FUNDAMENTOS DE FÍSICA CUÁNTICA. ASPECTOSHISTÓRICOS Y FILOSÓFICOS.

Notas, citas y referencias

“Con estas razones perdía el pobre caballero el juicio, y desvelábase por entenderlas y desentrañarles el sentido,que no se lo sacara ni las entendiera el mesmo Aristóteles, si resucitara sólo para ello”.

(Cervantes, Don Quijote de la Mancha, capítulo I).

“Physics is an honest trade; only after you have learned it have you the right to philosophize about it”.

(Heisenberg, citado por C.F. von Weizsäcker en “A Reminiscence from1932”, en Niels Bohr: a centenary volume, Harvard Univ., 1985).

1. Ciencia y Filosofía

1-1. “One can doubt the possibility of establishing philosophical truths by the methods of physics,but it is surely not outside the competence of physicists to demonstrate that certain statementswhich pretend to have a philosophical validity do not. And sometimes these 'negatives'philosophical discoveries by physicists are no less important, no less revolutionary for philosophythan the discoveries of recognized philosophers”.(London and Bauer, La Théorie de L'Observation en Mécanique Quantique, Hermann, Paris, 1939; cit. EnFrench S., “The Phenomenological Approach to Physics”, Studies in History and Philosophy of Modern Physics,30B (1999) 279).

1-2. “Tendremos que tener un cuidado especial con las conclusiones filosóficas derivadas de losresultados físicos. No hay duda de que cualquier vindicación filosófica debe ser reconciliada conlos mejores resultados de la ciencia física a nuestra disposición. Ni hay duda alguna de que elprogreso de la ciencia ha proporcionado un útil antídoto contra mucho dogmatismo en la filosofía.Pero al examinar lo que la física nos dice sobre las cuestiones filosóficas, debemos tener siemprepresente preguntar si se han insertado presuposiciones filosóficas en la propia teoría. Sidescubrimos que dichas presuposiciones han sido incluidas en la propia teoría, debemos de estarpreparados para examinar la cuestión de si esa forma de presentar la teoría es la única forma en laque sus resultados científicos podían haber sido acomodados o si pudiera haber otraspresuposiciones que nos llevarían a derivar conclusiones filosóficas bastante diferentes en caso deformar parte de la teoría”.([SKL-92], p. 24).

2. Desarrollo histórico de la Mecánica Cuántica

2-1. Muller, F.A.; “The Equivalence Myth of Quantum Mechanics. I”, Studies in Theory andPhilosophy of Modern Physics, 28B,1 (1997) 35-61.

2-2. Muller, F.A.; “The Equivalence Myth of Quantum Mechanics. II”, Studies in Theory andPhilosophy of Modern Physics, 28B,2 (1997) 219-247.

2-3. Heisenberg, W.; “Über quantentheoretische Umdeutung kinematischer und mechanischerBeziehungen”, Zeitschrift für Physik 33 (1925) 879-893.

2-4. Heisenberg, W.; Encuentros y conversaciones con Einstein y otros ensayos, Alianza Editorial,Madrid, 1980.

2-5. Born, M. and Jordan, P.; “Zur Quantenmechanik”, Zeitschrift für Physik 34 (1925) 858-888.

2-6. Born, M., Heisenberg,W. and Jordan, P.; “Zur Quantenmechanik. II”, Zeitschrift für Physik 35(1926) 557-615.

2-7. Pauli, W.; “Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik”,Zeitschrift für Physik 36 (1926) 336-363.

2-8. Born, M. and Wiener, N.; “A new formulation of the laws of quantization of periodic andaperiodic phenomena”, Journal of Mathematics and Physics (M.I.T.) 5 (1925-26) 84-98; “Eineneue Formulierung der Quantengesetze für periodische und nichtperiodische Vorgänge”, Zeitschriftfür Physik 36 (1926) 174-187.

2-9. Dirac, P.A.M.; “The fundamental equations of Quantum Mechanics”, Proceedings of theRoyal Society (London) A109 (1925) 642-653.

2-10. Dirac, P.A.M.; “Quantum Mechanics and a Preliminary Investigation of the Hydrogen Atom”,Proceedings of the Royal Society (London) A110 (1926) 561-579.

2-11. “At present one can form no picture of what a q-number is like... In order to be able to getresults comparable with experiments, we must have some way of representing q-numbers by meansof c-numbers, so that we can compare these c-numbers with experimental values. Therepresentation must satisfy the condition that one can calculate the c-numbers that represent x+y, xyand yx when one is given the c-numbers that represent x and y”. (Dirac, P.A.M., en ref. 10, ver [WAE-67], pp. 418-419).

2-12. Brillouin, M.; « Actions mécaniques à hérédité discontinue par propagation; essai de théoriedynamique de l'atom à quanta », Comptes Rendus 171 (1919) 1318-1320; « Actions à héréditédiscontinue et raies spectrales », Comptes Rendus 171 (1920) 1000-1002.

2-13. Broglie, L. de; « Recherches sur la théorie des quanta », Annales de Physique 10,3 (1925) 22-128.

2-14. Schrödinger, E.; « Quantisierung als Eigenwertproblem (Erste Mitteilung) », Annalen dePhysik 79 (1926) 361-376.

2-15. “Se puede, por tanto, intentar asociar la función con un proceso vibratorio en el átomo,proceso posiblemente más real que las órbitas electrónicas, cuya existencia está siendo muycuestionada actualmente.(...)El misterioso requisito de los números enteros no entra ya en las reglasde cuantización, sino que ha retrocedido, por así decir, un paso atrás, al demostrar que es elresultado de la finitud y univocidad de una cierta función espacial.(...)No hace falta decir cuántomás agradable sería concebir una transición cuántica como un intercambio de energía de un modode vibración a otro, que considerarla como un salto de electrones”.

(Schrödinger, en su primera comunicación, ref. 2-14, cit. en [ICA-91], p. 97).

2-16. Schrödinger, E.; “Quantisierung als Eigenwertproblem (Zweite Mitteilung)”, Annalen derPhysik 79 (1926) 489-527.

2-17. Schrödinger, E.; “Quantisierung als Eigenwertproblem (Dritte Mitteilung)”, Annalen derPhysik 80 (1926) 437-490.

2-18. Schrödinger, E.; “Quantisierung als Eigenwertproblem (Vierte Mitteilung)”, Annalen derPhysik 81 (1926) 109-139.

2-19. Schrödinger, E.; “Über das Verhältnis des Heisenberg-Born-Jordanschen Quantenmechanikzu der meinen », Annalen der Physik 79 (1926) 734-756.

2-20. Lanczos, K.; “Über eine feldmässige Darstellung der neuen Quantenmechanik”, Zeitschriftfür Physik 35 (1926) 812-830.

2-21. Eckart, C.; “The solution of the problem of the simple oscillator by a combination of theSchröndiger and the Lanczos theories”, Proceedings of the National Academy of Sciences 12 (1926)473-476.

2-22. Eckart, C.; “Operator calculus and the solution of the equations of the quantum dynamics”,Physical Review 28,2 (1926) 711-726.

2-23. Eckart, C.; “The hydrogen spectrum in the new quantum theory”, Physical Review 28,2(1926) 927-935.

2-24. Dirac, P.A.M.; “The Physical Interpretation of the Quantum Dynamics”, Proceedings of theRoyal Society (London) A113 (1927) 621-641.

2-25. Jordan, P.; “Über eine neue Begründung der Quantenmechanik”, Zeitschrift für Physik 40(1927) 809-838.

2-26. Hilbert, D., Von Neumann, J., Nordheim, L.; “Über die Grundlagen der Quantenmechanik”,Mathematische Annalen 98 (1927) 1-30.

2-27. Neumann, J. von; “Mathematische Begründung der Quantenmechanik”, GötingerNachrichten (1927) 1-57.

2-28. Neumann, J. von; “Wahrscheinlichkeitstheoretischer Aufbau der Quan-tenmechanik”,Göttinger Nachrichten (1927) 245-272.

2-29. Neumann, J. von; “Thermodynamik quantenmechanischer Gesamtheiten”, GöttingerNachrichten (1927) 273-291.

2-30. Neumann, J. von; “Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren”,Mathematische Annalen 102 (1929) 49-131.

2-31. Neumann, J. von; “Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik”,Zeitschrift für Physik 57 (1929) 30-70.

2-32. Neumann, J. von; Mathematische Grundlagen der Quantenmechanik, Springer, Berlín, 1932;Mathematical Foundations of Quantum Mechanics, Princenton Univ., Princenton, 1955,Fundamentos de la Mecánica Cuántica, C.S.I.C., Madrid, .

2-33. Rèdei, Miklós; “Why John von Neumann did not Like the Hilbert Space Formalism ofQuantum Mechanics (and What he Liked Instead)”, Studies in History and Phylosophy of ModernPhysics 27,4 (1996) 493-510.

2-34. “I would like to make a confession which may seem immoral: I do not believe absolutely inHilbert space any more”. (Von Neumann, cit. en Birkhoff, G.; “Lattice Theory”, American Mathematical Society Colloquium Publications,25 (1966), cit. en ref. 2-33).

2-35. “The situation in 1927 was therefore essentially this: The new formalism of wave mechanicswhich Schrödinger had established contained in its higher-level propositions a number ofuninterpreted terms, such as the wave function, but made it possible to deduce certain lower-levelpropositions that involved parameters which could be associated with empirically meaningfulconceptions such as energy or wave lengths. What was called for, apart from possibly additionalrules of correspondence for higher-level terms, was primarily some unifying explanatory principleor some model...”.(M. Jammmer, en [JAM-74], p. 24.).

2-36. “The concept 'energy' is something that we have derived from macros-copic experience andreally only from macroscopic experience. I do not believe that it can be taken over into micro-mechanics just like that, so that one may speak of the energy of a single partial oscillation. Theenergetic property of the individual partial oscillation is its frequency”.(Schrödinger, en carta a Max Planck, mayo 1926, cit. en [JAM-74], p. 29).

2-37. “Because of this unavoidable blurring a wave packet does not seem to me very suitable forrepresenting things to which we want to ascribe a rather permanent individual existence »(H.A. Lorentz, en carta a Schrödinger, mayo 1926, cit. en [JAM-74], p. 31).

2-38. “I could not follow him.(…) Every experiment by Franck and his assistants on electroncollisions appeared to me as a new proof of the corpuscular nature of the electron”.(M. Born, en Experiment and Theory in Physics, Cambridge Univ., Londres, 1943, cit. en [JAM-74], p. 39).

2-39. Heisenberg, W.; “Über den anschaulichen Inhalt der quantentheoretischen Kinematik undMechanik”, Zeitschrift für Physik 43 (1927) 172-198; “The physical content of QuantumKinematics and Mechanics”, en [WHE-83], pp. .

2-40. “Se ha insistido repetidamente que la función , en sí misma no puede ni debe en generalinterpretarse directamente en términos del espacio tridimensional, aunque el problema de unelectrón conduzca a esa interpretación, ya que en realidad es, en general, una función en el espaciode configuración y no en el espacio real”.(Schrödinger, en su cuarta comunicación, ref. 18, cit. en [ICA-91], p. 111).

2-41. “Es difícil dar a la onda de la Mecánica Ondulatoria la misma significación física quepodían poseer las ondas consideradas por la Física Clásica. En efecto, en Física Clásica lasmagnitudes que se propagan por ondas, están referidas a las vibraciones de un medio, cuyaexistencia es real o supuesta; deben, por consiguiente, representando un fenómeno real, expresarse

por una función real. Si a veces se juzga útil, como sucede en los cálculos de la óptica, reemplazarestas funciones reales por cantidades complejas, de las cuales son sus partes reales, ello no es másque un artificio de cálculo, del que se podría siempre prescindir. Por el contrario, en MecánicaOndulatoria, debido a la presencia de coeficientes imaginarios en la misma ecuación depropagación, el carácter complejo de la función de onda se manifiesta como esencial y se opone atoda tentativa que se realice para considerar la onda de la Mecánica Ondulatoria como una realidadfísica que corresponda a las vibraciones de algún medio”.(Broglie, L. de; La física nueva y los cuantos, Losada, Buenos Aires, 1941, cit. en [ICA-91], p. 110).

2-42. Regt, Henk W. de; “Erwin Schrödinger, Anschaulichkeit and Quantum Theory”, Studies inTheory and Philosophy of Modern Physics 28B,4 (1997) 461-481.

2-43. Schrödinger, E.; The interpretation of Quantum Mechanics. Dublin Seminars and otherunpublished essays, Ox Bow, 1995.

2-44. Heisenberg, W.; “Reminiscences from 1926 and 1927", en [FRE-85], pp. 163-171.

2-45. Born, M.; “Zur Quantenmechanik der Stossvorgänge”, Zeitschrift für Physik 37 (1926) 863-867, 38 (1926) 803-827.

2-46. “Schrödinger's quantum mechanics therefore gives quite a definite answer to the question ofthe effect of the collision; but there is no question of any causal description. One gets no answer tothe question, ‘what is the state after the collision’, but only to the question, ‘how probably is aspecified outcome of the collision’.(…)The motion of particles conforms to the laws of probability,but the probability itself is propagated in accordance with the law of causality”.(M. Born en ref. 45).

2-47. Born, M.; “Albert Einstein und das Lichtquantum”, Die Naturwissenschaften, 11 (1955) 425-435, cit. en [JAM-74], nota 33, p. 41.

2-48. Born, M.; “Das Adibatenprinzip in Quantenmechanik”, Zeitschrift für Physik 40 (1926) 167-192.

2-49. “The individual process, the `quantum jump', is therefore not causally determined in contrastto the a-priori probability of its ocurrence; this probability is ascertainable by the integration ofScrödinger's differential equation which is completely analogous to the corresponding equation inclassical mechanics, putting into relation two stationary time-intervals separated by a finitetemporal interval. The jump thus passes over a considerable abyss (“der Sprung geht also übereinen beträchtlichen Abgrund”); whatever occurs during the transition can hardly be describedwithin the conceptual framework of Bohr's theory, nay, probably in no language which lends itselfto visualizability”.(M.Born en ref. 48, cit. en [JAM-74], p. 42).

2-50. ”Summarizing Born's original probabilistic interpretation of the -function we may say that2d measures the probability density of finding the particle within the elementary volume d, theparticle being conceived in the classical sense as a point mas possesing at each instant both adefinite position and a definite momentum. Contrary to Schrödinger's view, does not representthe physical system nor any of its physical attributes but only our knowledge concerning the latter”.(M. Jammer, [JAM-74], pp. 42-43).

2-51. Hendry, J.; The creation of Quantum Mechanics and the Bohr-Pauli Dialogue, Reidel,Dordrecht, 1984; cit. en [ICA-91], p. 144.

2-52. “Here the whole problem of the determinism comes up. From the standpoint of our quantummechanics there is no quantity which in any individual case causally fixes the consequence of thecollision; but also experimentally we have so far no reason to believe that there are some innerproperties of the atom which condition a definite outcome for the collision. Ought we to hope laterto discover such properties (like phases or the internal atomic motions) and determine them inindividual cases? Or ought we to believe that the agreement of theory and experiment -as to theimpossibility of prescribing conditions for a causal evolution- is a preestablished harmony foundedon the no existence of such conditions? I myself am inclined to give up determinism in the world ofatoms. But that is a philosophical question for which physical arguments alone are not decisive”.(M. Born en ref. 45).

2-53. “Quantum Mechanics is very impressive. But an inner voice tell me that it is not yet the realthing. The theory produces a good deal but hardly bring us closer to the secret of the Old One. I amat all events convinced that He does not play dice. Waves in 3n-dimensional space whose velocityis regulated by potential energy (eg., rubber bands)...”(A. Einstein, en carta a M. Born, diciembre 1926, cit. en [PAI-82], p. 443).

2-54. Broglie, L. de; « Sur la possibilitè de relier les phènomènes d'interference et de diffraction á lathéorie des quante de lumière », Le Journal de Physique et le Radium 8 (1927) 225-241.

2-55. Broglie, L. de; « La structure atomique de la matière et du rayonnement et la Mècaniqueondulatoire », Comptes Rendus 183 (1926) 447-448.

2-56. Broglie, L. de; « La Mècanique ondulatoire et la structure atomique de la matière et durayonnement », Comptes Rendus 184 (1927) 273-274.

2-57. « (...)podrían quizás conducir también a comprender el papel que juegan las solucionescontinuas (Eigenfunktionen) en la nueva Mecánica. Estas soluciones no representarían realmentelos fenómenos atómicos, sino que el cuadrado de su amplitud daría, en la aproximaciónnewtoniana, las probabilidades de los estados y de transición como piensa M. Born. Para ladinámica de los sistemas, las ecuaciones admitidas por Schrödinger, que hacen intervenir la nociónabstracta de espacio de configuración, no serían verdaderas ecuaciones de propagación, sino queúnicamente determinarían probabilidades de presencia.(...) En Micromecánica como en óptica, lassoluciones continuas de las ecuaciones de propagación no deben proporcionar más que unarepresentación estadística, mientras que la descripción microscópica exacta de los fenómenos exigesin duda el empleo de soluciones con singularidades, que traducen la naturaleza atómica de lamateria y de la radiación.(Broglie, L. de, ref. 2-55, cit. en [ICA-91], p. 196).

2-58. Madelung, E.; “Quantentheorie in hydrodynamischer Form”, Zeitschrift für Physik 40 (1926)322-326.

2-59. Korn, A.; “Schrödingers Wellenmechanik und meine mechanischen Theorien”, Zeitschrift fürPhysik 44 (1927) 745-753.

2-60. Bohm, D. and Vigier, P.; “Model of the causal interpretation of quantum theory in terms of afluid with irregular fluctuations”, Physical Review 96 (1954) 208-216.

2-61. Mackinnon, E.; “Bohr on the Foundations of Quantum Theory”, en [FRE-85], pp. 101-120.

2-62. Heisenberg, W.; Physics and beyond, Allen and Unwin, Londres, 1971.

2-63. “Es la teoría la que decide lo que podemos observar”.(A.Einstein, en conversación con Heisenberg, ref. 2-4 y 2-44).

2-64. “Puede que en algún momento haya utilizado esa filosofía y que incluso haya escrito sobreella, pero no deja de ser un absurdo”.(A. Einstein, en conversación con Heisenberg, ref . 2-4, p. 123).

2-65. “I now concentrated all my efforts on the mathematical representation of the electron path inthe cloud chamber, and when I realized fairly soon that the obstacles before me were quiteinsurmountable, I began to wonder whether we might not have been asking the wrong sort ofquestion all along. But where had we gone wrong? The path of the electron through the cloudchamber obviously existed; one could easily observe it. The mathematical framework of quantummechanics existed as well, and was much too convincing to allow for any changes. Hence, it oughtto be possible to establish a connection between the two, hard though it appeared to be.

It must have been one evening after midnight when I suddenly remembered my conversationwith Einstein, and particularly his statement `It is the theory which decides what we can observe'. Iwas immediately convinced that the key to the gate that had been closed for so long must be soughtright here. I decided to go on a nocturnal walk through Faelled Park and to think further about thematter. We had always said so glibly that the path of the electron in the cloud chamber could beobserved. But perhaps what we really observed was something much less. Perhaps we merely saw aseries of discrete and ill-defined spots through which the electron had passed. In fact, all we do seein the cloud chamber are individual water droplets which must certainly be much larger than theelectron. The right question should therefore be: Can quantum mechanics represent the fact that anelectron finds itself approximately in a given place and that it moves approximately with a givenvelocity, and can we make these approximations so close that they do not cause experimentaldifficulties?

A brief calculation after my return to the institute showed that one could indeed representsuch situations mathematically, and that the approximations are governed by what would later becalled the uncertainty principle of quantum mechanics: the product of the uncertainties in themeasured values of the position and momentum cannot be smaller than Planck's constant. Thisformulation, I felt, established the much-needed bridge between the cloud chamber observationsand the mathematics of quantum mechanics. True, it had still to be proved that any experimentwhatsoever was bound to set up situations satisfying the uncertainty principle, but this struck me asplausible a priori, since the processes involved in the experiment or the observation had necessarilyto satisfy the laws of quantum mechanics. On this presupposition, experiments are unlikely toproduce situations that do not accord with quantum mechanics. ‘It is the theory which decides whatwe can observe’. I resolved to prove this by calculations based on simple experiments during thenext few days.

Here, too, I was helped by the memory of a conversation I had once had with BurkhardDrude, a fellow student at Göttingen. When discussing the difficulties involved in the concept ofelectron orbits, he had said that it ought to be possible, in principle, to construct a microscope ofextraordinarily high resolving power in which one could see or photograph the electron paths insidethe atom. Such a microscope would not, of course, work with ordinary light rays, but perhaps withgamma rays. Now this ran counter to my hypothesis, according to which not even the bestmicroscope could cross the limits set by the uncertainty principle. Hence I had to demonstrate that

the principle was obeyed even in this case. This I managed to do, and the proof strengthened myconfidence in the consistency of the new interpretation...” (Heisenberg, W.; “Reminiscences from 1926 and 1927”, en [FRE-85], pp. 169-170).

2-66. “When one wants to be clear about what is to be understood by the word ‘position of theobject’, for example of the electron (relative to a given frame of reference), then one must specifydefinite experiments with whose help one plans to measure the ‘position of the electron’; otherwisethis word has no meaning”.(Heisenberg en ref. 2-39).

2-67. « First we define the terms velocity, energy, etc. (for example, for an electron) which remainvalid in quantum mechanics. It is shown that canonically conjugate quantities can be determinedsimultaneously only with a characteristic indeterminacy. This indeterminacy is the real basis for theoccurrence of statistical relations in quantum mechanics”.(Heisenberg, en el abstract de la ref. 2-39).

2-68. “We have a consistent mathematical scheme and this consistent mathematical scheme tell useverything which can be observed. Nothing is in nature which cannot be described by thismathematical scheme”.(Heisenberg, W., “Quantum theory and its interpretation”, en Niels Bohr-His Life and Work as seenby his Friends and Colleagues, ed. Por Rozental, North-Holland, Amsterdam, 1967, p. 98, cit. en[JAM-74]).

2-69. “Mathematical clarity had in itself no virtue.(…)A complete physical explanation shouldabsolutely precede the mathematical formulation”.(N. Bohr, en ref. 2-68).

2-70. “All concepts which can be used in classical theory for the description of a mechanical systemcan also be defined exactly for atomic processes in analogy to the classical concepts. Theexperiments which provide such a definition themselves suffer an indeterminacy introduced purelyby the observational procedures we use when we ask of them the simultaneous determination oftwo canonically conjugate quantities”.(Heisenberg, en ref. 2-39).

2-71. “After the conclusion of the foregoing paper, more recent investigations of Bohr have led to apoint of view which permits an essential deepening and sharpening of the analysis of quantum-mechanical correlation attempted in this work. In this connection Bohr has brought to my attentionthat I have overlooked essential points in the course of several discussions in this paper Above all,the uncertainty in our observation does not arise exclusively from the occurrence of discontinuities,but is tied directly to the demand that we ascribe equal validity to the quite different experimentswhich show up in the corpuscular theory on one hand, and in the wave theory on the other hand.(…)I owe great thanks to Professor Bohr for sharing with me at an early stage the results of thesemore recent investigations of his-to appear soon in a paper on the conceptual structure of quantumtheory- and for discussing them with me”.(W. Heisenberg, addition in proof en ref. 2-39).

2-72. Landau,L., Peierls,R.; “Erweiterung des Unbestimmtheitsprinzips für die relativisticheQuantentheorie », Zeitschrift für Physik 69 (1931) 59-60.

2-73. Mandelstam, L.I., Tamm, I.; “The uncertainty relation between energy and time in non-relativistic quantum mechanics”, Journal of Physics (USSR) 9 (1945) 249-254.

2-74. Bohr, N.; “Das Quantenpostulat und die neuere Entwicklung der Atomistik”, DieNaturwissensenschaften 16 (1928) 245-257; “The Quantum Postulate and the Recent Developmentof the Atomic Theory”, Nature, 121 (1928) 580-590; rep. en [WHE-83], pp. 87-126.

2-75. Krylov, N.S., Fock,V.A.; “On the uncertainty relation between time and energy”, Journal ofPhysics (USRR) 11 (1947) 112-120.

2-76. Aharonov,Y., Bohm,D.; “Time in the quantum theory and the uncertainty relation for timeand energy”, Physical Review 122 (1961) 1649-1658.

2-77. Popper, Karl R.; The logic of Scientific Discovery, Hutchinson, Londres, 1980, cap. IX.

2-78. Condon, E.U.; “Remarks on uncertainty principle”, Science 69 (1929) 573-574.

2-79. “En la teoría cuántica, no se puede dar respuesta a ninguna cuestión referente a valoresnuméricos simultáneos para las q y los p. Se espera, sin embargo, poder responder a cuestiones enlas que sólo las q o los p tengan valores numéricos dados o, en general, cuando cualquier conjuntode constantes de integración que conmutan tomen valores numéricos dados”.(Dirac en ref.2-24).

2-80. “Para un valor dado de q, son igualmente posibles todos los valores de p”.(Jordan en ref. 2-25).

2-81. “Se puede mirar el mundo con el ojo p o con el ojo q, pero si se quiere tener los dos ojosabiertos al mismo tiempo, se comete error”.(Pauli en ref. 2-7).

2-82. Liechtenstern, Ch.R. von; “Die Beseitigung von Widersprüchen bei der AbleitungUnschärferelation”, Proccedings of the second International Union for the Phylosophy of Science,Zurich, 1954, Editions du Griffon, Neuchatel, 1955, pp. 67-70 (cit. en [JAM-74], p. 73).

2-83. « But what is wrong in the sharp formulation of the law of causality, ‘When we know thepresent precisely, we can predict the future’, is not the conclusion but the assumption. Even inprinciple we cannot know the present in all detail. For that reason everything observed is a selectionfrom a plenitude of possibilities and a limitation on what is possible in the future. As the statisticalcharacter of quantum theory is so closely linked to the inexactness of all perceptions, one might beled to the presumption that behind the perceived statistical world there still hides a `real' world inwhich causality holds. But such speculations seem to us, to say it explicitly, fruitless and senseless.Physics ought to describe only the correlation of observations. One can express the true state ofaffairs better in this way: Because all experiments are subject to the laws of quantum mechanics,and therefore to equation p q h, it follows that quantum mechanics establishes the final failureof causality”.(Heisenberg, W., ref. 2-39).

2-84. Heisenberg, W.; “Die Rolle der Unbestimmtheitsrelationen in der modernen Physik”,Monatshefte für Mathematik und Physik 38 (1931) 365-372.

2-85. “If at a certain time all data are known for a given system then there exist, at any later time,experiments the results of which can be exactly predicted, provided the system is subjected to noother disturbances than those necessary for the perfomance of the experiment. Whether a regularityof this kind may be still regarded as causality or not is purely a question of taste ».(Heisenberg en ref. 2-84.)

2-86. Held, Carsten; “The meaning of Complementarity”, Studies in History and Phylosophy ofScience 25,6 (1994) 871-893.

2-87. Folse, H.J.; The Philosophy of Niels Bohr: The Framework of Complementarity, North-Holland, Amsterdam, 1985.

2-88. Faye, J.; “Non-Locality or Non-separability? A Defense of Bohr's Anti-Realist Approach toQuantum Mechanics”, en Niels Bohr and Contemporary Philosophy, ed. por Faye, J. y Folse, H.,Kluwer, Dordrecht, 1994.

2-89. Honner, J.; The description of Nature: Niels Bohr and the Philosophy of Quantum Physics,Clarendon, Oxford, 1987.

2-90. Murdoch, D.; Niels Bohr's Phylosophy of Physics, Cambridge Univ., Cambridge, 1987.

2-91. Petersen, A.; “The Philosophy of Niels Bohr”, en [FRE-85], pp. 299-310.

2-92. “Notwithstanding the difficulties which, hence, are involved in the formulation of thequantum theory, it seems, as we shall see, that its essence may be expressed in the so-calledquantum postulate, which attributes to any atomic process an essential discontinuity, or ratherindividuality, completely foreign to the classical theories and symbolized by Planck's quantum ofaction.(…)This postulate implies a renunciation as regards the causal space-time co-ordination ofatomic processes. Indeed, our usual description of physical phenomena is based entirely on the ideathat the phenomena concerned may be observed without disturbing them appreciably.(…) Now, thequantum postulate implies that any observation of atomic phenomena will involve an interactionwith the agency of observation not to be neglected”.(N. Bohr, cit. en ref. 2-91).

2-93. “(The quantum postulate) forces us to adopt a new mode of description designated ascomplementary in the sense that any given application of classical concepts precludes thesimultaneous use of other classical concepts which in a different connection are equally necessaryfor the elucidation of phenomena”.(N. Bohr, cit. en ref. 2-91 y en [JAM-74], p. 95).

2-94. Einstein, A., Tolman, C. and Podolsky; “Knowledge of past and future in quantummechanics”, Physical Review 37 (1931) 780-781.

2-95. The Bohr postulate: “A quantum system has no dynamical properties of its own whatsoever(it is therefore meaningless to think of it as having some unknown ones). When associated with agiven experimental setup it can be said to have the dynamical property this setup is appropiate formeasuring. The, properly so-called, measurement event (The actual interaction with the instrument)then reveals the value this dynamical property has on the system ». (B. d'Espagnat, en [ESP-95], p. 223).

2-96. Principle of Complementarity: “The nonseparable whole constituted by the quantum systemand a definite instrument can be described by using a simplification of our language, according towhich some of the properties that the system and the instrument share with one another areconventionally attributed to the system. However, other properties, which in our classicalexperience leads us to think of, cannot then be attributed to the system. They are said to becomplementary to the first ones. They can also be attributed to a quantum system similar in type tothe one considered so far, but this is possible only if that system builds up an indivisible whole withsome new instrument, which is appropiate for a measurement of the new quantities.(B. d'Espagnat, en [ESP-76], p.252).

2-97. « Complementarity denotes the logical relation, of quite a new type, between concepts whichare mutually exclusive, and which therefore cannot be considered at the same time-that would leadto logical mistakes-but which nevertheless must both be used in order to give a completedescription of the situation”.(L. Rosenfeld, “Foundations of quantum theory and complementarity”, Nature 190 (1961) 384-388,cit. en [JAM-74], p. 105).

2-98. García Alcaine, G.; “Complementaridad, coherencia, dualidad”, Revista Española de Física6,3 (1993) 8-9.

2-99. Holton, G.; Ensayos sobre el pensamiento científico en la época de Einstein, Alianza,Madrid, 1982.

2-100. “There is no quantum world. There is only an abstract physical description. It is wrong tothink that the task of physics is to find out how nature is. Physics concerns what we can say aboutnature”.(Bohr, cit. en ref. 2-91).

2-101. “(The) fundamental limitations, met with in atomic physics, of the objective existence ofphenomena independent of their means of observation”.(N. Bohr , Atomic Physics and Human Knowledge, Science Editions, New York, 1961).

2-102. “These conditions (which define the possible types of predictions regarding the futurebehaviour of the system) constitute an inherent element of the description of any phenomenon towhich the term physical reality can be properly attached”.(N. Bohr, en ref. 2-21).

2-103. “We have to remember that what we observe is not nature in itself but nature exposed to ourmethod of questioning”.(W. Heisenberg, Physics and Philosophy, Penguin, London, 1989).

2-104. “It is possible to ask whether there is still concealed behind the statistical universe ofperception a `true' universe in which the law of causality would be valid. But such speculationsseem to us to be without value and meaningless, for physics must confine itself to the description ofthe relationship between perceptions”.(W. Heisenberg en ref. 2-39).

2-105. “(A subsequent measurement deprives to a certain degree the information obtained througha previous measurement of its predictive significance... these facts...) not only set a limit to theextent of the information obtained by measurement, but they also set a limit to the meaning which

we may attribute to such information. We meet here in a new light the old truth that in ourdescription of nature the purpose is not to disclose the real essence of the phenomena but only totrack down, so far as it is possible, relations between the manifold aspects of our experience”.(N. Bohr en Atomic Theory and the description of nature, Cambridge Univ., Cambridge, 1934).

2-106. “That we have been forced step by step to forego a causal description of the behavior ofindividual atoms in space and time, an to reckon with a free choice on the part of nature betweenvarious possibilities to which only probability considerations can be applied”.(N.Bohr, ibid.)

2-107. “(In a measurement of position, for example, as performed with the gamma-ray microscope),the electron is forced to a decision. We compel it to assume a definite position; previously it was, ingeneral, neither here nor there; it had not yet made its decision for a definite position. (…)Weourselves produce the results of measurement [Wir selber rufen die Tatbestände hervor]”(Jordan, P.; Quantenphysikalische Bemerkungen zur Biologie und Psychologie, Erkenntnis 4(1934) 215-252).

2-108. « Our task is to communicate experience and ideas to others. We must strive continually toextend the scope of our description , but in such a way that our messages do not thereby lose theirobjective or unambiguous character”.(N. Bohr, cit. por Petersen en [FRE-85], ref. 91).

2-109. “We are suspended in language in such a way that we cannot say what is up and what isdown. The word `reality' is also a word, a word which we must learnt to use properly”.(N. Bohr, ibid.).

3. El debate Einstein-Bohr y el teorema Einstein-Podolsky-Rosen

3-1. “That question of causality worries me also a lot. Will the quantum absorption and emission oflight ever be grasped in the sense of complete causality, or will there remain a statistical residue? Ihave to confess, that I lack the courage of a conviction. However I should be very, very loath toabandon complete causality...”.(A. Einstein, en carta a Born, enero 1920, cit. en [JAM-74], p.122).

3-2. Bohr, N., Kramers, H.A., Slater, J.C.; “The Quantum Theory of Radiation», PhilosophicalMagazine 47 (1924) 785-822.

3-3. « Bohr's opinion of radiation interests me very much. But I don't want to let myself be driven toa renunciation of stricty causality before there has been a much stronger resistance against it than upto now. I cannot bear the thought that an electron exposed to a ray should by its own free decisionchoose the moment and the direction in which it wants to jump away. If so, I'd rather be a cobbleror even an employee in a gambling-house than a physicist. It is true, my attempts to give the quantapalpable shape have failed again and again, but I'm not going up hope for a long time yet”.(Einstein, en carta a Born, abril 1924, cit. en [JAM-74], p.124).

3-4. “(…)a final abandonment of strict causality is very hard to me to tolerate”.(Einstein, en carta a Ehrenfest, 1924, cit. en [JAM-74], p.124)

3-5. Bohr, N., “Discussion with Einstein on epistemological problems in Atomic Physics”, enAlbert Einstein: Philosopher-Scientist, ed. Schilpp, Library of Living Philosophers, Evanston,1949.

3-6. “Nowhere in the literature can a better access to (Bohr's) thinking be found, and it is a must forall students of Quantum Mechanics, now or later”.(A. Pais “Reminiscence from the post-war years”, p. 225, en Niels Bohr- His life and work as seenby his friends and colleagues, ed. Rozental, Wiley, Nueva York, 1967, en referencia al artículo deBohr de la ref. 3-5, cit. en [JAM-74]).

3-7. Electrons et Photons-Rapports et discussions du Cincième Conseil de Physique tenu àBruxelles du 24 au 29 octobre 1927 sous les Auspices de l'Institut International de PhysiqueSolvay, Gauthier-Villars, Paris, 1928.

3-8. Bonk, Thomas; “Why has de Broglie's Theory been Rejected?”, Studies in History andPhilosophy of Modern Physics 25,3 (1994) 375-396.

3-9. Ferrero, M. et al., eds., Fundamentos de Física Cuántica, Curso de verano de El Escorial,Complutense, Madrid, 1996.

3-10. Ortoli, S. y Pharabod, J.P.; El cántico de la cuántica. ¿Existe el mundo?, Gedisa, Barcelona,1997, pp. 48-49.

3-11. “Me parece que esta dificultad no puede resolverse a no ser que la descripción del proceso entérminos de la onda de Schrödinger se suplemente con alguna especificación detallada de lalocalizaciónde la partícula durante su propagación. Creo que de Broglie investiga correctamente en estadirección. Si uno trabaja con ondas de Schrödinger, esta interpretación (II) de Ψ, creo, viola elpostulado de relatividad”.(A. Einstein, en el 5º Congreso Solvay, cit. en ref. 3-9).

3-12. “(El experimento de la doble rendija)encierra en sí el corazón de la Mecánica Cuántica, enrealidad, contiene el único misterio”.(R. Feynman, en The Feynman Lectures on Physics, Addison-Wesley, 1965).

3-13. Le magnètisme-Rapports et Discussions du Sixième Conseil de Physique tenu à Bruxelles du20 au 25 octobre 1930 sous les Auspices de l'Institut International de Physique Solvay, Gauthier-Villars, Paris, 1932

3-14. “During the whole evening he was extremely unhappy, going from one to the other and tryingto persuade them that it couldn't be true, that it would be the end of physics if Eisntein were right;but he couldn't produce any refutation”.(L. Rosenfeld, sobre la reacción de Bohr tras la intervención de Eintein, en Proceedings of the 14thSolvay conference, Interscience, New York, 1968, cit. en [BAG-92], p. 94).

3-15. Paty, M.; Einstein en la tempestad, en [DEL-90], pp. 51-62.

3-16. “Einstein, I am ashamed of you, you are arguing against the new quantum theory just as youropponents argue about relativity theory”.(Ehrenfest, cit. por Heisenberg en ref. I44).

3-17. “El Einstein de hoy ha cambiado”.(M.Born en Schilpp, ed., Albert Einstein: Philosopher-Scientist, Library of Living Philosophers,Evanston, 1949).

3-18. Criterio de realidad (condición suficiente)}: “If, without in any way disturbing a system, wecan predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, thenthereexists an element of physics reality corresponding to this physical quantity”.(Einstein, A., Podolsky, B. and Rosen, N.; “Can Quantum-Mechanical description of physicalreality be considered complete?”, Physical Review 47 (1935) 777-780).

3-19. Criterio de completitud (condición necesaria)}: “Every element of the physical reality musthave a counterpart in the physical theory”.(ibid.)

3-20. “One could object to this conclusion on the grounds that our criterion of reality is notsufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two ormore physical quantities can be regarded as simultaneous elements of reality only when they can besimultaneously measured or predicted. On this point of view, since either one or the other, but noboth simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real.This makes the reality of P and Q depend upon the process of measurement carried out on the firstsystem, which does not disturb the second system in any way. No reasonable definition of realitycould be expected to permit this”.( ibid.).

3-21. “From our point of view we now see that the wording of the above- mentioned criterion ofreality proposed by Eisnstein, Podolsky and Rosen contains an ambiguity as regards the meaning ofthe expression `without in any way disturbing a system'.(...)...there is essentially the question of aninfluence on the very conditions which define the possible types of predictions regarding the futurebehavior of the system. Since these conditions constitute an inherent element of description of anyphenomenon to which the term `physical reality' can be properly attached, we see that theargumentation of the mentioned authors does not justify their conclusion that quantum-mechanicaldescription is essentially incomplete.”(N. Bohr, “Can Quantum-Mechanical description of physical reality be considered complete?”,Physical Review 48 (1935) 696-702).

3-22. It is rather discomforting that the theory should allow a system to be steered or piloted intoone or the another type of state at the experimenter's mercy in spite of his having no access to it.//(Schrödinger, E.; “Discussion of probability relations between separated systems”, Proceedings ofthe Cambridge Philosophical Society 31 (1935) 555-562).

3-23. Principio de separabilidad: “Let U and V be two systems that have once interacted but areno longer interacting. Then ‘the real factual situation of the system V is independent of what is donewith the system U, which is spatially separated from the former'”.(A. Einstein, cit. en [ESP-76], p.112).

3-24. Principio de divisibilidad por pensamiento: Any extended physical system-be it particlelikeor fieldlike or partly both-can be thought of as composed of elements or parts localized in differentregions of space, an exhaustive knowledge of which is concevaible; and if the Hamilton function

for systems of the same general type as this particular one is known, exact complete knowledge ofthe values of the physical quantities attached to each one of these parts constitutes all by itself anexhaustive knowledge of the whole composite system”.(D'Espagnat, [ESP-95], p.112).

3-25. Einstein, A.; “Quatenmechanik und Wirklichkeit”, Dialectica 2 (1949) 320-324.

3-26. Chiao, R.Y., Kwiat, P.G. and Steinberg, A.M.; “¿Más veloz que la luz?”, Investigación yCiencia, octubre 1993, pp. 15-23.

3-27. Bell, J.S.; “Bertlmann's socks and the nature of reality”, Journal de Physique, Colloque C2,42,3 (1981) C241-C246, trad. en [BEL-90].

3-28. Espagnat, B. d'; “La no-separabilidad o la realidad incomprensible”, en [DEL-90], pp. 127-131.

3-29. Bergia, S.; “Desarrollo conceptual de la Física Cuántica”, en Navarro, L., ed., El siglo de laFísica, pp. 87-126, Tusquets, Barcelona, 1992.

3-30. “La Mecánica Cuántica conduce a resultados precisos en lo que concierne a los valoresmedios, pero no da ninguna información sobre los detalles de cada proceso individual. Eldeterminismo, hasta hoy considerado como la base de las ciencias exactas, debe ser abandonado”.(Born y Heisenberg en el quinto Congreso Solvay, cit. en [Gal-89]).

3-31. “Mantenemos que la Mecánica Cuántica es una teoría completa, cuyas hipótesisfundamentales, físicas y matemáticas, no son susceptibles de modificación”.(Born y Heisenberg, ibid.).

3-32. “Me siento por consiguiente inclinado a creer que la descripción de la Mecánica Cuántica (...)ha de considerarse como una descripción incompleta e indirecta de la realidad, a ser sustituida porotramás directa y completa”.(A. Einstein, Dialectica 320 (1948), en The Born-Einstein Letters, cit. en ref. 3-27).

4. Variables ocultas, realismo local y teoremas de Bell. Experimentos.

4-1. Bell, J.S.; “Einstein-Podolsky-Rosen experiments”, Proceedings of the Symposium on FrontierProblems in High Energy Physics, Pisa, junio 76, pp.33-45, trad. en [BEL-90].

4-2. Bell, J.S.; “On the Einstein-Podolsky-Rosen paradox”, Physics 1 (1964) 195-200, trad. en[BEL-90].

4-3. Belousek, D.W.; “Einstein's 1927 Unpublished Hidden-Variable Theory: Its background,Context and Significance”, Studies in History and Philosophy of Modern Physics, 27B,4 (1996 )437-461.

4-4. Bell, J.S.; “On the impossible pilot wave”, Foundations of Physics 12 (1982) 989-999, trad. en[BEL-90].

4-5. Bell, J.S.; “On the problem of hidden variables in quantum mechanics”, Reviews of ModernPhysics 38 (1966) 447-452, trad. en [BEL-90].

4-6. Jauch, J.M. and Piron, C.; Helv. Phys. Acta 36 (1963) 827.

4-7. Gleason, A.M.; J. Math. Mech. 6 (1957) 885.

4-8. Santos, E.; “The Search for Hidden Variables in Quantum Mechanics”, en [SEL-88], pp. 365-390.

4-9. “One should no more rack one's brains about the problem of whether something one cannotknow anything about exists all the same than about the ancient question of how many angels areable to sit on a point of a needle. But it seems to me that Einstein's questions are ultimately alwaysof this kind.”(Pauli, comentario a Born, cit. por N.D. Mermin en “A Bolt from the Blue: The E-P-R Paradox”, en[FRE-85], p. 144).

4-10. Castiel, A.; “La virtud de la desigualdad”, en [DEL-90], pp. 109-115.

4-11. Clauser, J.F., Horne, M.A., Shimony, A. and Holt, R.A.; “Proposed experiment to test localhidden-variable theories”, Physical Review Letters 23 (1969) 23.

4-12. Aspect, A., Grangier, P. and Roger, G.; “Experimental Tests of Realistic Local Theories viaBell's Theorem”, Physical Review Letters 47 (1981) 460-463; “Realization of E-P-R-BohmGedankenexperiment: A New Violation of Bell's Inequalities”, Physical Review Letters 49 (1982)91-94.

4-13. Aspect, A., Dalibard, J. and Roger, G.; “Experimental Test of Bell's Inequalities Using Time-Varying Analyzers”, Physical Review Letters 49 (1982) 1804-1807.

4-14. Davies, P.C.W. and Brown, J.R., ed.; The ghost in the atom, Cambridge Univ., Cambridge,1986.

4-15. “It is a very important experiment, and perhaps it marks the point where one should stop andthink for a time, but I certainly hope it is not the end. I think that the probing of what quantummechanics means continue, and in fact it will continue, whether we agree or not that it is worthwhile, because many people are sufficiently fascinated and perturbed by this that it will go on”.(J.S. Bell en Davies, P.C.W. and Brown, J.R., ed.; The ghost in the atom, Cambridge Univ.,Cambridge, 1986).

4-16. Kennedy, P.J.; “Delayed-Choice Experiments”, en [FRE-85], pp. 148-152.

4-17. Wheeler, J.A.; “The `Past' and the `delayed-choice' Double-Slit experiment”, en R.Marlow,ed., Mathematical Foundations of Quantum Theory, Academic, New York, 1978.

4-18. “The extent to which renunciation of the visualization of atomic phenomena is imposed uponus by the impossibility of their subdivision is strikingly illustrated by the following example, towhich Einstein very early called attention and often has reverted. If a semi-reflecting mirror isplaced in the way of a photon, leaving two possibilities for its direction of propagation, the photonmay either be recorded on one, and only one, of two photographic plates situated at great distances

in the two directions in question, or else we may, by replacing the plates by mirrors, observe effectsexhibiting an interference between the two reflected wave-trains. In any attempt of a pictorialrepresentation of the behavior of the photon we would, thus, meet with the difficulty: to be obligedto say, on the one hand, that the photon always chooses one of the two ways and, on the other hand,that it behaves as if it had passed both ways. (…) It is just arguments of this kind which recall theimpossibility of subdividing quantum phenomena and reveal the ambiguity in ascribing customaryphysical attributes to atomic objects.(...) It may also be added that it obviously can make nodifference as regards observable effects obtainable by a definite experimental arrangement, whetherour plans of constructing or handling the instruments are fixed beforehand or whether we prefer topostpone the completion of our planning until a later moment when the particle is already on itsway from one instrument to another”.(N. Bohr en ref. 2-5, cit. en ref. 4-17).

4-19. Wheeler, J.A.; “Genesis and Observership”, en R.E. Butts and J. Hintikka, eds., FoundationalProblems in the Special Sciences, Dordrecht, Netherlands, D.Reidel, 1977; cit. en ref. 4-16)

4-20. “(...) in a loose way of speaking, we decide what the photon shall have done after it hasalready done it. In actuality it is wrong to talk of the ‘route’ of the photon. For a proper way ofspeaking we recall once more that it makes no sense to talk of the phenomenon until it has beenbrought to a close by an irreversible act of amplification: ‘No elementary phenomenon is aphenomenon until it is a registered (observed) phenomenon.” (J.A. Wheeler en The American Philosophical Society and The Royal Society: papers read at ameeting, June 5, American Philosophical Society, Philadelphia, cit. en [BAG-92], p. 156.

4-21. “Registration equipment operating in the here and now has an undeniable part in bringingabout that which appears to have happened. Useful as it is under every-day circumstances to saythat the world exists ‘out there’ independent of us, that view can no longer be upheld. There is astrange sense in which this is a ‘participatory universe’”.(J.A. Wheeler, en “Law without law”; J.A. Wheeler and W.H. Zurek eds., Quantum Theory andMeasurement, Princenton Univ. , Princenton, 1983).

4-22. “La novedad que los experimentos de este tipo nos aportan es la certidumbre por finconseguida de que esta idea (la de que pueden considerarse como válidos los conceptos juntos derealismo y separabilidad no obstante la excelencia de las reglas de cálculo cuánticas) no es válida.Más exactamente (porque no hay que dejar de tener en cuenta los diversos riesgos de losexperimentos), lo que nos aportan es la idea de que la afirmación ‘Realismo y separabilidad sonverdaderos’ es irreconciliable no solamente con tal o cual a priori filosófico o epistemológico, sino,más significativamente, con los hechos”.(Espagnat, B. d'; “La no-separabilidad o la realidad incomprensible”, en [DEL-90], pp. 127-131).

4-23. Ferrero, M. y Santos, E.; “Contraste de las desigualdades de Bell con fotones de cascadasatómicas”; Ferrero, M. et al., ed., Fundamentos de Física Cuántica, Curso de verano de El Escorial,Complutense, Madrid, 1996.

4-24. Sanz, A.L. y Sánchez-Gómez, J.L.; Europhysics Letters 3 (1987) 519.

4-25. Santos, E.; Physical Review Letter 66 (1991) 1388; Physical Review A46 (1992) 3646.

4-26. Huelga, S.F., Ferrero M. y Santos, E.; Physical Review A51 (1995 )5008.

4-27. Kwiat, P.G., Eberhard, P.H., Steinberg, A.M. y Chiao, R.Y.; Physical Review A49 (1994)3209.

4-28. Fry, E.S.; en Proceedings of the International Conference on Lasers' 93, Wang, C.P. ed., STSPress, McLean, 1993.

4-29. Marshall, T.W. and Santos, E.; Foundations of Physics 18 (1988) 185; Marshall, T.W.,Foundations of Physics 21 (1991) 209; 22 (1992) 363.

4-30. Vuillemin, J.; “Física Cuántica y Filosofía”, en [DEL-90], pp.177-196.

4-31. Buonomano, V.; Lett. Nuovo Cimento B57 (1986)146; The Proceedings of the InternationalConference on Microphysical Reality and Quantum Formalism, Urbino, Italy, 1985, Tarozzi, G.and Merwe, A. van der, eds., Reidel, Dordrecht, 1988; “Bell's Inequality and the Non-ergodicInterpretation of Quantum Mechanics », en [SEL-88], pp. 327-344

4-32. Mückenheim, W.; Lett. Nuovo Cimento 35 (1982) 300; “An Extended- Probability Responseto the Einstein-Podolsky-Rosen Argument”, en [SEL-88], pp. 345-364.

4-33. Ivanovic, I.D.; Lett. Nuovo Cimento 22 (1978) 14.

4-34. Vigier, J.P.; Lett. Nuovo Cimento 24 (1979) 258,265; Kyprianidis, A. y Vigier, J.P.,“Quantum Action-at-a-Distance: The Mystery of Einstein-Podolsky-Rosen Correlations”, en [SEL-88], pp. 273-299.

4-35. Stapp, H.P.; Tarozzi, G. and Merwe, A. van der, eds., The Proceedings of the InternationalConference on Microphysical Reality and Quantum Formalism, Urbino, Italy, 1985; Reidel,Dordrecht, 1988; “Are Faster-Than-Light Influences Necessary?”, en [SEL-88], pp. 63-85.

4-36. Costa de Beauregard, O.; Nuovo Cimento B42 (1977) 41.

4-37. Carroll, Alley, Jakukowicz, Steggerda, Wickes; "A delayed Random Choice QuantumMechanical Experiment with Light Quanta", Kamefuchi Proceedings, pp. 158-164.

5. El problema de la medida

5-1. “La teoría de la medida es un razonamiento concerniente a S+M, y debería describir cómo estárelacionado el estado de S con ciertas propiedades del estado de M (en particular con las posicionesde un índice, puesto que esto es lo que el observador lee). Además, es totalmente arbitrario incluir ono al observador en M.” (von Neumann en ref. 2-32)

5-2. London, F. et Bauer, E.; La Théorie de l'Observation en Mécanique Quantique, Hermann,París, 1939, trad. al inglés en [WHE-83], pp. 217-259.

5-3. “No es, pues, una interacción misteriosa entre el aparato y el objeto la que produce, durante lamedida, un nuevo estado del sistema. Es la conciencia de un `yo' que puede separarse de la función$\psi$ antigua y constituir, en virtud de su observación, una nueva objetividad, atribuyendo alobjeto, a partir de ese momento, una nueva función Ψ”.

(London y Bauer, en ref. ant.).

5-4. Margenau, H.; “Quantum Mechanical description”, Physical Review 49 (1936) 240-242.

5-5. Pauli; “Die allgemeinen Prinzipien der Wellenmechanik”, Geiger y Scheel, eds., Handbuchder Physik, 2ª ed., vol. 24, Springer, Berlin, 1933.

5-6. Jordan, P.; “On the process of measurement in Quantum Mechanics”, Philosophy of Science 16(1949) 269-278.

5-7. Schrödinger, E.; “Die gegenwärtige Situation in der Quantenmechanik”, DieNaturwissenschaften 23 (1935) 807-812, 824-828, 844-849,trad. al inglés en [WHE-83], pp. 152-167.

5-8. “It must be admitted that most physicists are not bothered by the Schrödinger's cat case. Theytake the standpoint that the cat's being or not being electrocuted should itself be regarded as ameasurement. Thus in their view, the reduction of the wave packet takes place (...) when the cateither feels or not feels the jolt of electric current hitting its body. More precisely, the reduction orthe wave packet takes place precisely when if it ad not taken place a superposition of differentstates of some macro-observable would have been predicted. What this shows is that workingphysicists accept the principle that macro-observables always retain sharp values (by macroscopicstandard of sharpness) and deduce when measurement must take place from this principle. But theintellectual relevance of the Schrödinger cat's case is not thereby impaired. What the case shows isthat the principle that macro-observables retain sharp values at all times is not deduced from thefoundations of quantum mechanics, but is dragged in as a additional assumption”.(H. Putnam, “A philosopher looks at quantum mechanics”, en Colodny, ed., Beyond the Edge ofCertainty, Prentice Hall, Englewwod Cliffs, N.J., 1965, pp.75-101, cit. en [JAM-74], pp. 217-218).

5-9. Bell, J.S.; “Against measurement”, en [MIL-90].