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%%%%%% -------------------- F I N E ________________
%%%%%% ---------------------------------------------
%%%%%% ----- B I B L I O G R A F I A ---------------
%%%%%%
%----------------------------------------- abbreviazioni --------------
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\def\nc {\sl Nuovo Cimento\ }
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\def\cpc {\sl Comp. Phys. Commun.\ }
\def\cm {\sl Celestial Mechanics\ }
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%
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%
\biblitem{nelascona}
E. Nelson, {\it Field theory and the future of stochastic mechanics}, in
S. Albeverio, G. Casati, D. Merlini eds., {\sl Stochastic processes in
classical and quantum systems}, pages 438--469, Lecture Notes in Physics N. 262,
Springer (Berlin, 1986).
\biblitem{nelqf}
E.~Nelson, {\it Quantum fluctuations}, Princeton U.P. (Princeton, 1985).
\biblitem{coleman}
S. Coleman, R.E. Norton, {\it Runaway modes in model field theories},
{\sl Phys. Rev.} {\bf 125}, 1422--1428 (1962).
\biblitem{arai}
A. Arai, {\it Rigorous theory of spectra and radiation for a model in
quantum electrodynamics}, {\sl J. Math. Phys.} {\bf 24}, 1896--1910
(1983).
\biblitem{tunnel}
A.~Carati, P.~Delzanno, L.~Galgani, J.~Sassarini, {\it Nonuniqueness
properties of the physical solutions of the Lorentz--Dirac equation},
{\sl Nonlinearity} {\bf 8}, 65--79 (1995).
\biblitem{dardie}
D.~Bambusi and D.~Noja, {\it Classical electrodynamics of point
particles and mass renormalization. Some preliminary results}. {\sl
Lett. Math. Phys.} {\bf 37}, 449--460 (1996).
\biblitem{darioabl}
D. Bambusi, {\it A proof of the Lorentz--Dirac equation for charged
point particles}, preprint.
\biblitem{andreadiego}
D. Noja, A. Posilicano, {\it The wave equation with one point
interaction and the linearized classical electrodynamics of a point
particle}, {\sl Ann. Inst. H. Poincar\'e-- Phys. Th.}, in print.
\biblitem{rohrlich73}
F.~R\"ohrlich, {\it The electron: development of the first elementary particle
theory}, in J.~Mehra ed., {\it The phisicist's conception of nature}, D.Reidel
(Dordrecht, 1973).
\biblitem{ytzykson}
C. Ytzkinson, J,--B. Zuber, {\sl Quantum field theory}, Mc Graw--Hill
(New York, 1980). See pages 43--44.
\biblitem{erber}
T.~Erber, {\it The classical theory of radiation reaction},
{\sl Fortschr. der Phys.} {\bf 9}, 343--392 (1961).
\commento{una delle piu' belle rassegne}
\biblitem{bell1}
J.S. Bell, {\it Introduction to the hidden--variable question},
Proc. Int. School of Phys. E. Fermi, course 49, Academic (New York,
1971); reprinted in ref. 1 (see page 33).
\biblitem{bell2}
J.S. Bell, {\it Einstein--Podolsky--Rosen experiments},
Proc. Sympos. Frontier Probl. in High En. Phys., Pisa (1976); reprinted
in ref. 1 (see note 24).
\biblitem{belllibro}
J.S. Bell, {\sl Speakable and unspeakable in quantum mechanics},
Cambridge U.P. (Cambridge, 1987).
\biblitem{stapp}
H.P. Stapp, {\it Theory of reality}, {\sl Found. Phys.} {\bf 7},
313--323 (1977).
\biblitem{hale}
J.K.~Hale and A.P.~Stokes, {\sl J. Math. Phys.} {\bf 3}, 70 (1962).
\commento{Some physical solutions of Dirac--type equations}
\biblitem{haag}
R.~Haag, {\it Die Selbstwechselwirkung des Elektrons}, {\sl Z. Naturforsch.} {\bf 10 A}, 752--761 (1955).
\biblitem{haag2}
R. Haag, {\it An evolutionary picture for quantum physics}, {\sl Comm.
Math. Phys.} {\bf 180}, 733--744 (1996).
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F.~Bopp, {\sl Ann. der Phys.} {\bf 42}, 573--608 (1943).
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J. Baez, I. Segal, Z. Zhan, {\it Introduction to algebraic and
constructive quantum field theory}, Princeton U.P. (Princeton, 1992).
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J.F. Clauser, A. Shimony, {\it Bell's theorem: experimental facts and
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P.A.M.~Dirac, {\sl Proc. Royal Soc. (London)} A{\bf 167}, 148--168 (1938).
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N.D. Mermin, {\it Bringing home the atomic world: quantum mysteries for
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E. Schroedinger, {\it Discussion of probability relations between
separated systems}. {\sl Proc. Cambridge Phil. Soc.} {\bf 31}, 424--431
(1935).
%%% -------------- F I N E -------------------------------------------
\autobibliografia
%%%%%%%%%%%%%%%%%%%%%%%%%%% DEFINIZIONI (dette macro) %%%%%%%%%%%%
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\def\eps{ { \varepsilon } }
\def\={ = }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
~
\vskip 2 truecm
\centerline{ \bf NONLOCALITY OF CLASSICAL ELECTRODYNAMICS}
\vskip 1 truemm
\centerline{\bf OF POINT PARTICLES,}
\vskip 1 truemm
\centerline{\bf AND VIOLATION OF BELL'S INEQUALITIES}
\vskip 3 truecm
\centerline{Andrea CARATI, \ Luigi GALGANI}
\vskip 1 truemm
\centerline{Universit\`a di Milano, Dipartimento di Matematica}
\vskip 1 truemm
\centerline{Via Saldini 50, 20133 MILANO (Italy)}
\vskip 3 truecm
\centerline {ABSTRACT}
\vskip 1 truemm
\newbox\boxabstr
\newdimen\ampiezza
\ampiezza=\hsize
%%%%%%%%%%%%%%% Per stringere o allargare l'abstract cambia il
%%%%%%%%%%%%%%% numero 2 nella riga seguente (adesso e' 1 cm per lato)
\advance\ampiezza by -2 truecm
%%%%%%%%%%%%%%% Questo prepara l'abstract. Non modificarlo.
\setbox\boxabstr\vtop{\hsize=\ampiezza\parindent 0pt\ignorespaces
\noindent {We show that Bell's inequalities are violated in a model of
two charged particles interacting with two potential barriers, which
mimic the measuring instruments; the motion of each particle is
described by the \ALD equation in the
nonrelativistic version, and the role of the hidden variables is played
by the initial accelerations. The essential nonlocality property of the
system is induced by the celebrated Dirac's nonrunaway condition, which
makes the measuring instruments have a certain influence on
the observed system, by determining the domain of definition of the hidden variable
(the Bopp--Haag phenomenon).
So this model strongly supports E. Nelson's
suggestion, namely
that nonlocality properties suited to violate Bell's inequalities
appear in classical field theories when regularizing cutoffs are
removed.
}}
\centerline{\box\boxabstr}
\vskip 2. truecm
\noindent PACS numbers: 03.65.B, 03.50.D
\vskip .5 truecm
\noindent
Running title: BELL'S INEQUALITIES AND CLASSICAL ELECTRODYNAMICS
\vskip 1 truecm
\vfill\eject
1. {\bf Introduction}\quad It is very well known that the proof of
Bell's inequalities
relies essentially on the assumption of some sort of
locality, for example on what Bell calls the ``principle
of local causality''\upccite{belllibro}{clauser}. So, as experiments seem to violate
Bell's inequalities in the way predicted by quantum mechanics, one
concludes on the one hand that it is nature itself that seems to present
nonlocal properties, and on the other hand that the formalism of quantum
mechanics implicitly contains in itself some appropriate nonlocal
features.
Finally, one concludes that classical hidden--variable
theories cannot correctly describe nature, at least if they are
conceived, as is usually done, as local theories.
But clearly the same cannot be said of classical theories of nonlocal
type.
Now, it is obvious that one could produce some strange
model with just the nonlocal properties required to
violate Bell's inequalities, but this would be too
artificial. In Bell's words: ``{\it On the other hand, if no restrictions
whatever are imposed on the hidden variables, ...., it is trivially clear
that such schemes can be found to account
for any experimental results whatever}''\upcite{bell1}.
So, the significant problem for hidden variable theories is rather whether
nonlocal properties might occur
in classical theories in some natural way. This was apparently first
stressed by Nelson\upccite{nelqf}{nelascona}. What he had in mind is that the appropriate nonlocal
features might arise naturally in classical field theories in the
limit in which one removes space and momentum
cutoffs, previously introduced in order to
regularize the theory. Nelson gave indeed in support of his thesis some
heuristic and qualitative arguments, which led him to state that
(\cite{nelascona}, page 438) ``{\it A discussion of Bell's theorem leads
to the conclusion that it is no
obstacle to the description of quantum phenomena by classical random
fields}''; but neither a general proof nor a concrete example were available
to him. However, in \cite{nelqf} he suggested
that it should be of interest to study preliminarly some classical
models
describing the interactions of fields with particles; in such a case,
the removal of momentum cutoffs would be equivalent to taking the limit of
point particles.
In the present paper we point out that classical electrodynamics of point
particles in the so--called dipole approximation is indeed nonlocal,
just in virtue of the celebrated Dirac's nonrunaway
condition\upcite{dirac38} (see also \cite{coleman} and \cite{arai}), which
plays in fact a constitutive role in the definition of the point limit
itself. We then show how such a nonlocality is effective in leading to a
violation of Bell's inequalities, although not exactly in the way conjectured
by Nelson; this is obtained in a particular model which exploits a
striking phenomenon, namely that of Bopp\upcite{bopp} and Haag\upcite{haag}
(see also \cite{tunnel}), occurring in classical
electrodynamics of point particles as a consequence of Dirac's
condition.
Let us briefly recall what is meant by classical electrodynamics of point
particles in the dipole approximation, and which are the main results
for it. Classical electrodynamics of a
point particle is in principle nothing but the familiar Maxwell--Lorentz system, i.e.,
that having for unknowns the electromagnetic field and the
particle's position with minimal coupling, namely: the field obeys Maxwell equations having for source the
current due to the particle, while the particle satisfies the
relativistic Newton equation with the Lorentz force due to the field.
But for a point particle the system is ill defined, because of the infinite
``self--force'' on the particle (think of the Coulomb self--force in the
static case), and so needs a regularization. This can be performed
by imposing space and momentum cutoffs, or just
momentum cutoffs, and then one remains with the problem of
studying the limit in which the cutoffs are removed.
Such a program is still unaccopmplished; in Nelson's words
(\cite{nelqf}, page 65--66):
``{\it Is it an exaggeration to say that nothing whatever is known about
the behavior of this system as the cutoffs are removed, that there is
not one single theorem that has been proved?}''.
There are however partial results. First of all there are
the classical old results for the nonrelativistic case which
go back to Lorentz and Abraham and are concerned with the dipole
approximation; this is a linearization in which the current
$\vett j(\vett x,t)=\dot {\vett q}\delta({\vett x}-{\vett q}(t))$,
where $\vett q(t)$ is the particle motion, is
approximated by $\dot {\vett q}\delta({\vett x})$, while in the Lorentz
force the magnetic term is neglected. The limit was
performed in a way which today might be considered as heuristic, but in
any case the most relevant result was that the naive point limit
(i.e. that with a fixed bare mass) leads to a trivial dynamics, with the
particle and the field actually decoupled. So mass renormalization was
first introduced, with the bare mass diverging to $-\infty$, and in
such a way it was shown that the particle obeys in the limit the well
known third--order \ALD equation. The
relativistic case was dealt with in the year 1938, still in a
not completely rigorous way, by Dirac\upccite{dirac38}{haag}, who found the relativistic
version of the particle's equation; we will refer to both equations
(i.e. the relativistic and the nonrelativistic ones) generically as the \ALD
equation. Such classic results were fully confirmed, at
least in the nonrelativistic case and in the dipole approximation, by
the recent works \cite{dardie}, \cite{darioabl} and
\cite{andreadiego}, where the problem was studied with the present
standard of rigour; in particular, in \cite{andreadiego} the limit equation for the field was
found for the first time.
In any case, for the aim of the present paper the most striking feature of the
point limit is the generic appearence of the absurd runaway solutions,
which apparently were first
discussed by Dirac in connection with his relativistic equation: for
example, it turns out that the free particle accelerates exponentially fast as time
increases, and analogous divergences occur also for ther field.
Now, faced with such an apparently absurd situation
one might be tempted to simply
throw classical electrodynamics away. Another possibility, suggested
by Dirac himself, consists of imposing from outside for the limit system
a new prescription, which consists in restricting the phase space to those initial data leading to
nonrunaway motions; we recall that, in the typical case of scattering,
such motions are defined by the property that the acceleration vanishes
for $t\to +\infty$. It is thus clear that Dirac's
prescription assumes a constitutive role in the definition itself
of the the theory, radically changing its mathematical structure:
namely, classical electrodynamics of point particles as
defined by mass renormalization and Dirac's prescription is a new theory, structurally
different from standard electrodynamics of macroscopic (as opposed to
point) particles. From the mathematical point of view, the difference
consists in the fact that
the nonrunaway prescription leads to a problem
which resembles more to a Sturm--Liouville than to a Cauchy problem.
This is just the reason why many people appear to dislike the Dirac
prescription, blaming it of being, as they say,
acausal\upcite{ytzykson}.
The point we make is instead that such a theory is rather non locally causal,
more or less in the sense of Bell; this is indeed
a characteristic nonlocal feature of classical electrodynamics of point particles
which ultimately turns out to lead to a violation of Bell's inequalities, a fact that strangely
enough seems not to have been noticed up to now (see however \cite{rohrlich73}
).
This will be exhibited below in a very simple model, conceived within a
setup typical of the {\it gedankenexperiments} related to
Bell's inequalities. The model consists of two charged particles
which, after having somehow interacted, separate away along two opposite
directions, and proceed with no further mutual interaction.
Then, each of the two particles interacts with a measuring instrument,
which we take to be an external potential barrier,
the measurement consisting in observing whether the
particle crosses the barrier or is reflected from it. A
dicothomic variable is thus defined, which takes for example the value $+1$
in the former, and $-1$ in the latter case. As usual in this kind of
problems, we allow each measuring instrument to be prepared in one of
some (typically two or three) different settings, which here are just three
different heights of the barrier.
Now, the measurement would be trivial in the purely mechanical case, because
the particle would certainly cross the barrier or be reflected from it,
according to the value of its energy. But things are completely different
if the self--interaction with the electromagnetic field is taken into account, what
we do by
assuming that the particle's motion is a solution of the nonrelativistic
\ALD equation in
presence of the external potential barrier. Indeed, use is made here of a relevant,
highly nontrivial, property of the
\ALD equation which, although being known already to
Bopp\upcite{bopp} and Haag\upcite{haag}
and somehow adumbrated in a theorem of Hale and Stokes\upcite{hale},
was particularly appreciated quite recently\upcite{tunnel}; namely, that in general the
nonrunaway solutions of the \ALD equation in presence of an external
barrier are not uniquely defined by the initial mechanical state
(position and velocity) of the particle. As a consequence, the initial acceleration has really to
be assigned as an additional variable, not uniquely defined by the
mechanical state, and thus plays here
the role of the hidden variable, for which
a probabilistic description is needed. It turns out, as proven in
\cite{tunnel}, that according to the value of the hidden variable the
particle crosses the barrier or is reflected from it, which is a
situation somehow reminiscent of the tunnel effect.
Moreover, such a Bopp--Haag phenomenon occurs for initial mechanical
states in a domain which actually depends on the setting of the
measuring instrument (i.e. on the height of the barrier), and this
is the feature that turns out to attribute indeed an essential nonlocal character to the system.
In section 2 the relevant notions concerning the \ALD equation and the
Bopp--Haag nonuniqueness phenomenon are
recalled, and it is discussed how the particle's aceleration plays the
role of the hidden variable; in section 3 the two--particle model is
discussed, and it is shown that
Bell's inequalities are violated for certain unfactorized probability
distributions of the hidden variables; some further discussions of a
general type are deferred to section 4, and the conclusions then follow.
\vskip 3truemm
{\bf Acknowledgement.} The present work has been developed under the EC
contract ERBCHRXCT940460 for the project `` Stability and universality in
classical mechanics''.
\vskip .5truecm
\noindent
2. {\bf Relevant features of the \ALD equation: Dirac's condition
and the Bopp--Haag
nonuniqueness phenomenon}.\quad
Limiting our attention to the case considered below in our
model, namely that of a particle moving on a line (the $x$ axis)
in the presence of an external potential energy $V$ in the
nonrelativistic approximation, the \ALD equation has the form
$$
\ddot x = -\frac 1m V'(x) +\eps \tdot x \, , \autoeqno{1}
$$
where dot and prime denote derivatives with respect to time $t$ and
position coordinate $x$ respectively,
$m$ is the particle's mass,
while $\eps=\frac23\, e^2/mc^3$ is a parameter (with the dimensions of
a time)
depending on the speed $c$ of light and on the charge $e$ of the particle. The
natural phase space for the equation is
$\reali^3$, referred to coordinates $(x,v,a)$
defining the particle's position $x$, velocity $v=\dot x$ and acceleration
$a=\dot v$. But it turns out that
generic initial data in such a space give rise to
runaway motions. This is immediately seen in the simplest example, i.e.
that of the free
particle characterized by $V=0$, because equation \eqrefp{1} then
reduces to a closed
equation for the acceleration, namely $\epsilon \dot
a =a$; the general solution $a(t)=a_0 \exp(t/\epsilon)$ thus leads to
absurd self--accelerating motions for all initial data $a_0$, with the only
exception of the initial data on the
manifold $a=0$, which lead to the natural motions $a(t)=0$. Such an
invariant subset of phase space, defined by $a=0$ and
constituted by orbits not having runaway
character, can be called the {\sl physical manifold} or {\sl Dirac manifold}.
More in general, let us consider a scattering problem, with the force vanishing
sufficiently fast at infinity. The problem of existence of the
Dirac manifold can be stated in the following way: given an initial
mechanical state $(x_0,v_0)$, one asks whether there exists
an initial acceleration
$a_0$ such that the corresponding motion, with initial data
$(x_0,v_0,a_0)$, has a nonrunay character, i.e.
satisfies the condition
$a(t)\to 0$ for $t\to+\infty$. This clearly is a kind of
Sturm--Liouville problem. Existence was proven by Hale and Stokes for a
large class of potentials, but they couldn't prove
uniqueness; mathematically, this is due to the circumstance
that the existence problem turns
out to be reduced to a fixed point
problem not involving a contraction. Thus, for
a given mechanical state one can a priori expect that there exist several allowed values for
the initial acceleration; in other terms, the Dirac manifold is not
a priori the graph of a function $a=f(x,v)$, and can in general be folded
above the mechanical $(x,v)$ plane.
As a matter of fact, a case of nonuniqueness was already known to
Bopp and Haag, who could find by elementary methods two solutions corresponding to the same
mechanical state in a certain domain,
for a potential step. But such a nonuniqueness property did not arouse much
interest, and apparently was not even known to Hale and Stokes.
In addition, the presence of just two nonrunaway solutions (for a
given mechanical state) in the case discussed
by Bopp and Haag led some authors (see the discussion in \cite{erber}) to
conceive that only one of them should be retained as
physically meaningful, while the other one should be discarded, although
it is not obvious which criterion of selection should be adopted.
On the other hand, in \cite{tunnel} it was shown that such a nonuniqueness
phenomenon is indeed a common fact for a large class of potentials,
and was in particular proven to occur
essentially for all potential barriers with a sufficiently sharp
maximum; moreover, it was found that there occur in general not just
two, but an arbitrary number of solutions. This goes as follows.
For a given barrier and an initial position $x_0$,
there exists an interval of initial velocities, and thus an interval
$I$ of initial energies (located about
the maximum of the barrier), such that the corresponding initial acceleration
leading to nonrunaway motion is not unique. More precisely, for any
positive integer $n$ there exists an interval $I_n\subset I$ with
$n$ nonrunaway solutions crossing the barrier and $n$ nonrunaway
solutions reflected from it.
It is thus
clear that the allowed initial accelerations corresponding to a given
mechanical state are all apparently on the same
footing, and there is no hope to find a natural criterion
for selecting a particular one among them as privileged.
It rather appears that, given
an initial mechanical state with energy in the interval $I_n$,
one should instead more naturally be led to
assign some probability distribution to the allowed values for the
``nonmechanical variable'', i.e. for the acceleration.
Such a qualification of nonmechanical
variable for the initial particle's acceleration seems to be
appropriate. Indeed, in \cite{darioabl} and \cite{andreadiego} it was shown that the initial acceleration to
be inserted in the Cauchy data for the \ALD equation is a certain
definite function of
the initial data of the original \ML
system describing the complete system particle plus field; in other
words, the particle's initial acceleration in the
Cauchy problem for the \ALD equation is just a trace of the initial data
for the field in the complete \ML system.
Furthermore, a property of the allowed
initial accelerations (or of the initial field in the complete system,
according to what just said) which naturally favours the interpretation
of the acceleration as the hidden variable with respect to the
mechanical ones is the dependence of the allowed initial accelerations
on the initial position $x_0$ as $|x_0|$ is taken farther and farther
away from the barrier, which is the
case of interest for the description of scattering processes. Indeed, while
the energy strip $I$ (where the nonuniqueness phenomenon occurs)
becomes essentially independent of $x_0$, it turns out that the
allowed initial accelerations corresponding to a given mechanical state
collapse to zero exponentially fast as $|x_0| \to
+\infty$; and this makes the different accelerations, leading to
nonrunaway motions for a
given mechanical state, essentially undistinguishable, as should be expected
of variables to be qualified as hidden. In physical terms, with
reference to the complete \ML system, it is thus actually
impossible to prepare the initial state with a concrete control of the
initial electromagnetic field required to discriminate
whether the particle will cross the barrier or not. This seems indeed
a property to be expected of a hidden variable, namely of a variable
that, according to Bell, should be rather called {\sl uncontrolled},
{\it`` for these variables, by hypothesis, for the time being, cannot be
manipulated at will by us''}\upcite{bell2}.
\vskip .5truecm
\noindent
3. {\bf The model, and the violation of Bell's inequalities}.\quad
In all {\it gedankenexperiments} concerned with Bell's inequalities, one
deals first of all, following Einstein, Podolsky and Rosen
themselves, with two equal
subsystems which initially interact in some way and then
separate away along two opposite directions, evolving as free subsystems;
then, measurements of some dicothomic physical quantity are
separately performed on each of them by some instrument which can be
prepared in a certain number of different settings. In our
model, the system is constituted of two equal charged particles which,
after having initially interacted in some way that doesn't concern us
here, separate away along opposite directions on a straight line, and then proceed, say
for $x_1\ge L$ and $x_2\le -L$, with
no mutual interaction; the measuring
instruments are just two potential barriers
located on opposite sides with respect to the
origin (i.e. with respect to the source) very far away from it,
with heights that can assume three
different values, and the measurement consists in observing whether
each particle crosses its barrier or is reflected from it.
The dicothomic variable is defined as taking
the value $+1$ in case of crossing and the value $-1$ in the
opposite case. The motion of each
independent particle is described as a solution of the nonrelativistic
\ALD equation
with the given potential. It is assumed that the heights of the
barriers are such as to allow for the nonuniqueness phenomenon described
above to occur; so the mechanical state $(x,v)$ of a particle does not
uniquely define its motion, and the role of the hidden variable uniquely defining the
motion is played by the acceleration, which takes values in a
domain depending on the height of the barrier, i.e. on the setting of
the measuring instrument.
Consider a time, which following Einstein--Podolsky--Rosen we call $T$, at which the two particles are outside the
interaction region, i.e. have positions
$x_1(T)=x_{1}^*>L$, $x_2(T)=x_{2}^*<-L$, and assume that the
velocities $v_1(T)=v_1^*>0$, $v_2(T)=v_{2}^*<0$
are such that the corresponding initial energies of the two particles
belong to the intervals where the nonuniqueness phenomenon occurs,
for all the three possible heights of the barrier; assume moreover that
the barriers are so far away from the
interaction region that the location of the nonuniqueness intervals
is practically
independent of the precise values of the positions $x_{1}^*$, $x_{2}^*$.
Fix then the heights of the barriers in the following way:
choose three positive numbers $n_\mu$
$(\mu=1,2,3)$, and fix the height of the first barrier
in such a way that the allowed values for the acceleration
$a_1(T)$ of the first particle are in number of $n_\mu$
if the barrier is at $\mu$--th height; analogously for the second
particle, having chosen three positive numbers $m_\nu$ $(\nu=1,2,3)$.
Denote by $a_1^{i,\mu}$ ($i=1, \cdots, n_\mu$) and by
$a_2^{j,\nu}$ ($j=1, \cdots, m_\nu$) the allowed values
of the hidden variables $a_1(T)$, $a_2(T)$, when the instruments are in
settings $\mu$ and $\nu$ respectively ($\mu,\nu=1,2,3$).
For a given setting of the barriers,
i.e. of the measuring instruments, the physical or Dirac manifold of the
two--particle system turns out to be a well defined four--dimensional
manifold in the
six--dimensional phase space $\reali^3\times \reali^3$ with coordinates
$(x_1,v_1,a_1,x_2,v_2,a_2)$.
Indeed, in the non--interaction region the Dirac manifold
is just the product of the two--dimensional Dirac manifolds of the
uncoupled particles, while in the interaction region
the Dirac manifold is simply
defined by prolongation, i.e. by letting the system evolve backward in time
according to the coupled dynamics, whose precise definition is not of
interest here. Concerning the global Dirac manifold, notice in particular that
changing even just one of the barriers
produces a change in the complete manifold itself, and that the manifold
does not have a product structure in the interaction region, which is
the one where the initial data are in principle
asssigned.
According to the three possible choices for each of the
barriers we have nine distinct Dirac manifolds, say $D_{\mu\nu}$
($\mu,\nu=1,2,3$), and the section of each such manifold $D_{\mu\nu}$
with the two--dimensional
plane $x_1=x_{1}^*$, $x_2=x_{2}^*$, $v_1=v_{1}^*$, $v_2=v_{2}^*$ is just a
finite set of points, namely the set of points
$(x_{1}^*,v_{1}^*,a_{1}^{i,\mu},x_{2}^*,v_{2}^*,a_{2}^{j,\nu})$,
with $i=1,\cdots, n_\mu$;
$j=1,...,m_\nu$; $\mu,\nu=1,2,3$. So, for any choice of the heights of the two barriers
one has a discrete space of events $\Omega_{\mu\nu}$,
i.e. the space of the pairs of allowed initial accelerations
$(a_1^{i,\mu},a_2^{j,\nu})$,
whose cardinality depends on the height of both barriers.
Now, in order to discuss the outcomes of our {\it gedankenexperiment},
we have to assign a probability to the initial states, i.e. we have to
assign a probability measure ${\rm Pr}_{\mu\nu}$ on each
space $\Omega_{\mu\nu}$ $(\mu,\nu=1,2,3)$. It is clear that in such a
way one thus assigns an invariant probability measure on each Dirac
manifold $D_{\mu\nu}$, and conversily that every invariant probability measure on
$D_{\mu\nu}$ defines a probability measure ${\rm Pr}_{\mu\nu}$ on
$\Omega_{\mu\nu}$. On the other hand there seems to be no reason to
privilege any particular invariant probability measure on $D_{\mu\nu}$,
and consequently any particular
probability measure on $\Omega_{\mu\nu}$;
in particular, as the Dirac manifolds don't have a product structure,
there is no reason to privilege the factorized measures, i.e. the
measures assigning independent probabilities to the
accelerations of the two particles. In consideration of this,
we seem to be authorized to
assume that all possible choices of the probability measures
${\rm Pr}_{\mu\nu}$ on $\Omega_{\mu\nu}$ are on the same footing.
In particular, the nine probability measures
${\rm Pr}_{\mu\nu}$ (each defined on the corresponding space $\Omega_{\mu\nu}$)
can be assigned independently from each other.
This is what we do here; further comments comments will be given below.
Now Bell's theorem, which we take in Nelson's version (see \cite{nelascona},
page 445), says that, with suitable assumptions of locality to be
recalled in the next section: {\it There do not exist
random variables $\alpha_\mu$ and $\beta_\nu$
(for $\mu,\ \nu=1,2,3$) such that $\alpha_\mu$ and $\beta_\nu$
is equal $\pm 1$ and }
$$
\eqlabel{2}
\eqlabel{3}
\eqalignno
{
&{\rm Pr}_{\mu\mu}( \alpha_\mu\beta_\mu=-1) =1 &\eqrefp{2}\cr
&{\rm Pr}_{\mu\nu}( \alpha_\mu\beta_\nu=-1) <\fraz12\ \quad (\mu\neq\nu)
\, &\eqrefp{3}\cr
}
$$
(think of $\alpha_\mu$ as the dichotomic variable corresponding to the
first particle crossing or not the barrier at $\mu$--th height, and
analogously of $\beta_\nu$ for the second particle).
So a violation of
Bell's inequalities occurs if one finds random variables $\alpha_\mu$,
$\beta_\nu$ satisfying relations \eqrefp{2} and \eqrefp{3}. We show
now that our dichotomic variables do indeed satisfy them, for certain
probability measures ${\rm Pr}_{\mu\nu}$. To be concrete, make the
following choice: for barriers of the
same height (i.e. for $\mu=\nu$) assign zero probability to all events in
which both particles are reflected or both particles cross their
barriers, so that \eqrefp{2} is
satisfied; instead, for barriers of different heights (i.e. for $\mu\neq\nu$)
assign arbitrary
probabilities to all events of the set in which both particles are reflected
or both particles cross their barriers, with the only constraint that their sum be less than
$1/2$, and distribute arbitrarily the remaining probabilities
in the complementary set. It would be a rather simple exercise to take into
consideration Bell's inequalities in their usual version involving
correlations, and prove that they can be violated too.
Thus the counterexample to Bell's theorem was obtained
here in a completely trivial way, just by exploiting the
complete arbitrariness of the probability measures ${\rm Pr}_{\mu\nu}$.
This seems to be in agreement with the quotation from Bell reported at the
beginning of the introduction ({\it it is trivially clear ...}).
However, the point we make is that the violation was obtained here for a
model which is not a strange
{\it ad hoc} one, but just classical
electrodynamics of point particles in the dipole approximation,
when due consideration is given to
mass renormalization and to its main manifestation,
namely the generic occurrence of runaway solutions, which is dealt with
through Dirac's nonrunaway condition.
We now add some further comments, trying to emphasize the relevant
features of our model wich led to the violation
of Bell's inequalities.
1. We start by pointing out that nonlocality comes about in
our model in two ways. The first one, which we discuss presently,
manifests itself already in the case of just one particle, when the
interaction with the measuring instrument is taken into account.
This is related to the fact that the domain of
definition of the hidden variable depends on the setting of the
instrument, no matter how far away it is situated, and is ultimately a
consequence of the nonlocal character of Dirac's nonrunaway condition
(the Bopp--Haag phenomenon).
Then, when the two--particle system is considered, this first
nonlocality property leads to the fact that the probability spaces
$\Omega_{\mu\nu}$ themselves depend on the settings of both instruments;
so one should expect that the probability distributions of the hidden
variables too depend on the settings of both instruments. And this is
completely at variance with the hypotheses used by Bell (``{\it ... we
supposed that the experimental settings could be changed without
changing the probability distribution of the hidden parameter}'', see
\cite{belllibro}, page 154, and note 21). In particular, the fact that the
domain of definition of the hidden variable depends on the settings of
both instruments is sufficient to completely invalidate the
argument by which, following Mermin\upcite{mermin}, Nelson showed
(see \cite{nelqf} sec. 23, especially page 120) that classical hidden
variables should be ruled out.
By the way, it seems to be of interest to point out that the situation
described here for classical electrodynamics, namely that the setting of
the instruments has a certain ``influence'' on the observed system,
is quite similar to that occurring in quantum mechanics. In the words of
Bell: ``{\it Since quantum phenomena indicate that
the experimental devices must be regarded as integral parts of the whole
experimental situation, not separable from the system being studied,
there is no reason to expect that there should be any quantities that
can be held fixed as the experiments are changed.}'' (see also the
interesting remark at page 154 of \cite{belllibro}).
\vskip .3 truecm
2. Let us now come to the second nonlocality property; this refers
altogether to the global two--particle system (each particle being
in the presence of the corresponding measuring instrument), and is
related to the unfactorization of the initial state.
Let us describe this point in a greater detail. As indicated above,
when the measuring instruments are in settings $\mu$, $\nu$,
one has to consider for the complete
system the probability space $\Omega_{\mu\nu}$, a point of which is constituted by the pair of hidden variables
(the accelerations at the ``initial'' time $T$), and an initial state is just a probability
measure ${\rm Pr}_{\mu\nu}$ on $\Omega_{\mu\nu}$. Now, there exist
first of all the states which are factorized (i.e. assign independent
probabilities to each particle's acceleration),
and one immediately proves (see the appendix) that for them one has
$$
{\rm Pr}_{\mu\nu}( \alpha_\mu\beta_\nu=-1) >\fraz12 \ ;
$$
thus the violation of Bell's inequalities, as in the example above,
can be obtained only for unfactorized states. In
other words, the system decouples into two independent
subsystems if the initial probability is factorized, while
in the opposite case one has
a correlation which is essential for the violation of the
inequalities. Here too there is a strong analogy with quantum mechanics,
where unfactorized states are required to violate Bell's inequalities;
the corresponding correlation was discussed by
Schroedinger\upcite{schroedinger}, who called
it {\it entanglement}. Now, why should the unfactorized
(or entangled) states be preferred, as being the generic ones? To this we can give two
answers. The first one is exactly the same given by
Schroedinger in the case of quantum mechanics,
namely that the initial (i.e. at time $T$)
states are generically unfactorized (or entangled) just because of
the previous mutual interaction of the particles.
The second answer is related to the fact
that, as one immedialtely proves (see
again the appendix), the singlet (i.e. satisfying \eqrefp{2}) states which are factorized are
necessarily trivial, i.e. are such that each particle either
certainly crosses its barrier or is certainly reflected from it;
consequently, in a sense ``genuine'' singlet states are necessarily
unfactorized.
\vskip .5truecm
\noindent
4. {\bf Further comments.}\quad
So we have shown how nonrelativistic classical electrodynamics of point
particles in the dipole approximation in general violates Bell's inequalities.
On the other hand the fundamental problem raised by such inequalities
is the connection between causality and relativity, and the
interesting problem would be to know whether the inequalities are violated also
for the relativistic (and nonlinear) version of our model.
Now, in order to discuss this point we would first of all need a
rigorous deduction of the dynamics of point particles interacting with
the electromagnetic field in the relativistic case, which is still lacking.
So we limit ourselves to
express here our personal conjecture, which is as follows. We think that
very probably such a rigorous discussion will eventually confirm
the result of Dirac himself, namely that the
particle obeys Dirac's relativistic version of the \ALD equation. Now, for
such an equation
the situation with respect to runaway and nonrunaway motions
is essentially the same as in the nonrelativistic case.
Thus, all the requirements imposed by relativity should already be
there, and Dirac's prescription of restricting the phase space to the
nonrunaway solutions should be at all compatible with relativity and
causality. As a
consequence, no substantial changes should occur in the relativistic
case,
and the situation would be essentially the same as
discussed in the present paper.
To check whether this is the case or not is a very interesting open
problem.
Now we address the following question: if it can be shown that the
correct relativistic theory in the point limit is that of Dirac,
with the essential ingredient of the nonrunaway prescription, with which form
of locality would this be compatible ?
In this connection let us recall
the properties of passive and active locality as introduced by Nelson,
who verbally describes them as follows (\cite{nelascona}, page 446). Let $A$ and $B$ be space--like separated
bounded open sets in space--time, and $A^+$ the future cone of $A$; define
by {\sl slice} an open subset of space--time bounded by two parallel
space--like hyperplanes, and let $X$ be a slice disjoint from
$A^+\bigcup B^+$. Passive locality is the property: {\sl if the
field is known in the slice $X$, then an observation in one of $A$ or
$B$ gives no additional information about an observation in the other}.
Instead, active locality is: {\sl an experiment in $A$ affects
the field only in $A^+$}.
Nelson shows (in the proof of the theorem quoted above) that at least one of
the two locality properties has to be abandoned, if Bell's
inequalities are to be violated, and says he is inclined to think
that passive locality should be abandoned. We are rather inclined to think
that neither active nor passive locality are imposed by relativity, at
least if one takes for granted that the relativistic version of the \ALD
equation is correct. This is essentially due to the Bopp--Haag effect.
Indeed, consider first active locality, which is in fact concerned with
just one particle. This requires that an experiment (i.e. the
setting of the instrument) in $A$ affects the field only in its future
cone $A^+$; but this is not the case (i.e. active locality does not
hold) with the \ALD equation, because
the setting of the instrument affects also the past cone $A^-$,
inasmuch as it determines by Dirac's condition, through the initial
particle's acceleration, the domain of the possible initial data of the
field (such a phenomenon is
often referred to as the phenomenon of {\sl preacceleration}).
Neither does passive locality hold, again because
the domain of definition of the field in
$X\bigcap(A^- \bigcap B^-)$ is determined by the setting of both
instruments.
These are the reasons that lead us to think that neither active
nor passive locality in Nelson's sense should hold in
relativistic theories. Which form of locality, in some suitable weak
sense, should then be appropriate in relativistic theories is a very
interesting
question of principle, discussed by many authors (see for example
\cite{stapp}, \cite{haag2}, \cite{segal}), on which we are unable
to say anything conclusive at the moment.
\vskip .5truecm
\noindent
5. {\bf Conclusions.}\quad
In conclusion, we have pointed out that, in classical electrodynamics of point
particles in the nonrelativistic and dipole approximation, the
setting of the measuring instruments has a certain ``influence'' on the
observed system, inasmuch as it determines the possible range of the
parameters playing the role of hidden variables; this is indeed the
essence of the Bopp--Haag phenomenon, and is ultimately due
to Dirac's nonrunawy condition. Then we have shown how this property
leads in a particular model, for some unfactorized initial
states, to a violation of Bell's inequalities. Furthermore, we have
pointed out that analogous results might be expected to hold in the full
relativistic nonlinear version of the model.
In such a way we believe we have given a strong indication in favour of
the correctness of the idea suggested by Nelson, namely that a nonlocality
property suited to violate Bell's inequalities
might appear in classical field theories when the regularization cutoffs
are removed.
\vskip 2 truecm
\centerline{\bf REFERENCES}
\vskip .5 truecm
\insertbibliografia
\vskip 2 truecm
\centerline{\bf APPENDIX}
\vskip .5 truecm
We prove here two simple lemmas concerning probabilities for
factorized states. The formal description of the situation is the
following one, referring to a given setting of the measurement
instruments (so that the indices $\mu$, $\nu$ will be omitted here).
We have two random variables $a_1$ nad $a_2$ (in our
model, the hidden variables, i.e. the accelerations of the first and of the second particle at the
``initial'' time $T$) whose possible values are $a_{1i}$, $a_{2.j}$,
($i=1,\cdots,n$, $j=1,\cdots,m$). The space of the elementary events is the set
$\Omega$ of pairs $(a_{1i}, a_{2j})$. A state is a probability
measure in $\Omega$, i.e. a probability distribution
$p_{ij}$ (with $p_{ij}\ge 0, \sum_{ij}p_{ij}=1$), and a state is
factorized by definition if
$$
p_{ij}=p_i q_j\ , \quad (p_i\ge 0\ ,\ q_j\ge 0\ , \ \sum_ip_i=\sum_jq_j=1)\ .
$$
Let $\alpha$, $\beta$ be two dichotomic random variables, i.e. real valued
functions on $\Omega$, taking values $+ 1$ or $-1$; moreover, let
$\alpha$ depend just on $a_1$ and $\beta$ depend just on $a_2$
(in our case, whether
the first particles crosses the barrier or not just depends on the
value of $a_1$, and analogously for the second particle), i.e. assume
$$
\alpha_{ij}=\alpha_i\ , \quad \beta_{ij}=\beta_j\ , \quad
(i=1,\cdots,n,\quad j=1,\cdots,m)\ ,
$$
Then, denoting by
${\rm Pr} (A)$ the probability of an event $A$, one has
\vskip .2 truecm
\vskip .2 truecm
\noindent {\sl Lemma 1}:\quad For factorized states it is
$$
{\rm Pr} (\alpha \beta=-1) > \frac 12\ .
$$
\noindent {\sl Proof.} \quad One has ${\rm Pr} (\alpha \beta=-1)
=pq+(1-p)(1-q)$, where $p= {\rm Pr} (\alpha =1)$, and $q= {\rm Pr} (\beta=-1)$.
One thus has to look for the minimum, in the unit square, of the function $f(p,q)=
pq+(1-p)(1-q)$, which is immediately found to be $\frac 12$.
\vskip .1 truecm
Consider now a singlet state, i.e. one such that
${\rm Pr} (\alpha \beta=-1) =1$. Say furthermore that a state is trivial
if each of the variables $\alpha$, $\beta$ takes just one of the two
possible values $+1$, $-1$ (i.e. each particle either certainly crosses
the barrier or is certainly reflected from it). Then we have
\vskip .2 truecm
\noindent {\sl Lemma 2}: Singlet states which are factorized are trivial.
\noindent {\sl Proof.} \quad Let
$$
\eqalign{
\alpha_i&=+1\quad {\rm for}\quad i=1,\cdots,n^* \ ,\quad\quad
\alpha_i=-1\quad {\rm for}\quad i=n^*+1,\cdots,n \ , \cr
\beta_j&=+1\quad {\rm for}\quad j=1,\cdots,m^* \ ,\quad\quad
\beta_j=-1\quad {\rm for}\quad j=m^*+1,\cdots,m \ .\cr
}
$$
Then the singlet condition requires
$$
\eqalign{
p_iq_j&=0\quad {\rm for} \quad i=1,\cdots, n^*\ ,\ j=1,\cdots, m^*\cr
p_iq_j&=0\quad {\rm for} \quad i=n^*+1,\cdots, n\ ,\ j=m^*+1,\cdots, m .\cr
}
$$
One of the probabilitities $q_1,\cdots, q_n$ has to be nonvanishing, and
assume for example $q_1\neq 0$. Then necessarily one has
$p_i=0$ for $i=1,\cdots,n^*$, which means that $\alpha$ takes only the value
$-1$, and consequently $\beta$ just the value $+1$.
\bye