Foundations of Quantum Mechanics, an Empiricist Approach | |||

de Muynck (Willem M.) | |||

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**BOOK ABSTRACT: **__Preface___{1}

- In this book old and new problems of the foundations of quantum mechanics
^{2}are viewed from the new perspective provided by a generalization of the mathematical formalism of that theory encompassing so-called positive operator-valued measures. At its inception the standard formalism of quantum mechanics^{3}as developed by Dirac and von Neumann seemed to yield the natural mathematical framework for describing the microscopic world of atoms and subatomic objects. In particular, Hermitian operators seemed to be able to replace the phase space functions of classical mechanics as mathematical representations of physical quantities. For a large part of the century axiomatic systems were set up, based on this latter idea. Only in the second half of that century, starting with the pioneering work by Davies and Lewis it was gradually realized that Hermitian operators constitute too narrow a framework to encompass all experiments possible in the atomic domain. - The generalization of the mathematical framework meant here is often referred to as the ‘operational approach’. One objective of the present book is to demonstrate the crucial role the generalized formalism plays in fundamental issues as well as in practical applications, and to contribute to its development. The fundamental insight inherent in an acknowledgment of the necessity of the generalized formalism is that the interaction between microscopic object and measuring instrument should be duly taken into account when assessing the meaning of quantum mechanics
^{4}. This insight might seem not to be new at all, since it was shared by the Copenhagen interpretation already at a very early stage. It should be realized, however, that this sharing is only a partial one because recognition of the necessity of a quantum mechanical description of the measurement interaction draws a dividing line between the Copenhagen and operational approaches. As will be seen from a thorough analysis of the Copenhagen principles of ‘correspondence’ and ‘complementarity’, these principles were tuned to the standard formalism, quantum mechanical measurement being thought not to be analyzable beyond a classical description of measurement phenomena. - Although the Copenhagen principles stressed the importance of the measurement arrangement, they missed the important point that transfer of information from a microscopic object to a measuring instrument is a microscopic process, to be described by quantum mechanics
^{5}. It seems that in criticizing the Copenhagen interpretation Einstein has been perfectly right when characterizing the professed unanalyzability of the measurement process as a “tranquilizing philosophy”. By ignoring Bohr’s dogmatic ban on applying quantum mechanics^{6}to quantum measurement it has become evident by now that such a paradigmatic ‘thought experiment’ as the double-slit experiment is even outside the domain of application of the standard formalism, requiring the generalized formalism for its description. It is hardly surprising, then, that a discussion of the ‘thought experiments’ remaining within the confines of the standard formalism has produced so many confusing and paradoxical conclusions. Clarifying these is a second objective of this book. In particular, the so-called “measurement problem” will be discussed at some length, revealing as a second shortcoming of the Copenhagen interpretation its neglect of a sufficient distinction between the physical procedures of ‘measurement’ and ‘preparation’. - A third objective is to provide a critical assessment of interpretations of the quantum mechanical formalism. As is well known, Einstein’s main criticism of the Copenhagen interpretation was directed against its claim that quantum mechanics
^{7}is a complete theory. However, what does ‘completeness’ mean? For Einstein and Bohr ‘completeness’ had very different meanings. For Einstein it was associated with a choice between an interpretation of the state vector either as a description of an individual object (‘completeness’) or as a description of an ensemble (‘incompleteness’). For Bohr the reason to claim ‘completeness’ had quite a different source, namely, the mutual exclusiveness of measurement arrangements of incompatible observables, causing momentum to be ill-defined in the context of a position measurement (and vice versa). This, once again, regards the interaction with a measuring instrument, thus connecting the ‘completeness’ question to the principles of ‘correspondence’ and ‘complementarity’. - Physicists are generally reluctant to deal with interpretation as they are used to associate this with metaphysical speculation. It should be realized, however, that without an interpretation the mathematical formalism of quantum mechanics
^{8}would be just mathematics. In order to make it physics, an interpretation in the sense of a mapping of entities of the mathematical formalism into reality is indispensable. A physicist will have to specify the physical meaning of his formalism if he wants to be able to compare the results of his calculations with what happens in reality. Pragmatic approaches, in which such a meaning is not specified and the resulting vagueness is employed to circumvent problems, may appear to be successful in solving specific problems but will be detrimental in the end because they tend to prevent a consistent overall account. In clarifying a number of possible interpretations the generalization of the mathematical formalism referred to above will once again be illuminating by relaxing the urge to adhere to a particular interpretation suggested by the standard formalism. - With respect to the interpretation of the quantum mechanical formalism our main concern will not be the question of ‘completeness’ in the sense of whether the wave function describes the microscopic object either completely or incompletely; the question will rather be whether the wave function describes the microscopic object at all. This amounts to a choice between a realist interpretation and an empiricist one. In the first interpretation the mapping is into the microscopic world; Hermitian operators are thought to represent properties of microscopic objects, either in the objectivistic sense as Einstein would like to have it, or in the contextualistic one adhered to by Bohr. In an empiricist interpretation the mapping is into the (macroscopic) world of preparation and measurement devices, observables being seen as labels of measurement procedures; a description of the microscopic world is relegated to subquantum theories to be developed if need be. The difference between a realist and an empiricist interpretation of quantum mechanics
^{9}might be characterized in terms of Plato’s allegory of the cave by asking whether the theory is either referring to the real (quantum) world, or just to the shadows on the wall actually observed by the cave dwellers. In an empiricist interpretation justice is done to the difference between the shadows and the real objects. The generalized formalism of quantum mechanics^{10}gives ample occasion to endorse the view that this theory does not yield Plato’s ideal description of reality but just describes the shadows projected onto our measuring instruments. - We should note here that interpretations cannot be proven. They can only be made plausible by showing that they satisfy reasonable criteria like absence of inconsistencies, both internally and with respect to experimental evidence. This implies the possibility of different interpretations, acceptable to different persons for different reasons. On the other hand, one interpretation may be more liable to paradoxical consequences than another. Our conclusion will be that virtually all the paradoxes plaguing quantum mechanics
^{11}are caused by adhering to a realist interpretation, and can be circumvented by relaxing to an empiricist one. In particular, the extension of the domain of applicability of the theory by taking into account the generalized formalism is very helpful in reaching this conclusion. - Is a choice for an empiricist interpretation of quantum mechanics
^{12}a “betrayal of the great enterprise” as seems to be implied by John Bell’s assertion? It would be, if quantum mechanics^{13}is not only “our most fundamental physical theory”, but if it were the most fundamental theory we will ever be able to think of, the “theory of everything”. It is not, if quantum mechanics^{14}is just a phenomenological theory describing certain phenomena occurring within its domain of application, but liable to be superseded by a still more advanced theory as new domains of experimentation are explored. Perhaps we can learn from history here. On earlier occasions physicists as well as philosophers have thought that they had reached the boundaries of knowledge, and were on the brink of “knowing the mind of God”. However, each time God turned out to be more sophisticated than man had ever dreamed of. Is there any reason to think that it will be different this time? Of course, we do not know. But it is far too early to assume that quantum mechanics^{15}gives all answers to our questions, the more so as the necessity of the generalization of the quantum mechanical formalism, referred to above, already demonstrates failure of the standard formalism to give a comprehensive account of our experience even at this moment. - John Bell’s great enterprise seems to be inspired by the idea that the physicist’s task is to design a blueprint of reality. In view of the necessity of experimental testability, felt by many physicists to be a necessary condition to be satisfied by any scientific theory, the necessary interaction of object and measuring instrument seems to make such an endeavor self-defeating. As will be seen, even Einstein’s less exacting ideal that quantum mechanics
^{16}yield a description of an objective reality, independent of any influence exerted by an observer (including his measuring instruments), cannot be upheld. It seems that on this count Bohr was right: knowledge about reality obtained from quantum mechanical measurements has only a contextual meaning. It is knowledge about a microscopic object as it is “colored” by its environment, of which the measuring instrument is an important component. Although the empiricist interpretation of quantum mechanics^{17}developed in this book is fundamentally different from the “orthodox” Copenhagen interpretation, its acknowledgement of the impossibility of ignoring the measurement arrangement might justify referring to it as a neo-Copenhagen interpretation. - It seems that at this moment realist interpretations of quantum mechanics
^{18}are the more fashionable ones. Many-worlds interpretations, modal interpretations, Bohm’s interpretation and even the experimentalist’s classical way of speaking are working together to weave a picture of reality that is in agreement with the quantum mechanical formalism. Admittedly, it is not impossible that quantum mechanics^{19}describes some features of microscopic reality, analogously to the way the classical theory of rigid bodies describes some properties of billiard balls. However, billiard balls are not rigid bodies. They consist of atoms needing a different theory (viz, quantum mechanics)^{20}for their description. By the same token electrons are not wave functions, which in an empiricist interpretation just describe macroscopically observable features of a preparation procedure. It is not at all impossible that an attempt at understanding microscopic reality on the basis of a realist interpretation of quantum mechanics^{21}is comparable to an attempt at understanding the rigidity of a billiard ball on the basis of a model of densely packed rigid atoms. In particular, the widely discussed notions of ‘nonseparability’ and ‘nonlocality’ might very well turn out to be of this kind, attributing to microscopic reality properties of (macroscopic) phenomena observed in measuring instruments. The related problem of the Bell inequalities will be discussed in the last two chapters. Here, too, the generalized formalism of quantum mechanics^{22}will play its part in analyzing the problem of violation of the Bell inequalities, and in suggesting an alternative explanation to the widely accepted explanation of such a violation on the basis of ‘nonlocality’. This book could not have been realized but for the inspiration and support from the part of students, friends and colleagues. First of all I would like to thank the Faculty of Applied Physics of Eindhoven University of Technology and the Department of Theoretical Physics for allowing me to pursue a subject seemingly removed so far from direct application. Hopefully, the insights obtained from a study of the generalized formalism will contribute to future developments of such advanced technologies as quantum computation and quantum communication which are nowadays contemplated as possibilities. In any case, a critical assessment of the way theory is interpreted in this field won’t hurt. On the other hand, it is my conviction that a technological environment can be very beneficial when studying interpretations of physical theories by stimulating a down-to-earth attitude. Awareness of practical realizability constantly reminds the philosopher of what it means to test a physical theory, thus stimulating an operationalist attitude and pushing him into the direction of an empiricist interpretation. - This book is an overview of research done in collaboration with many students. Of these I want to mention in particular Sander Santman and Peter Janssen who started off my interest in the foundations of quantum mechanics
^{23}by being intrigued by the problem of joint measurement of position and momentum, thus establishing incompatibility of quantum mechanical observables as the driving force of my research. Over the years they have been succeeded by a large number of students of applied physics who did not have foundations as their main interest but were nevertheless sufficiently motivated to be engaged with it during some time. Many of them have made substantial contributions. Hans Martens should particularly be thanked here. His contributions evoke a suspicion that foundations might have looked different by now if he had been able to pursue his efforts in this field. I also want to thank Willy DeBaere for a fruitful collaboration, convincing me of the necessity to take into account subquantum theories in understanding quantum mechanics^{24}. My special thanks are due to the participants of the Quantum club started by Jan Hilgevoord as an informal forum for discussion of the foundations of quantum mechanics^{25}more than 25 years ago. Like everyone else I started on a realist interpretation. In contrast to most participants I have decided that it will be more fruitful to view quantum mechanics^{26}in an empiricist way, thus more or less solving a conundrum, unresolved during more than 70 years of discussion, by cutting a Gordian knot. This book gives the reasons for such a decision. It is up to the reader to decide whether these reasons are convincing enough.

→ Willem M. de Muynck, Eindhoven, June 2002

- Full text, but historical quotations omitted.

- Kluwer Academic Publishers, 2002
- Downloaded during Springer promotion, Dec. 2015
- Sadly, the mathematics is impenetrable, though the words might offer some light.

- Blue: Text by me; © Theo Todman, 2020
- Mauve: Text by correspondent(s) or other author(s); © the author(s)

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