Abstract
In the context of the EPR paradox and Bell’s inequality (see Chap. 20), we came across the question of whether quantum mechanics as a physical theory is complete.
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Notes
- 1.
In any case, this is apparently the prevailing view, e.g.: “Values cannot be ascribed to observables prior to measurement; such values are only the outcomes of measurement.” K. Gottfried and T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd Edition, 2003, p. 42.
- 2.
However, also in approaches based on objective chance or tending towards the many-worlds interpretation, one assumes that states refer to individual systems and not merely to ensembles. The crucial element for the following considerations is requirement (2).
- 3.
As a Hamlet-style question, so to speak: ‘To be’ or ‘to be found’?
- 4.
Contextuality states that the measurement outcome of an observable depends on the set of compatible observables that are measured at the same time. Thus, nonlocality can be considered as a reflection of contextuality in spatially separated systems.
- 5.
In this context, Gleasons’s theorem is of interest (see Appendix T, Vol. 2). It deals in fact with the question of how to introduce probabilities into quantum mechanics, but also refers to a contradiction in the assignment of properties of a quantum system. This contradiction is addressed by the Kochen-Specker theorem.
- 6.
We note that these projectors commute.
- 7.
Strictly speaking, the statement does not apply to the states \(\left| a_{n}\right\rangle \), but to the corresponding rays; the phase factors cancel each other in the expression \(\left| a_{n}\right\rangle \left\langle a_{n}\right| \). For reasons of clarity, we accept this imprecision.
- 8.
It is the set of these 18 basic vectors (or yes-no-tests) in a four-dimensional space with which recently the first experimental implementation of a Kochen-Specker set was performed; see Vincenzo D’Ambrosio et al., ‘Experimental Implementation of a Kochen-Specker Set of quantum Tests’, Phys. Rev. X 3, 011012 (2013).
- 9.
G. Kirchmair et al., ‘State-independent experimental test of quantum contextuality’, Nature 460, 494–497 (2009).
- 10.
In fact, in this paper four spin-1/2 systems are used. The simplification to the three quantum objects considered here was introduced some time later by Mermin.
- 11.
An attempt at a treatment of this topic suitable for schools was given for example by : ‘EPR Pardoxon in school—Absolute and relative, and Bertelsmann’s socks’, by K. Jaeckel and J. Pade, in: H. Fischler (Ed.), Quantum physics in school, IPN 133, (1992) (text in German).
- 12.
Since we have only three observers, we use the shorthand notation \( \left| \psi \right\rangle =\frac{\left| h,h,h\right\rangle +\left| v,v,v\right\rangle }{\sqrt{2}}\equiv \frac{\left| 1:h,2:h,3:h\right\rangle +\left| 1:v,2:v,3:v\right\rangle }{\sqrt{2}}\).
- 13.
As in the case of Bell’s inequality, the problem is that one cannot measure the six quantities \(L_{1}^{\prime },L_{2}^{\prime },L_{3}^{\prime },C_{1},C_{2},C_{3}\) simultaneously; these are the eigenvalues of operators that do not all commute with each other. The measurement of the six variables is counterfactual: in one experiment, one cannot measure more than three of them.
- 14.
Jian-Wei Pan et al., ‘Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement’, Nature, Vol. 403, pp. 515–519 (2000).
- 15.
A.J. Leggett, ‘Nonlocal hidden-variable theories and quantum mechanics: an incompatibility theorem’, Found. Phys. 33 (2003) 1469–1493. was awarded the 2003 Nobel Prize for his work in the field of superfluidity.
- 16.
S. Gröblacher et al., ‘An experimental test of non-local realism’, Nature 446 (2007), pp. 871–875.
- 17.
M. Socolovsky, ‘Quantum mechanics and Leggett’s inequalities’, Int. J. Theor. Phys. 48 (2009), pp. 3303–3311.
- 18.
A.J. Leggett, ‘Realism and the physical world’, Rep. Prog. Phys. 71 (2008), 022001.
- 19.
In principle, one cannot exclude for example that the rules of ordinary logic do not apply and/or need to be expanded in the realm of quantum mechanics—rules which are tacitly applied in the derivation of Bell’s inequality and the other arguments. Here, Gödel’s theorem comes into play, according to which, roughly speaking, any theory that is proposed as the basis for mathematics, including logic, is necessarily inadequate, incomplete or contradictory. It is not clear at this point, however, what should be changed in conventional logic in order that Bell’s inequalities not be violated by quantum mechanics.
- 20.
This is done by comparing experimental results with an extension of Bell’s inequality, the CSCH inequality, proposed in 1969 by Clauser, Horne, Shimony and Holt.
- 21.
In connection with delayed-choice experiments (see Appendix M, Vol. 1) also, the idea of a time-reversed effect is discussed. In fact, the fundamental laws of physics are time-reversal invariant, i.e. time-symmetrically causal, and do not reflect the time-asymmetrical idea of cause and effect.
- 22.
Though the idea that events obey a definite causal order is deeply rooted in our understanding of the world, causal order needs not be a required property of nature. For instance, it was recently shown that in quantum mechanics, there are correlations that cannot be understood in terms of definite causal order; see Ognyan Oreshkov et al., ‘Quantum correlations with no causal order’, Nature Comm. 3, 1092 (2012), doi:10.1038/ncomms2076.
- 23.
“I believe that in our present picture of physical reality, especially regarding the nature of time, a huge upheaval is imminent, it may be even greater than the revolution that has already been triggered by relativity theory and quantum mechanics.” Roger Penrose, British mathematician and physicist, in New Mind. The emperor’s new clothes or the debate over artificial intelligence, consciousness and the laws of nature.
- 24.
In Appendix U, Vol. 2, some quotes from philosophers, artists, etc. are compiled; they show illustratively that the classical notion of ‘reality’ does not exist and has never existed in the past.
- 25.
A certain skepticism about the concept of ‘reality’ is in the tradition of modern science. More than 200 years ago, Georg Christoph Lichtenberg stated in one of his physics lectures: “We care little about whether the bodies have an objective reality apart from us or not. It would always be possible that at least some would not have one. We have to imagine the things; the idea does not depend on us, but of those things that make an impression on us, the impression cannot act on us as in another way than our abilities admit. At least is that what we feel of the bodies apart from us, not always objectively real.” Gottlieb Gamauf, in Physics lectures, from the memoirs of Gottlieb Gamauf.
- 26.
“The development of quantum mechanics early in the twentieth century obliged physicists to change radically the concepts they used to describe the world. The main ingredient of the first quantum revolution, wave-particle duality, has led to inventions such as the transistor and the laser that are at the root of the information society. Thanks to ideas developed by Albert Einstein and John S. Bell, another essential quantum ingredient, entanglement, is now leading us through the conceptual beginnings of a second quantum revolution—this time based on quantum information.” Alain Aspect, ‘Quantum mechanics: To be or not to be local’, Nature 446, pp. 866–867 (19th April, 2007).
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Pade, J. (2014). Is Quantum Mechanics Complete?. In: Quantum Mechanics for Pedestrians 2: Applications and Extensions. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00813-4_27
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