Abstract
Quantum mechanics is a marvelous theory; its towering successes and amazing predictive power are beyond the slightest doubt. Yet we are still amid the great quantum muddle, as Popper (1967) put it. Physicists and philosophers become puzzled by the deep conceptual and philosophical problems that emerge as soon as they dig for a profound understanding of what quantum theory says about the world and how it says it.
True, the finite interaction between the object and the measuring devices... implies... the necessity to renounce the classical idea of causality, and a radical revision of our attitude toward the problem of physical reality. (Bohr 1928)
J cannot seriously believe in [quantum mechanics] because it cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance. (Einstein 1947)
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The example is given in a letter by Einstein to Schrödinger of June 1933, as recounted in Moore (1989), chapter 8. A detailed account of this classic problem can be found in Deltete and Guy (1990), so the present discussion will be rather schematic.
It is normally accepted that every physical theory contains at least two components: 1) an abstract formalism and 2) a set of semantic rules, that may be called collectively the interpretation (or the semantics; they are also called operative definitions, epistemic correlations or rules of correspondence). The formalism is the logical skeleton of the theory; though it contains nonlogical, descriptive terms (such as mass, electric field, and so on), it is merely an abstract mathematical structure devoid of empirical meaning. It acquires physical meaning by means of the interpretation that correlates the nonlogical terms in the theoretical model with the empirical quantities or operations they are supposed to represent. The usual textbook views on quantum mechanics are based on some variant of the Copenhagen or orthodox interpretation. Since this interpretation (and indeed, all interpretations of quantum mechanics) contains in an essential way Born’s (1926) probabilistic interpretation of the wave function, and in addition it was strongly influenced by Heisenberg, it would be more properly called Copenhagen-Göttingen interpretation, though Wigner (1963) proposed to apply the term ‘orthodox’ more specifically to the view adopted by von Neumann, as reshaped by London and Bauer (1939). One could also call the former the customary or regular interpretation, although it is not so clear that the present-day practicing physicist adheres to it in his daily endeavours as tightly as such names may fancy. In a broad sense we call it normally (but not necessarily) the conventional interpretation. One should bear in mind, however, that such terms do not refer to a sharp set of precepts, since an ample range of tenets with respect to some of the central interpretative issues can be distinguished among its practitioners. An introductory account of the different interpretations of quantum mechanics and their variants can be found in Bunge (1956), and more advanced expositions by professional philosophers of science are found, among others, in Bunge (1973) and Redhead (1987).
The term measurement is common in this context within the conventional interpretation; however, it is an ill-defined concept. It may refer to a perturbation of the system, or to the creation of a result by the measurement, or to a measurement of what was already there. Sometimes it refers to a real physical action on the system, sometimes to a mere change in our knowledge of it; further, the measurement may or may not require an observer, or even his or her conciousness, and so on: quot homines tot sententiae.
Realism is a philosophical term to which there correspond many nonequivalent notions; an idea of the rich variety of its meanings may be obtained from Harré (1986). In its broad ontological meaning, (objective) realism postulates that independently of our theories and prior to them, there is an objective reality; in other words, it posits the existence of an independent reality which precedes any effort to disclose it. The task of scientific endeavour is just to disclose the nature of this reality and the laws of behaviour of its things. On the epistemological plane, realism opposes subjectivism; however, there is a rich variety of epistemologic versions of realism. The empiricist viewpoint (adopted by the great majority of writers of conventional texts on quantum mechanics) postulates that our knowledge of the external world originates exclusively in our sensorial perceptions, and that it is not possible to go beyond them (and make inferences on unobservable entities); so, the whole question of an objective reality evaporates as mere speculation. Logical positivism (also known by other names, such as logical empiricism) is represented by a specific empiricist school (the Vienna Circle) that had a deep and extended influence in the development of the interpretation of quantum mechanics. It holds that only the propositions analysable with the tools of logic into elementary propositions that are either tautological or empirically verifiable, are meaningful. (We comment that rational knowledge should not be reduced to a mere logical process, since the creative acts of the mind more often than not fall outside the domain of logic; thus the advance of knowledge rests frequently on (rational) ideas and innovations, the genesis of which cannot be the object of logical analysis. The fact that several logics coexist —traditional, fussy, multivalued, various brands of quantum logic, and so forth—, all of them rational, shows clearly that what is rational may not be logical.) In the text we use a restricted notion of physical realism which originates in the famous EPR paper [Einstein, Podolsky and Rosen 1935], namely, that the values determined (for the elements of reality) without disturbing the individual system exist prior to the determination. It is a realism of possessed values, according to which the individual systems (to be taken as the fundamental concern of physics) are at all times in well-defined objectively real states [Deltetc and Guy 1990]. Thus, for instance, individual systems have objectively real trajectories, even if unknown, and their space-time description should be possible in principle. The meaning given to the terms determinism and causality is discussed in §1.1.1.
Needless to say, traces of quantum particles are observed in cloud chambers and photographic films, from which the trajectories can be determined; but even these are then interpreted as produced by a measurement. To overcome this objection, experiments specifically designed to verify the existence of trajectories have been suggested [see, e.g., Cufaro-Petroni and Vigier 1992].
The ensemble interpretation was first adumbrated by Slater (1929) in the very beginnings of quantum theory; a considerable contribution to its development was given by Einstein from 1935 on [Einstein 1949, 1953a, 1953b]. Among other physicists who embraced it are Langevin (1934), Blokhintsev (1953, 1965), Margenau (1958, 1978), Mott (1964), Lamb (1969, 1978), etc. Expositions of it at different levels may be found in Ballentine (1970), Belinfante (1975) —where it is called ‘objective interpretation’—, Ross-Bonney (1975), Newton (1980); a more recent, detailed and comprehensive discussion of the ensemble interpretation is Home and Whitaker (1992). The great majority of textbooks on quantum mechanics are written following the conventional interpretation, mostly implicitly. Among the rare exceptions are the above mentioned book by Blokhintsev (1965), the Russian edition of Quantum Mechanics by Sokolov et al. (1962) (in the English translation the relevant parts were omitted), de la Peña (1979), and Ballentine (1989). Despite its conceptual advantages (resolution of the known paradoxes, disappearance of the collapse and of the measurement problem, etc.), but probably due to its limitations (to be discussed below), the ensemble interpretation is frequently neglected by philosophers of science [see, e.g., Hanson 1959, Putnam 1965, Fine 1973]. Others instead take it seriously, as, e.g., Powers (1982) and Harré (1986). A detailed analysis of the conventional interpretation is given in Stapp (1972). A discussion on why Einstein considered quantum mechanics an essentially statistical theory, as opposed to statistical physics, is given in Einstein and Infeld (1938), p.280.
In the EPR article it is shown that the requirement that quantum mechanics give a local realistic description of nature implies that it is incomplete. Essentially the argument goes as follows. The definition of completeness used is that each pertinent element of reality should have its counterpart in the theory. Since, according to the criteria used to identify the elements of reality, in the EPR gedankenexperiment both the position and the momentum of the particle are real, they should have well-defined simultaneous values in the description, which is obviously not the case in quantum mechanics. In an immediate answer to this paper, Bohr (1935) concludes that the EPR results are inadmissible because they are based on a conceptual scheme that violates the quantum rules. In contrast, Schrödinger (1935a, b) takes the paradox further and even transfers it to the macroscopic scale with his cat paradox (§1.4.1), to show the nonrealistic nature of the conventional description. The subject has been discussed with unending interest since the initial publication of the EPR paper, and the related literature is huge; digested accounts are given in Jammer (1974) and Sellcri (1988).
Reviews or reprints of important work, as well as ample lists of references to papers dealing with this subject matter, can be found in De Witt and Graham (1971), Belinfante (1973), Jammer (1974), Nilson (1976), Wheeler and Zurek (1983) and Ballentine (1988).
For our discussion we follow the treatment given in Brody (1993), where the present points of view are developed in more detail. Other points of view, arbitrarily selected from among the vast and contradictory literature on these subjects, may be seen in Bunge (1959, 1973), Bhaskar (1975), Powers. (1982), Lucas (1984), Harré (1986), Sosa and Tooley (1993).
The determinism of modern digital computers refers to a unidirectional procedure, since in the memory operations previous values are replaced by new ones in an irrecoverable form. This is why they can be used to generate series of random numbers, using only causal processes. With a Laplacian machine this would be inconceivable. Also, in the physical world, even for relatively simple (but nonlinear) systems, governed by strictly causal laws, the possibility of prediction is lost more frequently than not. Although it has become customary to talk in such cases of deterministic chaos, it would perhaps be more correct to call it ‘causal chaos’.
Some supporters of the conventional interpretation of quantum mechanics have considered an undermined notion of causality, referred to as statistical causality. Rosenfeld (1953) even asserts that the statistical causality that he sees in quantum theory is already present in thermodynamics. Some realistic physicists have expressed analogous ideas, e.g., Bitsakis (1983) under the name of statistical quantum determinism or Selleri (1987, p. 125), who calls it predictability with some degree of inductive probability. These concepts represent attempts to reintroduce causality into the conventional quantum view, indeed, but at the price of interpreting limitations of a description as ontological properties.
Though not absolutely devoid of sense, the term hidden variables is not the most fortunate one [Bell (1987) calls it ‘a piece of historical silliness’]. The subject has been amply reviewed in the literature, some well-known monographies being Belinfante (1973), Jammer (1974) and Nilson (1976). Related theorems are those of Gleason (1957) and Kochen and Specker (1967). In section 13.5 some questions concerning the related Bell theorem are studied.
As is well known, the classical limit is defined in quantum mechanics as the result obtained when ħ → 0 or when the quantum numbers n → ∞. However, frequently both limiting processes are simultaneously required under constraints such as that J = n ħ remains fixed and equal to its classical value. For example, the formula for the energy levels of the H atom, E n = -Z 2 me 4 /2 ħ 2 n 2, goes to the classical solution E = -Z 2 me 4 /2J 2 in the last instance, but to ∞ or 0 in the first two cases, respectively. This is conventional quantum folklore; reality has proven to be more complex, and which is the proper classical limit remains as yet unclear. It is possible, for instance, to show that the chaotic features of classical nonlinear systems cannot be recovered from the corresponding quantum systems by a limiting procedure. An illustrative example of such behaviour has been given by Mantica and Ford (1992).
Other discussions can be found, e.g., in Bunge (1970), Lucas (1970), Rédei and Szegedi (1989), Home and Whitaker (1992).
For particularly clear detailed discussions of these important topics see Mermin (1993) and Peres (1993).
A more complete discussion can be found in Cohen and Zaparovanny (1980).
The first phase-space description of a quantum system was made by Weyl (1927); the Wigner function was introduced some years later by Wigner (1932). The theory of the latter was substantially developed by Moyal (1949); ample reviews of it can be seen in Tatarskii (1983) and Hillery et al. (1984). General formulas for the quantum phase-space distribution (which apply to the Wigner function as a special case) are given in Cohen (1976) and Cohen and Zaparovanny (1980). Some aspects of the theory of quantum distributions (mainly from the point of view of quantum optics) are discussed in chapter 13. The problem of the sign of the Wigner distribution is sometimes avoided by judiciously (but arbitrarily) smearing it over phase-space cells of volume of order ħ; a Gaussian coarse-graining leads to the so-called Husimi distribution [Husimi 1940; Cartwright 1976]. However, the time evolution of the resulting wave packets may be at variance with some quantum laws [see, e.g., Nauenberg and Keith 1992].
Julg (1988) has remarked that the dissociation energy of rigid diatomic molecules depends on the dispersion of the ground-state energy, which might allow for an empirical decision between the different ordering rules.
The acceptance of negative probabilities implies a fundamental change in the axioms of probability theory. 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 (1941); see also Feynman (1982, 1987) and the detailed discussion in Mückenheim (1986), where they are called extended probabilities], they tend to be pragmatically accepted, even if this renders the meaning of probability unintelligible. Once this door is open, anything may happen; thus, for instance, imaginary probabilities have been considered to reconcile quantum theory with locality [Ivanovic 1978].
Field theory, from Faraday onwards, was the answer of theoretical physics against actions at a distance; as Maxwell puts it (1873): “Faraday saw a medium where [the mathematicians] saw nothing but distance”. The introduction of the notion of field was perhaps one of the most revolutionary steps of 19th century physics. Analogously, it was the deep dissatisfaction with actions at a distance in the Newtonian theory of gravity that gave Einstein his strong drive to look for a more convincing theory of gravitation, thus eradicating all kinds of actions at a distance from mechanics. One could speculate that the requirement of absence of actions at a distance could have been the cornerstone for Einstein’s construction of special relativity. Their resurrection by the conventional interpretation of quantum mechanics was one of Einstein’s strong motivations against this interpretation (see epigraph to the chapter).
The notion of entangled states was introduced by Schrödinger (1935a) during his search for a realistic description of the quantum world; he found them particularly distressing. Accessible discussions on the present status of the problem of entanglement are given in Greenberger et al. (1993) and Mermin (1994).
Recall that this is just the criterion proposed in EPR to identify the elements of reality. So the spin down of the partner particle is identified as an element of reality. However, neither |Ψ0) nor ρ̂ P allow one to predict this value, but only to assign to it a probability 1/2. Thus, the description provided by the wave function is incomplete, and we come back to the substance of the EPR argument.
The operation of averaging changes the nature of the quantity being averaged; thus, the average citizen of a country has a noninteger number of parents, of brothers and sisters, and even of eyes and legs. Upon averaging, the kinetic energy of the molecules of a gas becomes a temperature, a concept that cannot be applied to an individual molecule and is even foreign to mechanics. The process of averaging may even change the nature of the theory itself, giving way to new quantities and concepts and new laws relating them. In the case of the gas, for instance, it leads to the concepts of equilibrium, pressure (average momentum transfer), temperature (average kinetic energy), and so on, and then to the relations between them (ideal gas law, law of distribution of velocities, etc.). Even if quite obvious, this fact is frequently neglected, thus giving rise to paradoxical conclusions.
We are of course referring to the causal or ‘hidden-variables’ interpretation developed by Böhm from his famous paper of 1952, based on the set of equations just derived. The quantum potential is usually a rapidly varying and sensitive function of the coordinates, which results in a complex (and uncontrollable) motion of the otherwise classical particle. A most valuable outcome of the theory was that it served as an explicit counterexample against the impossibility proof of von Neumann (1932). For an introductory account see Albert (1994), and for recent discussions on the subject see Böhm and Hiley (1993) and Holland (1993). In section 2.2 a set of equations similar to that characteristic of Bohm’s theory will be obtained from a statistical perspective and within a phenomenological description, according to which the quantum potential represents an energy due to the diffusive motions.
A most important example of this kind is the state preparation process, in the sense given to this term by Margenau (1963). For a discussion see, e.g., Home and Whitaker (1992).
The episode is narrated in detail in Moore (1989), chapter 8. To explain his point of view to Schrödinger, Einstein used the example of a mass of gunpowder that would probably explode spontaneously in the course of the next year, so that during this interval the wave function would describe a superposition of exploded and unexploded gunpowder. One fancies here the seed of the cat paradox. Jammer (1974), chapter 6, contains a more technical account of Schrödinger’s work on the EPR argument.
Other characteristic antirealistic views nourished by the conventional interpretation can be seen in Rigden (1986) and Adler (1989).
An ample and commented selection of papers on the traditional measurent problem is found in Wheeler and Zurek (1983); a particularly strong defense and detailed study of the reduction postulate is given in Primas (1983), section 3.5. For more detailed studies of natural decoherence and related topics, the interested reader may consult the popular account in Zurek (1991) or the reviews by Zurek (1982), Walls et al. (1985), Busch et al. (1991), etc. Two recent reviews on the subject from different viewpoints are Chan-Pu (1993) and Pearle (1993).
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de la Peña, L., Cetto, A.M. (1996). Quantum Mechanics and the Real World. In: The Quantum Dice. Fundamental Theories of Physics, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8723-5_1
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