Pre-print
a cura dell’autore. Originale pubblicato su:

IL NUOVO
CIMENTO VOL. 83B, N. 2 11
Ottobre 1984

**A Newtonian Separable Model which Violates Bell’s
Inequality (*)**

*Istituto Dipartimentale di Fisica dell’Università *- *Catania*

*Gruppo Nazionale di Struttura della Materia *- *Catania*

*Istituto Nazionale di Fisica Nucleare *- *Sezione di Catania*

* *

* *

(ricevuto il 16 Novembre 1983; manoscritto revisionato ricevuto il 20 Agosto 1984).

**Summary**. — On the basis of the Mackey’s axiomatization
of quantum mechanics an argument is given which allows, in determinate circumstances,
the violation of Bell’s inequality also in a «classical mechanics»*
*and a «classical probability» context. A mechanical model made out of two
separate subsystems of coupled oscillators is studied by computer experiments
to illustrate the point. In fact, the model violates Bell’s inequality. The
hypothesis is put forward that the principal reason for this violation is due
to the special kind of « detectors» introduced in the model which give a
«count» every time a given dynamical variable of the mechanical system crosses
an assigned threshold.

PACS. 03.65. — Quantum theory; quantum
mechanics.

**1. — Introduction.**

In two well-known papers, BELL (^{1,2})** **succeeded in demonstrating that previous impossibility proofs of
«hidden variables» in quantum mechanics were wanting because of too restrictive
assumptions on their mathematical structure (^{1}).** **However,
he proposed (^{2})** **an inequality which must be satisfied
by any hidden-variable theory, under certain separability requirements, in
order to restore causality in a gedanken experiment of EPR type (^{3})** **in the Bohm-Aharonov (^{4})** **version.

The inequality is violated by
quantum mechanics in certain instances.

Those papers have stimulated many
researches both theoretical and experimental, as can be seen in the reviews
contained in ref. (^{5-8}).

Most experiments favour quantum
mechanics and hence seem to refute local physical theories which would continue
to mantain the existence of «elements of reality» as defined in ref. (^{3}),* *unless we are willing to introduce certain nonlocal featuires
which, however, would «resolve the EPR paradox in the way which EINSTEIN would
have liked least»,* *to quote BELL (^{1}).** **Many physicists believe that the conclusions to be drawn from the
present experimental results «are philosophically startling: either one must
totally abandon the realistic philosophy of most working scientists, or
dramatically revise our concept of space-time»,* *as, *e.g.*,* *we read in the
abstract of ref. (^{7}).

A way-out of this dilemma may be to
question the significance of Bell’s inequality.

This way has already been pursued by
many authors on different grounds.

One is the argument, maintained in
ref. (^{9,10}),** **that the hypothesis of Bell which
assigns a unique probability distribution to the hidden parameters in the
different observational directions is too restrictive for not considering that
the measurement process will affect in some way the probability distribution by
changing the state of the system.

In the present paper I shall analyse
the above argument on a different basis.

The conclusion will be that the
argument is correct because of different reasons than those connected with the
change of state introduced by the measurement process.

We shall see that the reason resides
in the interpretation that one presupposes, for the probabilistic concepts as
applied to quantum and classical physics.

This point is illustrated in a more
general context by FINE (^{11}).

** **

** **

**2. — Paradigmatic inequality in EPR physical
situations.**

It is often believed that «quantum
mechanics purports to be a description of physical reality which deliberately
eliminates from theory all features not demanded by experiment»(^{12}).

However, it is not so easy te pursue
this aim and other people think it is not worth and even misleading; but, in
this limited context, in order to clarify the issue without unnecessary
metaphysical presumptions, I shall try te describe the physical situation
avoiding any term which cannot be reconducted to experimental observations or
manipulations.

In such terms, the paradigm in which
Bell’s inequality is derived is the following: there is a «physical system» *S*
defined by certain physical manipulations. On both sides of *S*, in a
given spatia1 direction which we call the *z*-axis, we have a pair ef
objects P_{1} and P_{2 }which we call «polarizers»,*
*each characterized by a unit vector **θ**_{1} (**θ**_{2})
in the plane normal to the *z*-axis.

The directions of these vectors can
be varied by opportune knobs.

We have, in series to the
«polarizers»,* *two «detectors» D_{1 }and D_{2
}which register «counts» in the presence of *S *.

We have also a «coincidence system»
which gives a «count» if and only if, separately, two «counts» will be
registered in D_{1 }and D_{2 }within a fixed interval of time
which we call the «resolving time».

The «polarizers» affect the counting
intensity of the two «detectors» *n(***θ**_{1}*) *and *n(***θ**_{2}*) *and,
hence, of the «coincidence», *n(***θ**_{1 }, **θ**_{2}*) *in dependence of the setting of the knobs.
Now we abandon the realm of «observations» and hypothesize a set *Λ*** **of
variables λ associated with a probability density

Each value of λ determines
univocally the registering or not of a count in either of the two detectors
(and hence also in the coincidence system) at a given instant of time. This for
any given set of directions of the polarizers.

The separability (locality)
requirement is expressed (^{2,14})
by pretending that the registering of a count on one side is not
influenced by the setting of the knob on the other side.

Moreover, we assume that, whenever
the two polarizers are set in the same spatial direction, we have* *with certainty a count in the coincidence system every time we
have a count in one of the two detectors. Under this hypothesis we are able te
know, for a given direction, what will happen on one side if we know what ought
to happen on the other side in the chosen common direction, so that we may
limit ourselves to consider the possible outcomes, determined by the values of
the λ’s, on one side ouly.

Thus, if we choose three arbitrary
possible directions **θ**_{1 }, **θ**_{2},
**θ**_{3 },_{ }for a polarizer they will partition
the *Λ*-space in eight disjoint subsets of measure *P*({*A*_{θ}_{1
}*, A*_{θ}_{2 }*, A*_{θ}_{3}}) with *A*_{θ}_{i}**_{
}**a variable which assumes the values unity or zero according to
the registering or not of a count in the direction

We suppose that the counting
intensities are proportional to the corresponding probabilities, this
hypothesis is necessary to connect theory to possible experiments (^{5,7}),** **thus we write (following an analogous reasoning in ref. (^{5,14}))

(1) _{}

let us define

(2) _{}

By solving the linear system with respect to *P*({*A*_{θ}_{1
}*, A*_{θ}_{2 }*, A*_{θ}_{3}}) , we get

(3) _{}

* *

By summation

(4) _{}

by dividing by *m*_{3} ³ 0,

(5) _{}

or, in terms of the counting intensities,

(6) _{}

* *

This last inequality is violated
when we identify the system *S* with a pair of photons in a *J ***=** 0 ® 1 ® 0 atomic cascade, analysed by two ideal
polarizers P_{1 }and P_{2 }and detected by two photomultipliers
D_{1 }and D_{2}; in this case we have (^{7})

(6) _{};

if we choose θ_{1}= 0°, θ_{2}**= **60°, θ_{3}**= **30°,

we get

(8) cos^{2 }30° **+ **cos^{2 }30° £ 1+ cos^{2 }60°,

*i.e.*

(9) _{} which is false.

There is nothing te object to this
line of reasoning. But there is something to object to the supposed generality
of the assumptions made for the significance entailed by the probabilistic
structure of the hydden variables. This structure cannot encompass even quite
simple physical situations in a «classical mechanics» and «classical
probability» context.

I shall give an example of such a situation in
the next section, by now I want to analyse the issue from the theoretical
viewpoint by resorting to a wellknown axiomatization of a general physical
statistical theory (^{15})* *(see also ref. (^{11})).** **In
this axiomatization (^{11,15}),** **the probability measure is defined as
a function *p* which assigns a real number 0 £ *p(A , α , E) *£* *1 to each triple *A , α , E, *where *A *is an observable, *α*
is a physicad state and *E *is a Borel
subset of the real line *R.*

We have *p(A , α , φ) *= 0 (*φ*
represents the empty set), *p(A, α, R) *= 1,

* p(A, **α, E _{1
}UE_{2 }U ...)*

This first axiom is enough, for the
moment, to draw some conclusions.

We note, first, that this
probability measure defined on the Borel subsets of the real line is
Kolmogorovian and hence is «a perfectly ordinary variety of statistical theory»
(^{11}).

As a second point we note that the
observable *A *and the state *α* are to be thought as parameters
(or sets of parameters) which fix the probability measure, so that, if the
state and/or the observable change, then they will induce a change also on the
Borel measure *p*.

Until now we are dealing with an
abstract setting. Of course a good axiomatization is very useful to clarify
the formal relations among the objects of our discourse but is not a substitute
for the elucidation of the meaning of concepts. This can be accomplished by the
interpretation of the abstract formalism in concrete physical situations.

In classical statistical mechanics,
the observable *A *will correspond to a
Borel function, on the phase space of the given dynamical system, which takes
values on the real line *R. *The state *α* will be identified with a unique
Borel measure on the family of the Borel subsets of the phase space of the
system and is determined by the macroscopic constraints. This Borel measure
will, univocally, determine the function *p* on the Borel subsets, *E*,* *of the range of the possible values of
the dynamical variable *A*,*
*namely *p(A, α, E) *= *α (A*^{-}^{1}*(E)) *(^{15}).

In quantum mechanics the observable *A*
will correspond to a self‑adjoint operator in the Hilbert space of the
quantum system. The state *α* will
be represented by a unique density operator in that space. The probability
will be *p(A, α, E) *= *tr* [*α A(E)*] ,* *with *A(E) *the spectral
measure of the Borel set *E *(^{15}).

The previous examples are already
given in ref. (^{11,15}),* *we recall them f or easy reference in
the following discussion and, to this end, we give one more example which will
be, too, useful later and take it from the theory of random noise (^{16}).

Consider a function of the time *f(t). *Under certain circumstances it can
be thought as the time hystory of a stochastic process (*f(t) *may be the
voltage at each instant of time at the output of a photomultiplier). The
observation of such a function over a finite time interval is called a «sample
record» of a stochastic process *f(t* - *τ)*** **with *τ* a random variable. Sometimes an investigation of zero
crossings or arbitrary level crossings of such sample records (^{16})** **is required.

The statistical setting set forth
before allows us an interpretation also in this case. Here the state *α*
may be represented by the function *f(t).* The observable *A *may be the number of level crossings
in a fixed time interval, starting from the random instant of time *τ*.
In this case the probability measure will be concentrated over the integers.

We must stress here that, in all
previous examples, according to the original definition, the probability
measures are always defined on the Borel subsets of the real line (and hence
perfectly «classical»), but such Borel measures are determined jointly by the
state *α *and by the observable *A.
*Thus it is not legitimate, in general, to use the same Borel measure as a
probability measure for different observables.

This point will be relevant for an assessment
of the significance of Bell’s inequality which is just derived from the
hypothesis of the existence of such a unique Borel measure over the space *Λ**.*

All this will be illustrated by a
classical (Newtonian) system which will fill in the «black box» of our
paradigmatic EPR physical situation.

Our system *S* will be
«separable» and described in a «classical» probabilistic structure, partaking
of all the three examples just mentioned. However, it violates the inequality
derived before.

**3. — Systems of linear, transversal
oscillators.**

Now we suppose that our system *S*
is constituted by two identical sub-systems S_{1} and S_{2}
each made of a one-dimensional lattice of *n* **+ **2 classical particles with linear interactions between
nearest neighbours only.

The equilibrium positions of all the
particles in both subsystems are aligned along the *z‑*axis,* *but the particles are free to oscillate
in the (*x*, *y*)-plane.

We suppose that, in both subsystems,

*a)
*the
elastic constants in the longitudinal direction are stiff enough so that we
have, practically, only transverse vibrations;

*b)
*the
transverse couplings and masses are identical for all particles;

*o)
*the
particles with index *ρ* **=** 0
and *ρ ***= ***n ***+ **1 are constrained to their equilibrium positions.

Under such assumptions, for each
subsystem, the equations of motion for two orthogonal directions, normal to the
*z*-axis, are (by choosing a natural time unit *T* =* *1/ω)

(10) _{}

The eigenfrequencies are given by (^{17})

(11) _{}

We give identical initial conditions to both
subsystems:

(12) _{}

* *

The solutions may be easily found:

the vibration state in the (*x*, *y*)-plane
may be represented by the complex vector

(13) _{}

with

(14) _{}

__ __

If we introduce the unit vector

* *

(15) _{} ,

the vibration amplitude along any given
direction _{} of the (*x*, *y*)-plane for particle
ρ will be given by the real part of the scalar product between the state _{} and the polarization
state _{} corresponding to the
projector _{} , * i.e.*

(16) Re
_{}.

Now we specify the kind of
«polarizers» we want to introduce in our «black box». These will be constituted, in each subsystem, by constraints
ori the particles with ρ = σ (1 < σ < *n*)* *which impede vibrations along a
specified direction in the (*x*, *y*)-plane by letting the particle
vibrate only along the orthogonal direction, say _{},* *which we call the «easy direction»,*
*this implies the equation of the constraints: *x*_{σ }sin _{}*= y _{σ} *cos

It is obvious that, if we give
initial conditions different from zero only at particles with
ρ < σ ,* *the vibration state of all particles
will change with respect to the state without constraints but of same initial
conditions; however, the vibration component along the «easy direction» will not
be affected by the constraints.

Moreover, for ρ ³ σ it will be the only component
actually present and, for such particles, we have a state of linear
polarization along the «easy direction» _{},* *so that our constraint may be represented
by the projecter _{}*.*

It remains to us to specify the
«detectors»: we assume that the detectors are such that, without perturbing in
any sensible way the system, they give a «count» every time some appropriate
dynamical variable crosses a given threshold with positive slope. As particular
instance of such dynamical variables, we have chosen the vibration amplitude
or its square modulus, along the «easy direction» of two specified particles,
one for each side, namely the particles with ρ = *n.*

This system simulates the
paradigmatic EPR situation.

There are many ways te randomize our
system: we may choose to treat as random variables the initial constants *a*
and *b *in formulae (12) and (14), in
this case we have the standard situation in classical statistical mechanics if,
for instance, we look for the probability distribution of a given dynamical
variable, as the vibrating amplitude of a particle in a given direction
(formula, (16)) and so on.

However, if we are interested in the
number of level crossings or the number of coincidences in a given time
interval, the situation is more like to the random‑noise example. The
probability measure, in this case, is not so easily related to the probability
measure, over the phase space, which defines the statistical state of the
system but is altogether well defined.

There are, of course, other ways to
randomize the system: by randomizing any of the constants which define the
system as the number of particles, the elastic constants and so on; the simplest
way, actually chosen in our model, is to fix the initial conditions and
randomize the instant of time in which we enter into the process by counting
the number of levol crossings and the number of coincidences in a fixed
interval of time.

Unfortunately it is not easy to
write down a usable algebraic expression which gives in a closed form the
number of level crossings as a function of the fixed parameters of the systems
but we may resort to the computer. Thus it has been done by means of a program
which finds out the zeros of a generic function. In our case the problem was
that of finding the number of level crossings for two of the functions defined
by formula (16), or its square moduli, one for each subsystem, and the number
of coincidences within a fixed resolving time relative to same pair of
polarization directions _{}* *and _{}.

In different computer experiments we
varied the number *n *of particles, the
initial constants *a* and *b *of
formula (14), the resolving time of the coincidence, the «detection threshold», the duration of the counting time
interval and the epoch *τ* of entering into the process.

We observed that the vibrating
amplitude (16) behaved in an apparently very random way after a short time of
about a factor two the interval for the first reflection of the initial wave
pulse which was of about *n *natural
time units. Also the interarrival times of threshold crossings appeared perfectly
random. Their statistical distribution was stationary in the explored time
range **(**@10^{6}
natural time units) and almost independent of the number of particles *n.*

The mean value of such interarrival
times was of about three natural time units and the standard error of about one
time unit.

Choosing, as crossing quantity, the
square modulus of the vibrating amplitude with initial constants (12) *a* *= *1, *b ***= **_{}**
(these initial conditions would impress a uniform circular motion to the particle
with ρ ****= **1 in case we should constrain at rest
the particle with ρ **= **2 with a threshold for both detectors
of 0.001 and a resolving time of the coincidence of 0.63), we obtained, for the
ratio defined in formula (7), a function quite near to the _{}* *law.

Actually it was almost identical for
_{}**= **30° and *n *>* *20; it
deviated somewhat in the other directions (the system is not rotationally
invairiant) or for smaller number of particles.

However, by defining the ratio

(17) _{}

as a measure for the degree of violation of
inequality (6), we found that this ratio was under the limit fixed by
inequality (6) for a very small number of particles, but increased steadily
remaining at the value of the cos^{2 }law from *n* **= **20 to *n* **= **100 which was the maximum number
explored.

When we took the vibrating
amplitude, instead of its square modulus, the violation was ever greater but
the ratio (7) was not any longer similar to the cos^{2} law. We have already
noted that our system, for the particular initial conditions, is not
rotationally invariant; however, the number of «counts» as a function of the
direction, *i.e. *_{}*,** *varied of less than 10%** **with a minimum value at 90°. It was instead quite constant by
choosing initial constants *a = *0.345
and *b* **= **0.345_{}, but in this case the ratio (7) was not so near to the cos^{2
}law, nevertheless it violated all the same inequality (6) by the same
amount of the cos^{2 }law by choosing other three directions, namely _{}= 45°, _{}= 105°,_{}= 75°.

Let us pause al this point and note
that in our model we have a situation similar to our random-noise paradigm.
However, given the peculiarity of our system we have many points in common also
with the paradigm of quantum mechanics. In fact, for each particle of each
subsystem we have a state space of vectors _{} which is isomorphic
to the state space of the vectors which represents the polarization states of
a photon in quantum mechanics; moreover, we have linear operators as the
projector _{};* *we, analogously, may construct a
product space for the two subsystems and so on.

In our case, too, the probability
measure for a given physical quantity is determined jointly by the state _{} and the operator _{} as far as the simple
subsystem is concerned or by the product state and product operator for the composite
system.

It may not be the case that such a
probability measure will be determined by the same rules as in quantum
mechanics but, in other respects, the system is perfectly analogous.

However, in our Newtonian separable
(actually «separate») model we have found initial conditions, values of
detector threshold and of coincidence resolving time which simulate a quantum
system also in probability measure.

Notwithstanding we must stress here
that we are not proposing in any way a model for any quantum system but only
give an instance in which Bell’s inequality is violated in order to illustrate,
with a concrete example, the theoretical point we made before.

As a last point we shall consider
the empirical implications of our fìndings. Let us remind that in our Newtonian
system the projection of the amplitude given by formula (16) is not modified by
the «polarizer»,* *so it is a property of the system
independent of any particular observation made through a pair
«polarizer-detector».* *(In fact, we can do away altoghether
with the «polarizers» by easily devicing opportune «detectors» which do not
perturb in any way the physical state of the system.)

Thus it has a meaning also in the
particular case with a single vibrating oscillator for each side (^{18}).

In this simple case one can, more
perspicuously, see the reason why Bell’s inequality is violated; in fact, in
this case the square modulus of the amplitude at an angle _{}* *will be

_{}

If *T** _{h}
*is the value of the detector threshold, the crossing times will be

_{}

with *c(T _{h})
*not depending on

For three directions _{} , _{} and _{} and for a given *r*

_{} .

By choosing a resolving time for the
coincidence circuit

_{} or _{}

inequality (6) will be trivially satisfied.

In fact, in the first case, all
correlated events in the three channels will stay inside the resolving time,
while, in the second case, only one for each channel will be inside the
resolving time. In this case the ratio corresponding to that given by formula
(17) is unitary. But, if we take,

_{} ,

* *

the corresponding ratio will be *r ***= **2, grossly violating inequality (6), because we have a
coincidence in channel pairs _{} and _{},* *but not in channel pair _{}, in spite of the fact that the three events in the different
channels are totally correlated.

By opportunely weighting these two
extreme cases, as, in practice, it was achieved by our model, we can obtain
also the QM value, _{}.

The finite resolving time of the
coincidence eliminates the correlations between certain pairs of channels and,
*vice versa*,* *by the same
mechanism, it may introduce spurious correlations between uncorrelated channels
(^{19}).

Because these effects are not connected with nonlocality,
in fact, an ideal experiment can be easily deviced which conforms to all
requisites of the more detailed «black box» proposed recently by BELL (^{20});** **in any experiment intended to test quantum mechanics in
connection with the EPR paradox (^{3}), one must, beforehand, check the
invariance of the statistical distribution of the coincidence time intervals
against the variation of the relative angle of the polarizers also in the case
in which particular devices aro being used to test nonlocality as, for
instance, in the experiment reported in ref. (^{21}).

**4. — Conclusions.**

The theoretical considerations made
in sect. **2** supported by the concrete example of sect. **3 **convince us that Bell’s inequality is
not so general as commonly believed. Thus it cannot represent an impediment to
a completion of quantum mechanics according to the original ideas of EPR (^{3}).

The advantages of a «realistic»
interpretation of quantum mechanics, also from a heuristic point of view, has
been often stressed by eminent physicists and philosophers (see, in particular,
ref. (^{22,26}) ).

It is opportune at this point to try
an assessment of the limits of validity of Bell’s inequality.

To do so, it is useful to consider
some more elements of Mackey’s axiomatics (^{15}): let us call an observable
*Q*** **a «question», if
in every state *α** *the
measure *p(Q, α,* *E) *is concentrated in the points 0 and
1, that is *p(Q, α, {*0*, *1*}) = *1. The functions

A question may always be constructed
from an observable *A*,*
*given a Borel subset *E*,*
*by taking the question *φ** _{E}(A)
*with

In classical statistical mechanics a
question is simply a Borel function that takes on only the values 0 and 1; it** **is uniquely determined by the Borel subset of the phase space on which
it takes the value 1; the function *m _{α}
*mapping questions into numbers becomes the measure which defines the state
on the phase space.

For a given state *α* the
Borel measure is unique, according to the definition _{} (ref. (^{15}))** **and we may think of classical phase space as the Boolean algebra
of all its Borel subsets; in this case, obviously, Bell’s inequality is
generally valid.

This may not be so if we consider,
as a physical observable quantity, the number of level crossings of a classical
dynamical variable. In this case, by fixing the numerical value of the
threshold, we shall have different Borel measures for different dynamical
variables because, by equating these variables to a constant value, different
hypersurfaces are determined and our elementary events are just given by the
points in which the dynamical trajectory of the system crosses such
hypersurfaces.

In this case the lattice of the
questions may not be any longer Boolean (^{27}) (we remind that the
partial ordering in the present axiomatization is determined by the
probability measure (^{15})).

It is interesting at this point to
remind that it has been demonstrated (^{28})* *that with the usual axioms a lattice
must necessarily be Boolean if a Kolmogorovian probability measure is supposed
as a unique function on the elements of the lattice.

This allows us to understand why in
our system of oscillators we have noncommuting operators, like the projectors _{}, and, however, we can make simultaneous physical measurements
of any corresponding quantities. It can be said that, in our model, two
observables are incompatible when they cannot be described statistically by a
unique Kolmogorovian probability measure over a Boolean algebra of the Borel
subsets of some opportune phase space, without reference to the physical
possibility of performing a measurement.

We can ask ourselves if the
preceding notation could be an hint in order to give a meaning to the concept
of incompatible observables in a future «realistic and causal» interpretation
of quantum mechanics.

By now we must have this question
open, but only remind that a purely statistical interpretation has already been
advocated (^{29}).

We, however, remind that in our
model we have a two‑dimensional linear vector space (the space of the
polarization states), we have associated experimental questions to the system
through opportune «polarizers» and «detectors». However, the lattice of those question is not Boolean,
because the corresponding projectors of the linear vector space do not
commute, but simply modular (^{27}).** **We have defined joint probabilities _{}, but we do not have a subtending phase space of variables *λ*** **with a unique Borel measure over it
(moreover, our system is not metrically transitive (^{30}),** ***i.e. *in our model it is possible to
decompose the phase space into invariant parts of positive measure (^{30}));** **but we have «many» Kolmogorovian measures, one for each «observational
state» different from the «physical state» of the system.

Here we stress the fact that neither
the «polarizers» nor the «detecters» need, «physically», affect the «physical state»
of the system.

Of course, a «generalized
probability measure» (^{31})** **could be defined over the lattice of
the experimental questions and, in other contexts, it may result very usefull
and important; however, in our limited context it does not help in clarifying
our particular issue, where, by quoting one of the referee, probability spaces
are more important than more probability measures.

* * *

Thanks are due to Profs. J.**
**S. BELL, G.* *FONTE, D. GUTKOWSKI, G.*
*LOCHAK, M. SACHS and Dr. G. C. SCALERA for useful comments on a
preliminary draft of the present paper; thanks are due also to one of the
referees for useful suggestions.

**• RIASSUNTO**

L’assiomatizzione della meccanica quantistica di Mackey permette, in determinate circostanze, la violazione della disuguaglianza di Bell anche in un contesto di «meccanica classica» e «probabilità classica». Si studia al calcolatore un modello meccanico costituito da due sottosistemi separati di oscillatori accoppiati che violano la diseguaglianza di Bell e si fa l’ipotesi che la principale ragione di tale violazione è da imputare ai particolari «rivelatori» introdotti nel modello che forniscono un «conteggio» ogni qual volta una determinata variabile dinamica del sistema superi una soglia assegnata.

(*) To speed up publication, the author of this paper has agreed
to not receive the proofs for correction.
TORNA

(1) J. S. BELL: *Rev. Mod. **Phys.*,* ***38**, 447 (1966).

(2) J. S. BELL: *Physics*,*
***1**, 195 (1964).

(3) A. EINSTEIN, N. ROSEN and B. PODOLSKY: *Phys.
Rev.*,* ***47**, 777 (1935).

(4)* * D. BOHM arid Y. AHARONOV: *Phys. Rev.*,* ***108,
**1070 (1957).

(5) E. J. BELINFANTE: *A Survey of
Hidden-Variables Theories *(Oxford, 1973).

(6) B. D’ESPAGNAT: *Conceptual
Foundations of quantum Mechanics *(Reading, Mass., 1976).

(7) J. F. CLAUSER and A. SHIMONY: *Rep.
Prog. Phys.*,* ***41**, 1881 (1978).

(8)** **F. M. PIPKIN: *Adv. At. Mol. **Phys.*,*
***14**, 281 (1978).

(9) L. DE LA PENA, A. M.
CETTO and T. A. BRODY: *Lett. Nuovo
Cimento*,* ***5,
**177 (1972).

(10) G. LOHAK: *Found. Phys.*,* ***6**,* *173 (1978).

(11) A. FINE: in *Problems in the Foundation of Physics*,* *edited by M. Bunge
(Berlin, 1971), p. 79.

(12)** **F. A.
KAEMPFFER: *Concepts in Quantum Mechanics *(New
York, N.Y., 1965), p. 1.

(13) W. FELLER: *Introduction to Probability Theory, *Vol. 2 (New York, N.Y., 1966),
p. 136.

(14) E. P. WIGNER: *Am. J. Phys.*,*
***38, **1005 (1970).

(15) G.* *W. MACKEY: *Mathematical Foundations of Quantum
Mechanics *(New York, N.Y., 1963).

(16) J. S. BENDAT: *Principles and Applications of Random Noise Theory *(New York, N.Y.,
1968).

(17)** **H C. CORBEN
and P. STEHLE: *Classical Mechanics *(New
York, N.Y., 1950), p. 136.

(18)** **A model formally
equivalent to this case has been independently described by
G. C. SCALERA in a note to appear in *Lett. **Nuovo Cimento. *Private
commmnication.

(19) G. FARACI, S. NOTARRIGO and A. R.
PENNISI: *Nucl. Instrum. Methods, ***165, **325 (1979).

(20) J. S. BELL: *Comments At. Mol. Phys*., **4**,* *121 (1980).

(21) A. ASPECT, J. DALIBARD and G.
ROGER: *Phys. **Rev.
Lett., ***49**, 1804 (1982).

(22) A. EINSTEIN: *Mécanique quantique et
réalité *(reproduced in *Ann. Found. De
Broglie, ***3**, 81 (1978)).

(23) L. DE BROGLIE: *Reflexions sur la
causalité *(reproduced in *Ann. Fond.
De Broglie, ***2**, 69 (1977)).

(24) E. SCHRÖDINGER: *Are there quantum jumps? *(reproduced in *Ann. **Fond. De Broglie, *2, 51 (1977)).

(25) K. R. POPPER: in *Quantum Theory and Reality*,*
*edited by M. Bunge (Berlin, 1967), p. 7.

(26) M. BUNGE: in *Quantum Theory and Reality, *edited by M. Bunge (Berlin, 1967), p.**
**105.

(27) A. S. HOLEVO: *Probabilistic and Statistical Aspects of* *Quantum Theory *(North
Holland Pub. Co., Amsterdam, 1982), p. 20.

(28) K. R. POPPER: *Nature (London), ***219, **682 (1968).

(29) L. E. BALLENTINE: *Rev. Mod. Phys., ***42**, 358 (1970).

(30) C. TRUESDELL: *Proc. S.I.F., *Course XIV, edited by P. Caldirola (Academic Press,
New York, N.Y., 1961), p. 21.

(31) D. G. HOLDSWORTH and C. A. HOOKER: *A critical survey of quantum logic*,*
*in *Logic in the 2O ^{th}
Century, Scientia, *Suppl. to Vol.