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Problems in Quantum Physics, GDANSK 1987, Editors World Scientific

**ON
THE PHYSICAL MEANING OF BELL’S INEQUALITY**

**AND
THE RELATED EXPERIMENTAL TESTS**

Dipartimento di Fisica
–Università di Catania – 95129 Catania, Italy

ABSTRACT: It
is shown that the experiments performed so far in order to test Bell’s
inequality are not suited to test for determinism and locality, because of an
intrinsic impossibility of factorizing the inputs of the required coincidence
circuit. On the other hand the factorizability hypothesis is a necessary
ingredient for proving the inequality and implies the need for ideal
apparatuses which, however, cannot be easily realized in practice without a
detailed deterministic model which is, in principle, rejected by the usual
interpretation of quantum mechanics.

After Bell’s result ^{1]}
and the experimental verification of the quantum mechanical previsions, ^{23]}
it seems that the following propositions can be safely accepted:

Proposition A: *Any classical separable physical system must
obey Bell’s inequality*

and because quantum physical systems
do exist which violates Bell’s inequality and, moreover, such violations are
experimentally well confirmed, we are entitled to assert the following:

Proposition B: *No local hidden variable theory can
reproduce all the results of quantum theory which are experimentally well
verified.*

We shall clarify, in the sequele,
the precise meaning of the terms used in propositions A and B, but by now we
accept their usual meanings and observe that, certainly, A implies B but, of
course, if A should not be true then B may not be true or, at any rate, Bell’s
inequality cannot be used to assert B. In order to falsify proposition A, given
its logical structure, it will be enough to exhibit a single concrete example
of a separable classical physical system which violates Bell’s inequality. Such
an example, in fact, has already been given in a previous paper.^{4]}
Here, we shall briefly summarize it in order to examine some logical consequences
which may have some bearing on the physical interpretation of quantum
mechanics.

In order to present the essential
point without introducing unnecessary metaphysical presumptions, we shall try
to describe the paradigmatic physical situation avoiding any term which cannot
be reconducted to experimental observations or manipulations. In such terms we
consider a *black box*,*
*S, representing our physical system, defined by certain physical
manipulations. On opposite sides of S, in a given spatial direction which we
call the *z‑*axis,* *we have a pair of objects *P _{1} *and

Given such an experimental
situation, it is easy, by using Bell’s hypotheses, to derive his
inequality ^{4]} and to verify that all experiments performed so
far to test it ^{2,3] }fall under this scheme. However, we may
also fill in the *black box *with a
classical separable (actually: *separated*)*
*physical system S, which reproduce the quantum correlation curve and so
violates Bell’s inequality. Such a system is constituted by two identical
subsystems 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. *It is supposed, for simplicity,
that in both subsystems: a) the elastic constants in the longitudinal
directions are stiff enough so that we have, practically, only transverse
vibrations; b) the transverse couplings and masses are identical for all
particles; c) the particles with index ρ = 0 and ρ = *n + *1 are constrained to their equilibrium positions. The effect of
the *polarizers *can be realized by
putting constraints on the two particles (one for each subsystem) with
ρ = σ ,
being σ a specified value of the index, comprised in the range 1 £ σ £ *n. *These constraints will impede vibrations along a specifìed
direction *(hard direction) *in the (*x*, *y*)*‑**plane*,* *letting the particle
vibrate only along the orthogonal direction *(easy
direction)*, say _{}; this implies the following equation for the constraints: x_{σ} sin_{} = y_{σ}* *cos_{},* *on each side. Such constraints will act
on the propagating motion of the particles like a calcite cristal acts for
light waves. In fact, if we give non-zero initial conditions only to particles
with ρ *< *σ,
the vibration state of all particles will be modified with respect to the
non-constrained state with same initial conditions but the vibration component
along the *easy direction *will not be
affected by the constraints and, by observing the particles with ρ *> *σ, it wiIl be the only
one actually present, so that the constraints generate, for such particles, a
state of linear polarization along the *easy
direction *_{}*. *As *detectors*,*
*we may assume devices such that, without perturbing in any sensible way
the system, will give a *count *every
time some appropriate dynamical variable crosses a given threshold with
positive slope (in our particular case we have chosen the square modulus of the
vibration amplitude along the *easy
direction *of the two particles, one for each side, with ρ *= n*).* *Of course, such combined
system of *polarizer-detectors *obeys
the separability (locality) requirement, used to derive Bell’s inequality, namely:
the registering of a count on one side is not influenced by the setting of the
knob (constraint) on the other side. Here, of course, there is no problem with
any possible superluminal communication because the two subsystems are
actually separated at whatever relative distance of systems and detectors.

The motion of each particle results
in a sovrapposition of circular motions whose frequencies have irrational
ratios. It is obvious that, given the stationarity of the motion, our system
can be replaced by a symmetrical statistical ensemble of couples of identical,
physically independent but correlated, spinning particles with identical
precession motion (but for the initial phases) realized by suitable external
forces; in such a way, we can better simulate an experiment of the
Bohm-Aharonov type.^{5]} By a computer simulation of the described
experiment, ^{4]} by choosing particular values for the various
parameters of the system, we were able to obtain the quantum correlation curve
for an ideal experiment realized with an atomic cascade, ^{2,3]}
thus violating Bell’s inequality.

In order to see more perspicuously
the reason for such a violation we may consider a much simpler classical
separable system constituted by two classical particles describing identical
simple circular motions (also here, we can think of a statistical ensemble of
couples of precessing classical spins) and observe the coincidences between the
threshold crossings of the square modulus of the amplitude at an angle _{},* *which in this case is given, for each
of the two particles, by *A ^{2}cos^{2}(ωt *- θ

In fact, to describe the working of
a coincidence circuit,^{6]} as those used so far,^{2,3]}we may
consider two trains of electrical pulses incoming on the two inputs of the
coicidence system: let *A(t) *and *B(t) *be the electrical voltages, as
functions of the time, at the two inputs of the coincidence system; let *α* º [ *t*_{1},* t*_{1}* *+ Δ_{1}]
and *β* º [ *t*_{2},* t*_{2}* *+ Δ_{2}] be two time intervals; let *f _{A}(α,χ*

But let us see if we can find out a
more convenient experimental situation which may allow us to verify Bell’s
inequality. In order to explore such a possibility it is convenient to analyse
more carefully Bell’s mathematical hypotheses and compare them with the usual
definitions of the concepts of determinism or *causality *on one
side and *separability *or *locality *on the other side (the two
terms in each couple are, usually, treated as synonymous in the physical
literature).

In the words of Heisenberg, which
were intended to express the old conception of determinism as mantained by
Einstein but to be substituted by a new one compatible with Quantum Mechanics,
the law of *causality *is formulated as
follows: ^{8]} *“If at a
certain time* *all data are known for a given system then
it is possible to predict unambiguously the physical behavior of the system
also for the future.”*

EPR added to this a request for *realism*:^{7]}* “If,
without in any way disturbing a system, we can predict with certainty (i.e.,
with probability equal to unity) the value of a physical quantity, then there
exists an element of physical
reality corresponding to this physical quantity.”*

The preceding definitions imply also
the principle of *statistical determinism *as
expressed by d’Espagnat: ^{9]} *“If
two statistical ensembles of physical systems are submitted to identical
treatments, and if subsequent observation reveals significant statistical
difference between them, the implication is that the two ensembles were not
identical at the start.”*

The notion of *locality, *referring to two physical systems which are correlated by
some physical interaction which they may have undergone in the past but anyway
no longer interacting, is introduced by Einstein with the following
words: ^{10]} *“The factual
situation of the system S*_{2 }*is independent of what is done with the system S*_{1}*, which is spatially separated from the former.”*

In the following we shall use the
term *classical *for any system or
observable compatible with the previous standard definitions.

Bell’s mathematical formulation of *determinism *is stated at p. 195-6 of his
paper ^{1]} by pretending that the Q.M. operators, one of which
yields the value +1 upon a measurement of the component _{} of the spin of the
first particle, where _{} is some unit vector,
when the other yields the value -1 upon a measurement of the spin
component _{} of the second particle
and viceversa, are represented, in a *more
complete speciflcation *(i.e. *deterministically)*,*
*by two observable _{}, _{}, λ being
the *hidden-variable. *So he is
postulating a correspondence between the deterministic observables _{} and _{} and the quantum
operators _{}and _{} (we have indicated
with the symbol *f *the mapping which
gives the right correspondence, from the set of the bounded quantum operators
to its counterpart represented by the set of the deterministic
observables ^{11]}). On p.195 he says: *“The result A of measuring *_{} *is then determined by *_{}^{ }and
λ, and the result B of measuring _{} *in the same instance is determined by *_{}* and λ”*, so
that the measured values of the deterministic observables _{} for each possible
value of the unit vector _{}* *must belong to the spectrum _{} of the corresponding
quantum operators _{};* *we summarize this assertion by the
formula:^{11] }

(1) _{}

Bell’s mathematical formulation of *locality *is explicitly worded at p.196 by
pretending *“that the result of B for
particle *2 *does not depend on the
setting *_{}*, of the magnet for particle *1,** ***nor A on *_{}*”*; but implicitly assuming in his formula
(2) that the elementary event, determined by a specified element λ in the
statistical ensemble Λ of the hidden variables, is given by the product _{}, or, in his own words:* “If ρ(λ) is the probability
distribution of λ* *then the expectation
value of the product of the two components *_{} *and *_{} *is*

_{}

(formula (2) of Bell’s paper). *This should equal the quantum mechanical
expectation value, which for the singlet state is*

* *

_{} .*”*

in other words the deterministic observable
corresponding to the quantum operator given by the product _{}corresponds also to the product of the two deterrninistic
observables *A *and *B *corresponding respectively to the
quantum operators _{}* *and _{}* *or, in a formula: ^{11] }

(2) _{}

Besides: formula (2) of Bell’s paper implies
that, for the quantum operator _{},* *its quantum expectation
value is equal to the expectation value of the corresponding deterministic
observable, for any quantum state _{}, thus also for any distribution density of the hidden
variables _{} i.e.

_{}

with ρ independent on _{}* *and *f
*independent on _{}. All this implies that _{}* *is (modulo an unobservable * functional F, such that _{}, for any _{}* *and any _{}) a linear functional of _{}*. *We remind here
that, according to a generalization of the Gleason‑Kahane‑Zelazko
theorem, for any invertible operator either in the Banach or in the Jordan algebra
of the bounded quantum operators, formula (2) implies (1) and also ^{11]}

(3) _{}

which corresponds to von Neumann’s definition
of hidden variables as the determinants of the dispersion free states. ^{13]}

But now we can see that many *classical
*observables exist which do not obey Bell’s axioms and may violate his
inequality. We remind that it has always been asserted that the formalism of
Quantum Mechanics takes into account the role of the observing apparatus on the
observed physical system. There are many ways in which such a role can be
introduced in a classical model. The simplest way is to suppose that any
classical observing apparatus performs a time averaging on any classical dynamical variable ** and that the quantum operator corresponds to such *phenomenological observable *and not
directly to the classical dynamical variable. We may represent such a
phenomenological observable by:

_{} ,

with *x(χ)
*a point in phase space at time χ *.
*We remind, here, that such an integration is always performed in any
experimental observation by the high fequency cut introduced by any concrete
physical measuring device. Obviously such classical observables obey all the
standard definitions of determinism and locality given before but cannot obey
conditions (1), (2), or (3). In
fact, given any two classical dynamical variables *f *and *g*, for the corresponding *classical *phenomenological observables we have, in general
(excluding exceptional cases as, for instance, the case in which one or both
variables take constant values in the course of the time), _{} (we may suppose,
here, that the observable _{} is being measured by a simple correlator).
Also (3) and, for classical observables corresponding to bounded quantum
operators, (1) are, obviously, violated. We note that such violations apply
also to the dichotomic variables envisaged by Bell ^{1] }and the
violation may be very strong in dependence of the average time duration of the
two alternating values of the variables, as compared with the range of
integration, as determined by the parameter *τ* ;* *but such detailed knowledge can only be obtained by means of a
fully deterministic model which, however, is, in principle, excluded by the
usual interpretation of Quantum Mechanics. So that, in general, Bell’s
hypotheses can be applied only for ideal observations not realized in practice.

Of course, many other ways can be
introduced, classically, to take into account the role of the observing
apparatus as the number of *zero-crossings
*of a fluctuating but deterministic time function, with the same effect on
Bell’s inequality as considered in the previous example, whose detailed study
has been described in Ref.4, showing the intrinsic non-factorizability of any
coincidence circuit used so far in order to test Bell’s inequality.

In conclusion, from what we bave
shown, it emerges: Bell’s inequality is, of course, a correct mathematical
theorem under his hypotheses. Two of such hypotheses (namely, determinism and
locality) have been exphicitly considered by Bell; a third hypothesis
(factorizability) was implicitly advanced, certainly because he was considering
*ideal *measuring apparatuses. The
theorem is physically relevant on the theoretical side because it forbids any
local classical interpretation of Q.M. which does not take into account the
role of the finite resolution of any physical device. However, on the empirical
side, Bell’s theorem cannot forbid a *local
*classical interpretation of Q.M., when the role of the measuring device is
taken into account for a more complete specification, as envisaged by EPR, of
the experiments. Of course, our considerations cannot tell how such a complete
specification could be achieved but simply tell that the experiments devised
for this purpose, at date, cannot forbid it.

**Notes and References**

** **

*** **See axiom two of Mackey
axiomatics ^{12] } TORNA

**** **This hypothesis is a generalization
of the *phenomenological derivative *introduced
to cope with Loschimdt and Zermelo paradoxes in classical statistical mechanics;
see ^{14]} TORNA

1] Bell, J. S., Physics **1**, 195 (1964).

2] Clauser, J. F. and Shimony, A., Rep. Progr. Phys.
**41, **1881 (1978).

3] Aspect, A., Dalibard, J., and Roger, G.,
Phys. Rev. Lett., **49**, 1804 (1982).

4] Notarrigo, S., Nuovo Cimento **83B**,
173 (1984).

5] Bohm, D., and Aharonov, Y., Phys. Rev. **108**,
1070 (1957).

6] Faraci, C., Notarrigo, S., and Pennisi, A.
R., Nucl. Instr. and Meth. **165**, 325 (1979).

7] Einstein, A., Podolsky, B., and Rosen, N.,
Phys. Rev. **47**, 777 (1935).

8] Jammer, M., *The Philosophy of Quantum Mechanics *(Wiley, New York, 1974)

9] d’Espagnat, B., *Conceptual Foundations of Quantum Mechanics *(Addison‑Wesley,
Reading Ma., 1976).

10] Einstein, A., in: *Albert Einstein, Philosopf her Scientist, *Ed. P. A. Schilp, (Libr.
of Living Philos., Evanston Il., 1949) p.85.

11] Bach, A., Lett. Nuovo Cimento **35**,
377 (1982); W. Cuz, Found. Phys.
**15**, 121 (1985).

12] Mackey, G. W., *Mathematical Foundations of Quantum Mechanics *(Benjamin, New York,
1958).

13] v. Neumann, J., *Mathematical Foundations* *of Quantum
Mechanics *(Princeton, N.J., 1955).

14] Katz, A., *Principles of Statistical Mechanics *(Freeman, San Francisco,1967).