ON THE ROLE OF V. HEISENBERG, E. SCHRÖDINGER, K.F. WEIZSACKER, N. BOHR AND M. HEIDEGGER IN THE DEVELOPMENT OF THE LOGICAL AND MATHEMATICAL FOUNDATIONS OF QUANTUM PHYSICS
The development of the logical and mathematical foundations of quantum physics and, first of all, quantum mechanics is the work of a large team of physicists, mathematicians and philosophers who have contributed to the solution of this problem. The title of the article lists the names of four scientists who, according to the author, paved the way for the creation of quantum logic. This must be understood in such a way that from the very beginning (Heisenberg and Schrödinger) it was established that the properties of quantum objects are described not by numbers, but by operators. In the relationships of the operators themselves, one could see an analogue of Boolean algebra of classical logic. Weizsäcker raised the question of how the physical quantity of time, in describing the motion states of quantum objects, can be represented: an operator or a state vector? The author of the article shows that the development of quantum mechanics has gone through two stages. The first stage differs in that complex numbers and functions were introduced into physics, in other words, elements of the theory of functions of a complex variable. At the second stage, spinor calculus penetrated into quantum physics. If at the first stage methods were presented for describing the external degrees of freedom (motion) of elementary particles and other quantum objects, then the second stage was marked by the description of internal degrees of elementary particles. It became possible to understand that the measured mass of an electron or any other fermion is an averaged quantity, it consists of two ingredients: a positive mass and a negative mass (two values of the corresponding operator). An important role in this discovery played the operation of dialectical negation (privation), borrowed from Heidegger's fundamental ontology.
The development of the logical and mathematical foundations of quantum physics and, first of all, quantum mechanics is the work of a large team of physicists, mathematicians and philosophers who have contributed to the solution of this problem. The title of the article lists the names of four scientists who, according to the author, paved the way for the creation of quantum logic. This must be understood in such a way that from the very beginning (Heisenberg and Schrödinger) it was established that the properties of quantum objects are described not by numbers, but by operators. In the relationships of the operators themselves, one could see an analogue of Boolean algebra of classical logic. Weizsäcker raised the question of how the physical quantity of time, in describing the motion states of quantum objects, can be represented: an operator or a state vector? The author of the article shows that the development of quantum mechanics has gone through two stages. The first stage differs in that complex numbers and functions were introduced into physics, in other words, elements of the theory of functions of a complex variable. At the second stage, spinor calculus penetrated into quantum physics. If at the first stage methods were presented for describing the external degrees of freedom (motion) of elementary particles and other quantum objects, then the second stage was marked by the description of internal degrees of elementary particles. It became possible to understand that the measured mass of an electron or any other fermion is an averaged quantity, it consists of two ingredients: a positive mass and a negative mass (two values of the corresponding operator). An important role in this discovery played the operation of dialectical negation (privation), borrowed from Heidegger's fundamental ontology.
In this introductory chapter, which actually is the core of the entire book, we make a serious effort to bring together a variety of ideas from philosophy and physics. We first lay the epistemological foundations for our approach to particle physics. These philosophical foundations, combined with tools borrowed from the amazingly sophisticated, but still not entirely satisfactory, mathematical apparatus of present-day quantum field theory and augmented by some new ideas, are then employed to develop a mathematical representation of fundamental particles and their interactions. In this process, also the relation between “fundamental particle physics” and “quantum field theory” is going to be clarified. This first chapter consists of two voluminous sections: a number of philosophical contemplations followed by a discussion of mathematical and physical elements . The reader might wonder why these massive sections are not presented as two separate chapters. The reason for keeping them together is the desire to emphasize the intimate relation between these two sections: the selection and construction of the mathematical and physical elements is directly based on our philosophical contemplations. The beauty of the entire approach is a fruit of this intimate relationship. The presentation of the material in this chapter is based on the assumption that the reader has a basic working knowledge of linear algebra and quantum mechanics. If that is not the case, the reader might want to consult the equally entertaining and serious introduction Quantum Mechanics: The Theoretical Minimum by Susskind and Friedman [4]. The book Quantum Mechanics and Experience by Albert [5] offers a fascinating introduction with a strong focus on the measurement problem. An almost equation-free discussion of the history and foundations of quantum mechanics can be found in the book Einstein, Bohr and the Quantum Dilemma: From Quantum Theory to Quantum Information by Whitaker [6]. Complex vector spaces, Hilbert space vectors and density matrices for describing states of quantum systems, bosons and fermions, canonical commutation relations, Heisenberg's uncertainty relation, the Schrodinger and Heisenberg pictures for the time evolution of quantum systems, as well as a basic idea of the measurement process are all referred or alluded to in Section 1.1; these basics of quantum mechanics are recapped only very briefly in Section 1.2. The crucial construct of Fock spaces is explained in a loose way in Section 1.1.3 and later elaborated in full detail in Section 1.2.1.
‘Sprache und Wirklichkeit in der modernen Physik’ is the title of a lecture Heisenberg gave in 1960 at the Bayerische Akademie der Schönen Künste. The polemics that had accompanied the establishment of quantum mechanics were over by then and the so-called problems of its foundations had lost much of their initial interest. Einstein had been dead for some years. Right up to the end he had expressed his profound dissatisfaction with a theory which, in his opinion, had something unreasonable about it. With the volume dedicated to him by the community of physicists and philosophers, the long debate between Bohr and Einstein had come to an end. It had been a disappointing conclusion, marked by the reaffirmation of the respective viewpoints of two scientists who by now found great difficulty even in defining a common code of communication. As though seeking to narrow the gap between them, Heisenberg took the opportunity to underline the existence of a common feature in their contributions to 20th century physics. Their discoveries had in fact made it possible to recognize that ‘even the fundamental and most elementary concepts of science, such as space, time, place, velocity, have become problematic and must be re-examined’. The conceptual and cognitive implications of relativity and of quantum mechanics were obviously different but Heisenberg maintained that both theories asked the same question: ‘Does the language we use when we speak of experiments correspond to the artificial language of mathematics which, as we know, describes real relationships correctly; or has it become separated from it so that we must be content with imprecise linguistic formulations and only return to the artificial language of mathematics when we are forced to express ourselves with precision?’.
The ideas in Heisenberg's paper on quantum mechanics in 1925 mark the beginning not only of a new phase in modern physics, but also of Heisenberg's own philosophical journey. This thesis examines that journey between 1925 and the late 1950s by situating Heisenberg's philosophy of quantum mechanics in the context of his encounters with his contemporaries as well as the context of various strands of thought in the German-speaking world at the time. Heisenberg's early philosophical critique of the 'classical' viewpoint between 1925 and 1927 bears the decisive influence of Einstein's theory of relativity, more specifically, the positivism he saw as underpinning Einstein's emancipation from Newtonian physics. The positivist influence on Heisenberg's early attitude to quantum mechanics is evident in three ways: (a) his invocation of an observability principle in 1925 to justify the renunciation of the concept of the electron orbit, (b) an instrumentalist conception of understanding, which characterised Heisenberg's response to Schrodinger's demand for classical visualisation in space and time in 1926-7, and (c) the introduction of an operational definition of concepts such as position and velocity in 1927, in an attempt to replace the concepts of classical physics. But after discussions with Bohr and Einstein in 1926-7, Heisenberg soon recognised what we might term his `empiricist' viewpoint was problematic. In 1927 Heisenberg's thought undergoes a shift away from the `empiricist' viewpoint that had underpinned his early philosophy of quantum mechanics. The nature and scope of this transformation, which forms the central theme in this thesis, has, up until now, been poorly understood and often completely neglected. Through his discussions with Bohr, Heisenberg came to the realisation that despite their limitations, classical concepts were conditions for the possibility of the description of all experience. This marked the abandonment of his earlier attempt to replace classical concepts with quantum concepts. The recognition of the primacy of classical language forms the point of departure for much of Heisenberg's later thought, which brought him into contact with the attempts in the German-speaking world in the 1920s to reconstruct Kantian epistemology. By the mid-1930s, Heisenberg advocated a 'pragmatic transformation' of Kantian philosophy, in which classical concepts were held to be a priori in the sense that they remained the conditions for the possibility of experience, but were no longer held to be necessary or universal in a strict Kantian sense. After 1940 Heisenberg saw the paradoxes of quantum mechanics under the aegis of what can be termed a 'transcendental conception of language', according to which language is not a mere tool, but actively shapes, gives form, and objectifies, our 'reality'. The limits of a classical 'description' in quantum mechanics therefore came to signify for Heisenberg, the limits of 'objective reality'. While Bohr exerted perhaps the most important philosophical influence on Heisenberg, their intellectual relationship was characterised by disagreement and misunderstanding. This is most strikingly displayed in their respective views on wave-particle duality and complementarily. While after 1927 Heisenberg accepted Bohr's basic insight that our knowledge of the quantum world is mediated through classical language, he did not share Rohr's interpretation of complementarity. While Heisenberg certainly used terms such as 'complementarity' and wave-particle duality' in his writings, a close reading reveals that these terms had very different meanings for the two physicists. This is particularly evident in the contrast between Heisenberg's notion of wave-particle equivalence and Bohr's idea of complementarity. In bringing to light these divergences between Bohr and Heisenberg, this thesis lends further weight to the view - already advocated by scholars such as John Hendry and Mara Beller - that the so-called Copenhagen interpretation of quantum mechanics should not be thought of as a unified philosophical position, but actually comprises a number of different strands.
Standard Boolean logic is based on ideas taken from classical, first-order set theory. The first-order logic of quantum mechanics introduced by von Neumann is not a classical one. It is therefore natural to ask whether it is possible to formulate the corresponding theory of quantum sets, their (quantum) topology and higher-order structures. Once we have quantum sets and quantum topologies we can try to quantize essentially anything: phase space, groups, spacetime, etc. In his book Finkelstein reports results of some 40 years of studies in this direction. The book is not addressed to specialists, but to `thoughtful students of physics, mathematics and philosophy', which may justify the fact that this review is written by a person who is not actively working in the field of quantum logic. From this layman's position I must say that the book does not read very well. First of all, the standard language of physics is not naturally suited for speaking about such problems, so Finkelstein invents a new one. Obviously, it is not easy to read a difficult text which, in addition, is written in a language we do not know. It seems that the author should have tried to depart as little as possible from the standard terminology, but he could not resist coining new terms even for things we already know under different names. When I first looked into the table of contents I felt both curiosity and some intimidation since I did not understand the words. To give an example, we find there spinors, cospinors, anti-spinors, anti-cospinors, ambi-spinors, ambi-cospinors, etc. It turned out that they are simply upper- and lower-index, primed or unprimed ordinary spinors, or their second-rank tensor products. Fortunately higher-rank spinors were not needed. The notation also does not help to understand the text. For example, on p 148 the symbol denotes a spacetime index, but three lines later, in equation (11), it is a gauge group parameter, and no comment is made on this. On the other hand, the same object is often denoted in different ways (see, for instance, equation (1) on p 44 and (7) on p 47) and if we add a number of typographical errors the first reading of the book is really tiring. A general philosophy advocated in the book is to use a kind of two-time description which replaces continuous dynamics with discrete algebraic `activity'. The construction begins with a bi-modal logic of initial and final actions and the preclusion (or `occlusion' ) relation ` ' (essentially iff a transition between and is forbidden). A Hilbert space and a field of coefficients are derived from . Discretization of time goes hand-in-hand with the idea that spacetime should be quantized and, finally, an outline of its construction is given. An intriguing consequence of the Finkelstein formalism is the appearance of internal local gauge groups of the standard model just from the spacetime structure. It seems that some results presented in the book are subject to limitations which are not mentioned by the author. For example, the construction based on will not work correctly if the dynamics between the initial and final acts is nonlinear. This is so because mutually precluding acts do not have to correspond to orthogonal vectors if we use nonlinear eigenvalue problems. So details of evolution between initial and final acts may be more important than one would expect naively. The same concerns the proof that transition probabilities must be of the form . Such subtleties can be ignored in standard quantum mechanics but here we aim at generalization. I find some parts of the text misleading. A canonical example illustrating mathematical and physical meanings of the Finkelstein formalism is a Malusian experiment (i.e. the one involving photons and polarizers), but the Bell theorem given on p 251 cannot distinguish between its single-photon and two-photon versions. Physically the two systems are essentially different and a purely classical Aerts model that gives the Malus law for a single particle is known, even though the theorems and discussion on pp 251 - 3 suggest that this is impossible. Perhaps the original work the theorems were taken from clarifies this point but a reader will not find it for two reasons. First, the name of its author is misprinted. Second, there is no reference to the paper. This is quite typical of the book; the results which are regarded as inessential from the viewpoint of the main argument are oversimplified and quoted without references to original works. The clarity of the presentation would be improved if the author did not try to briefly summarize what he knows about physics (and certainly he knows a lot) but, instead, were more detailed about the material that is really new. In spite of all these minor critical remarks I think the book may be inspiring and opens an entirely new perspective on physics. Many of the problems discussed by Finkelstein remain unsolved but the strategy he proposes looks at least promising and, hopefully, some readers will join the author in his brave attack on the most difficult questions of modern science.
Born, de Broglie, Wigner, Bondi, Popper, and others of comparable stature here express their esteem for and his lifelong dedication to his science in a particularly appropriate way: by contributing to this book articles in which they operate at their deepest level of ongoing concern rather than by offering simple tributes or reminiscences. No better salute could be accorded to a man who, in his eighties, continues to operate at such a level himself.Some of the papers deal with the specific areas in which made important contributions to the theory of atomic structure in his younger days, such as the Lande g-factor so important to spectral analysis. Most of them, however, take up the larger issues of the nature of quantum reality and the consistency and completeness inherent in our descriptions of it.Lande was an early supporter of the Copenhagen of quantum phenomena, developed in the late 1920s by Bohr and Heisenberg. This depends on a fundamental duality in which the wave-like and particle-like aspects of matter and radiation are each regarded as necessary but partial views of reality, mutually complementing one another.In his later years, however, became increasingly dissatisfied with this interpretation and began to develop a unitary approach does not depend on this dualism. This was to lead to his demonstration that quantum theory, instead of being a set of enigmatic, though most successful, rules of calculation, can be understood as the logical consequence of a few almost self-evident postulates of symmetry and invariance. To Lande's own scientific conscience, the quantum riddle has found its solution. Even those who take a different point of view, including some of the contributors to this book, treat this solution with the respect its seriousness has earned.The contributors and their topics are as follows: Max Born: An Open Letter to Lande; Louis de Broglie: A New Interpretation Concerning the Coexistence of Waves and Particles; Hans-Jurgen Treder: The Einstein-Bohr Box Experiment; Eugene P. Wigner: Quantum-Mechanical Distribution Functions Revisited; Margenau and James L. Park: The Logic of Noncommutability of Quantum-Mechanical Operators--and Its Empirical Consequences; Alfred Kastler: How the Factor of an Atom Can Be Changed by Putting the Atom in a Radiofrequency Bath; David Bohm: Space-Time Geometry as an Abstraction from Spinor Ordering; Helmut Honl: A Contribution to the Thermodynamics of the Universe and to 3K Radiation; Dennis Caldwell and Henry Eyring: Quantum-Mechanical Rate Processes; O. Costa de Beauregard: Statistical Irreversibility and Quantized Wave Retardation; Fritz Bopp: The Internal Symmetries of Elementary Particles Resulting from the Geometric Structure of Lattice Space; Jean-Pierre Vigier: Hidden Parameter Theory; Karl R. Popper: Particle Annihilation and the Argument of Einstein, Podolsky, and Rosen; Walter M. Elsasser: Philosophical Dissonances in Quantum Mechanics; Leon Rosenfeld: Unphilosophical Considerations on Causality in Physics; Hermann Bondi: Logical Foundations in Physics; Andre Mercier: Forms of Determinism, Objectivity, and the Classification of Sciences; and Paul Bernays: Causality, Determinism, and Probability.
Most philosophical work on quantum physics has concerned simple systems. And for good reason. Even one or two particle systems can exhibit the striking features we have come to associate with quantum physics—features such as entanglement, interference, and nonlocality. But as far as physics goes, such systems barely scrape the surface. In particular, they have only a finite number of degrees of freedom, which means their states are characterized by a finite number of independent parameters. The systems studied in fields such as high-energy particle physics and many-body physics, meanwhile, have uncountably many degrees of freedom. They are far more complicated than the usual philosophical fare.This is not troubling as long as the conceptual heart of quantum physics can be effectively condensed down. But what if philosophers' toy systems are fundamentally different from realistic ones? That would be a problem. And indeed, as Laura Ruetsche convincingly argues in Interpreting Quantum Theories, systems with uncountable degrees of freedom—what Ruetsche collectively calls QM∞—are fundamentally different from the systems that have become the stock and trade of philosophy of physics, in ways that change the interpretational project.Ruetsche's book does two things exceedingly well. The first is to provide a (comparatively) accessible and philosophically oriented introduction to the algebraic approach to quantum physics, which is essential for understanding QM∞. This is a valuable contribution. The algebraic approach is the setting for much recent philosophical work. And yet learning it is notoriously difficult, as even pedagogical texts are aimed at research-level mathematicians. Ruetsche's book is technically demanding, but it works to bring nonmathematicians along. It is, without a doubt, the best place to enter this literature.Second, the book sets the agenda for future work on QM∞, by effectively and judiciously drawing important foundational questions out of a sprawling and difficult physics literature. In this regard, it is a close cousin to John Earman's (1995) excellent Bangs, Crunches, Whimpers, and Shrieks, which has had a significant influence on subsequent philosophy of spacetime physics. Philosophers of physics will be addressing the questions Ruetsche has raised, in many cases for the first time, for years to come.Interpreting Quantum Theories centers on what might be called the problem of inequivalent representations. The quantum theories that philosophers are accustomed to are set in a mathematical structure known as a (separable) Hilbert space, which is an at-most countably infinite dimensional complex vector space. Rays in a Hilbert space represent possible states of a physical system, and self-adjoint operators on that Hilbert space correspond to properties of the system.To treat QM∞, meanwhile, physicists pass to a generalization of this formalism. Now one begins with an abstract algebra of operators corresponding to the various properties associated with the system. States in this more general framework are linear functionals on the algebra whose value at each operator is interpreted as the expectation value of a measurement of that property in that state.The trouble with this more general framework is that the interpretational tools of Hilbert space quantum theories do not directly apply. Some solace comes from recognizing that any (appropriate) algebra of operators admits a representation in terms of the operators on a Hilbert space. This suggests that one can always move back to the more familiar Hilbert space formalism. But this is a pyrrhic victory. In general these Hilbert space representations are neither unique nor empirically equivalent. In particular, these inequivalent representations, interpreted in the familiar way, yield disagreements about which algebraic states are physically realizable.Faced with this situation, Ruetsche presents several principled options—what she calls “pristine” interpretations of QM∞. The two most significant are what she calls “Hilbert Space Conservatism” and “Algebraic Imperialism.” The Conservative claims that, for a given system, there is one true Hilbert space representation, which should be interpreted in the familiar ways. The Imperialist, meanwhile, insists that only the abstract algebra matters: the physical states are the algebraic states, properties correspond to the operators in the algebra, and if our old methods do not apply, so much the worse for our methods.At this level of description, Imperialists may appear to have the advantage, since they need not fret over which representation to choose. But there's a catch. As we have seen, there is a sense in which Imperialists generally recognize a broader range of possible physical states. But Conservatives allow for more properties. This is because when one represents the abstract algebra as operators on a Hilbert space, the operators in the representation naturally approximate additional operators on that Hilbert space, in a way that is unavailable in the abstract setting. Thus each interpretation suffers from limitations that the other does not.To adjudicate between the interpretations, Ruetsche takes them to the trenches, to see which provides the resources to support the actual explanatory and representational practices of physics. After a masterful survey, she concludes that neither interpretation has all of the requisite resources. In particular, she argues, physicists regularly make essential use of operators that are only available to Conservatives, while making equally essential use of multiple inequivalent representations—which only Imperialists can support. More, the particular mix of Conservatism and Imperialism varies between applications, apparently ruling out a stable interpretation of the formalism that can be articulated independently of the messy details. We are left with what Ruetsche calls “adulterated” interpretations.Philosophers' practice notwithstanding, it might seem unsurprising that we cannot separate our interpretation of physical theories from the pragmatics of their application to the world. After all, we are interested in these theories only insofar as they are successfully applied. But Ruetsche believes the problem is more acute. It is not merely that pragmatic considerations enter the debate between Conservatives and Imperialists. Ruetsche argues that there may not be any consistent, general interpretation of QM∞ compatible with practice. And this, she claims, makes realism about QM∞ untenable.The idea that the applications of a broad class of our best physical theories undermine their consistent interpretation in a way that renders realism incoherent is striking. Insofar as Ruetsche's case is convincing, it should have serious repercussions for general philosophy of science—both because it bears on important debates and because it shows how generalist questions depend on the details of particular scientific theories and their applications. Indeed, Ruetsche's skepticism about the terms of the realism debate and her radical particularism should influence general philosophy of science. But that said, her argument ultimately leaves room for disagreement.Ruetsche's basic claim is that Conservatism and Imperialism each support practices that the other cannot and that both kinds of practice are essential. But there is an important asymmetry here. Conservatives suffer because they fail to represent putative physical possibilities appealed to in practice. Imperialists, meanwhile, need to justify that in certain applications, they may limit attention to a subset of the physical possibilities their interpretation admits, and then use resources that only make sense in those limited regimes.The trouble is that the Imperialists' opportunism is neither unusual in physics nor difficult to motivate. In spacetime physics, for instance, properties such as the total mass of the universe or global time can be defined only in very special cases. And yet such properties are used in practice, because often the salient spacetimes are precisely the ones in which these quantities make (approximate) sense. To make her case that adulteration in QM∞ undermines realism, Ruetsche would need to establish that the representation-dependent quantities used in applications of QM∞ are different in kind from analogous quantities that appear throughout physics. And here it is not clear that she succeeds.None of this undercuts the importance of the book, however. Indeed, Ruetsche acknowledges that her argument may depend on how one understands idealization in physics. And even if one takes the view just described, there is a great deal of work to do on how Imperialism can support the practices Ruetsche identifies—work that will surely proceed within the framework Ruetsche has laid out, by addressing questions Ruetsche has raised. And so, wherever our study of QM∞ takes us, Ruetsche's book is sure to play a central role.