\%. (The 50\% comes from the Eberhard/Rosselet version: Hardy's optical case
gives 67\%)}
\end{figure}
For example, the solid line from $b$ to $g$ represents the prediction that:\\
``If $L1$ is performed in {\bf L} and the outcome there is $b$, then if $R2$ is
performed in {\bf R} the outcome there will be $g$.''
This prediction can be written symbolically as:
$$
(L1\wedge b) \Longrightarrow (R2 \Longrightarrow g),
$$
where $\Longrightarrow$ stands for {\it implies}, the strict conditional, and
$\wedge$ stands for conjunction. Equivalently, this prediction can be written
in the form of line $8$ of the proof.
\newpage
Informally, the general line of argument goes as follows. Suppose we consider a
frame in which the measurement in region {\bf L} is chosen and performed
earlier than the measurement in region {\bf R}. Suppose experiment L2 is
chosen in {\bf L}. Then the LOC condition that the outcome at the earlier time
cannot be one outcome if the later choice in {\bf R} is to perform experiment
R2, but a different outcome if the later choice is to perform experiment R1,
combined with the two QM predictions $ g \rightarrow c$ and $c \rightarrow f$,
allows one to conclude, under this condition that the earlier measurement is L2,
that $g \rightarrow f$. Following the symbolic conventions of modal logic we
express this conclusion in the following way:
$$
(L2\wedge R2\wedge g)\Longrightarrow
[ R1 \Box \rightarrow (L2\wedge R1\wedge f)],
$$
which is line $4$ of the proof. It says that: ``If L2 and R2 are performed
and the result g=$R2+$ appears in {\bf R} then if R1 had been performed there,
instead of R2, then the outcome f=$R1-$ would have appeared.''
Statements of this general kind are commonplace in physics: theory often
allows one to deduce from the outcome of certain measurements on a system
what the outcome of some alternative possible measurement would necessarily be.
That is why modal logic really had to be developed: to cope with the fact that
statements of this general kind are endemic in science.
The conclusion obtained above can be formulated as a statement that if
measurement L2 is performed then $g \rightarrow f$, where the consequence
is a statement referring only observables in region {\bf R}. [See line $5$
of the proof for the complete statement.]
Now the truth of this statement cannot depend on how we select the frame of
reference, which is merely a notational convention. So let us shift to another
frame, where the measurements in region {\bf L} are chosen and performed later.
The statement $g \rightarrow f$ must still be true. But the truth of this
statement, which refers only to observables in {\bf R}, cannot, according to
the putative locality principle LOC, depend on whether measurement L2 or L1 is
chosen to be performed at the later time. Yet if this implication $g \rightarrow f$
holds also under the condition that L1 is performed in {\bf L} then we can
combine that implication with $b \rightarrow g$ to get $b \rightarrow f$,
which contradicts the prediction of QM that $b$ entails $e$ some nonzero
fraction of the time.
\newpage
\noindent {\bf 3. The Proof}
1. LOC1: $(L2\wedge R2\wedge c)
\Longrightarrow [R1\Box \rightarrow (L2\wedge R1\wedge c)].$
2. QM: $(L2\wedge R2\wedge g) \Longrightarrow (L2\wedge R2\wedge c).$
3. QM: $ (L2\wedge R1\wedge c) \Longrightarrow (L2\wedge R1\wedge f).$
4. LOGIC: $(L2\wedge R2\wedge g)\Longrightarrow [ R1 \Box
\rightarrow (L2\wedge R1\wedge f)].$ [From1, 2, 3.]
5. LOGIC: $(L2)\Longrightarrow [(R2\wedge g) \Longrightarrow (R1\Box
\rightarrow f)].$ [Logically equivalent to 4.]
6. LOC2: $(L1) \Longrightarrow [(R2\wedge g) \Longrightarrow (R1\Box
\rightarrow f)].$
7. LOGIC: $(L1\wedge R2) \Longrightarrow [ g \Longrightarrow
(R1 \Box\rightarrow f)].$ [Equivalent to 6.]
8. QM: $(L1\wedge R2) \Longrightarrow [ b\Longrightarrow g].$
9. LOGIC: $(L1\wedge R2) \Longrightarrow [ b\Longrightarrow
(R1 \Box\rightarrow f)].$ [From 7 and 8.]
10. LOC3: $(L1\wedge R2) \Longrightarrow
[R1 \Box\rightarrow (b\Longrightarrow f)].$
11. LOGIC: $R2\Longrightarrow [L1\Longrightarrow [R1\Box \rightarrow
(b\Longrightarrow f)]].$ [Equivalent to 10.]
12. QM: $L1\Longrightarrow [R1 \Longrightarrow \neg (b\Longrightarrow f)].$
13. LOGIC: $L1\Longrightarrow [R1\Box \rightarrow \neg (b\Longrightarrow f)].$
[Entailed by 12.]
14. LOGIC: $R2\Longrightarrow [L1\Longrightarrow [R1\Box \rightarrow
\neg (b\Longrightarrow f)]].$ [Entailed by 13.]
The symbol $\neg$ is negates the proposition that follows it.
The conjunction of $11$ and $14$ contradicts the assumption that the
experimenters in regions {\bf R} and {\bf L} are free to choose which
experiments they will perform. Thus the incompatibility of the assumptions
of the proof is established.
\noindent {\bf 4. Conclusion}
A contradiction has been obtained. Thus some assumption of the proof must be
false. The only assumptions were: (1) the validity of some simple predictions
of quantum theory, (2) the explicitly stated locality conditions, (3) the
general idea that physical theories can cover a variety of special instances
that can be imagined to be created by free choices of experimenters, and (4)
the {\it general} principles of modal logic, which merely formalize what we
mean by modal language. The latter two assumptions do not contravene the
principles of quantum mechanics as they are normally understood: Bohr often
emphasized that the freedom of experimenters to examine one aspect or another
of a quantum system is in no way compromised by quantum theory: indeed his idea
of complementarity rested on the idea that we enjoy that freedom. And he
embraced the use of ordinary language and logic to describe the observable
aspects of things.
An advantage of the present proof over those of references 4 and 5 is that
there is no appeal here to any special technical assumption or to any of the
special rules of ``closeness of worlds'' that philosophers have introduced to
extend the meaning of modal language beyond what is strictly entailed by the
strict laws of nature and other conditions, such as our locality conditions,
that allow conclusions to be drawn simply from the basic meanings of the words,
as formalized by the {\it general} principles of modal logic.
It should be emphasized that no conflict has been established here between
quantum theory and the general causal principle that the present facts are
independent of free choices to be made at later times. The contradiction arises
only when one tries to extend this principle by asserting the relativistic
notion that the principle can be assumed to hold in all Lorentz frames. There
is no difficulty with the assumption that this causality condition holds in
some single preferred frame of the universe, such as the cosmic black-body
radiation rest frame.
\noindent {\bf 5. Concluding Comment}
The theory of relativity was created within the context of deterministic
physical theories. In deterministic theories the entire course of the history
of the universe is determined by the initial conditions, or by early
conditions. Thus an account of the actual process of the physical unfolding of
this history is not really essential to the physical theory. That was
Einstein's point. The data that scientists use are what can be written down in
a log book that lists what was observed. Some observations may include the
reading on some clock when the observation was made. But this latter reading is
just part of the observation. This log-book sort of information is timeless,
even though the observations do include readings on clocks. There is no remnant
in this data of the order in which the data from different regions ``came into
being'', beyond the logged-in evidence itself. But the log-book sort of
physical data is fixed by a deterministic theory without mentioning any process
of the ``coming into beingness'' of reality: one can simply adopt an overall
spacetime point of view.
In quantum theory the evolution of the state in accordance with the
Schroedinger equation is deterministic. But that is only half of the dynamical
story. There is, in practice, also a second process, represented by the
collapse of the wave function. This second process is, within contemporary
theory, not deterministic: at any point in the development of reality the
future is not yet fixed by the deterministic part of the dynamics, but depends
on what the yet-to-be-determined future collapses will select as the actual.
The original ``Copenhagen'' interpretation was epistemological in character,
and the collapse of the wave function was construed to be a consequence of the
increased knowledge of the observer/scientist. As such it could not be
considered to cause any actual transfer of information faster than light. The
system should, therefore, inherit from the Schroedinger equation (in its local
field theory form) the property that causal influences propagate only into the
forward light cone.
This Copenhagen strategy of bringing the observer and his knowledge into the
physical theory was opposed by Einstein and others in the early days, and it
seems again to be troubling physicists. Einstein believed that the future
theory would not only remove the observer and his knowledge from center stage,
but would also save the relativistic principle that causal influences propagate
no faster than light. Saving this principle has been shown here to be
impossible. However, it is still possible to save the relativistic principles
that there is no preferred frame of reference, and no propagation of influence
backward in time, where ``backward'' now means into the (closed) backward
light-cone
In an effort to remove ``the observer'' from physical theory I proposed many
years ago a ``theory of events''[10]. A similar idea has recently been proposed
independently by Rudolf Haag[11]. Without going into details let me only say
that the basic idea is to postulate that there are real events associated with
finite regions in spacetime, and with real (objective) collapses of the
(objective) wave function. If, as a special case, one takes these regions to be
confined to the (open) backward light-cones from spacetime points, then one
obtains a picture of the process of generating reality as a succession of steps
each of which augments the ``past'' by adding the backward lightcone from some
new spacetime point, and moves forward the three-dimensional manifold ``now''
that separates the past from the unfixed future. Each new event is associated
with a collapse of the wave function that defines the state on the new surface
``now''. No special frame is singled out as preferred. The surface ``now''
advances always into the future, in finite steps. Each event influences,
immediately, only things at points spacelike separated from itself, in the
sense that each old point of the new surface ``now'' is spacelike separated
from every point in the region associated with the new event. Although each
quantum event can, therefore, influence things at spacelike separated points,
the model is otherwise in line with the ideas of the theory of relativity.
The apparent violation established in this paper of the putative relativistic
causality condition lends support to an an objective evolutionary picture of
this general sort, in which the collapses are real actual events that can
influence the objective propensities for subsequent (to ``now'') spacelike
separated events.
\begin{center}{\bf Acknowledgement}\end{center}
I thank Philippe Eberhard and an anonomous referee for helpful suggestions.
\begin{center}{\bf References}\end{center}
1. A. Einstein, B. Podolsky, and N. Rosen, ``Can Quantum Mechanical Description
of Physical Reality be Complete?'' {\it Physical Review 47}, 777-780 (1935).
2. J.S. Bell, ``On the Einstein-Podoksky-Rosen Paradox'', {\it Physics 1},
195-200 (1964).
3. J.F. Clauser and A. Shimony, Bell's theorem: experimental tests and
implications, {\it Reports on Progress in Physics 41}, 1881-1927 (1978);
D.M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, Bell's theorem
without inequalities {\it American Journal
of Physics 58}, 1131-1143 (1990); Lucien Hardy, Quantum mechanics, local
realistic theories, and Lorentz-invariant realistic theories, {\it Physical
Review Letters 68}, 2986-2984 (1992); Lucien Hardy, Nonlocality for two
particles without inequalities for almost all entangled states, {\it
Physical Review Letters 71}, 1665-1668 (1993); P.H. Eberhard and P.
Rosselet, Bell's theorem based on a generalized EPR criterion of reality,
{\it Foundations of Physics 25}, 91-111 (1995).
4. H.P. Stapp, ``Noise-induced reduction of wave packets and faster-than-light
influences'', {\it Physical Review 46 A}, 6860-6868 (1992).
5. H.P. Stapp, Reply to `Stapp's algebraic argument for nonlocality', {\it
Physical Review A49}, 4257-4260 (1994).
6. Lucien Hardy, ``Quantum mechanics, local realistic theories, and
Lorentz-invariant realistic theories'', {\it Physical Review Letters 68},
2981-2984 (1992); ``A quantum optical experiment to test local realism'',
{\it Physics Letters A 167}, 17-23 (1992).
7. P.H. Eberhard and P. Rosselet, ``Bell's theorem based on a generalized EPR
criterion of reality'', {\it Foundations of Physics 25}, 91-111 (1995)
8. N. David Mermin, ``Quantum mysteries revisited'', {\it American Journal of
Physics 58}, 880-887 (1994).
9. David Lewis, {\it Counterfactuals,} Blackwell Press, Oxford; {\it
Philosophical Papers Vol II}, Oxford University Press, Oxford, 1986;
D. Bedford and H.P. Stapp, ``Bell's Theorem in an Indeterministic
Universe'', {\it Synthese 102}, 139-164 (1995).
10. Henry P. Stapp, ``Bell's Theorem and World Process'', Nuovo Cimento 29,
270-276 (1975); ``Theory of Reality'', Foundations of Physics 7, 313-323
(1977); ``Whiteheadian Approach to Quantum theory and the Generalized Bell's
Theorem'', Foundations of Physics 9, 1-25 (1979).
11. Rudolph Haag, {\it Local Quantum Physics} Springer-Verlag, Berlin,1996,
Second Edition, Chapter VII; ``An Evolutionary Picture for
Quantum Physics'', Commun. Math. Phys. 180, 733-743 (1996).
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