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

        1) Introduction

        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 (E-M) 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 E-M 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).

        2) Mermin's Apparatus

        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.

        3) Two relevant features

        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 above-mentioned 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.

        4) Synchronization

        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^[5-8]), 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.

        5) Aspect's experiment

        The actual physical experiment involving quantum mechanics, has been done by Aspect.  In his experiments^[5-8], 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 photo-detectors) 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.

        6) Description of our apparatus

        6a) The classical light source "C"

        Let "C" emit classical and identical pulses of polarized E-M 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 E-M 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 ).

        6b) The classical detecting units "A" and "B"

        The two classical detecting units "A" and "B" are identical.  The E-M 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 E-M radiation passes through polarizers making 120^o between them as suggested by Mermin^[2].  Detectors of E-M 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 E-M radiation, we know that the polarizer oriented in the same plane as the direction of the incident light, will produce no attenuation of the E-M signal.  However, the polarizers at 120^o or 240^o , will produce an attenuation of (Cos² (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).

        7) Other classical considerations

        One can see that the considerations mentioned above are incomplete. In fact, we know experimentally, that when the E-M 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 E-M 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 E-M 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 E-M pulse.  We will see now how to solve this problem and satisfy both features perfectly and simultaneously.

        8) A Realistic Solution

        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 E-M pulse polarized at 0^o, one finds that 50% of the energy is coupled with a detector at 0^o degrees while 25% (Sin² 120^o or Sin² 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.

        9) Conclusion

        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; E-M 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.

        10) Acknowledgment

        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. 38-47 1985
  [3]  Anupam Garg, N. D. Mermin Phys. Rev. D, 35, 3831-3835 1987
  [4]  Philip M. Pearle, Phys. Rev. D, 2, P. 1418-1425, 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. 14-21 Vol 149-150, July 1981.
  [9]  J. S. Bell, Physics 1, 195-200, 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.