Note: After publication of this paper, an experiment was performed with a detection efficiency near unity, thus eliminating the 'detection efficiency' loophole. This paper no longer offers a realistic solution to violations of Bell's inequality.
See: M.A. Rowe, D. Kielpinsky, V. Meyer, C.A. Sackett, W.M. Itano, D.J. Wineland, "Experimental violation of a Bell's Inequality with efficient detection", Nature 409, p. 791, 2001.
 The estate of Paul Marmet
The
probabilistic interpretation of quantum mechanics is
incompatible with physical reality. It is claimed that no
realistic interpretation can be given to quantum physics without
involving absurd^{[1]} statements like infinite velocities or "spooky^{[2]} actions at a
distance." A thought (gedanken)
experiment is described by Mermin^{[2]} in which he claims to prove that no conceivable
realistic mechanism can actually explain some particular
observations. The experiment suggested by Mermin^{[2]} is a challenge to the realistic interpretation.
We show here that the results of this famous experiment,
believed to be unexplainable classically, can be described
without the quantum mechanical interpretation. Only the
electromagnetic (EM) theory compatible with Maxwell's equations is
required.
A
classical apparatus is described here, giving results identical
to the ones supposedly requiring a quantum interpretation.
In order to keep all our attention on the basic aspect of a
realistic interpretation, we will use the same "gedanken
experiment" as described by Mermin^{[2]}. We refer the reader to Mermin's article.
Some authors^{[3,4]}, using other considerations, point out that Mermin
has overlooked the inefficiency of the detectors, but no clear
or detailed description of a way of solving Mermin's gedanken
experiment is given.
A real
classical solution to the challenge can be guaranteed only if
one gives a complete detailed physical description of the
apparatus. We show here how the classical EM theory,
described by Maxwell's equations, can provide a complete and
realistic description of Mermin's^{[2]} gedanken experiment. We do not claim that the
solution given here is the final interpretation to quantum
mechanics. The solution given here simply shows that it is
incorrect to claim that there is a valid proof in favor of non
realistic physics.
We also
believe that there is another way (than the one described here)
to explain quantum phenomena using nothing but a realistic
model. The solution is in the "relativistic effects" and
the relevant frame of reference (light) where the phenomenon
takes place. In that frame, moving near the velocity of
light, clocks are stopped and physical lengths are unlimited, so
that there is a coupling (in local "apparent" time which has
stopped) between the emitter and the receiver even if they are
at large distances. This last description is explained in
more recent books on quantum
mechanics^{(A1)} and on relativity^{(A2)}.
The full description given by Mermin is not repeated
here. The reader should refer to the original paper^{[2]}. Let us recall here
the main features.
Mermin
describes his experimental setup made of:
1) A
source "C" emitting something (particles or photons or whatever)
and,
2) Two
similar detecting units "A" and "B". As described by
Mermin^{[2]},
each detecting unit has a switch having three possible settings
(1, 2 and 3). Each detecting unit has also two
lights. They can be built so that one can detect which one
of the three directions (0^{o} ,
120^{o} or 240^{o} ) the signal detected is polarized. Let us call
these three directions D1, D2 and D3 respectively.
Shortly
after one pushes a button on source "C", each detecting unit
flashes one of its lights (red or green). The detecting
unit can be built, so that the green light flashes "G" if the
switch setting number is the same as the number of the observed
direction (D1, D2 or D3). In other cases, when the number
is different, the red light "R" flashes.^{[10]}
Any
obstacle (for example a brick) between the source and one of the
detecting units prevents that detecting unit from
flashing. The switch settings on each individual detecting
unit are varied randomly from one run to another. There
are no connections among the three parts of the apparatus other
than via whatever is passing from C to A or from C to B.
After
each experimenter (at A or B) has chosen independently and
randomly a switch setting (1, 2 or 3), each experimenter records
his switch setting and the color of the lamp that went on.
Then the experiment is repeated many times.
Since
either a red "R" or a green "G" lamp on one detecting unit can
be turned on, each experimenter can then record one of the
following possibilities: 1G, 2G, 3G, 1R, 2R, or 3R. After
all runs are completed, the two observers combine their list of
data, so that their combined results are, 32RG, 12RR, 11GG, . .
. etc.
Mermin^{[2]}
reports that "there are just two relevant features" in
the experimental results. They are:
a) When switches of the two detecting
units have the same settings (1, 2 or 3), one finds that lights
always flash the same color.
b) When one examines all runs, without
any regard to how the switches are set, then the pattern of
flashing is completely random. In particular, half the
time, lights flash the same color and half the time different
colors.
Mermin^{[2]} concludes:
"the
data
described
above
violate
Bell's
inequality^{[9]} and therefore there can be no instruction sets that
could produce classically the abovementioned features."
Mermin^{[2]} argues that the
results obtained with his apparatus can be explained only by non
realistic quantum mechanics. We will show here that such a
claim is erroneous.
When
describing his gedanken apparatus, one notices that Mermin has
neglected some important considerations. Since, in
practice, the quantum efficiency is smaller than 100%, some
counts are missing on one of the detectors, therefore changing
the number of pairs. Furthermore, in actual experiments
(as in Aspect's experiment^{[58]}), there is a random background and therefore other
unpaired (non correlated) photons that would certainly render
independent synchronization impossible. In the case of an
imperfect quantum efficiency of the detectors, when one of the
detectors has not detected the particle, it is not possible for
the operator to choose a new switch setting since he does not
know whether or not an event took place.
Mermin
assumed three hypotheses that are not realized in practice.
a) A
quantum efficiency of 100%,
b) A
perfect collection of all the photons (or whatever) generated,
and
c) No
background.
One
knows that those perfect conditions do not exist
experimentally. It is for that reason that Mermin believes
that there is no realistic solution to his gedanken
experiment. No light detector has ever existed with
perfect efficiency. In the frequency range used by
Aspect, the quantum efficiency is definitely less than
100%. Furthermore, an optical system does not collimate
all photons. Moreover, a background signal can never be
completely avoided. These phenomena lead to important
difficulties if they are ignored. Since the two observers
cannot communicate, they cannot find the way to synchronize
their two lists of data. The synchronization method
described by Mermin, is then impossible. Clearly, Aspect's
observations could not be done exactly as described by Mermin^{[2]}. One
acceptable way to solve that difficulty is to inform (by sending
a signal) the operators A and B, every time something
happens. This is the hypothesis we use in this paper.
The
actual physical experiment involving quantum mechanics, has been
done by Aspect. In his experiments^{[58]}, calcium atoms are
excited by two laser beams. After excitation, there is a
transition such that two correlated "photons" are emitted in
random directions. Some of these correlated "photons" are
statistically emitted in opposite directions and are then
detected by each detecting unit A and B as described above by
Mermin^{[2]}.
The photon beam is switched very rapidly from one
position to another using mobile mirrors moved by an ultrasonic
generator. This way, the moving mirrors (represented by
switches) are not given their random setting until after the
particles have departed from their common source.
The
fundamental principles involved in the apparatus used by Aspect
can be found in Mermin's gedanken apparatus. Since the two
photons emitted simultaneously are polarized in the same plane,
they flash lamp G (Green) when the settings of the two switches
on each detecting unit are at the same position (1, 2 or 3) and
R (Red) when settings are different.^{[10]}
When
the synchronization problem (resolved by an outside signal as
done by Mermin and Aspect's photodetectors) is solved, feature
a (realized by our experimental setup
above) is satisfied exactly as required.
However, one finds then that the relevant feature b is not satisfied now, because
statistically, lights will flash the same color 5/9 (0.5555) of
the time, instead of 0.50 that should be obtained. Bell's
inequality theorem has been applied here. Since the
denominator, (that is the possible number of settings in the
calculation of probability), is an odd number (number 9) and the
numerator is an integer, it is absolutely impossible to obtain
this way, the exact fraction 0.5 required by the quantum
mechanical calculation.
Let "C" emit classical and identical pulses of polarized EM radiation (light). As allowed in Mermin's^{[2]} gedanken experiment, the emitted radiation is polarized randomly in each of the three directions (at 120^{o} degrees). In fact, one could prove that the problem can be solved with completely random directions of polarization. In order to be more specific here, let us use a source in "C" generating square EM pulses having all the same shape, amplitude, duration (coherence length). Pulses generated in "C" would differ only by the direction of polarization (at 120^{o} ).
The
two classical detecting units "A" and "B" are identical.
The EM pulse entering each detecting unit (via space) is
divided into three equal parts. This can be done, using
fractionally reflecting mirrors as used by Aspect.^{[11]} Each of the three beams of
EM radiation passes through polarizers making 120^{o} between them as suggested by Mermin^{[2]}. Detectors of
EM radiation are located at each beam (having polarizations at
120^{o} ). Directions D1, D2 and
D3 are as described in section 2 above.
Since
we have EM radiation, we know that the polarizer oriented in
the same plane as the direction of the incident light, will
produce no attenuation of the EM signal. However, the
polarizers at 120^{o} or 240^{o} , will produce an attenuation of (Cos^{2} (120 ) = 1/4), therefore transmitting one quarter of
the signal. This reduced amplitude can be measured by the
detectors.
Since
each operator of detecting units "A" and "B" can determine (for
each individual pulse) what is the initial direction of
polarization of the pulse (D1, D2, and D3 defined above), one
can build the detecting units in such a way that the green light
will go on, when the switch setting number (1, 2, or 3) is the
same as the direction of polarization D1, D2 and D3. If
the direction of polarization is not identical, then the red
light goes on.
One can
see then that every time the two operators have chosen the same
(number) setting, both operators will find the same color.
This classical apparatus satisfies Mermin's feature a. However, one finds, as in
Mermin's apparatus that the relevant feature b is not satisfied because statistically,
5/9 (~ 0.5555) of the time, lights will flash the same color
(and not half the time as one expects).
One
can see that the considerations mentioned above are incomplete.
In fact, we know experimentally, that when the EM pulse is sent
to a polarizer at 120^{o} with
respect to the direction of polarization, 1/4 of the light is
passing through the polarizer. This is an experimental
fact that is compatible with the classical description of
Maxwell's equations. Therefore, it takes four of those
pulses (filtered at 120^{o} ) to
give as much energy to the detector as one pulse having a
direction of polarization parallel (0^{o} ). This fact has been neglected in section 6
above. Since detectors are receiving one quarter of the
energy, one must take it into account according to EM
theory. One has to take into account then that every time
one has counted four pulses at 120^{o} (that corresponds to 4 red lights), one has the same
energy as one full pulse (therefore one green light).
The
correction required to take into account that "neglected energy"
can be done by changing one "R" (red light) (corresponding to
1/4 of the energy) into one "G" (green light) (corresponding to
the full energy of the pulse) every time one has counted
(accumulated) four red lights. This is quite normal when
considering EM theory. This can be done automatically
when designing the detecting unit.
Statistical distribution following this correction show now that
half the time lights will flash the same color and half the time
different colors. The pattern of flashing, can also be
completely random as required. This is in perfect
agreement with relevant feature b,
as expected.
Consequently, we have seen that on the one hand, when one does
not take into account the fact that one quarter of the energy is
transmitted through the polarizer at 120^{o} , only relevant feature a
is satisfied. On the other hand, when one considers the
energy transmitted through the polarizer at 120^{o} (changing an "R" for a "G" once every four "R"), then
half the time lights flash the same color and half the time
different colors as in feature b.
However, now feature a is no longer
satisfied because it is not known which "R" must be changed into
a "G". One finds that the relevant feature a is now satisfied only about 92.5% of
the time. Therefore this preliminary result is not
completely satisfactory but it helps to understand what is going
on when one considers a real EM pulse. We will see now
how to solve this problem and satisfy both features perfectly
and simultaneously.
There is another way to take into account the energy of the
detector at 120^{o} and to avoid
the possible substitution of a green light"G" in the data at the
wrong time. This alternative is achieved by eliminating
some chosen pieces of data that has received only a fraction of
a full pulse (considering that the detector is not always
sensitive to such a fraction of a pulse) and considering them as
"blanks". In order to achieve that desired result, the
detectors are programmed so that one red signal "R" is changed
for one "blank" once every second red signal "R". No
change is needed when green lights "G" are flashed.
This
recipe is deduced from considerations of classical
electromagnetic theory. It can be shown that the removal
of one red signal (for a blank) out of two is required by the
fact that during an EM pulse polarized at 0^{o}, one finds that 50% of the energy is coupled with a
detector at 0^{o} degrees while
25% (Sin^{2} 120^{o} or Sin^{2} 240^{o} ) of energy is coupled to each receiver at 120^{o} and 240^{o}.
Table I
shows samples of data received by each observer A and B, their
usefulness, and finally in the forth column, data as they appear
in the format presented by Mermin. As suggested above, one
finds that a piece of data becomes a "blank" every time it is
the second red signal. These classical instruction sets
produce exactly and completely Mermin's features reported in
section 3 above.
























































Statistical calculations show that Mermin's features a and b are
now completely satisfied. Furthermore, we have made
computer simulation of those experimental conditions and we have
been able to verify that the elimination of 50% of red light
data (regularly or statistically) leads to a perfect agreement
with both conditions (a and b) described by Mermin^{[2]}. Even the
distribution obtained is random as required.
One
must conclude that the classical description given above is
undistinguishable from the quantum interpretation of experiments
of Mermin and Aspect. This description proves that it is not
necessary to use the "spooky" interpretation of quantum
mechanics to explain those experiments. Consequently,
neither Mermin's nor Aspect's experiments can prove the validity
of interpretation of quantum mechanics since the results can be
explained by classical considerations as described above.
In fact, those two experiments cannot prove at all what it was
hoped for.
The
present study has even shown us that the solution presented here
is not unique and that it is possible to conceive a realistic
classical solution without (at all) requiring any reduced
quantum efficiency. This is outside the scope of this
paper. Books have been published on the subject^{[A1, A2]}.
However, for the time being, Mermin's statement^{[2]}: "Alas, this
explanation, the only one, . . . is untenable" is erroneous; EM
theory can provide a realistic description. The
mathematics of quantum mechanics gives an excellent prediction
of the physical mechanism described above but a classical
interpretation is certainly possible here.
The author wishes to acknowledge the financial collaboration of the National Research Council of Canada and National Science and Engineering Research Council.
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References
[A1] P.
Marmet "Einstein"s Theory of
Relativity versus Classical Mechanics" Ed. Newton Physics Books 200 pages,
1997, Ogilvie Road, Gloucester, Ontario, Canada K1J 7N4
[A2] P.
Marmet, "Absurdities in
Modern Physics: A Solution" Ed. Les Éditions du
Nordir, C/O Y. Yergeau, 165 Waller Street, Simard Hall,
Ottawa, Ontario, Canada K1N 6N5.
[1] Richard
Feynman "QED The Strange Theory of Light and Matter", P. 10,
Princeton University Press, pp. 158 1985
[2] N. D. Mermin,
Physics Today 38, P. 3847 1985
[3] Anupam Garg, N.
D. Mermin Phys. Rev. D, 35, 38313835 1987
[4] Philip M.
Pearle, Phys. Rev. D, 2, P. 14181425, 1970
[5] A. Aspect, P.
Granger and G. Roger, Phys. Rev. Letters 47, 460 1981
[6] A. Aspect, P.
G. Granger and G. Roger, Phys. Rev. Letters, 49, 91, 1982
[7] A. Aspect, J.
Daligard and G. Roger Phys. Rev. Letters, 49, 1804 1982
[8] M. de
Pracontal, A. Gedilaghine Physique, Science et Vie No: 766 P.
1421 Vol 149150, July 1981.
[9] J. S. Bell,
Physics 1, 195200, 1964
[10] Note added in 2012: This is not a
correct description of Mermin's argument, see Ref. [2].
[11] Note added in
2012: This is not a correct description of Aspect's experiment,
see Ref. [7].
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About the Author
Résumé
Une
expérience spéculative a été décrite par Mermin dans le but de
prouver qu'aucune description classique n'est compatible avec
les résultats observés. Il croît avoir prouvé l'impossibilité‚
de trouver une interprétation réaliste aux observations
physiques décrites, ce qui démontrerait que la nature ne peut
être expliquée d'une façon rationelle mais exigerait
l'interprétation probabiliste de la mécanique quantique.
Contrairement à cette opinion, et en utilisant uniquement la
théorie électromagnétique, l'on donne ici une description
complète permettant d'expliquer classiquement et même de
construire l'appareil pouvant résoudre l'énigme de Mermin.