The classical roots of wave mechanics: Schrödinger's transformations of the optical-mechanical analogy

In the 1830s, W. R. Hamilton established a formal analogy between optics and mechanics by constructing a mathematical equivalence between the extremum principles of ray optics (Fermat's principle) and corpuscular mechanics (Maupertuis's principle). Almost a century later, this optical-mechanical analogy played a central role in the development of wave mechanics. Schrödinger was well acquainted with Hamilton's analogy through earlier studies. From Schrödinger's research notebooks, we show how he used the analogy as a heuristic tool to develop de Broglie's ideas about matter waves and how the role of the analogy in his thinking changed from a heuristic tool into a formal constraint on possible wave equations. We argue that Schrödinger only understood the full impact of the optical-mechanical analogy during the preparation of his second communication on wave mechanics: Classical mechanics is an approximation to the new undulatory mechanics, just as ray optics is an approximation to wave optics. This completion of the analogy convinced Schrödinger to stick to a realist interpretation of the wave function, in opposition to the emerging mainstream. The transformations in Schrödinger's use of the optical-mechanical analogy can be traced in his research notebooks, which offer a much more complete picture of the development of wave mechanics than has been previously thought possible.     

Pluralism and anarchism in quantum physics: Paul Feyerabend's writings on quantum physics in relation to his general philosophy of science

This paper aims to show that the development of Feyerabend's philosophical ideas in the 1950s and 1960s largely took place in the context of debates on quantum mechanics.In particular, he developed his influential arguments for pluralism in science in discussions with the quantum physicist David Bohm, who had developed an alternative approach to quantum physics which (in Feyerabend's perception) was met with a dogmatic dismissal by some of the leading quantum physicists. I argue that Feyerabend's arguments for theoretical pluralism and for challenging established theories were connected to his objections to the dogmatism and conservatism he observed in quantum physics.However, as Feyerabend gained insight into the physical details and historical complexities which led to the development of quantum mechanics, he gradually became more modest in his criticisms. His writings on quantum mechanics especially engaged with Niels Bohr; initially, he was critical of Bohr's work in quantum mechanics, but in the late 1960s, he completely withdrew his criticism and even praised Bohr as a model scientist. He became convinced that however puzzling quantum mechanics seemed, it was methodologically unobjectionable – and this was crucial for his move towards ‘anarchism’ in philosophy of science.     

Laws of nature and natural laws

The relationship between conceptions of law and conceptions of nature is a complex one, and proceeds on what appear to be two distinct fronts. On the one hand, we frequently talk of nature as being lawlike or as obeying laws. On the other hand there are schools of philosophy that seek to justify ethics generally, or legal theory specifically, in conceptions of nature. Questions about the historical origins and development of claims that nature is lawlike are generally treated as entirely distinct from the development of ethical natural law theories. By looking at the many intersections of law and nature in antiquity, this paper shows that such a sharp distinction is overly simplistic, and often relies crucially on the imposition of an artificial and anachronistic suppression of the role of gods or divinity in the worlds of ancient natural philosophy. Furthermore, by tightening up the terms of the debate, we see that the common claim that a conception of ‘laws of nature’ only emerges in the Scientific Revolution is built on a superficial reading of the ancient evidence.     

Non-integrability and mixing in quantum systems: On the way to quantum chaos

In spite of the increasing attention that quantum chaos has received from physicists in recent times, when the subject is considered from a conceptual viewpoint the usual opinion is that there is some kind of conflict between quantum mechanics and chaos. In this paper we follow the program of Belot and Earman, who propose to analyze the problem of quantum chaos as a particular case of the classical limit of quantum mechanics. In particular, we address the problem on the basis of our account of the classical limit, which in turn is grounded on the self-induced approach to decoherence. This strategy allows us to identify the conditions that a quantum system must satisfy to lead to non-integrability and to mixing in the classical limit.     

The quantitative content of statistical mechanics

I give a brief account of the way in which thermodynamics and statistical mechanics actually work as contemporary scientific theories, and in particular of what statistical mechanics contributes to thermodynamics over and above any supposed underpinning of the latter׳s general principles. In doing so, I attempt to illustrate that statistical mechanics should not be thought of wholly or even primarily as itself a foundational project for thermodynamics, and that conceiving of it this way potentially distorts the foundational study of statistical mechanics itself.     

Ten reasons why a thermalized system cannot be described by a many-particle wave function

It is widely believed that the underlying reality behind statistical mechanics is a deterministic and unitary time evolution of a many-particle wave function, even though this is in conflict with the irreversible, stochastic nature of statistical mechanics. The usual attempts to resolve this conflict for instance by appealing to decoherence or eigenstate thermalization are riddled with problems. This paper considers theoretical physics of thermalized systems as it is done in practice and shows that all approaches to thermalized systems presuppose in some form limits to linear superposition and deterministic time evolution. These considerations include, among others, the classical limit, extensivity, the concepts of entropy and equilibrium, and symmetry breaking in phase transitions and quantum measurement. As a conclusion, the paper suggests that the irreversibility and stochasticity of statistical mechanics should be taken as a real property of nature. It follows that a gas of a macroscopic number N of atoms in thermal equilibrium is best represented by a collection of N wave packets of a size of the order of the thermal de Broglie wave length, which behave quantum mechanically below this scale but classically sufficiently far beyond this scale. In particular, these wave packets must localize again after scattering events, which requires stochasticity and indicates a connection to the measurement process.     

Holism, physical theories and quantum mechanics

Motivated by the question what it is that makes quantum mechanics a holistic theory (if so), I try to define for general physical theories what we mean by `holism'. For this purpose I propose an epistemological criterion to decide whether or not a physical theory is holistic, namely: a physical theory is holistic if and only if it is impossible in principle to infer the global properties, as assigned in the theory, by local resources available to an agent. I propose that these resources include at least all local operations and classical communication. This approach is contrasted with the well-known approaches to holism in terms of supervenience. The criterion for holism proposed here involves a shift in emphasis from ontology to epistemology. I apply this epistemological criterion to classical physics and Bohmian mechanics as represented on a phase and configuration space respectively, and for quantum mechanics (in the orthodox interpretation) using the formalism of general quantum operations as completely positive trace non-increasing maps. Furthermore, I provide an interesting example from which one can conclude that quantum mechanics is holistic in the above mentioned sense, although, perhaps surprisingly, no entanglement is needed.     

A close examination of the pseudo-Aristotelian Mechanical Problems: The homology between mechanics and poetry as technē

The pseudo-Aristotelian Mechanical Problems is the earliest known ancient Greek text on mechanics, principally concerned with the explanation of a variety of mechanical phenomena using a particular construal of the principle of the lever. In the introduction, the author—thought to be an early Peripatetic—quotes the tragic poet Antiphon to summarise a discussion of the technē-physis (art-nature) relationship and the status of mechanics as a technē. I argue that this citation of a poet is an Aristotelian cultural signature, intended to guide its readers towards a better understanding of the nature of mechanics as expounded in the Mechanical Problems. By analysing several instances where Aristotle cites Antiphon (as well as other tragic poets) in the Aristotelian corpus, I propose that both the author of the Mechanical Problems and Aristotle use poets for the purpose of persuasion. This is in turn explained by understanding the homologous relationship between mechanics-as-technē (according to the author of the Mechanical Problems) and poetics-as-technē (according to Aristotle) in terms of their shared status as poiētikē technē (productive art) and claims to universal knowledge. A final facet of the proposed relationship between mechanics and poetry is hypothesised on the grounds of their mimetic nature.     

Is the classical limit “singular”?

We argue against claims that the classical ? → 0 limit is “singular” in a way that frustrates an eliminative reduction of classical to quantum physics. We show one precise sense in which quantum mechanics and scaling behavior can be used to recover classical mechanics exactly, without making prior reference to the classical theory. To do so, we use the tools of strict deformation quantization, which provides a rigorous way to capture the ? → 0 limit. We then use the tools of category theory to demonstrate one way that this reduction is explanatory: it illustrates a sense in which the structure of quantum mechanics determines that of classical mechanics.     

Kant and the nature of matter: Mechanics,chemistry, and the life sciences

Kant believed that the ultimate processes that regulate the behavior of material bodies can be characterized exclusively in terms of mechanics. In 1790, turning his attention to the life sciences, he raised a potential problem for his mechanically-based account, namely that many of the operations described in the life sciences seemed to operate teleologically. He argued that the life sciences do indeed require us to think in teleological terms, but that this is a fact about us, not about the processes themselves. Nevertheless, even were we to concede his account of the life sciences, this would not secure the credentials of mechanics as a general theory of matter. Hardly any material properties studied in the second half of the eighteenth century were, or could have been, conceived in mechanical terms. Kant's concern with teleology is tangential to the problems facing a general matter theory grounded in mechanics, for the most pressing issues have nothing to do with teleology. They derive rather from a lack of any connection between mechanical forces and material properties. This is evident in chemistry, which Kant dismisses as being unscientific on the grounds that it cannot be formulated in mechanical terms.     

Hertz's Mechanics and a unitary notion of force

Heinrich Hertz dedicated the last four years of his life to a systematic reformulation of mechanics. One of the main issues that troubled Hertz in the customary formulation of mechanics was a ‘logical obscurity’ in the notion of force. However, it is unclear what this logical obscurity was, hence it is unclear how Hertz took himself to have avoided it.In this paper, I argue that a subtle ambiguity in Newton's original laws of motion lay at the basis of Hertz's concerns; an ambiguity which led to the development of two slightly different notions of force. I then show how Hertz avoided this ambiguity by deriving a unitary notion of force, thus dispelling the obscurity that lurked in the customary representation of mechanics.     

Experimental : The double standard in the quantum-information approach to the foundations of quantum theory

Among the alternatives of non-relativistic quantum mechanics (NRQM) there are those that give different predictions than quantum mechanics in yet-untested circumstances, while remaining compatible with current empirical findings. In order to test these predictions, one must isolate one's system from environmental induced decoherence, which, on the standard view of NRQM, is the dynamical mechanism that is responsible for the ‘apparent’ collapse in open quantum systems. But while recent advances in condensed-matter physics may lead in the near future to experimental setups that will allow one to test the two hypotheses, namely genuine collapse vs. decoherence, hence make progress toward a solution to the quantum measurement problem, those philosophers and physicists who are advocating an information-theoretic approach to the foundations of quantum mechanics are still unwilling to acknowledge the empirical character of the issue at stake. Here I argue that in doing so they are displaying an unwarranted double standard.     

A remark on Fuchs’ Bayesian interpretation of quantum mechanics

Quantum mechanics is a theory whose foundations spark controversy to this day. Although many attempts to explain the underpinnings of the theory have been made, none has been unanimously accepted as satisfactory. Fuchs has recently claimed that the foundational issues can be resolved by interpreting quantum mechanics in the light of quantum information. The view proposed is that quantum mechanics should be interpreted along the lines of the subjective Bayesian approach to probability theory. The quantum state is not the physical state of a microscopic object. It is an epistemic state of an observer; it represents subjective degrees of belief about outcomes of measurements. The interpretation gives an elegant solution to the infamous measurement problem: measurement is nothing but Bayesian belief updating in a analogy to belief updating in a classical setting. In this paper, we analyze an argument that Fuchs gives in support of this latter claim. We suggest that the argument is not convincing since it rests on an ad hoc construction. We close with some remarks on the options left for Fuchs’ quantum Bayesian project.     

On the verge of <Emphasis Type="Italic">Umdeutung</Emphasis> in Minnesota: Van Vleck and the correspondence principle. Part one

In October 1924, The Physical Review, a relatively minor journal at the time, published a remarkable two-part paper by John H. Van Vleck, working in virtual isolation at the University of Minnesota. Using Bohr’s correspondence principle and Einstein’s quantum theory of radiation along with advanced techniques from classical mechanics, Van Vleck showed that quantum formulae for emission, absorption, and dispersion of radiation merge with their classical counterparts in the limit of high quantum numbers. For modern readers Van Vleck’s paper is much easier to follow than the famous paper by Kramers and Heisenberg on dispersion theory, which covers similar terrain and is widely credited to have led directly to Heisenberg’s Umdeutung paper. This makes Van Vleck’s paper extremely valuable for the reconstruction of the genesis of matrix mechanics. It also makes it tempting to ask why Van Vleck did not take the next step and develop matrix mechanics himself. This paper was written as part of a joint project in the history of quantum physics of the Max Planck Institut für Wissenschaftsgeschichte and the Fritz-Haber-Institut in Berlin.     

“A dedicated missionary”. Charles Galton Darwin and the new quantum mechanics in Britain

In this paper I discuss the work on quantum physics and wave mechanics by Charles Galton Darwin, a Cambridge wrangler of the last generation, as a case study to better understand the early reception of quantum physics in Britain. I argue that his proposal in the early 1920s to abandon the strict conservation of energy, as well as his enthusiastic embracement of wave mechanics at the end of the decade, can be easily understood by tracing his ontological and epistemological commitments to his early training in the Cambridge Mathematical Tripos. I also suggest that Darwin's work cannot be neglected in a study of quantum physics in Britain, since he was one of very few fellows of the Royal Society able to judge and explain quantum physics and quantum mechanics.     

Is there a stability problem for Bayesian noncommutative probabilities?

A stability condition for Bayesian statistical inference, which Redei [(1992). When can non-commutative statistical inference be Bayesian? International Studies in the Philosophy of Science, 6, 129–132; (1998). Quantum logic in algebraic approach. Dordrecht: Kluwer Academic Publishers] formulated as a rationality constraint holding in classical probability theory, is shown to fail in quantum mechanics. That allegedly challenges a Bayesian interpretation of quantum probabilities. In this paper we demonstrate that Redei's argument does not apply to quantum mechanics. Moreover, we provide a solution to the problem of Bayesian noncommutative statistical inference arising from the violation of stability condition in general probability spaces.     

Rheological behavior of mammalian cells

Rheological properties of living cells determine how cells interact with their mechanical microenvironment and influence their physiological functions. Numerous experimental studies have show that mechanical contractile stress borne by the cytoskeleton and weak power-law viscoelasticity are governing principles of cell rheology, and that the controlling physics is at the level of integrative cytoskeletal lattice properties. Based on these observations, two concepts have emerged as leading models of cytoskeletal mechanics. One is the tensegrity model, which explains the role of the contractile stress in cytoskeletal mechanics, and the other is the soft glass rheology model, which explains the weak power-law viscoelasticity of cells. While these two models are conceptually disparate, the phenomena that they describe are often closely associated in living cells for reasons that are largely unknown. In this review, we discuss current understanding of cell rheology by emphasizing the underlying biophysical mechanism and critically evaluating the existing rheological models. Received 25 May 2008; received after revision 19 June 2008; accepted 1 July 2008     

In defence of Copenhagenism

We rebut the objections to the Copenhagen interpretation of quantum mechanics presented by Park [9,10], Margenau [10], and Popper [11]. It seems to us that these authors, having adopted different interpretations of quantum mechanics, have been unable to grasp the perspective of the Copenhagenist. They therefore miss the points which the Copenhagenist is making when he: (a) accords a special status to observations in quantum theory; (b) attributes a state vector to an individual system; (c) places restrictions on the simultaneous measurability of non-commuting observables; (d) hesitates to use his measurements for retrodictions. In our opinion, the arguments of the above authors reflect their incomprehension of Copenhagenism. Elsewhere [5,6] we have discussed two alternative interpretations of quantum mechanics which we have called Copenhagenism and Popperism. We have there shown how the dispute between the schools stems from disparate uses of the word ‘state’. We continue the discussion here within the context of the above points and show how these disparate notions of state are related to diverse notions of ‘behaviour’.