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It is extremely difficult to discover an element of physical reality that might solve the “spooky action at a distance” formulated by Einstein-Podolsky-Rosen known as the EPR paradox. One very promising candidate was recently discovered in the letter of Malus addressed to Lancret in 1800. In this letter Malus (the discoverer of the polarization of light in 1808) modelled color as the composition of light and “caloric”. In the modern notation we can formulate the color and heat of polarized photons as the ratio of the ordinary and extraordinary wavefronts of that polarized photon in the Descartes’ model of colors caused by the rotation of spin-orbit of photons. Laser photons pass through the half waveplate where they get color and “heat content”, and then in the process of the spontaneous parametric down conversion, they create two entangled photons. In the pleochroic polarizers of Alice and Bob entangled colored photons modify individually their colors and their “heat content”. Pleochroism from Greek words πλÅLεω (pléon) and χρωμα (khrôma) means “more colors” and in the geological analysis describes dependence of color variation on the orientations of polarizer, analyzer, and the sample. This independent local color change in Alice and Bob polarizers can be mathematically described by the haversine and havercosine formulae. The havercosine describes the probability of a particle to stay on the same latitude. The haversine describes the probability of a particle to stay on the same longitude. In this model the “spooky action at a distance” is interpreted as the “local pleochroism” of entangled photons. This model can be further tested in the “Herschel-type” experiments where the polarized photons heat thermometers in the dependence of their “caloric” content. Bell four states mathematically describe the entanglement of two particles but without the physical interpretation based on the local pleochroism.

Introduction

The “spooky action at a distance” formulated by Einstein, Podolsky, and Rosen as the EPR paradox was intensively debated after the year 1935, e.g., [1]–[10]. Bell in 1964 formulated the mathematical formulae in order to experimentally compare the models with “hidden variables” with the predictions of quantum mechanics, e.g., [11]–[20]. The sophisticated experiments confirmed the mathematical predictions of quantum mechanics, e.g., [21]–[44] and excluded all local hidden-variable theories. However, there are still some attempts to discover those hidden variable models, e.g., [45]–[57].

Now, there is a generally accepted statement that all local hidden-variable theories are excluded forever. Is there an existing overlooked loophole hidden in works of Old Masters?

Malus Thermochromatic Polarized Photon

In 2008 Kahr and Claborn rediscovered the overlooked letter of Malus addressed to Lancret in the year 1800 [58], [59]. It was the period when many researchers measured the temperature of heated thermometers by the prismatic light [60]–[73]. The most known researcher was Herschel with his discovery of the infrared light. Malus originally interpreted the heat content of photons and their color as the composition of light with “caloric”. Color depended on the relative proportion in light of caloric, as red light is more heating than violet: where a expresses an invariant portion of light and 1, 2, 3, 4, 5…the quantity of caloric as it is depicted in Fig. 1.

b l u e s g r e e n s y e l l o w r e d s a + 1 , a + 2 , a + 3 , a + 4 , a + 5 , a + 6 , a + 7 , a + 8 , a + 9 , a + 10

Fig. 1. Malus interpretation of colors as the composition of light and “caloric”.

Recently, Stávek [74] compared the color theories of Descartes (based on the rotation energy of spin-orbit of photons) with the Newtonian model (based on the relation color = wavelength). It is now generally accepted fact that the Newtonian model cannot correctly describe the color appearance in many situations. There are many experiments when the same wavelength of photons can have different immutable colors.

We propose to use the Descartes’ scenario for Malus polarized photons with a given wavelength to describe their color and heat content based on the ratio of their ordinary and extraordinary wavefronts in the dependence of the analyzer azimuth (Fig. 2).

Fig. 2. The hypothesis of the Malus’ polarized photons: the color and heat content depend on the orientation of the analyzer.

The “heat content” and “color appearance” of polarized photons transmitted through the analyzer can be measured in the “Herschel” type experiment based on the proposed (1): where h is the Planck constant, ν is the frequency of the polarized photon and θ is the orientation of the analyzer towards the ordinary wavefront of that polarized photon.

E h e a t t r a n s = h ν cos 2 θ

The “heat content” and “color appearance” of polarized photons reflected or refracted on the analyzer can be measured in the “Herschel” type experiment based on the proposed (2):

E h e a t r e f = h ν sin 2 θ

We can model the action of the analyzer on the polarized photon as the “pleochroic action” modifying locally the “heat content” and “color appearance” of that photon. (Pleochroism from Greek words πλέω (pléon) and χρωμα (khrôma) means “more colors” and in the geological analysis describes dependence of color variation on the orientations of polarizer, analyzer, and the sample, e.g., [75]). Fig. 3 shows schematically the “pleochroic analyzer”.

Fig. 3. The pleochroic action of the analyzer to modify locally the heat content and the color of the polarized photon.

Bell’s “Black and White” Schema and Malus‘ “Pleochroic” Schema

In 1964 Bell derived his famous conditions for unknown local hidden variables and predictions of quantum mechanics. Fig. 4 depicts the Bell’s schema in the “black and white” colors because there are not defined hidden variables. This schema in Fig. 4 is familiar to all scholars in quantum mechanics. Fig. 5 describes the Malus’ “pleochroic” schema.

Fig. 4. Bell’s schema with an un-known hidden variable λ leading to the impossibility of local hidden variables to describe events with entangled photons, Alice and Bob do not know the “hidden-variable” acting in the system.

Fig. 5. Malus’ pleochroic schema: entangled photons have their color and heat content λ. Alice and Bob operate independently with their pleochroic analyzers where local modifications of photon colors and heat content occur.

Fig. 6 shows the spontaneous parametric down conversion of two entangled photons. In the “black and white” schema there is not defined the missing hidden variable. In the “pleochroic” schema is an extra information describing the color and heat content of entangled photons. These “colored” photons later locally react in the pleochroic analyzers at Alice and Bob sites.

Fig. 6. The spontaneous parametric down conversion of entangled photons. In the “black and white” entanglement formation the hidden variable is not known. In the Malus’ pleochroic schema: entangled photons have their color and heat content. Alice and Bob operate independently with their pleochroic analyzers where local modifications of photon colors and heat content occur. Based on reference [20].

Alice and Bob Individual and Joint Correlations in the Malus’s “Pleochroic” Schema

Alice and Bob independently measure the color of their photons. They rotate independently their analyzers and will obtain locally modified colors of their photons. Fig. 7 shows schematically colors at individual analyzers. They both observe individually P+ = orange = red + yellow = 1/2 and P- = cyan = green + blue = 1/2.

Fig. 7. Alice and Bob individual observation of photon colors: P+ = orange = red + yellow, P− = cyan = green + blue.

If Alice and Bob evaluate their joint correlations, they will get four colors: red for P++ = ½ cos2 θ, blue for P– = ½ cos2 θ, chartreuse green = yellow + green for P+− = P−+ = ½ sin2 θ. This situation is given by Fig. 8.

Fig. 8. Alice and Bob joint correlations of photon colors: P++ = red, P−− = blue, P+− = P−+ = chartreuse green = green + yellow.

Havercosine and Haversine at Individual Analyzers of Alice and Bob

Alice and Bob independently measure the color of their photons by their “pleochroic” analyzers. They rotate independently their analyzers and will obtain locally modified colors and heat content of their photons. Fig. 9 depicts the local color modification at the individual analyzers.

Fig. 9. Alice and Bob separately modify the color of photons (their spin orbit) by rotating of their analyzers by the angle θ (picture taken from Wikipedia).

This independent rotation of analyzers with the individual entangled photon can be described by the trigonometric spherical functions: havercosine and haversine. Fig. 10 brings schema of the spheres for the determination of havercosine and haversine and the rotation of entangled photon in the analyzer.

Fig. 10. Spheres for the determination of trigonometric spherical functions havercosine and haversine for individual entangled photons. These functions describe the rotation of polarized photons at the analyzer (picture taken from Wikipedia).

Individual correlations at analyzers can be evaluated using the havercosine and haversine formulae. Havercosine describes the probability of the particle to stay on the same latitude–(3) and (4). Haversine describes the probability of the particle to stay on the same longitude–(5) and (6):

P N o r t h = sin 2 θ θ 2 + cos θ cos θ sin 2 π / 2 2 = 1 2 cos 2 θ
P S o u t h = sin 2 θ θ 2 + cos θ cos θ sin 2 π / 2 2 = 1 2 cos 2 θ
P W e s t = sin 2 θ θ 2 + sin θ sin θ sin 2 π / 2 2 = 1 2 sin 2 θ
P E a s t = sin 2 θ θ 2 + sin θ sin θ sin 2 π / 2 2 = 1 2 sin 2 θ

The total probability for the particle to stay on the same latitude is given by the joint contributions of both the North and South poles of that sphere–(7):

P N o r t h + P S o u t h = 1 2 cos 2 θ + 1 2 cos 2 θ = cos 2 θ

The total probability for the particle to stay on the same longitude is given by the joint contributions of both the East and West poles of that sphere–(8):

P E a s t + P W e s t = 1 2 sin 2 θ + 1 2 sin 2 θ = sin 2 θ

The joint contributions of havercosine and haversine of individual particles leads to the mathematical description that “looks like the spooky action at the distance” known from the mathematical description of quantum mechanics derived for particles separated after their joint formation.

The famous quote of Étienne Louis Malus is still actual [76]: “We find that light acquires properties which are relative only to the sides of the ray, –which are the same for the north and south sides of the ray, (using the points of the compass for description’s sake only) and which are different when we go from the north and south to the east or to the west sides of the ray. I shall give the name of poles to these sides of the ray, and shall call polarization the modification which gives to light these properties relative to these poles.”

Conclusion

This contribution is based on the overlooked letter of Malus to Lancret in 1800 where Malus (the discoverer of the light polarization in 1808) explained heating effect of light colors as the composition of light with “caloric”. There are two main schools dealing with the physical color theory. The Newtonian school and the Descartes’ school. Both schools collected many experimental evidences on the color properties. We are now in the situation without a general physical color theory to explain all these experimental evidences with colors. Therefore, we should search for a more general physical model of color properties and “heat content” of polarized photons.

  1. The Descartes’ model of rotating globules (based on the spin-orbit of photons) was selected as the potential candidate to interpret the missing “hidden-variable” in quantum optics.
  2. The process of the spontaneous parametric down conversion gives to entangled photons “heat content” and colors.
  3. The separated entangled photons react at the Alice and Bob analyzers and independently modify their “heat content” and color based on the analyzer azimuth.
  4. The spherical trigonometric function havercosine describes the probability of a particle to stay on the same latitude.
  5. The spherical trigonometric function haversine describes the probability of a particle to stay on the same longitude.
  6. In this thermochromatic model of polarized photons we study the “pleochroic” properties of entangled polarized photons as the local effect at individual analyzers.
  7. The “heat content” of polarized photons can be studied in the “Herschel” type experiments with illuminated thermometers.

References

  1. Einstein A, Podolsky B, Rosen N. Can quantum-mechanical description of physical reality be considered complete? Phys Rev. 1935;47(10):777–80.
     Google Scholar
  2. Bohr N. Can quantum-mechanical description of physical reality be considered complete? Phys Rev. 1935;48(8):696–702.
     Google Scholar
  3. Schr˝odinger E. Probability relations between separated systems. Math Proc Camb Philos Soc. 1936;31(4):555–63.
     Google Scholar
  4. Schr˝odinger E. Discussion of probability relations between separated systems. Math Proc Camb Philos Soc. 1936;32(3):446–52.
     Google Scholar
  5. Bohm D. Quantum Theory. Englewood Cliffs: Prentice Hall; 1951. Chapter 5 and 22.
     Google Scholar
  6. Bohm DA. A suggested interpretation of the quantum theory in terms of hidden variables I. Phys Rev. 1952;85(2):166.
     Google Scholar
  7. Bohm DA. A suggested interpretation of the quantum theory in terms of hidden variables II. Phys Rev. 1952;85(2):180.
     Google Scholar
  8. Peres A. Einstein, Podolsky, Rosen, and Shannon. Foundations of Physics. 2005;35(3):511–4.
     Google Scholar
  9. Wiseman HM, Jones SJ, Doherty AC. Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox. Phys Rev Lett. 2007;98(14):140402.
     Google Scholar
  10. Blaylock G. The EPR paradox, Bell’s inequility, and the question of locality. Am J Phys. 2010;78(1):111–20.
     Google Scholar
  11. Bell JS. On the Einstein-Podolsky-Rosen paradox. Phys Phys Fiz. 1964;1(3):195–200.
     Google Scholar
  12. Bell JS. On the problem of hidden variables in quantum mechanics. Rev Mod Phys. 1966;38(3):447–52.
     Google Scholar
  13. Clauser JF, Horne MA, Shimony A, Holt RA. Proposed experiment to test local hidden-variable theories. Phys Rev Lett.1969;23(15):880–4.
     Google Scholar
  14. Aspect A. Proposed experiment to test the nonseparability of quantum mechanics. Phys Rev D. 1976;14(8):1944–51.
     Google Scholar
  15. Fine A. Hidden variables, joint probability, and the Bell inequalities. Phys Rev Lett. 1982;48(5):291–5.
     Google Scholar
  16. Shimony A. Contextual hidden variable theories and Bell’s inequalities. Brit J Philos Sci.1984;35(1):25–45.
     Google Scholar
  17. Bell JS. Speakable and Unspeakable in Quantum Mechanics. Cambridge: Cambridge University Press; 1987. ISBN: 9780521368698.
     Google Scholar
  18. Mermin ND. Hidden variables and two theorems of John Bell. Rev Mod Phys. 1993;65(3):803–15.
     Google Scholar
  19. Dehlinger D, Mitchell MW. Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory. Am J Phys. 2002;70(9):903–10.
     Google Scholar
  20. Jensen G, Marsh Ch, To S. Entangled photons and Bell’s inequality. 2013. Available from: http://www2.optics.rochester.edu/workgroups/lukishova/QuantumOpticsLab/2013/OPT_253/GroupC_Lab1_Chris.pdf.
     Google Scholar
  21. Freedman SJ, Clauser JF. Experimental test of local hidden-variable theories. Phys Rev Lett. 1972;28(14):938–41.
     Google Scholar
  22. Fry ES, Thompson RC. Experimental test of local hidden-variable theories. Phys Rev Lett. 1976;37(8):465–8.
     Google Scholar
  23. Clauser JF, Shimony A. Bell’s theorem–experimental tests and implications. Rep Prog Phys. 1978;41(12):1881–927.
     Google Scholar
  24. Aspect A, Grangier P, Roger G. Experimental tests of realistic local theories via Bell’s theorem. Phys Rev Lett. 1981;47(7):460–3.
     Google Scholar
  25. Aspect A, Dalibard J, Roger G. Experimental test of Bell inequalities using time-varying analyzers. Phys Rev Lett. 1982;49(25):1804–7.
     Google Scholar
  26. Aspect A, Grangier P, Roger G. Experimental realization of Einstein-Podolsky-Rosen Gedankenexperiment–A new violation of Bell inequalities. Phys Rev Lett. 1982;49(2):91–4.
     Google Scholar
  27. Shih YH, Alley CO. New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion. Phys Rev Lett. 1988;61(26):2921–4.
     Google Scholar
  28. Tapster PR, Rarity JG, Owens PCM. Violation of Bell’s inequality over 4 km of optical fiber. Phys Rev Lett. 1994;73:1923.
     Google Scholar
  29. Kwiat PG, Mattle K, Weinfurter H, Zeilinger A. New high-intensity source of polarization-entangled photon-pairs. Phys Rev Lett. 1995;75:4337.
     Google Scholar
  30. Weihs G, Jennewein T, Simon C, Weinfurter H, Zeilinger A. Violation of Bell’s inequality under strict Einstein locality conditions. Phys Rev Lett. 1998;81(23):5039–43.
     Google Scholar
  31. Aspect A. Bell’s theorem: the naive view of an experimentalist. 2004. Arxiv: quant-ph/0402001.
     Google Scholar
  32. Giustina M, Mech A, Ramelow S,Wittmnn B, Kofler J, Beyer J, et al. Bell violation using entangled photons without the fair-sampling assumption. Nature. 2013;497:227.
     Google Scholar
  33. Christensen BG, McCusker KT,Alepeter JB, Calkins B,Gerrits T, Lita AE, et al. Detection-loophole-free test of quantum nonlocality, and applications. Phys Rev Lett. 2013;111:130406.
     Google Scholar
  34. Gallichio J, Friedman AS, Kaiser DI. Testing Bell’s inequality with cosmic photons: closing the setting-independence loophole. Phys Rev Lett. 2014;112(11):110405.
     Google Scholar
  35. Giustina M, VersteeghMA,Wengerowsky S, Handsteiner J, Hochrainer A, Phelan K, et al. Significant-loophole-free test of Bell’s theorem with entangled photons. Phys Rev Lett. 2015;115:250401.
     Google Scholar
  36. Hensen B, Bernien, Dréau AE, Reiserer A, Kalb N, Blok MS, et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometers. Nature. 2015;526:682–6.
     Google Scholar
  37. Aspect A. Closing the door on Einstein and Bohr’s quantum debate. Phys. 2015;8:123.
     Google Scholar
  38. Shalm LK,Meyer-Scott E, Christensen BG, Bierhost P, Wayne MA, Stevens MJ, et al. Strong loophole-free test of local realism. Phys Rev Lett. 2015;115:250402.
     Google Scholar
  39. Abellán C, Amaya W, Mitrani D, Pruneri V, Mitchell MW. Generation of fresh and pure random numbers for loophole free Bell tests. Phys Rev Lett. 2015;115:250403.
     Google Scholar
  40. RosenfeldW, Burchardt D, Garthoff R, Redeker K, Ortegel N,Weinfurter H. Event-ready Bell test using entangled atoms simultaneously closing detection and locality loopholes. Phys Rev Lett. 2017;119:010402.
     Google Scholar
  41. Yin J, Cao Y, Li YH, Liao SK, Zhang L, Ren JG, et al. Satellite-based entanglement distribution over 1200 kilometers. Sci. 2017;356:1140–4.
     Google Scholar
  42. Li MH, Wu C, Zhang Y, Liu WZ, Bai B, Liu Y, et al. Test of local realism into the past without detection and locality loopholes. Phys Rev Lett. 2018;121(8):80404.
     Google Scholar
  43. Rauch D, Handsteiner J, Hochrainer A, Gallichio J, Friedman AS, Leung C, et al. Cosmic Bell test using random measurement setting from high-redshift quasars. Phys Rev Lett. 2018;121(8):80403.
     Google Scholar
  44. Freire O Jr. Alain Aspect’s experiments on Bell’s theorem: a turning point in the history of the research on the foundations of quantum mechanics. Eur Phys J D. 2022;76:248.
     Google Scholar
  45. Genovese M. Research on hidden variable theories: a review of recent progresses. 2007. Arxiv: quantph/070107v1.
     Google Scholar
  46. Bacciagaluppi G, Crull E. Heisenberg (and Schr˝odinger, and Pauli) on hidden-variables. Stud Hist Philos Sci part B: Stud Hist Philos Modern Phys. 2009;40:374–82.
     Google Scholar
  47. Boozer AD. Hidden-variable theories and quantum nonlocality. Eur J Phys. 2009;30(2):355–65.
     Google Scholar
  48. Freire O. The Quantum Dissidents: Rebuilding the Foundations of Quantum Mechanics (1950–1990). With a forward by S.S. Schweber. Berlin, Springer; 2015. ISBN-10:9783662446614.
     Google Scholar
  49. Budroni C, Moroder T, Kleinmann, Guhne O. Bounding Temporal Quantum Correlations. Phys Lett. 2013;111,020403.
     Google Scholar
  50. Khrennikov A. Get rid of nonlocality from quantum physics. Entropy. 2019;21:806–15.
     Google Scholar
  51. Kupczynski M. Closing the door on quantum nonlocality? Entropy. 2018;20:877–90.
     Google Scholar
  52. Jung K. Polarization correlation of entangled photons derived without using non-local interactions. Front Phys. 2020;8:170.
     Google Scholar
  53. Khrennikov A. Quantum versus classical entanglement: eliminating the issue of quantum nonlocality. Found Phys. 2020;50(12):1762–80.
     Google Scholar
  54. Khrennikov A. Two faced Janus of quantum nonlocality. Entropy. 2020;22(3):303.
     Google Scholar
  55. Vatarescu A. Polarimetric quantum-strong correlations with independent photons on the Poincaré sphere. Quantum Beam Sci. 2022;6:32.
     Google Scholar
  56. Hess K. Einstein-local counter-arguments and counter-examples to Bell-type proofs. J Mod Phys. 2023;14:89–100.
     Google Scholar
  57. Hess K. Malus-law models for aspect-type experiments. J Mod Phys. 2023;14:1167–76.
     Google Scholar
  58. Kahr B, Claborn K. The lives of Malus and his bicentennial law. ChemPhysChem. 2008;9:43–58.
     Google Scholar
  59. Chappert A. Étienne Louis Malus (1775–1812) et la Théorie Corpusculaire de la Lumière. Paris: Librairie Philosophique J. Vrin; 1977.
     Google Scholar
  60. Rochon AM. Sur la degré de chaleur des rayons colorés. English trans. An essay on the degree of warmth of coloured rays. Philos Mag. 1783;45:410–4.
     Google Scholar
  61. Wünsch CE. Versuche über die Vermeinte Sonderung des Lichtes der Sonnenstrahlen von der Wärme derselben. German: Magazin der Gesselschaft Naturforschende Freunde zu Berlin; 1807. pp. 185–207.
     Google Scholar
  62. Seebeck TJ. Ueber die ungleiche Erregung der Wärme im prismatischen Sonnebilde. In: Abhandlungen der königlichen Akademie der Wissenschaften in Berlin aus den Jahren 1818–1819. Berlin, German: Georg Reimer; 1820. pp. 305–50.
     Google Scholar
  63. Seebeck TJ. On the unequal evolution of heat in the prismatic spectrum. Philos Mag. 1825;66:330–43, 445–55.
     Google Scholar
  64. Cornell ES. The radiant heat spectrum from Herschel to Melloni I. The work of Herschel and his contemporaries. Ann Sci. 1938;3:119–37.
     Google Scholar
  65. Cornell ES. The radiant heat spectrum from Herschel to Melloni II. The work of Melloni and his contemporaries. Ann Sci. 1938;3:402–13.
     Google Scholar
  66. Barr ES. Historical survey of the early development on the infrared spectral region. Am J Phys. 1960;28:42–54.
     Google Scholar
  67. Barr ES. The infrared pioneers–I. Sir William Herschel. Infrared Phys. 1961;1:1–10.
     Google Scholar
  68. Hentschel K. Mapping the spectrum. In: Techniques of Visual Representation in Research and Teaching. Oxford: Oxford University Press; 2009. 21–72. Ch. 2.
     Google Scholar
  69. Herschel W. Investigations of the powers of the prismatic colours to heat and illuminate objects; with remarks that prove the different refrangibility of radiant heat. Philos Trans R Soc, Lond. 1800;90:255–283, 284–292, 293–326, 437–538.
     Google Scholar
  70. Lowell DJ. Herschel’s dilemma in the interpretation of thermal radiation. Isis. 1968;59(1):46–60.
     Google Scholar
  71. Hilbert M. Herschel’s investigation of the nature of radiant heat: the limitations of experiment. Ann Sci. 1999;56:357–78.
     Google Scholar
  72. Minkina W. How infrared radiation was discovered–range of this discovery and detailed, unknown information. Appl Sci.2021;11:9824.
     Google Scholar
  73. Stávek J.What is hidden in the Planck distribution function and theWien’s peaks? I. Three features of the Solar photons. Eur J Appl Phys. 2023;2:1–8.
     Google Scholar
  74. Stávek J. A new interpretation of the physical color theory based on the Descartes rotation energy of visible, ultraviolet, and infrared photons. Europ J Appl Phys. 2;5(5):29–37.
     Google Scholar
  75. Delly JG. Essential of Polarized Light Microscopy and Ancillary Techniques.Westmont: McCrone, Group; 2017. ISBN-10:0692979471.
     Google Scholar
  76. Malus EL. Available from: https://mathshistory.st-andrews.ac.uk/Biographies/Malus/quotations/ (Accessed 28 September 2023).
     Google Scholar


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