Abstract
This initial chapter is devoted to a brief, critical review of the major conceptual difficulties that permeate the whole of quantum mechanics, especially when it is interpreted within the framework of the (mainstream or orthodox) Copenhagen school. The text is written for a reader who is interested in these issues and ready to accept that no interpretationInterpretation of quantum mechanics is free of conceptual difficulties, which require some repair. A short overview of the contents of the book is included at the end of the chapter, guided by the leitmotif of the theory presented, namely, that quantization can be understood as an emergent phenomenon arising from a deeper stochastic process. Specifically, the permanent interaction between matter and the zero-point radiation field is shown, chapter by chapter, to give rise to quantum features of both, field and matter. An appendix to the chapter provides a concise introduction to the Probability interpretation!ensembleensemble Ensembleinterpretation ofProbability interpretation probability, a much extended Probabilities!extendednotion among physicists, but hardly discussed in the literature.
...[quantum-mechanical] vagueness, subjectivity, and Indeterminism indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice.
Bell (1987, page 160)
... that today there is no Interpretation interpretation of quantum mechanics that does not have serious flaws, and that we ought to take seriously the possibility of finding some more satisfactory other theory, to which quantum mechanics is merely a good approximation Weinberg, S. .
Weinberg (2013, page 95)
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Notes
- 1.
In statements about superluminal influences, it is difficult to know which kind of influences are being considered. Anyhow, detailed analysis shows that special relativity and quantum mechanics have still a peaceful coexistence (see e.g. Shimony, A. Shimony 1978; Redhead M. L. G. Redhead 1983, 1987).
- 2.
- 3.
For example, a given system of linear differential equations can represent a mechanical, an acoustical, an electrical or an electromagnetic system, or even an analog computer as well. There is ample conceptual space to accommodate the interpretation.
- 4.
Virtually all science philosophers have received with approval the philosophical conclusions arrived at from (orthodox) quantum mechanics, despite its nonrealistic (even antirealistic) and subjective Probability interpretation!subjectivetrends. Far from helping to drive quantum physics towards a more realistic conception, this of course has contributed to reinforce such trends.
- 5.
Determinism Determinismmust be clearly distinguished fromLocality!and causality causalityCausality, the latter referring to an ontological property of the system. The notion of Indeterminismindeterminism wavers in the literature from ontological to epistemic connotations, and from Probability interpretation!objectiveobjective to subjective meanings. In this book we understand by (physical) determinism a property of the description of a physical system, not of the system itself, and thus of epistemological nature. Although many different meanings are ascribed also to causalityCausality, this term refers to a direct genetic connection among the elements of the description, i.e. to an ontological property of the underlying physical realityReality. We could say that causality Causalityrefers to the hardware of nature, determinism to our software about it.
- 6.
Whether the Indeterminismindeterminism is ontic or merely manifests itself at the observational or descriptive level is a controversial issue, to which every decoder adds his own preferred interpretation (see BungeBunge, M. 1956 for examples). Still, the attempts to construct a fundamental and deeper deterministic theory from which qm could emerge through an appropriate mechanism to generate Indeterminismindeterminism, speak to the existing conviction in some circles that quantum indeterminism demands explanation. For example, t’Hooft T’Hooft G.has envisioned a process of local informationInformation loss leading to equivalence classes that correspond to the quantum states Elitzur, A.(T’Hooft G.’t Hooft 2002, 2005, 2006).
- 7.
The textbook (and historical) explanation of the Heisenberg inequalitiesHeisenbergHeisenberg, W. inequalities as a result of the perturbation of, say, the electron by the observation cannot be taken as the last word, at least because the inequalities follow (as a theorem) from the formalism without introducing observersObserver and measuring apparatus.
Within the statisticalCausality!statistical interpretation of qm (see Sect. 1.2.2 ) they indeed refer to the product of the (objective) variances of two noncommuting dynamic variables in a given stateBallentine, L.E. (see e.g. Ballentine 1998, Sect. 8.4).
- 8.
The interpretative difficulties are even greater with the energy-time inequality, because this inequality (in its usual form) does not belong to the customary formal apparatus of the theory. There are of course various proposals to replace it (see e.g. BungeBunge, M. 1970; Jammer, M. Jammer 1974, Sect. 5.4). Also the introduction of a time operator has been explored by several authors (see e.g.Muga, J. G. Sala-Mayato R. Egusquiza, I. L. Muga et al. 2008, in particular the contribution by P. Busch; see alsoAtomic stability!against surroundings Hilgevoord, J. Hilgevoord and Atkinson 2011).
- 9.
The acceptance of negative Probabilities!negativeprobabilitiesNegative probabilities implies a fundamental change in the axioms of probability Brody!and probability theorytheory. Since “they are well-defined concepts mathematically, which like a negative sum of money ...should be considered simply as things which do not appear in experimental results” ( Dirac 1942; see also Feynman 1982, 1987; D’Espagnat, B.d’Espagnat 1995, 1999; and the detailed discussion in Mückenheim W. Bartlett, M. S. Mückenheim et al. 1986 Ludwig, G., where they are called extended probabilities), Probabilities!extendedthey tend to be pragmatically accepted, even if this renders the meaning of probability obscure. Once this door is open, anything may step in; thus, for instance, imaginary probabilities have been considered to reconcile quantum theory with localityLocality (Ivanović 1978).
InKhrennikov, A. Khrennikov (2009) the probabilistic machinery of quantum mechanics is extended Probabilities!extendedwithin a realist Description!realistpoint of view, to the description of any kind of contextual contingencies, which leads to a theory that finds application in several fields of inquiry, including economics and psychology.
- 10.
- 11.
One should add that a theory of measurement (i.e., of our methods to interrogate nature) cannot be part of a fundamental (thus general) description of nature, because the former must be quite specific and detailed in every instance to have any predictive capacity.
- 12.
The notion of reduction or collapse Collapseof the wave functionCollapse!wave function was introduced as a quantum postulate byVon Neumann, J. von Neumann (1932) and Pauli, W.Pauli (1933)Geiger, H.. There is no clear definition of the qualities of the perturbation of the physical system that demarcate the two ways of evolution (the causal one and the collapse). Thus, “[T]he observed system is required to be isolated in order to be defined, yet interacting to be observed”Stapp H. P. (Stapp 1971). Within the single-system interpretationInterpretation the collapse Collapseis avoided by means of the ‘many-worlds interpretation’ (or ‘relative-state formulation’) of qm Everett, H.(Everett 1957, from Everett, H.Everett’s thesis 1956), according to which the world splits into as many independent worlds as different results of the measurement can occur. We will not discuss here this (extreme, even if logical) interpretation.
- 13.
It is of course possible in principle to include the measurement Measurementapparatus in the Hamiltonian; a well known example of this is Bohm, D.Bohm’s theory (see Chap. 8). This helps to express the measurement problem in more realistic terms. Another well-known exampleVan Kampen, N. G. is van Kampen (1988).
- 14.
An argument against the observerObserver, aimed at recovering objectivity in the quantum ‘potentialities’, has been advanced from cosmology. According to inflationary theory, the early classical inhomogenities in the cosmic microwave background originated in earlier quantum fluctuations. This quantum-to-classical transition took place much before even galaxies existed. It follows that the measurement problem in cosmologyValentini, A. is of a different kind (Perez et al. 2006 Sahlmann, H.; Valentini 2008).
- 15.
- 16.
However, the possibility to construct quantum trajectories Trajectories!quantum(by considering additional elements into the usual quantum description) has received special attention since the times of de BroglieDe Broglie, L.. The best known example of quantum trajectory is perhaps the one afforded by Bohm’s theory (discussed in Chap. 8).
- 17.
An early introductory account of the different interpretations of qm and their variants can be found in BungeBunge, M. (1956). More advanced expositions, also by professional philosophers of science, are found, among others, in BungeBunge, M. (1973) and Redhead (1987). A more recentAuletta, G. monograph by a physicist is Auletta (2000).
- 18.
Since this interpretation (as indeed all interpretations) contains in an essential way Born’sBorn, M. (1926) probabilistic notion of the wave function, and in addition it was strongly influenced by Heisenberg,Heisenberg, W. it would be more properly called Copenhagen-Göttingen interpretation. Wigner (1963) proposedWigner, E.P. Quantum distributions!Wigner to apply the term ‘orthodox’ more specifically to the view adopted by von NeumannVon Neumann, J., as reshaped byLondon, F. Bauer, E. London and Bauer (1939).
- 19.
Margenau, H.More recent advocates are Margenau (1958, 1978), Sokolov, A. A. Sokolov et al. (1962),Mott, N. F. Mott (1964), Marshall, T. W. Marshall (1965), Lamb, Jr. W. E. Lamb (1969, 1978), Belinfante, F. J. Belinfante (1975), NewtonNewton, R. G. (1980), Santos E. Santos (1991), de MuynckDe Muynck W. M. (2002), Laughlin (2005), Khrennikov (2009), NieuwenhuizenNieuwenhuizen, Th. M. (2005) (in Adenier et al. 2006), Adenier, G.etc. For an important defense of the ensemble Probability interpretation!ensembleinterpretation of qm see the old paper byBallentine, L. E. Ballentine (1970), or his more recent books (1989, 1998); BallentineBallentine, L. E. takes, however, an indeterministic viewHome, D.. Whitaker M. A. B. Home and Whitaker (1992) contains a detailed discussion, from a Description!realistrealist point of view, of the different versions of the Interpretation!ensembleensemble interpretation of qm. Further, an interesting analysis is that of Rylov Yu. A. Rylov (1995) who demonstrates on general arguments that qm (including Dirac’s theory) necessarily refers to an ensemble of Extended particleparticles.
- 20.
It is not too difficult to find openly antirealistic views nourished by the Interpretation!conventionalconventionalRigden J.S. interpretation of qm. See e.g. Rigden (1986), Adler (1989). Adler, C. G.There are also some researchers that go as far as to consider that the universe itself is not real; see e.g. Henry (2005).
- 21.
We are using here the term realism with the meaning of gnoseologic realism Realism!gnoseologic(BungeBunge, M. 1985), i.e. ontologically as the belief in an external world, independent of our theories and observations, and epistemologically as the conviction that it is possible to know that world, part by part. However, in some places we use a restricted notion of physical realism Realismwhich originates in the famous EPR theoremEPR Correlations!EPR1935 paper, namely that if a value can be determined for a variable without disturbing the individual system, there exists an element of reality associated with it, even prior to the measurementMeasurement. According to this notion, the individual systems are at all times in objectively real states (Deltete Deltete, R.and GuyGuy, R. 1990), even if unknown, and should in principle be amenable to a space-time descriptionSpace-time description.
- 22.
An introductory discussion of scientific realismScientific realism by a realist can be seen in BoydBoyd, R. N. (1983). The author shows, in particular, how the educated (expressly in science) common sense is a good guide towards scientific realismScientific realism.
- 23.
A word of caution is needed here. The measured value may or may not preexist, it suffices to consider that some feature or property related to the measured value preexists. The clearest example is perhaps the measurement Moment!magneticof a spinSpin Magnetic moment!of the spinwith a Stern-Gerlach apparatus, which obviously may reorient the spin. Thus, a realist theory is compatible with both possibilities; it all depends on the nature of the measured variable. SeeAllahverdyan, A. E. Allahverdyan et al. (2013).
- 24.
In a letter to Physics Today by HenryHenry, R. C. (2004, p. 14) discussing why physics understanding is so poor in the United States, the author ends by saying: “We know from quantum mechanics that nothing is real, except for the observations themselves.” Another typical example reads: “one cannot consider quantum properties as being ‘real,’ in the sense of ‘objective Realism!objectivereality” (PaulPaul, H. 2008).
- 25.
As is the case with other quantum paradoxes, the collapse of theCollapse!wave function wave function becomes understandable within the Interpretation!ensembleensemble interpretation. The fact that an individual observation is made does not change the (original) ensemble, it only changes our knowledge by giving us an extra piece of Informationinformation. We add this information to construct a new ensemble that corresponds to the updated situation, a quite normal statisticalCausality!statistical procedure. The ‘collapsed’ state vector describes the new situation.
- 26.
- 27.
We find trajectories in Feynman’s method of path integrals, but they are virtual and attain arbitrary velocities, and besides all possible trajectories Trajectoriesare considered with equal amplitude, not only those (unknown) related to the actual motion followed by a given electron travelling from point \(A\) to point \(B\).
- 28.
This was precisely one of the persistent arguments put forward byLocality!EinsteinEinstein against the Copenhagen interpretation.
- 29.
By contrast, Shimony, A. Shimony (1989) contends that the formalism of qm may have to be modified so that the theory meets certain metaphysical constraints. He even suggests the need to modify qm to save physical realism. By way of example he points out a possible modification of the topology of space-time at a subquantum scale. He alerts the reader, remarking that “[t]his proposal is the antithesis of [his] attempt to draw philosophical consequences from scientific results, for it indicates rather a reliance on philosophical considerations to supply the heuristics for a scientific investigation.” (page 34).
As can be surmised, the conceptual problems associated with the violation of the Bell, J. S.Bell inequalitiesBell!inequalities have led some authors to even question qm as a fundamental theory of nature [see e.g. Howard (1989)].
- 30.
More precisely, that local realism Local realism Local realistic theories Sed!linear theory Linear sed!and local realismand quantum theory are incompatible. This can be argued, as summarized by Ferrero, M. Ferrero (1987), as follows: It is possible to demonstrate that the following four statements are incompatible:
a) Realism; b) Locality; c-EPR) Quantum mechanics is a complete theory; c-Bell) Quantum mechanics accepts hidden variablesHidden variables (it is not a complete theory); d) Quantum mechanics is a valid theory of Nature.
a, b, d and c-EPR are the assumptions in the EPR paper;
a, b, d and c-Bell are the assumptions in the early derivation of Bell’s theoremBell!theorem.
Thus, independently of theCompleteness completeness of qm (i.e., of c-EPR or c-Bell), a, b and d are incompatible. In Bell 1971 the demand c-Bell was eliminated.
- 31.
There exists a widespread belief that if two quantities cannot be measured simultaneously, they do not exist simultaneously. This (positivist) identification of existing (being) and being observed (measured) is of course merely a point of view; it is not part of the postulates of qm.
- 32.
A simple example may be illustrative of the ambiguity of the quantum description. Consider the state vector of two spin Spin1/2 particles in the singlet state (referred to a certain direction \(z\))
$$\begin{aligned} \left| 00\right\rangle _{z}=\tfrac{1}{\sqrt{2}}\left( \left| \uparrow \right\rangle \left| \downarrow \right\rangle -\left| \downarrow \right\rangle \left| \uparrow \right\rangle \right) . \end{aligned}$$A rotation of the system of reference to an arbitrary direction \( \hat{n}\) transforms this description into
$$\begin{aligned} \left| 00\right\rangle _{\hat{n}}=\tfrac{1}{\sqrt{2}}\left( \left| \hat{n}_{+}\right\rangle \left| \hat{n}_{-}\right\rangle -\left| \hat{n}_{-}\right\rangle \left| \hat{n}_{+}\right\rangle \right) . \end{aligned}$$Now the spins are referred to the arbitrary direction \(\hat{n}.\) Thus, the spins may be aligned in any direction whatsoever. In other words, the state vector gives absolutely no indication of the actual direction of the spins. From the ensemble point of view, the individual spin pairs are distributed uniformly in all directions.
- 33.
- 34.
A strong contention against the pragmatic and nonrealist views associated with the observer and his (hers in his language) measurements, reigns in the whole little (big) book of Bell on the foundations of quantum mechanics (Bell 1987). He even says that there are words that should not belong to the lingo of theoretical physics and should be banned from it, such as Measurement‘measurement’, ‘observation’, ‘observer’.
- 35.
Reviews or reprints of important work expressing differing views, as well as ample lists of references to papers dealing with this subject, can be found in de Witt and Graham (1974);Graham, R. N. Belinfante (1973); Jammer, M.Jammer (1974); NilsonNilson, D. R. (1976); Wheeler and Zurek (1983); Cushing and McMullin (1989); Ballentine (1989, 1998); OmnèsOmnés, R. (1994, 1999); Home, D. Home (1997); Auletta, G. Auletta (2000); Bertlmann, R. A. Bertlmann and Zeilinger (2002), etc. TheZeilinger A. list is endless.
- 36.
The latter is the name by which the theorem of these authors is commonly known, although a similar result was presented somewhat earlier in Bell (1966). For this reason some authors refer to it under the fairer acronym BKSBKS. There are not so many instances in which an almost simultaneous discovery by several authors is duly recognized—more often, science seems to have become a one-hundred meter steeplechase race.
- 37.
Interestingly, at present the zero-point fields are seen as possible sources of the conjectured dark energy. Even if for the moment this is not much more than a speculation (which carries its own problems), it brings to the fore the possible importance of zero-point fields (see e.g.Saunders S. Brown, H. R. Saunders and Brown 1991).
- 38.
A comprehensive account of the results obtained in sed up to 1995 is contained in the book The Quantum Dice, by L. de la PeñaDe la Peña, L. and A.M. CettoCetto, A. M. (1996), hereafter referred to as The Dice.
- 39.
We attribute this question to BoyerBoyer, T.H. by inferring it from his papers. In a private communication he has expressed himself in similar terms. See however BoyerBoyer, T.H. (2011).
- 40.
Among the many different perspectives on the subject within physics, the following cover a wide range of possibilities: Bunge Bunge, M.(1970)Rédei, E.; Lucas (1970)Lucas, J. R.; Gillies, D. A. Gillies (1973); Szegedi P. Rédei and Szegedi (1989)Bitsakis, E. I.; Home, D. Whitaker M. A. B. Home and Whitaker (1992). See also Interpretations of Probability in the online Stanford Encyclopedia of Philosophy.
- 41.
The most extended Probabilities!extendedsubjective views of probability are the individual degree of acceptability of a proposition (de Fenetti 1974), or its Bayesian version (Jeffreys (1939)Jeffreys, H.; Jaynes, E. T. Jaynes (1995); CatichaCaticha, A. (2008) as a measure of the informed personal opinion. According to the Bayesian views, any evaluation of a probability is conditional to some evidence that partially entails it; thus, Keynes, J. M. Keynes (1921) asserts that “the probability of the same statement varies with the evidence presented”. By contrast, the probability of decay of an atomic nucleus depends on the internal physical situation of the constituent nucleons, and is entirely independent of any personal information. This illustrates the different use of the concept of probability in physics and in other fields of knowledge. It should be considered that even if an assigned numerical probability is taken as depending on our degree of rational belief (or our degree of partial entailment), it contains some logical elements, since it is limited by rational constraints that ensure the possibility of using a mathematical apparatus Gillies, D. A.(see Gillies (1973), Introduction).
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Appendix A: The Ensemble Meaning of Probability
Appendix A: The Ensemble Meaning of Probability
Considering that probability is a somewhat obscure subject, about which all sorts of debates have taken place, the following observations—due in essence to BrodyBrody, T. A. (1975, 1993)Hodgson, P. E.—may be appreciated by some of our readers. The point is that several notions of probability coexist and are used in the physical literature, with their respective caveats. It would not be an overstatement to say that the personal grasp of the notion of probability plays an important role in the espousal of one or the other interpretation of qm. It therefore seems appropriate to give some precision to the meaning given to it in the present work.Footnote 40
Apart from the formal or axiomatic (Kolmogorovian) probabilities and theProbability!Kolmogorov subjective Probability interpretation!subjectiveinterpretation of probability,Footnote 41 there are two interpretations of probability popular among the practitioners of physics. One of them isProbability interpretation!frequentist the frequentist or objective ( empirical) interpretation. According to this interpretation, proposed byVenn J. Venn (1880), and developed by Reichenbach H. Reichenbach (1949) andVon Mises R. von Mises (1957), among others, a series of observations is made and the relative frequency of an event is thus determined; its probability is taken as the value attained in the limit when the number of cases in the series tends to infinity. Here we are dealing with events (not with propositions as in the formal rendering, or with opinions or beliefs as is the case with the subjective Probability interpretation!subjectiveinterpretation), and the determination of the relative frequency is an empirical, objective (although approximate) process. There are however some problems that hamper a strict formulation of this probability: if experimental frequencies are used, the infinite limit is unattainable; if the relative frequency is a theoretical estimate, then the limit is probabilistic and the frequentist Probability interpretation!frequentistdefinition becomes circular. Again, the existence of the limit value should be assumed. Moreover, the theoretical structure Structurelacks an experimental counterpart: why should the experimental relative frequencies Relative frequenciescorrespond to the theoretical estimates? Notwithstanding such difficulties, this interpretation constitutes a widely used practical tool. As Bunge Bunge, M.(1970) puts it: “All we have is a frequency evaluation of probability”.
Let us turn our attention to another important view on probability, much extended among physicists, namely theProbability interpretation!ensemble ensemble interpretation. We follow here the discussion on the subject by Brody Brody, T. A.(1975, 1975), particularly Chap.10), and start by recalling the usual concept of ensemble . Each theoretical model of reality should be in principle applicable to all cases of the same kind, i.e., to all cases where the properties of the system considered by the model are equal; the factors neglected by the model may vary or fluctuate freely, but in consistency with the applicable physical laws. The set of all these cases constitutes the ensembleEnsemble of interest. The notion of ensemble as a set of theoretical constructs can thus be established without recourse to the concept of probability, and can be structuredStructure so as to possess a measure, which is then used to define averages over the ensemble. The ensembleEnsemble concept of probability can then be introduced as follows. Let \(A\) be a property of interest and let \(\chi _{A}\) be the Indicator functionindicator function of \(A\), i.e., \(\chi _{A}(\omega )=1\) if the member \(\omega \) of the ensemble has the property \(A\), \(\chi _{A}(\omega )=0\) otherwise. Then the probability of \(A\) is the expectation over the ensemble of \(\chi _{A}(\omega ),\)
where \(\mu (\omega )\) is the measure function for the ensemble, usually normalized over \(\Omega \), the range of the events \(\omega \). It is possible to show that this definition satisfies all the axioms of Kolmogorov, A. N. Kolmogorov (1956), so that indeed the ensembleEnsemble can become the basic tool for probabilistic theorization.
The experimental counterpart of this probability is the relative frequency as measured in an actual (and of course finite) series of experiments. If the relative frequencies Relative frequenciesthus measured do not correspond to the theoretical estimates, the ensemble (the measure) should be redefined until agreement is reached through the appropriate research work. Here there is no global recipe. Of course, as is the case with any other physical quantity, theoretical probabilities and their experimental values need not necessarily be exactly the same.
The ensembleEnsemble definition of probability does not allow the application of the notion of probability to a singular case (there is no ensemble). Thus, for example, the philosophical problem of the probability of a given theory being true, becomes meaningless. To give meaning to the assertion about the probability of a single event, it must be translated into a statement about its relative frequency.
The most interesting aspect of the ensembleEnsemble notionProbability interpretation!ensemble of probability is its direct correspondence with the concept used by physicists in their daily undertakings, so that we adhere to it in the present work, even though it is not entirely free of conceptual and philosophical problems—asKhrennikov, A.Yu. any other interpretation of probability.
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de la Peña, L., Cetto, A.M., Valdés Hernández, A. (2015). Quantum Mechanics: Some Questions. In: The Emerging Quantum. Springer, Cham. https://doi.org/10.1007/978-3-319-07893-9_1
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