Review of "The Collected Works of Eugene Paul Wigner", Volume I, III, and VI. Excerpt from the Conclusions: Many of Wigner’s papers on mathematical physics are great classics. Most famous is his work on group representations which is of lasting value for a proper mathematical foundation of quantum theory. The modern development of quantum theory (which is not reflected in Wigner’s work) is in an essential way a representation theory (e.g. representations of kinematical groups, or representations of C*-algebras). This view owes very much to Wigner’s seminal papers on the unitary representations of compact and noncompact groups. Wigner showed much courage in relating the then unresolved questions of the measurement problem to the much deeper problem of consciousness. In view of this very unorthodox proposal it is astonishing that Wigner was very reactionary with respect of the dogmas of orthodox quantum mechanics. In contrast to von Neumann himself, he took the old von-Neumann codification of quantum mechanics as authoritative and not to be questioned. Much of the efforts to interpret the meaning of this codification and to prove no-go theorems, such as the insolubility of the measurement problem or the impossibility of a quantum theory of individual objects, are physically irrelevant since they are based on a codification of quantum mechanics that is valid only for strictly closed systems with finitely many degrees of freedom. However, in nature there are no such systems. Every material system is coupled to the gravitational and to the electromagnetic field – systems which require in a Hamiltonian description infinitely many degrees of freedom. A deeper insight into the conceptual problems of quantum theory is possible only if the modern development of a quantum theory of infinite systems is taken into account.
Review of "The Collected Works of Eugene Paul Wigner", Volume I, III, and VI. Excerpt from the Conclusions: Many of Wigner’s papers on mathematical physics are great classics. Most famous is his work on group representations which is of lasting value for a proper mathematical foundation of quantum theory. The modern development of quantum theory (which is not reflected in Wigner’s work) is in an essential way a representation theory (e.g. representations of kinematical groups, or representations of C*-algebras). This view owes very much to Wigner’s seminal papers on the unitary representations of compact and noncompact groups. Wigner showed much courage in relating the then unresolved questions of the measurement problem to the much deeper problem of consciousness. In view of this very unorthodox proposal it is astonishing that Wigner was very reactionary with respect of the dogmas of orthodox quantum mechanics. In contrast to von Neumann himself, he took the old von-Neumann codification of quantum mechanics as authoritative and not to be questioned. Much of the efforts to interpret the meaning of this codification and to prove no-go theorems, such as the insolubility of the measurement problem or the impossibility of a quantum theory of individual objects, are physically irrelevant since they are based on a codification of quantum mechanics that is valid only for strictly closed systems with finitely many degrees of freedom. However, in nature there are no such systems. Every material system is coupled to the gravitational and to the electromagnetic field – systems which require in a Hamiltonian description infinitely many degrees of freedom. A deeper insight into the conceptual problems of quantum theory is possible only if the modern development of a quantum theory of infinite systems is taken into account.
Von Neumann's theory of measurement in Quantum Mechanics was spelled out in the last chapter of his book, which was published in 1932. This book was highly mathematical for its time, and in 1939 London and Bauer provided a simplified account of the measurement theory part of it (London and Bauer, 1939). Von Neumann died in 1957. Thirty years after the publication of von Neumann's book, Wigner published a review containing his views on the shortcomings of von Neumann's theory, but omitting any discussion of its mathematical core, namely von Neumann's analysis of composite systems (Wigner, 1963). He also published a set of lecture notes entitled Interpretation of Quantum Mechanics (Wigner, 1983) in which some of his concerns were spelled out in greater detail, and an article addressed ‘to an audience of non-physicists’ (Wigner, 1964). Wigner's own contributions to measurement theory were discussed by Shimony in a talk at the Wigner centennial conference (Shimony, 2002). The English translation of von Neumann's book was published in 1955.1 The account that follows is based on these sources. We shall assume that the reader is acquainted with notions such as wave function collapse and the Heisenberg cut, but we shall not assume familiarity with the technicalities of von Neumann's theory. This chapter is organized accordingly. Section 8.1 explains what we mean by the term von Neumann's measurement theory and gives an overview of the subject. It is followed by Sections 8.2 and 8.3, in which the theory is spelled out in detail. Section 8.4 recounts Wigner's reservations.
Quantum theory is the soul of contemporary physics. It was discovered in an adventurous way, under the urge to solve the puzzles posed by atomic spectra and blackbody radiation. But after its invention, it immediately became clear that it was not just a theory of specific physical systems: it was rather a new language of universal applicability. Already in 1928, the theory had received solid mathematical foundations by Hilbert, von Neumann, and Nordheim, and this work was brought to completion in the monumental work of von Neumann, in the form that we still study nowadays. The theory is extraordinarily successful, and its predictions have been confirmed to an astonishing level of precision in a large spectrum of experiments. However, almost 90 years after von Neumann's book, quantum theory remains mysterious. Its mathematical formulation – based on Hilbert spaces and self-adjoint operators – is far from having an intuitive interpretation. The association of physical systems to Hilbert spaces whose unit vectors represent pure states, the representation of transformations by unitary operators and of observables by self-adjoint operators – all such postulates look artificial and ad hoc . A slightly more operational approach is provided by the C * -algebraic formulation of quantum theory – still, this formulation relies on the assumption that observables form an algebra , where the physical meanings of the multiplication and the sum are far from clear. In short, the postulates of quantum theory impose mathematical structures without providing any simple reason for this choice: the mathematics of Hilbert spaces is adopted as a magic blackbox that “works well” at producing experimental predictions. However, in a satisfactory axiomatization of a physical theory the mathematical structures should emerge as a consequence of postulates that have a direct physical interpretation. By this we mean postulates referring, e.g., to primitive notions like physical system, measurement, or process, rather than notions like, e.g., Hilbert space, C * -algebra, unit vector, or selfadjoint operator. The crucial question thus remains unanswered: why quantum theory? Which are the principles at the basis of the theory? A case that is often invoked in contrast is that of Special Relativity theory, which directly follows from the simple understandable principle of relativity.
Abstract The book starts with a description of classical mechanics then discusses the quantum phenomena that require us to give up our commonsense classical intuitions. We consider the physical and conceptual arguments that led to the standard von Neumann-Dirac formulation of quantum mechanics and how the standard theory explains quantum phenomena. This includes a discussion of how the theory’s two dynamical laws work with the standard interpretation of states to explain determinate measurement records, quantum statistics, interference effects, entanglement, decoherence, and quantum nonlocality. A careful understanding of how the standard theory works ultimately leads to the quantum measurement problem. We consider how the measurement problem threatens the logical consistency of the standard theory then turn to a discussion of the main proposals for resolving it. This includes collapse formulations of quantum mechanics like Wigner’s extension of the standard theory and the GRW approach and no-collapse formulations like pure wave mechanics, the various many-worlds theories, and Bohmian mechanics. In discussing alternative formulations of quantum mechanics we pay particular attention to the explanatory role played by each theory’s empirical ontology and associated metaphysical commitments and the conceptual trade-offs between theoretical options.
Quantum mechanics is usually defined in terms of some loosely connected axioms and rules. Such a foundation is far from the beauty of, e.g., the `principles' underlying classical mechanics. Motivated, in addition, by notorious interpretation problems, there have been numerous attempts to modify or `complete' quantum mechanics. A first attempt was based on so-called hidden variables; its proponents essentially tried to expel the non-classical nature of quantum mechanics. More recent proposals intend to complete quantum mechanics not within mechanics proper but on a `higher (synthetic) level'; by means of a combination with gravitation theory (R Penrose), with quantum information theory (C M Caves, C A Fuchs) or with psychology and brain science (H P Stapp). I think it is fair to say that in each case the combination is with a subject that, per se , suffers from a very limited understanding that is even more severe than that of quantum mechanics. This was acceptable, though, if it could convincingly be argued that scientific progress desperately needs to join forces. Quantum mechanics of a closed system was a beautiful and well understood theory with its respective state being presented as a point on a deterministic trajectory in Liouville space---not unlike the motion of a classical N -particle system in its 6N -dimensional phase-space. Unfortunately, we need an inside and an outside view, we need an external reference frame, we need an observer. This unavoidable partition is the origin of most of the troubles we have with quantum mechanics. A pragmatic solution is introduced in the form of so-called measurement postulates: one of the various incompatible properties of the system under consideration is supposed to be realized (i.e. to become a fact, to be defined without fundamental dispersion) based on `instantaneous' projections within some externally selected measurement basis. As a result, the theory becomes essentially statistical rather than deterministic; furthermore there is an asymmetry between the observed and the observing. This is the point where consciousness may come in. Complemented by an introduction and several appendices, Henry Stapp's book consists essentially of three parts: theory, implications, and new developments. The theory part gives a very readable account of the Copenhagen interpretation, some aspects of a psychophysical theory, and, eventually, hints towards a quantum foundation of the brain--mind connection. The next part, `implications', summarizes some previous attempts to bridge the gap between the working rules of quantum mechanics and their possible consequences for our understanding of this world (Pauli, Everett, Bohm, Heisenberg). The last section, `new developments', dwells on some ideas about the conscious brain and its possible foundation on quantum mechanics. The book is an interesting and, in part, fascinating contribution to a field that continues to be a companion to `practical' quantum mechanics since its very beginning. It is doubtful whether such types of `quantum ontologies' will ever become (empirically) testable; right now one can hardly expect more than to be offered some consistent `grand picture', which the reader may find more or less acceptable or even rewarding. Many practicing quantum physicists, though, will remain unimpressed. The shift from synthetic ontology to analytic ontology is the foundation of the present work. This means that fundamental wholes are being partitioned into their ontologically subordinate components by means of `events'. The actual event, in turn, is an abrupt change in the Heisenberg state describing the quantum universe. The new state then defines the tendencies associated with the next actual event. To avoid infinite regression in terms of going from one state of tendencies to the next, consciousness is there to give these events a special `feel', to provide a status of `intrinsic actuality'. The brain of an alert human observer is similar in an important way to a quantum detection device: it can amplify small signals to large macroscopic effects. On the other hand, actual events are not postulated to occur exclusively in brains. They are more generally associated with the formation of records. Records are necessarily part of the total state of the universe: it is obvious that the state of the universe cannot undergo a Schrödinger dynamics and at the same time record its own history. `The full universe consists therefore of an exceedingly thin veneer of relatively sluggish, directly observable properties resting on a vast ocean or rapidly fluctuating unobservable ones.' The present ideas also bear on how the world should be seen to develop. While conventional cosmology encounters problems as to how to define the intial conditions, which would enter the governing equations of motion, here `the boundary conditions are set not at some initial time, but gradually by a sequence of acts that imposes a sequence of constraints. After any sequence of acts there remains a collection of possible worlds, some of which will be eliminated by the next act.' Connected with those acts is `meaning': there has always been some speculation about the special significance of local properties in our understanding of the world. One could argue that correlations (even the quantum correlations found, e.g., in the EPR-experiments) were as real as anything else. But also Stapp stresses the special role of locality: the `local observable properties, or properties similar to them are the natural, and perhaps exclusive, carriers of meaning in the quantum universe. From this point of view the quantum universe tends to create meaning.' This sounds like an absolute concept: meaning not with respect to something else, but defined intrinsically---not easy to digest. The role of consciousness in the developing quantum universe requires more attention. `The causal irrelevance of our thoughts within classical physics constitutes a serious deficiency of that theory, construed as a description of reality.' This is taken to be entirely different within quantum mechanics. `The core idea of quantum mechanics is to describe our activities as knowledge-seeking and knowledge-using agents.' `21st century science does not reduce human beings to mechanical automata. Rather it elevates human beings to agents whose free choices can, according to the known laws, actually influence their behaviour.' An example with respect to perception is discussed: `Why, when we look at a triangle, do we experience three lines joined at three points and not some pattern of neuron firings?' The brain `does not convert an actual whole triangle into some jumbled set of particle motions; rather it converts a concatenation of separate external events into the actualization of some single integrated pattern of neural activity that is congruent to the perceived whole triangle.' How convincing is this proposal? It is hard to tell. I think Henry Stapp did a good job, but there are tight limitations to any such endeavour. Quantum mechanics is often strange indeed, but it also gives rise to our classical world around us. For the emergence of classicality jumps and measurement projections (the basic phenomena connected with those fundamental events of choice) are not needed. Therefore, I doubt whether the explanation of the evolution of our world really allows (or requires) that much free choice. On the other hand, most scientist will agree that empirical science was not possible without free will: we could not ask independent questions if this asking was part of a deterministic trajectory. The fact that the result of a quantum measurement is indeterminate (within given probabilities) does certainly not explain free will. How about the type of measurment? The experimentalist will have to assume that he can select the pertinent observable within some limits. But given a certain design the so-called pointer basis (producing stable measurement results) is no longer a matter of free choice. `The main theme of classical physics is that we live in a clocklike universe.' Today it is often assumed that the universe was a big (quantum-) computer or a cellular automaton. Many would be all too happy to leave that rather restrictive picture behind. But where to go? Stapp suggests giving consciousness a prominent role: `The most profound alteration of the fundamental principles was to bring consciousness of human beings into the basic structure of the physical theory.' How far we are able to go in this direction will depend on the amount of concrete research results becoming available to support this view.
When investigating various interpretations of quantum mechanics one notices that each interpretation contains an element which escapes a complete and full description. This element is always associated with the stochasticity of the individual event in the quantum measurement process. It appears that the implications of this limit to any description of the world has not been sufficiently appreciated with notable exceptions of, for example, Heisenberg, Pauli and Wheeler. If we assume that a deeper foundation of quantum mechanics is possible, the question arises which features such a philosophical foundation might have. It is suggested that the objective randomness of the individual quantum event is a necessity of a description of the world in view of the significant influence the observer in quantum mechanics has. It is also suggested that the austerity of the Copenhagen interpretation should serve as a guiding principle in a search for deeper understanding. In recent years, a significant increase of interest in the foundations of quantum mechanics can be observed. This increase is certainly related to the immense progress made during the last two decades in experiments with individual quantum systems. This technological progress has made it possible to really perform more and more of the famous gedanken experiments which were so important in the early days of the theory. Not only did these experiments confirm the theory in every detail, they also opened up avenues for new experimental directions which might even lead some day to new technology. While such applications are certainly still far away, areas like quantum cryptography[1], quantum computation[2] and interaction-free measurement[3], just to name a few examples, certainly deserve the attention they receive. All this activity has also drawn more attention to the problems of understanding and interpreting quantum mechanics. It seems that there, a definitive consensus has not yet been reached. As witnesses quote authorities like Feynman as saying[4]: I think can safely say that nobody today understands quantum mechanics, whereby he apparently included himself, or like Roger Penrose[5], who describes his opinion remarking that while the theory agrees incredibly well with experiment and while it is of profound mathematical beauty, it makes absolutely no sense. Why did even physicists who have contributed so significantly to quantum theory Feynman was even rewarded the Nobel Prize for one of its mathematical formulations choose such emphatic and strong formulations? Why is it that the average physicist hears very little about such problems during his/her education? Why is the comprehension of the theory very often focused on formalism while questions that probe for deeper meaning are usually not tackled? To take a closer look at the situation, let us turn to a brief analysis of the interpretation of quantum mechanics. We note that there are at least two levels of interpreting a formalism, a physical theory. The first, basic, level supplies the rules that determine which element of the formalism corresponds to which measurable quantity or to which observable fact in a concrete experimental situation. Theses rules are a large, mostly not explicated but only implicit, set of instructions. These instructions concern the manner of working with the statements of the theory in order to obtain predictions for future experiments or events, and they concern the instructions on how to proceed in experiment in order to demonstrate or test a theoretical
The newly revised edition of this well-known book categorizes prevalent views on measurement as described by quantum mechanics, discusses basics agreed to by many who on other issues disagree, and describes some of the problems. Although many paragraphs rest unchanged from the first (1991) edition, results found since 1991 are used to reorganize and sharpen the focus of the book, so that even the old paragraphs are now understood differently and more clearly. One advance in the second edition is a new and more general no-go theorem; another is separation of formal derivations from interpretation. The dozen or so definitions essential to following the book even casually are stated clearly. By this means the authors succeed pretty well in achieving their stated desire to make the book comprehensible to those who do not share their philosophic predispositions. While I differ in philosophy from the authors, attending to their discussion uncovered weaknesses in my own thinking and improved it. The mathematical language of quantum mechanics expresses a system under measurement by states (as rays in a Hilbert space) and a hamiltonian operator, distinct from the apparatus that produces outcomes, which is expressed by a positive operator-valued (POV) measure. In this way quantum language separates state preparation from state measurement. Often the Hilbert space is viewed as a tensor product of subspaces, allowing further conceptual separations. The authors take the stance that quantum mechanics calls for splitting the empirical world into parts, the most essential being (1) objective systems S (to be observed) and (2) apparatus A (preparation and registration devices that produces outcomes (or, as the authors write, definite pointer values) of the observed system). In some interpretations, one or both of an observer and an environment appear as additional parts. What is to be included in the system S versus the apparatus A is not specified by the language, but is open to the theoretician; a probe, for example, can be analysed as either part of the apparatus or as part of the system, as was discussed by von Neumann in work cited by the authors. With what should a theory of measurement be concerned? A broad division of approaches is defined by how one answers the question: should one try to do away with the cut between measuring apparatus and system? The authors phrase this as a choice of REFERENT, a term which in my mind puts them on shaky semantic ground; anyway The Quantum Theory of Measurement is concerned with implications of the `yes'. The book unfolds with the power of a tragedy, in which the logical consequences lead, step-by-step, to the statement and proof of theorem 6.2.1, which, roughly speaking, asserts the non-existence of a solution to the problem of getting definite pointer values from measuring apparatus that exhibits the superposition demanded by its description as a quantum system. Possible responses to this theorem are discussed, among them the many-worlds interpretation and a modification of quantum mechanics that elevates decoherence to a principle. Citations of the literature are thorough, and many points of view are crisply summarized. The book is indispensable to both those who work in the broad direction chosen by the authors and to those who wish to set any other direction in context.