Quantum Mechanics: Some Questions

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)

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1.1 On Being Principled... At Least on Sundays

Tied to our microscopic place in the immensities of the Cosmos, we are beginning to unfold its mysteries with remarkable precision. Being as gigantic as we are compared to the atomic and subatomic worlds, we have been able nevertheless to uncover an important fraction of its workings. We do not know yet what an electron is made of, but we know already many of its secretsWilczek F. (see e.g. Wilczek 2002).

The remarkable scientific, technological, philosophicalDeterminism!philosophical, and even economic success of quantum mechanics is only the beginning. No physicist on Earth would question the numerically fitting description that quantum mechanics offers of the part of the world that pertains to its domains, which extend much beyond the atomic scale the theory originally was intended to cover, both towards the macroscopic and the ultramicroscopic. However, a nonnegligible portion of the practicing physicists would also acknowledge, either openly or reluctantly, that the mysteries of the quantum world have not been satisfactorily cleared or explained, after more than eighty years of successful existence of this most basic theory.

Such acknowledgment depends of course on what is meant by explanation. A historical example of what we have in mind follows from the Newtonian theory of gravitation: the clarity, universalityUniversality, simplicity and high precision of this theory made of it a grandiose paradigm; the theory reigned undisputed for over two centuries and became the ideological pedestal that supported the European Enlightenment. The universal gravitational force became the pivotal element to understand innumerable terrestrial and celestial facts, and a central element in the construction of a whole philosophy of nature. This occurred despite the known shortcomings of the theory in more than one essential aspect. Not only did it rest on the ageing concept of Action at a distanceaction at a distance, but the specific form of the force was selected ad hoc to lead to the Keplerian ellipses, introduced as a mere patch into the Newtonian system of mechanics, with no theoretical support or physical Determinism!physicalmechanism that would lead to it or explain it. From this more exacting point of view, one could say that the classical theory gives a precise and simple description of the facts, sufficiently good for all practical purposes (fapp ); but it hardly constitutes an explanation of what is going on in the real world. To find such an explanation the whole edifice of general relativity had to be put forth, allowing us to dispense with ad hoc elements or actions at a distance, and providing us instead with a causal rule. Indeed, general relativity explains the Newtonian theory.

Today we can calculate atomic transition frequenciesSed!linear theory Linear sed!and transition frequencies to within a billionth part, and use refined applications of the quantum properties of matter and the radiation field to construct marvelous and powerful devices that have become emblematic of our civilization. However, have we really got an understanding of what is happening deep-down in the quantum world? A glance at the quantum literature dedicated to the discussion of its fundamental aspects is sufficient to reveal the vast spread of meanings and uncertainties that beset current quantum knowledge. Of course, if the number predicted by the theory, or the use that is made of it, is taken as its test, just as was the case with Newtonian gravitation and the extended pragmatic viewpoint it prompted, the conclusion is that there is no problem at all. But we may be a bit more demanding and ask, for instance, for the physical Determinism!physical(rather than formal) explanation ofAtomic stabilityatomicSed!linear theory Linear sed!stability of stationary solutions stability, the origin of uncertainty or the Fluctuations!quantumquantum fluctuations. Again, are wave-particle duality and quantum nonlocalities the final word? Do superluminal influences really exist?¹ In short: the quantum formalism describes its portion of Nature astonishingly well and we do not know why. It would be difficult to express this kind of feelings about the status of present-day quantum theory more lucidly than Bell did in 1976: quantum mechanics is a fapp Fapp theorytheory. And Maxwell, N. Maxwell (1992) rightly asks: what is beyond fapp?

Since the creation of quantum mechanics (qm) there has been a flood of papers and essays discussing these and similar or deeper questions, and almost any conceivable (or inconceivable) argument or answer has been advanced, both from within physics and from the philosophy of science, ranging from a complete accord with quantum orthodoxy to a radical departure from it. Such extended and deep rumination has not been the endeavor of idle physicists and philosophers, since names such as Bohr, de BroglieDe Broglie, L., Dirac, Einstein,Locality!Einstein Heisenberg,Heisenberg, W. Landé, A.Landé, PopperPopper, K.,Detailed balance!and Schrödinger equation Schrödinger, do honor to an unending list of active participants.

Let us listen to some few big voices to get a better feeling of the magnitude of the quantum muddle, as Popper, K. Popper (1959) calls it. Feynman, R. P.Feynman writes:Leighton, R. B.

I think I can safely say that nobody understands quantum mechanics,

and goes on speaking of the [unsolved] mysteries of qm (Feynman et al. 1965). Referring to matter diffraction he asserts:

A phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality Realityit contains the only mystery...

How does it really work? What machinery is actually producing this thing? Nobody knows any machinery. Nobody can give you a deeper explanation of this phenomenon than I have given; that is, a description of it.

Gell-Mann, M. Gell-Mann (1981) in his turn qualifies:

In elementary particle theory one assumes the validity of three principles that appear to be exactly correct.

(1) Quantum mechanics, that mysterious, confusing discipline, which none of us really understands but which we know how to use. It works perfectly, as far as we can tell, in describing physical realityReality, but it is a ‘counter-intuitive discipline’, as social scientists would say. Quantum mechanics is not a theory, but rather a framework, within which we believe any correct theory must fit. (2) Relativity. (3) Locality!and causalityCausalityCausality.

In his turnDyson, F. J. Dyson (1958) observes:

...the student says to himself: ‘I understand QM’ or rather he says: ‘I understand now that there isn’t anything to be understood...’ .

And speaking about himself, he adds (Dyson 2007)

...the important thing about quantum mechanics is the equations, the mathematics. If you want to understand quantum mechanics, just do the math. All the words that are spun around it don’t mean very much.

Despite the hundreds of books and of international conferences discussing both physical and philosophical problems ofDeterminism!philosophical qm, the basic conundrums remain alive and as unresolved as they were eight decades ago. Fortunately nobody (to our knowledge) has blamed Bell, J.S.Bell of having been unable to understand qm, as was said about Einstein. He, Bell, solved the matter his own way: at the time of some lectures he explained that during the week he used the handy fapp theory. The weekends however he would regain his principles and search for something better (quoted in Gisin 2002).Gisin, N.

Experience shows that so far, neither physical Determinism!physicalnor philosophical arguments have been effective to get us out of the muddle. For the normal practicing physicist the philosophical arguments, when they have a meaning for science, are little more than an abstraction, an ethereal generalization of the truths already discovered by science. But if along its lines of reasoning, science has been unable to set foot on the profundities of the quantum world, we cannot expect philosophy to unfold them for us. Something of revealing importance can thus be extracted from these persistent discussions: as long as the issues are debated and the differing points of view defended from inside quantum theory, no definite conclusion can be reached. What is required then is to gain a look onto qm from outside it, to get a wider and clearer perspective. The work presented here represents precisely a systematicDerivative!systematic attempt to look onto qm from outside it, with the help of a deeper physical theory. This provides us with the possibility of getting answers from a wider perspective than that obtained by just interpreting (or reinterpreting, or misinterpreting) the formalism.

In fact, many of the difficulties with qm arise as a result of the interpretationInterpretation ascribed to its formalism. Though there have been claims that qm does not need interpretation,² the truth is that in no other place of physics do the theory and its formal content elicit such diverse and even contradictory meanings as in qm (see Sect. 1.2). And indeed, the formal apparatus of a theory is in general not enough to interpret it.³ If “nobody understands quantum theory” it is difficult to hold that the theory speaks for itself. Apart from the immediate problem that represents the lack of consensus on the Interpretationinterpretation of qm, the critical point is that many interpretations of it, particularly the dominant one, jeopardize (when not simply do away with) some principles that have been pillars of the whole edifice of physics. Even if—or precisely because—the principles of scientific philosophy are a distillate of the most fundamental discoveries of science, if qm demonstrates that Nature Sed!linear theory Linear sed!nature of its solutions(not a certain description of it) is incompatible with some of those principles, Localityas might be realismRealism, Determinismdeterminism, locality or objectivism, then Determinism!philosophicalthe philosophical framework must of course be modified accordingly, instead of forcing us to attune physics to worn presuppositions. It could be that the advances of science demand a revision of what is taken at a given moment for a firmly established general outlook; history is full of experiences of this nature. The central concerns and theories of the philosophy of science should be consistent with scientific discovery, and are therefore subject to revision, just as happens with science itself. When the scientific case is clear, science philosophy must adapt to what science tells us. But that requires an absolutely convincing demonstration, since principles as realismRealism, say, are just that, general principles extracted from a huge plurality of cases and circumstances, so their generality, universalityUniversality, solidity and soundness are utterly confirmed. Convincing demonstrations, not a mere interpretationInterpretation of the formal apparatus of qm, are thus required to abandon these solid principles.⁴

In the following section we present and comment on some of the most basic issues that beset qm, which originate when adopting a certain interpretation of the theory. By the same token, in this introductory chapter there is no attempt to resolve these issues or give answers to them. It is along the subsequent chapters, as we develop the theory, that we will be finding answers. This will allow us to summarize, in the final chapter, the insights afforded by the theory and discuss its outlook.

1.1.1 The Sins of Quantum Mechanics

Let us point out in brief some of the sins of qm—some venial, others capital—that are readily found and discussed in the scientific literature, particularly the one written under the spell of the Interpretation!orthodoxorthodox interpretation. It may seem amazing that two discussions on the subject written by physicists (one of whom later became a recognized philosopher of science) published almost half a century apart (BungeBunge, M. 1956; Laloë, F. Laloë 2002), touch essentially upon the same fundamental questions, of course with an emphasis that corresponds to the given moment.

• qm is an indeterministic theory. Indeed, though the quantum dynamic laws evolve deterministically, the theory is unable to predict individual events. The most the theory can offer are probabilistic predictions, whence the specific outcome of an experiment cannot be determined in advance. In itself, indeterminismIndeterminism is not a regrettable property of a physical theory. The statisticalCausality!statistical theories of classical physics are indeterministic (or, for some people, they obey statisticalCausality!statistical determinism) and this is not considered a shortcoming. The reason is that in such cases the origin of such indeterminacy is clear. Recall for instance the statisticalCausality!statistical description of a classical gas; there is a distributionDistribution of velocities of the molecules that calls for a statisticalCausality!statistical description with no practical alternative. The distributionDistribution of velocities of the molecules is a direct consequence of the fact that there is a myriad of microstates compatible with the macroscopicMacromolecule state under scrutiny, all of them having equivalent possibilities corresponding to the initial conditions. In other words, the indeterminacy is a feature of the description, not of the system itself. By contrast, in the usual rendering of qm we have no more explanation for the statisticalCausality!statistical indeterminism than the indeterminism of the theory. For some this means quantum indeterminismIndeterminism is Indeterminism!irreducibleirreducible.⁵ $^{\text {,}}$ ⁶

• qm has intrinsic Probability!intrinsic Probabilities!intrinsiclimitations to its predictive power. As stated above, the predictions of qm are only probabilistic. The specific reading of the meter is beyond what qm can predict, yet Nature gives in each instance a well-defined unique answer; we are therefore faced with two possibilities: (a) the predictions of qm are incomplete, or (b) the predictions are complete and God plays dice.

• qm is a noncausal theory. One of the most conspicuous Noncausalityexamples of noncausalitySed!linear theory Linear sed!and noncausality in qm (which is also a towering manifestation of indeterminism) are the Heisenberg inequalitiesHeisenberg inequalities, which imply the existence of unavoidable (quantum) Fluctuationsfluctuations. The cause for such fluctuations is alien to the theory (assuming that a cause must indeed exist), or is simply inexistent at all (assuming that no property of Nature escapes to the quantum description). There is a long list of schools and subschools, with different views on whether the Heisenberg inequalitiesHeisenberg inequalities refer to uncertainties (a measure of our ignorance), to (objective or ontic) indeterminacies, or to something else.⁷ $^{\text {,}}$ ⁸ In any case, Atkinson, D.the widespread attitude is that no cause for quantum fluctuations is considered to be required, and even less, investigated; they can happily remain ‘spontaneous’.

• qm is not a legitimate probabilistic theory. Though the predictions of qm deal with probabilities, no formulation of qm is fully consistent with a genuine probabilistic interpretation (in the classical sense). The use of probability amplitudes instead of probabilities implies a distinctive probability theory Brody!and probability theoryby itself. For example, negativeProbabilities!negative probabilitiesNegative probabilities appear in qm not only in connection with phase-space distributions, but also as a result of the superposition Superpositionprinciple. The amplitudes can interfere destructively and give rise to negative contributions to the probability densities, of a nonclassical nature. These results have led to a widespread acceptance of negative probabilities as a necessary trait of quantum theory.⁹

• qm is a nonlocal theory. Nonlocality is a major issue for quantum physics. It is inherent to the structure Structureof the theory, although subject to quite different connotations, some of which lead to the notion of Action at a distanceaction at a distance. LocalityLocality is a most fundamental physical demand; it pertains to the conceptual framework upon which theoretical physics is founded, yet it is apparentlyNonlocality!apparent contravened by all quantum systems, not only multipartite ones, in which Correlations!and entanglementthe entanglementEntanglement introduces the well-known nonlocal correlations between the subsystems. Thus, to understand the origin and meaning of quantum nonlocalityEntanglement!and nonlocality is a major task for a deeper understanding of present-day physics, one that has been put aside in favour of the development and expansions of its applications.

• qm is a theory of observables, not of beables. According to the moreBeable extended interpretation of qm, it is meaningless to speak of the value of a certain variable of a physical system until the corresponding measurementMeasurement has been performed. Therefore the theory refers to measured variables (observables) and not to preexisting, objective, individual properties of the system (beables). This is clearly a shortcoming from a realist Description!realistpoint of view.

• qm is a contextual theory. InBell, J.S. quantum theory (Bell’s) Contextualitycontextuality means that the result of measuring an observable $A$ depends both on the state of the system and the whole experimental contextContext. In particular, it depends on the result obtained in a previous (or simultaneous) measurement Measurementof another, commuting observable $B$. Thus the value attributed to $A$ depends on the whole contextContext.¹⁰

• qm requires a measurement theoryMeasurement theory. The pure statesPure state of the microworld are not realized in our everyday world. We need some means to reduce the former to mixturesMixture when passing to the macroscopic level. Traditionally the assumed agent is the observation (measurement); thus the observer Observerand his proxy break actively into the description in order to produce results.¹¹ It would not be an overstatement to say that the notion of measurement Measurementin qm raises more conceptual problems than those it is intended to solve.

• qm postulates a nonunitary evolution foreign to its formalism. In its usual Interpretationinterpretation, qm demands theCollapse collapse of the vector state (the projection onto a subspace associated with the observable under measurement) as a means to reduce all the possibilities encoded in the state into a single one, to account for the measurement Measurementprocess.¹² It is thus the observer Observerwho does the dirty task of suspending the unitary and Evolution!causalcausal evolution law to allow for the (nonunitary) collapse of the wave Collapse!wave functionfunction.¹³

• qm risks becoming subjective with the entry into scene of the observerObserver. The observer is an active intruder, the element that transforms the potentialProbability interpretation!potential into the real; however, he/she is not part of the libretto. For some people this is an opportunity to add subjective elements to the interpretation.¹⁴ Sudarsky, D.

• qm requires a boundary between the observed and the observerObserver, but the theory cannot define it. To avoid an infinite regression, the measuring instrument must be classical. Thus a part of the world is not described by qm, despite the fact that it is considered to be a fundamental theory, one that should apply to everything.¹⁵ Since quantum theory should lead to the description of the macroscopic world as a limiting process, in principle it cannot refer to elements of the latter in its foundations; yet it does precisely that.

• qm deals with objects of undefined nature. The theory does not embody an objective strict rule of demarcation that distinguishes between corpuscular and wave entities. Worse, even: whether these objects exhibit a corpuscle- or a wavelike behaviour is controlled by the free undertakings of the observerObserver. There is room for three quarks within a proton, but an electron may occupy the whole Interferometerinterferometer before hitting a single point on the screen.

• qm lacks of a space-time descriptionSpace-time description. In particular, the notion of trajectory is foreign to qm, presumably prevented by the Heisenberg inequalitiesHeisenberg inequalities. Thus, qm describes what the atomic electrons do in the abstract Hilbert space, but says nothing about what they do in common three-dimensional space.¹⁶

• qm is a nonrealist theory. The usual quantum description averts realismRealism from several sides, through the lack of a space-time descriptionSpace-time description, incomplete causalityCausality, unexplained indeterminism, nonlocality... (see Sect. 1.3 for a discussion on realism and quantum mechanics).

1.2 The Two Basic Readings of the Quantum Formalism

1.2.1 The Need for an Interpretation

The pure theoretical skeleton of a physical theory, its formalism, says nothing about the world; it is devoid of empirical meaning. To attribute physical meaningDe Broglie’s wave!physical meaning to the abstract mathematical apparatus, a set of semantic rules Semantic rules, collectively known as the interpretation, is required. The interpretation assigns a concrete empirical meaning to the nonlogical terms in the theoretical model (such as mass, force, charge, electric field, and so on). Physically, the model normally does not resemble what it models; the conformity resides in the functioning.

Which is the meaning we should ascribe to the different elements in the quantum formalism, e.g, the wave function, solution of the Detailed balance!and Schrödinger equationSchrödinger equation for a given problem? The answer is left to our ingenuity. And this is where the real problem starts... It is not difficult to count a dozen different interpretations of the same theory: Copenhagen interpretation (Bohr, Heisenberg, etc., from 1926 on); Ensemble Probability interpretation!ensembleensemble interpretation (Einstein, etc., from 1926 on); de Broglie–Causality!in Bohm theoryBohm De Broglie!theorytheoryBohm theory De Broglie, L. (de Broglie 1927; Bohm 1952a, b); quantum logic Birkhoff, G.(Birkhoff and von Neumann 1936); many worlds (Everett 1957); stochastic electrodynamicsMarshall, T.W. (Marshall 1936); stochastic mechanics Stochastic mechanics Nelson, E.(Nelson 1966); modal interpretationsVan Fraassen B. (van Fraassen 1972); propensities of smearons Smearons(Maxwell Maxwell, N. 1982); consistent histories (Griffiths 1984); quantum informationWheeler, J. A.|( (Wheeler 1983); transactional interpretation (CramerCramer, J. G. 1986); zitterbewegung Zitterbewegunginterpretation (Hestenes 1990);Hestenes, D. no-signaling plus some nolocalityPopescu, S. Rohrlich D. (Popescu and Rohrlich 1994); relational quantum mechanicsRovelli C. (Rovelli 1996); and so on. According to other authors, qm does not require an interpretation at all (PeresPeres, A. (2000)), or on the contrary, there is only one legitimate Omnés, R.interpretation (Omnès 1994), or even any interpretation goes Feyerabend, P. K.(Feyerabend 1978). We are further told that the description does not really describe the system, but merely our knowledge (or information) about it (Heisenberg 1958a, b, but seeMarchildon Marchildon 2004; Jaeger 2009)Jaeger, G.; or that the theory is about measurements and observables and not about beables (see Bell 1976, 1985); or that the awareness of our knowledge ‘actualizes’ the wave function, thus promoting us from external passive bystanders into active (although involuntary) participatorsPatton, C. M. Wheeler, J. A. (Patton and Wheeler 1975), without being included however in the formal structureStructure. A recent trend is to say that qm refers not to matter, but to bits of Informationinformation (see e.g. Vedral 2010). And so forth...

Thus we have a nice formal description of the quantum world, empirically adequate for our purposes, but we still lack of a real understanding of that world. No wonder that there are expressed recognitions of the need of a fundamental and deep amendment of our present quantum image (see e.g. Delta Delta scanScan 2008; Stenger V. J. Stenger 2010).

1.2.2 A Single System, or an Ensemble of Them?

A most basic and crucial question for any Interpretationinterpretation of qm relates to the meaning of the wave function: does it describe the dynamics of a single particle, or does it instead refer to an ensembleEnsemble of similarly prepared particles? The answer to this question distinguishes between the two mainstreams of the interpretation of quantum theory, the Copenhagen and the ensemble Probability interpretation!ensembleinterpretations.¹⁷

The usual textbook standpoint on qm is based on some variant of the Copenhagen (or orthodox) interpretationInterpretation!orthodox (CI).¹⁸ It might also be called the customary, mainstream or regular interpretation, although it is not so clear that the present-day practicing physicists (and physical and quantum chemists) adhere to it in their daily endeavours as tightly as such names may fancy. The founding fathers of the CI are of course Heisenberg (1930) andBohr, N. Bohr (1934), who were joined almost from the start by physicists like PauliPauli, W., Dirac (1930), Born, M. Born (1971), von Neumann (1932), and Landau.Landau, L. One should bearVon Neumann, J. in mind, however, that the name CI does not refer to a sharp set of precepts, since a wide range of tenets with respect to some of the central interpretative issues can be distinguished among its practitioners. Thus it encompasses a collection of variants of interpretation rather than a tight doctrine. In a broad sense one refers normally (but not necessarily) to any of the members of such collection as the conventional interpretation.Interpretation!conventional The basic tenet of the CI of qm is that a pure state Pure stateprovides a description as complete and exhaustive as possible of an individual system. So, qm goes as far as is possible in the knowledge of Nature, and physicists must renounce once and for all the hope for a more detailed description of the individual; Nature imposes upon us a limitation to our knowledge. This assumption has enormous consequences, some of which will be discussed in the following section.

A very different outlook ensues from the ensembleEnsemble (orCausality!statisticalstatistical) interpretation (EI) of qm. According to this interpretation the wave function refers to a (theoretical) Interpretation!ensembleensemble of similarly prepared systems, rather than to a single one. The earliest attempts to formulate an ensembleEnsemble interpretation of qm are found in Slater J. C. Slater (1929), Schrödinger (1932) and Fürth (1933)Fürth, R.. Other early advocates of this interpretation were Langevin, P. Langevin (1934), Popper, K. Popper (1959), Einstein (1936, 1949), Landé (1955, 1965), Blokhintsev (1964, 1965) (the original Russian version of 1949 was the first systematicAlbert, D. treatment of the Interpretation!ensembleensemble interpretation of qm).¹⁹ Being an intrinsically statisticalCausality!statistical description, for the advocates of the EI the description afforded by the wave function $\psi$ is neither complete nor exhaustive of the individual systems that conform the ensembleEnsemble (which in its turn gives significance to the different probabilities encoded in $\psi$). Chance enters into the picture in a fundamental way; the wave function does not “represent things themselves, but merely the probability of their occurrence” (Einstein 1933, slightly adapted).

1.3 Is Realism Still Alive?

“Quantum mechanics demolishes the view that the universe exists out there” (Wheeler 1979).

Quantum mechanics, or a certainWheeler, J.A.|) interpretation of it?

Such a view of qm is clearly nonrealist. This may not mean much to some, to others it may be unimportant, but to still others it may be of high significance, because philosophicalDeterminism!philosophical realism Realismis not a capricious free invention. As mentioned earlier, philosophers arrived at the notion of realism by distilling the works of creative scientists (and philosophers) along the centuries, and recognizing and extracting the essence of their diverse procedures. They have thus discovered that there are realist scientists, nonrealist scientists and anti-realistAntirealism scientists, and that the large majority of creative natural scientists are (spontaneously or consciously) realist and work under the assumption (or conviction) that the world they are studying is not an illusion, but exists by itself. This is the essence of scientific realismScientific realism Realism: the belief in a real world, external to us, independent of our attention to it, a world in which we act, which acts upon us, and upon which we act to know more about it. A nonrealist negates either the realityReality of the external world or its independence from us, or both; an antirealistAntirealism is more extreme and believes that the world is a result of our mental activity.²⁰ Henry, R. C. Along the centuries, science, with its remarkable development, has nourished and reinforced realismRealism. Shortly stated, realism is a synthetic result of the scientific venture.

Further to the general defining attributes of scientific realismScientific realism Realism—external realityReality, independent from our deeds, and the possibility to know the world—realism in physics embodies other demands of general validity. An obvious one isLocality!and causality causalityCausality, which lies at the basis of physical science. Another is the recognition that the phenomena occur in space and time, and thus should admit a space-time descriptionSpace-time description. A third one is that the causal relations are local, which means that there are no actions at a distance.²¹ $^{\text {,}}$ ²²

Let us look at some of the features of qm as seen from the CI and the EI, to make clear the position of these interpretations with regard to realismRealism. In doing so, we will touch upon some of the difficulties encountered in Sect. 1.1.1 and discusse them more at length.

As stated above, a most distictive quality of qm is its indeterminism, which in some instances is taken as noncausalityNoncausality.Linear sed!and noncausality In a situation commonly considered, a given observation can lead to one of a miscellany of possible results (e.g. a specific eigenvalue among a set of values). Which is the outcome is a matter of chance, and the CI grants that nothing, except chance, determines the result. The example of the decay of a single radioactive nucleus is illustrative: quantum theory can correctly assign a mean lifetime Lifetimeto the nucleus, but it cannot predict the precise moment or direction of the decay products. However, a nearby detector shows that such moment and such directions exist. The precise prediction escapes quantum theory. By considering the quantum description to provide the most complete attainable informationInformation about a given system, not unusually the CI declares that precise values of the physical variables cannot be predicted by qm simply because such variables do not have preexistent values; they do not exist until a measurement is performed, until a precise value is recorded).²³ Balian, R. Thus, for example, for the conventional school, the position of the particle is materialized or brought into being, as it were, as a result of its measurement. The values of the dynamical variables are thus objectively undetermined prior to their measurementMeasurement, and only probable values can be assigned to them; probabilities become irreducible.Indeterminism!irreducible Since the nonexistent cannot be measured, it is the measurement itself which fixes the measured value, giving reality Realityto it. It is here that the observerObserver (or the observer’s proxy) slips into the description; the realist fundamental principle that physics should refer to the world rather than to our knowledge of it (or information about it) is eroded, and with it the no less fundamental demand of a strictly objective rendering of the physical world. All this was clearly recognized (and accepted) by Bohr (1928) in his famous Como Lecture of September 1927, a characteristic sentence of which says:

...the finite interaction between the object and the measuring devices... implies... the necessity to renounce the classical idea of causalityCausality, and a radical revision of our attitude toward the problem of physical reality,

and by HeisenbergHeisenberg, W. in denying the existence of an underlying quantum realm (Heisenberg 1958a, page 129):

...the idea of an objective Probability interpretation!objectivereal world whose smallest parts exist objectively in the same sense as stones or trees exist, independently of whether or not we observe them ... is impossible...

or further (Heisenberg 1958b, page 15):

... the natural laws formulated mathematically in quantum theory no longer deal with the elementary particles themselves but with our knowledge of them. Nor is it any longer possible to ask whether or not these particles exist in space and time objectively...

The role of the observerObserver is not limited to bringing out a real physical variable out of a mere potentiality, it includes determining the very nature Sed!linear theory Linear sed!nature of its solutionsof the system. For instance, in an electron diffractionElectron diffraction experiment the electron suffers a series of transformations from being a (more or less) localized entity (with corpuscle-like properties) to becoming a structureStructure that fills a macroscopic volume (with wavelike properties)Wavelike properties and vice versa. It seems difficult to bring to terms this series of transmutations with the idea of a reality Realityindependent of our undertakings.²⁴

Along with the observer, a radical form of nonlocality is introduced into the theoretical framework: the collapse of the wave functionCollapse!wave function—instantaneous over the whole space—determined by a local measurement. Indeed, the collapseCollapse, which is the theoretical counterpart of the changes on the individual system brought about by the active observerObserver, becomes the inevitable mechanism by which a specific result is selected from among the various possibilities. The collapse disrupts the orderly causal development described by the evolution equation, introducing an abrupt fall to a lawlessly established state of a certain statisticalCausality!statistical mixture (these are the spookyAction at a distance!spooky actions at a distance, mentioned by Locality!EinsteinEinsteinEinstein, A. to Born;Born, M. see Born 1971). Thus two forms of evolution compete within the theory, and it is the observer —the ineluctable intruder—who determines with his actions which of them should operate. Of course, interpreting the collapse Collapseas merely a theoretical tool, without ascribing to it a sense of realityReality, becomes an acceptable pragmatic procedure. But this is not its usual grasp.²⁵

Since according to the spirit of the Copenhagen interpretation it is meaningless to attribute any existence to a certain physical variable until it is measured,²⁶ the quantum variables have been transformed into observables. Hence, the standard adumbration of qm demands from us to assume that the theory is not about existing objects of nature, but about our measurementsMeasurement and observations on them. Bohr states it clearly (as reported by Petersen, A. Petersen 1963):

There is no quantum world. There is only an abstract quantum mechanical description. It is wrong to think that the task of physics is to find out how Nature is . Physics concerns what we can say about Nature.

Heisenberg goes even farther (Heisenberg 1958b), by negating the reality of his very object of study:

...the atoms or the elementary particles are not as real [as any phenomena in daily life]; they form a world of potentialities or possibilities rather than one of things and facts.

Out of the frying pan into the fire, today we see a modern version of this idealistic vision of the world swiftly extending in connection with Informationinformation, which argues that the building blocks that constitute the world are not matter and energy, but ... bits of information (see e.g. Vedral 2010; BoribojeBoriboje, D. and BruknerBrukner, C. 2011, and references therein). A most fashionable formula for this was introducedWheeler, J. A. by Wheeler (1990): “It from bit”, where ‘bit’ stands for the unit of information; according to this dictum, the material world emerges from the (qu)bits of quantum information, not conversely.

As for the possibility to construct a space-time descriptionSpace-time description of quantum systems, the very idea was firmly negated by Heisenberg, W.Heisenberg, Bohr and other founders of qm, who declared the quantum world to be nonvisualizable. Thus, the concept of trajectory was taken as untenable in quantum theory since it is contrary to Heisenberg inequalitiesHeisenberg inequalities (and to the wavelike propertiesWavelike properties, many would add).²⁷ The view of a nonvisualizable world helped to do away with the need to explain some of the quantum paradoxes Jones, S.(Jones 2008, particularly Chap. 16). By 1927 quantum trajectories were so insistently negated—with the exception of de BroglieDe Broglie, L. and Einstein²⁸ Bacciagaluppi, G. (Bacciagaluppi and Valentini 2009)—thatValentini, A. at the closure of the Solvay 1927 Congress Lorentz, H. A.Lorentz felt obliged to make a declaration of principles:

... I should like to preserve this ideal of the past, to describe everything that happens in the world with distinct images. I am ready to accept other theories, on condition that one is able to re-express them in terms of clear and distinct images.

We are not longing for a past full of clear images, if that past is gone for ever. But, is it really gone? As Lorentz Lorentz, H. A.put it, we should be ready to accept the new theories, on condition that they are the result of transparent and definitive knowledge, free of free elections. Yet, by embracing the Copenhagen interpretation, we forsake the possibility not only of making precise predictions about individual trajectories, but entertaining that very notion.

The widespread conclusion that the violation of the Bell!inequalitiesBell inequalities by qm demonstrates Nature’s nonlocality represents one more argument against realism. As an exampleVan Fraassen B., van Fraassen (1989) contends that scientific realism Scientific realism Realismis invalidated at the microlevel by the violation of Bell’s inequalities, and therefore it cannot be valid more generally.²⁹ Howard, D.In fact, there is no need of these inequalities or any of their variants to demonstrate that qm corresponds indeed to a nonlocal description, as follows, for example, from Bohm, D.Bohm’s interpretation of qm. The point is that we must carefully distinguish between Nature being intrinsically nonlocal and a nonlocal rendering of the relevant portion of Nature.

To maintain a realist view of physics, either the definition of realism must be changed to accommodate for the new situation, or we must accept that qm cannot be the final tale. The standard lore purports the first alternative, which leads to consider that our current notion of realism is incompatible with science.³⁰ For example Stapp H. P. Stapp (1972) writes “If the statisticalCausality!statistical predictions of quantum theory are true, an objective Probability interpretation!objectiveuniverse is incompatible with the law of local causes.” It is interesting to compare this with Einsteins contention (in BornBorn, M. 1971, page 221):

I cannot seriously believe in [quantum mechanics] because it cannot be reconciled with the idea that physics should represent a reality Realityin time and space, free from spooky actions at a distance.

Clearly Einstein opted for the second alternative above, namely to admit that qm is not the final tale. As he expressed in Einstein (1949):

If in quantum mechanics we consider the psi-function as (in principle) a complete description of a real physical situation, we thereby imply the hypothesis of action-at-distance, a hypothesis that is hardly acceptable. If, on the other hand, we consider the psi-function as an incomplete description of a real physical situation, then it is hardly to be believed that, for this incomplete description, strict laws of temporal dependence hold.

By assuming that qm goes as far as possible in the knowledge of Nature, the CI forces us to admit a nonrealistic, irreductibly indeterministic, nonlocal and noncausal world. In contrast, once we concede that the quantum description is incomplete, the possibility of going beyond qm without having to renounce to realism opens in principle. A means to recover realism is thus offered by adhering to the ensemble Probability interpretation!ensemble Ensembleinterpretation. In particular, by recognizing that quantum theory is statisticalCausality!statistical and as such incomplete, the ensemble school allows for the possibility of understanding the indeterminismIndeterminism as due to such incompleteness, without necessarily assigning to it a more fundamental meaning, as could be that of an ontological property, or, perhaps, an Indeterminism!irreducibleirreducible indeterminism at the observational level. This leaves the door open to further studies at a deeper level, for the identification of the source of the indeterministic (or stochastic) behavior characteristic of quantum systems. For those who profess this credo this is a most important alternative. For a hard realist, who believes that each individual system has always a real state (may be unknown), and that among the tasks of physics an important one is to discover such real states, an essentially statisticalCausality!statistical theory cannot be taken as complete.

In an extended variant of the EI (also here there are variants, of course) the particle is assumed to have at each moment a set of well-defined, objectively real properties, even if these properties are not simultaneously described by the wave function.³¹ $^{,}$ ³² Thus for example, one thing is to say that the values of two variables associated with noncommuting operators cannot be simultaneously ascertained by resorting to $\psi$, and another one is to say that such values are not simultaneously defined, or simultaneously existent, even if distributed and unknown. Preexisting valuesPreexisting values thus may exist (Deltete Deltete, R.and Deltete and Guy 1990), Guy, R.yet the wave function $\psi$—a catalog of all the different possible outcomes—can only assign to each of them a certain probability. In the example of the decay of a single radioactive nucleus the fact that a precise prediction escapes to qm, does not mean by necessity that there are no precise (although unknown) factors precisely determining the result. Thus, the EI advocate distinguishes between the capabilities of our theories or descriptions, and what happens in the real world, at the ontological level. A particular, but immediate consequence of this is that the notion of trajectory, though recognized as foreign to the quantum description, is not forbidden in principle.

From an ontological point of view, what the EI and CI schools claim is the preexistence Preexistenceor not of features that lead to the observed value (see, however, footnote 23). Thus, referring to the observables of the CI, BellBell, J.S. contends: observables are not Beablebeables (Bell 1987, particularly articles number 5 and 7).³³ The transition from beables to observables—from preexisting valuesPreexisting values to undefined or nonexisting values—is one most important issue of quantum theory, which remains nevertheless unstudied. Out of the blue the observer enters the scene, although the quantum-mechanical formalism does not provide tools to establish where that boundary between the observed and the observerObserver lies, leaving room for an ambiguity and cloudiness that is totally strange to theoretical physics. Bell, J.S.Bell (1987, article 20) refers to this in unequivocal terms: “It is the toleration of such an ambiguity, not merely provisionally but permanently, and at the most fundamental level, that is the real break with the classical ideal. It is this rather than the failure of any particular concept such as ‘particle’ or Determinism‘determinism’.”³⁴

The pictures provided by the CI and the EI differ so widely—they in fact exclude each other—that at first glance it should be a simple matter to empirically demonstrate the fallacies behind one or the other. But almost eighty years have elapsed since the advent of quantum theory and the dichotomy remains, notwithstanding the endless discussions and enlightened studies on the subject.³⁵ The root of the difficultiesWheeler, J. A. is that the problem is deeply influenced by the personal philosophical stance.Determinism!philosophical There coexist several general outlooks about the world, and each one of us adopts one or another, consciously or unconsciously to different degrees. This is an (apparently) free personal selection, more or less as (apparently)Nonlocality!apparent free is the selection of a religious credo. Add to that the characteristic positivistic standpoint that pervades textbooks, entangled with their scientific content. The physics student is normally unprepared to recognize the presence of this mixtureMixture, and less so to disentangle it, so that he ends up assimilating as established knowledge what is far from that.

For a realist the CI is implausible, to say it mildly (other more belicose terms have been used), while a moderate orthodox considers the EI full of unnecessary metaphysics Fuchs, C.A. Fuchs and Peres (2000), or just dogmatic. For a more radical orthodox, the EI lacks the space needed to accommodate other elements demanded by his world view, such as the observer Observerand perhaps his mind. The pragmatic (fapp) physicist argues that the Copenhagen theory has been used successfully for many years without a single failure, which is a proof of its correctness, so we should derive from it our vision of the world and not the other way round. He therefore expects us to renounce our basic principles of physical thought in order to be able to understand physics Tambakis N. A.(Tambakis 1994) on the basis of a ‘quantum syllogism’, an attitude similar in nature to that required to give theological support to the theory of the epicycles, as Jaynes, E. T. Jaynes (1993) put it. Further, not few physicists add that qm describes what can be described, and that importing into the quantum domain knowledge that originated in the classical world leads to contradictions and paradoxes (see e.g. Lévy-Leblond 1973), as Bohr alerted usLévy-Leblond J.-M. since 1935.

It should be noted that, much as the strength of the EI lies in its essentially statisticalCausality!statistical nature, Ballentine, L.E.in it lies also its weakness. Indeed, the EI (as expounded e.g. in Ballentine 1970, 1989, 1998) is far from being free of difficulties on a very fundamental level. An immediate one is that the quantum-mechanical description is a very particular sort of statisticalCausality!statistical description, in terms, not of probabilities, but of amplitudes of probability, which have the peculiarity that they interfere among themselves. This is fundamental for qm; it is the basis for quantum interferenceInterference andCorrelations!and entanglement Entanglement entanglementEntanglement!and correlations, two most important and characteristic features of the quantum systems. This superposition Superpositionof amplitudes has at least two implications that go counter to the usual theory of probability: the occurrence of probabilities that depend on the context (contextualityContextuality, for short), and of negative Probabilities!negativeprobabilitiesNegative probabilities, as remarked in Sect. 1.1.1. Moreover, and connected to the latter, the quantum description Oscillator!joint distributiondoes not allow for a joint distributionDistribution for noncommuting variables, so it lacks of a true phase-space distribution of general applicability. The fact that joint Probability!jointprobability distributions do not exist for noncommuting variables puts into question the very definition of correlationsCommutator!and correlation between them. It should therefore not be surprising to find results such as those of Gleason (1957),Gleason, A. M. Bell (1966), Kochen, S. Specker, E. P. Kochen and Specker (1967),³⁶ showing that even if each observable is considered as a classical random variable, two incompatible observables (noncommuting operators) cannot be viewed simultaneously as classical random variables defined on the same space of events, with independence from the specific contextContext. The consequence of this is the nonexistence of a (context-independent) joint distributionOscillator!joint distribution of such variablesSuppes P. Zanotti M. (Suppes and Zanotti 1981). A particular sequel of such theorems is that any hidden-variables theory of qm is necessarily contextual.

Of course, such problems as negative probabilites and the lackProbabilities!negative ofNegative probabilities a phase-space description, being characteristic of the quantum formalism, are common to all interpretations of qm. However, the problem becomes more accute for the EI, precisely because it sees qm as a statisticalCausality!statistical theory. The widespread lack of clarity about this topic has led to a series of objections against the ensembleEnsemble interpretation of qm, with some authors claiming with conviction that such a formulation has been empirically disproved. About this there is still much to say.

1.4 What is this Book About?

Through the following eight chapters, a fundamental theory for quantum mechanics is constructed from first physical principles, disclosing quantization as an emergent phenomenon arising from a deeper stochastic process. The elements that sustain the pillars of the quantum-mechanical formalism are identified; hallmarks such as the mechanism responsible for Atomic stabilityatomic stabilityLinear sed!stability of stationary solutions, the nature of quantum fluctuations, the origin and meaning of quantum nonlocalities, as well as other central features of quantum theory, are elucidated. All this is carried out within a comprehensive and self-consistent theoretical framework that reaffirms fundamental scientific principles such as realismRealism,Locality!and causality causalityCausality, localityLocality, and objectivity. Thus, the theory developed in the present monograph hopefully may serve to show that those principles can survive their apparently Nonlocality!apparentunsurmountable adversities.

If one lesson can be drawn from the persistent but inconclusive enlightened studies on the meaning of the quantum laws, it is that the analysis of quantum theory from its inside leads to nowhere. Such studies may add richness, deepness and erudition to an interpretation, but the essentials remain the same. The virtue of the theory presented here is that it offers a perspective on the quantum world from outside it; one arrives at the quantum formalism from a distance, with a well-defined physical perspective. The interpretation comes from the physics, not the physics from the interpretation.

1.4.1 The Underlying Hypothesis

The fundamental hypothesis that is put to test and developed at length in this book is that every material system is an open stochastic system in permanent contact with the random zero-point radiation field Approximation!fixed zpf(zpf). TheCoherence!zpf modes existence of an all-pervading zpf follows quite naturally from the (classical) Maxwell equations, yet it is foreign to the classical realm, which graciously assigns zero energy to the field oscillators at zero temperature. The zpf Coherence!zpf modes is taken here as the athermal component of the radiation field, as real as any other solution of the Maxwell equations.

The most significant conclusion drawn from the present theory is that the quantum phenomenon, rather than being an intrinsic Probability!intrinsicproperty of matter or the radiation field, emerges from their interaction. A key element is found in the Fluctuationsfluctuations of the zpf, which correspond to Fluctuations!vacuum Vacuum!fluctuationsthe ‘vacuum fluctuations’ of quantum electrodynamics (qed). Vacuum fluctuations are commonplace in modern quantum theory, though some of their consequences seem not to be fully appreciated. The fluctuationsFluctuations of the best known vacuum field, the electromagnetic radiation field, are commonly considered to be (totally or partially) responsible for several physical phenomena, such as spontaneous radiation from excited systems (see e.g. Dalibard et al.Dalibard, J. Dupont-Roc, J. 1982),Cohen-Tannoudji, C. the CasimirBoyer!and Casimir forces forcesTer Haar, D. Casimir force(see, e.g., DavydovDavydov, A. S. 1965; Ballentine 1989), and the Lamb shiftRadiative corrections!Lamb shift Lamb shift (see e.g. Sokolov, A. A. Sokolov et al. 1962; Milonni, P. W. Milonni 1994). But apart from serving to explain these quantum corrections Quantum corrections, the vacuum field is mostly viewed as a nuisance, because it is responsible for several of the infinities that spoil the otherwise smooth quantum calculations.³⁷ Thus it is swept under the carpet as soon as possible (only to reenter through the back door) and reduced to a merely virtual field. In the theory presented here, rather than being a nuisance, the zpf Coherence!zpf modes becomes central for the understanding of the behavior of atomic matter. Thus, far from being considered as merely the origin of some small corrections or effects to be added on top of the quantum pattern of matter, the zpf is seen as the source of the quantum behavior of matter. This is the central premise of stochastic electrodynamics (sed)Linear sed!and entanglement, at least from the point of view of the present authors.

Naturally, since all vacuum fields may contribute in principle to the universal background noise, in line with our approach all of them could contribute to the fundamental stochastic behavior of matter on the microscopic level. However, at the scales to which qm is most frequently applied, or for systems basically of an electrodynamic nature, it is the electromagnetic vacuum that plays the pivotal role. At deeper levels or for systems of another sort, it may well be that other vacua become relevant; one can even speculate that all vacuum fields have similar statisticalCausality!statistical properties, so that a kind of universality holds, in the sense that the essential stochasticity of matter is basically independent of the nature of the dominant background field. One could also consider that the required random field is just a construct to simulate the effects of random fluctuationsCommutator!and correlation Fluctuations!of the metricof the metric, and take theseMetric fluctuations as the ultimate origin of the quantum phenomenon (a first heuristic approach to this idea has been given in Santos 2006)Santos E..

1.4.2 The System Under Investigation

Our system of study is composed of a material charged particle (rather, an ensemble of them) embedded in the zpf Coherence!zpf modes and having a dynamics that is initially described by a classical (stochastic) equation of motion. Due to the randomess of the system, the theory is statisticalCausality!statistical in essence. The system is then left to evolve. When, and if, it reaches a reversible regime Linear sed!detailed energy balancein which detailed energy balance Detailed balance!frequency mixing(i.e., at each frequency of the field) is attained in the mean between the field and matter, the radiative terms in the dynamical equations for the mechanical subsystem become mere corrections that can be neglected in a first approximation. Under these conditions the evolution turns out to be controlled by the quantum equations. Two independent and Complementaritycomplementary derivations of this fundamental result are presented, one in Chap. 4 (leading to the Detailed balance!and Schrödinger equationSchrödinger description)Detailed energy balance!and Schrödinger equation, and another in Chap. 5 (leading to the Heisenberg formalism). The ensuing classical-to-quantum transition could in a way evoke the usual textbook derivations in quantum field theory that start from a classical field theory and at some point incorporate an extra-classical (quantum) demand. Of course the converse transition, from quantum to classical, is theoretical commonplace—although not always based on conclusive arguments. Yet our procedure differs profoundly in essence and scope from such formal methods; here no quantum demand is introduced (neither a priori nor a posteriori). The zpf is the extra-classical physical entity that ultimately endows the system with its quantum properties, and in addition guarantees the internal consistency of the theory. The quantum is not the means, but the consequence.

The present theory should not be confused with a semiclassical theorySemiclassical theory, which treats matter quantum-mechanically but the field classically, or conversely (see e.g. Sokolov, A. A. Tumanov, V. S. Sokolov and Tumanov 1956). Quite the contrary, here we deal with an initially continuous radiation field (classical, but with its zero-point component) and a particle that initially satisfies classical equations of motion, and show that both end up being quantized.

As a prelude to the derivations in Chaps. 4 and 5, the phenomenological description of qm as a stochastic theory is discussed in Chap. 2, with the purpose of introducing the reader to some of the (old) methods that succeed in showing that it makes sense indeed to understand qm as a stochastic theory. In Chap. 3 we initiate the testing of our hypothesis, by analyzing the consequences of allowing for a zero-point contribution in the equilibrium radiation field. There it is shown that the zpf Planck distribution!zpf-field derivation has a decisive role in leading to the Planck distribution for the radiation in thermal equilibrium, and to the quantized spectrum for the oscillatorsOscillator of the field.

The treatment of matter and field as inseparable elements of a whole system makes it possible for the theory to go beyond qm in the most natural way. It provides the elements to study the radiation and absorption terms—a matter that is normally considered to belong to the domain of qed—which here appear as Fokker-Planck equation!radiative corrections Oscillator!radiative corrections Radiative correctionsradiative corrections Oscillator!Fokker-Planck equation(neglected in the previous approximation) to the quantum-mechanical description. In Chap. 6 it is shown that indeed, these terms are responsible for the finite lifetimes of excited atomic statesStability of atomic states, as well as for the absolute stability of the ground stateOscillator!ground state!stability in the sole presence of the zpf. A further radiative correctionOscillator!radiative corrections that appears quite naturally gives the Lamb, Jr. W.E.Lamb shiftRadiative corrections!Lamb shift Cavity effects!on the Lamb shift for isolated atoms, and the corresponding shifts in more complex situations. Of particular interest is the discussion, in the same Chap. 6, related to the origin of the electron spin Spinfrom the present perspective, as another consequence of the Fluctuationsfluctuations imposed on the particle by the field, in this case, those that give rise to rotational motions. We are thus faced with one more element that cannot be predicted from within theDetailed energy balance!and Schrödinger equation SchrödingerDetailed balance!and Schrödinger equation realm, but can be unfolded by recognizing the presence and action of the zpf Coherence!zpf modes. Moreover, being the spin of the charged particle the support for its magnetic Moment!magneticmomentMagnetic moment, it becomes clear that along with it, the theory determines the spinMoment!magnetic Magnetic moment!of the spin $g$-factor of the electron, predicting its correct value of 2.

When the theory is generalized to include systems of two particles, which is the subject of Chap. 7, a phenomenon expected in the present treatment appears, namely the emergence of correlations between (even otherwise noninteracting) nearby particles through common relevant modes of the vacuum field. The correlated motions of the particles attest to their entanglementCorrelations!and entanglement, induced by the zpf. Therefore, just as the zpf Coherence!zpf modesmay be capable of generating Decoherencedecoherence of the system, it also stands as the most important source of coherence Coherencein a significant class of bipartite systems. In particular, when the particles are identical and subject to the same external potential, our results disclose the mechanism underlying the PauliPauli, W. Pauli exclusion principleexclusion principle. More generally, the vacuum field is exhibited as an important source of nonlocality: when this field is ignored, the consequences of its action appear as nonlocal. Nonlocality is further studied in Chap. 8, both for the single-particle case and for a pair of correlated (entangled) particles; these studies unfold the important role played by the so-called diffusive Velocity!diffusivevelocity, just the one due to the quantum fluctuations, in providing the quantum system with its characteristic nonlocal descriptive features. In addition, in Chap. 8 we make a brief detour to the causal interpretation of qm, which among interesting features provides an opportunity to glance at a hidden-variables description and to take a fresh look at quantum nonlocality.

Attention is paid in Chap. 9 to the undulatory properties of matter; the de BroglieDe Broglie, L. wave is constructed and shown to originate in the radiation field around the moving particle. A well-defined physical wave is thus naturally associated to the moving corpuscle, yet both entities (particle and wave) are clearly distinguished from each other at all times. Further, a brief discussion is presented regarding the diffraction of electrons, which is explained by arguing that the electron diffraction pattern is but a trace of the pattern produced by the diffracted zpf Correlations!and zpf. A final section is devoted to a discussion on the relationship between atomic and Cosmological constantcosmological constants, with the zpf, of cosmic presence, acting as the bridge between these two realms of Nature. The final Chap. 10 contains an overview of the main results and implications for qm of the theory developed in the previous chapters. It further provides a brief account of several of its limitations and possible extensions, and ends with a brief discussion of sed Linear sed!and entanglement in the broader context Contextof theories of space-time metric fluctuationsMetric fluctuations.

It should indeed be noted from the start that the treatment given here to the quantum problem corresponds to a restricted theory in several senses. An obvious one is that the entire discussion is nonrelativistic. Further, the dynamics that takes place during the transition from the original classical state—in which the system is far from equilibrium—to the final state—the quantum regime,Quantum regime Sed!linear theory Linear sed!and the quantum regime controlled by the Quantum regime!and energy balance Energy balance!detaileddetailed balance Detailed balance!frequency mixingof energy—still needs to be worked out in detail; surely such studies will reveal a rich physics that so far remains hidden. Moreover, the entire treatment is limited here to the description of the dynamics of the material part of the system, while the field is considered as basically (though not entirely!) unperturbed. This excludes by construction the possibility of a full quantum-electrodynamic description. Consequently, the calculation of those phenomena that correspond to qed is everywhere limited in this volume to the lowest significative order of approximation. Within these limitations, nevertheless, the results derived Commutator!derivedare always the correct ones, appropriately coinciding with the corresponding predictions of either (nonrelativistic) qm or qed.

By looking at quantum theory from the perspective offered here, we hope that the reader will find a satisfactory explanation or answer to a number of the issues and puzzles mentioned in this chapter, and to others that may be boggling his mind. On the other hand, as discussed in the final chapter, it is clear that there are still many fundamental (and treacherous) facets to learn about the quantum world and its intriguing machinery. qm is a marvelous theory. Just because it is marvelous, it deserves to be better understood.

In concluding, we should note that the theory developed in this volume is an alternative, more advanced, complete and elaborate version of the previously developed theory of sed.³⁸ When it is necessary to distinguish between the traditional theory and the present version, the latter will be designated with lsed Matrix mechanics!and lsed(the l stands for linear; see the explanation in Sect. 5.2). The theory offers substantial answer to a fundamental question posed by T. H. BoyerBoyer, T.H.,³⁹ namely: which quantum problems can be explained using classical physics plus the zpf? A large collection of papers published in the past half century by different authors (by BoyerBoyer, T.H. himself, P. Claverie, D. C. Cole, França, H. M.H. M. França, T. Marshall, T.W.W. Marshall, A. Rueda, E. Santos, ourselves and several others) provided the ground for the construction of the present version and anticipated some of the results derived Commutator!derivedhere. Recent results obtained by some of these authors and others serve to legitimate or reinforce the ones presented here. We therefore wish, through the present work, to pay tribute to all those colleagues who have joined us in this exciting endeavour with the shared conviction that the quantum puzzle can be solved, and that the zpf Correlations!and zpf is a central part of the solution.

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.⁴⁰

Apart from the formal or axiomatic (Kolmogorovian) probabilities and theProbability!Kolmogorov subjective Probability interpretation!subjectiveinterpretation of probability,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 $\chi _{A}(\omega ),$

$$\begin{aligned} \Pr (A)=\int \nolimits _{\Omega }\chi _{A}(\omega )d\mu (\omega ), \end{aligned}$$

(A.1)

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.