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

This is the second installment of a two-part paper on developments in quantum dispersion theory leading up to Heisenberg’s Umdeutung paper. In telling this story, we have taken a 1924 paper by John H. Van Vleck in The Physical Review as our main guide. In this second part we present the detailed derivations on which our narrative in the first part rests. The central result that we derive is the Kramers dispersion formula, which played a key role in the thinking that led to Heisenberg’s Umdeutung paper. We derive classical formulae for the dispersion, emission, and absorption of radiation and use Bohr’s correspondence principle to construct their quantum counterparts both for the special case of a charged harmonic oscillator (Sect. 5) and for arbitrary non-degenerate multiply-periodic systems (Sect. 6). We then rederive these results in modern quantum mechanics (Sect. 7). 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. The authors gratefully acknowledge support from the Max Planck Institute for History of Science. The research of Anthony Duncan is supported in part by the National Science Foundation under grant PHY-0554660.     

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.     

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.     

On classical cloning and no-cloning

It is part of information theory folklore that, while quantum theory prohibits the generic (or universal) cloning of states, such cloning is allowed by classical information theory. Indeed, many take the phenomenon of no-cloning to be one of the features that distinguishes quantum mechanics from classical mechanics. In this paper, we argue that pace conventional wisdom, in the case where one does not include a machine system, there is an analog of the no-cloning theorem for classical systems. However, upon adjoining a non-trivial machine system (or ancilla) one finds that, pace the quantum case, the obstruction to cloning disappears for pure states. We begin by discussing some conceptual points and category-theoretic generalities having to do with cloning, and proceed to discuss no-cloning in both the case of (non-statistical) classical mechanics and classical statistical mechanics.     

Einstein’s quantum theory of the monatomic ideal gas: non-statistical arguments for a new statistics

In this article, we analyze the third of three papers, in which Einstein presented his quantum theory of the ideal gas of 1924–1925. Although it failed to attract the attention of Einstein’s contemporaries and although also today very few commentators refer to it, we argue for its significance in the context of Einstein’s quantum researches. It contains an attempt to extend and exhaust the characterization of the monatomic ideal gas without appealing to combinatorics. Its ambiguities illustrate Einstein’s confusion with his initial success in extending Bose’s results and in realizing the consequences of what later came to be called Bose–Einstein statistics. We discuss Einstein’s motivation for writing a non-combinatorial paper, partly in response to criticism by his friend Ehrenfest, and we paraphrase its content. Its arguments are based on Einstein’s belief in the complete analogy between the thermodynamics of light quanta and of material particles and invoke considerations of adiabatic transformations as well as of dimensional analysis. These techniques were well known to Einstein from earlier work on Wien’s displacement law, Planck’s radiation theory and the specific heat of solids. We also investigate the possible role of Ehrenfest in the gestation of the theory.     

From canonical transformations to transformation theory, 1926–1927: The road to Jordan's Neue Begründung

We sketch the development from matrix mechanics as formulated in the Dreimännerarbeit of Born, Heisenberg, and Jordan, completed in late 1925, to transformation theory developed independently by Jordan and Dirac in late 1926. Focusing on Jordan, we distinguish three strands in this development: the implementation of canonical transformations in matrix mechanics (the main focus of our paper), the clarification of the relation between the different forms of the new quantum theory (matrix mechanics, wave mechanics, q-numbers, and operator calculus), and the generalization of Born's probability interpretation of the Schrödinger wave function. These three strands come together in a two-part paper by Jordan published in 1927, “On a new foundation [neue Begründung] of quantum mechanics.”     

Quantum decoherence and the approach to equilibrium (II)

In a previous paper [Hemmo, M & Shenker, O (2003). Quantum decoherence and the approach to equilibrium I. Philosophy of Science, 70, 330–358] we discussed a recent proposal by Albert [(2000). Time and chance. Cambridge, MA: Harvard University Press. Chapter 7] to recover thermodynamics on a purely dynamical basis, using the quantum theory of the collapse of the quantum state of [Ghirardi, G, Rimini, A and Weber, T., (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review, D 34, 470–479]. We proposed an alternative way to explain thermodynamics within no collapse interpretations of quantum mechanics. In this paper some difficulties faced by both approaches are discussed and solved: the spin echo experiments, and the problem of extremely light gases. In these contexts, we point out several ways in which the above quantum mechanical approaches as well as some other classical approaches to the foundations of statistical mechanics may be distinguished experimentally.     

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.     

Between Viète and Descartes: Adriaan van Roomen and the <Emphasis Type="Italic">Mathesis Universalis</Emphasis>

Adriaan van Roomen published an outline of what he called a Mathesis Universalis in 1597. This earned him a well-deserved place in the history of early modern ideas about a universal mathematics which was intended to encompass both geometry and arithmetic and to provide general rules valid for operations involving numbers, geometrical magnitudes, and all other quantities amenable to measurement and calculation. ‘Mathesis Universalis’ (MU) became the most common (though not the only) term for mathematical theories developed with that aim. At some time around 1600 van Roomen composed a new version of his MU, considerably different from the earlier one. This second version was never effectively published and it has not been discussed in detail in the secondary literature before. The text has, however, survived and the two versions are presented and compared in the present article. Sections 1–6 are about the first version of van Roomen’s MU the occasion of its publication (a controversy about Archimedes’ treatise on the circle, Sect. 2), its conceptual context (Sect. 3), its structure (with an overview of its definitions, axioms, and theorems) and its dependence on Clavius’ use of numbers in dealing with both rational and irrational ratios (Sect. 4), the geometrical interpretation of arithmetical operations multiplication and division (Sect. 5), and an analysis of its content in modern terms. In his second version of a MU van Roomen took algebra into account, inspired by Viète’s early treatises; he planned to publish it as part of a new edition of Al-Khwarizmi’s treatise on algebra (Sect. 7). Section 8 describes the conceptual background and the difficulties involved in the merging of algebra and geometry; Sect. 9 summarizes and analyzes the definitions, axioms and theorems of the second version, noting the differences with the first version and tracing the influence of Viète. Section 10 deals with the influence of van Roomen on later discussions of MU, and briefly sketches Descartes’ ideas about MU as expressed in the latter’s Regulae.     

Quantum probability and many worlds

We discuss the meaning of probabilities in the many worlds interpretation of quantum mechanics. We start by presenting very briefly the many worlds theory, how the problem of probability arises, and some unsuccessful attempts to solve it in the past. Then we criticize a recent attempt by Deutsch to derive the quantum mechanical probabilities from the non-probabilistic parts of quantum mechanics and classical decision theory. We further argue that the Born probability does not make sense even as an additional probability rule in the many worlds theory. Our conclusion is that the many worlds theory fails to account for the probabilistic statements of standard (collapse) quantum mechanics.     

The meaning of πλασμαтιкό<Emphasis Type="Italic">ν</Emphasis> in Diophantus’ <Emphasis Type="Italic">Arithmetica</Emphasis>

Three problems in book I of Diophantus’ Arithmetica contain the adjective plasmatikon, that appears to qualify an implicit reference to some theorems in Elements, book II. The translation and meaning of the adjective sparked a long-lasting controversy that has become a nonnegligible aspect of the debate about the possibility of interpreting Diophantus’ approach and, more generally, Greek mathematics in algebraic terms. The correct interpretation of the word, a technical term in the Greek rhetorical tradition that perfectly fits the context in which it is inserted in the Arithmetica, entails that Diophantus’ text contained no (implicit) reference to Euclid’s Elements. The clause containing the adjective turns out to be a later interpolation, that cannot be used to support any algebraic interpretation of the Arithmetica.     

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.     

Quantum sidelights on The Material Theory of Induction

John Norton's The Material Theory of Induction bristles with fresh insights and provocative ideas that provide a much needed stimulus to a stodgy if not moribund field. I use quantum mechanics (QM) as a medium for exploring some of these ideas. First, I note that QM offers more predictability than Newtonian mechanics for the Norton dome and other cases where classical determinism falters. But this ability of QM to partially cure the ills of classical determinism depends on facts about the quantum Hamiltonian operator that vary from case to case, providing an illustration of Norton's theme of the importance of contingent facts for inductive reasoning. Second, I agree with Norton that Bayesianism as developed for classical probability theory does not constitute a universal inference machine, and I use QM to explain the sense in which this is so. But at the same time I defend a brand of quantum Bayesianism as providing an illuminating account of how physicists' reasoning about quantum events. Third, I argue that if the probabilities induced by quantum states are regarded as objective chances then there are strong reasons to think that fair infinite lotteries are impossible in a quantum world.     

Reading Gauss in the Computer Age: On the U.S. Reception of Gauss’s Number Theoretical Work (1938–1989)

C.F Gauss’s computational work in number theory attracted renewed interest in the twentieth century due to, on the one hand, the edition of Gauss’s Werke, and, on the other hand, the birth of the digital electronic computer. The involvement of the U.S. American mathematicians Derrick Henry Lehmer and Daniel Shanks with Gauss’s work is analysed, especially their continuation of work on topics as arccotangents, factors of n ² + a ², composition of binary quadratic forms. In general, this strand in Gauss’s reception is part of a more general phenomenon, i.e. the influence of the computer on mathematics and one of its effects, the reappraisal of mathematical exploration. I would like to thank the Alexander-von-Humboldt-Stiftung for funding this research. For their comments I would like to thank Catherine Goldstein, Norbert Schappacher and especially John Brillhart.     

Ehrenfest’s adiabatic theory and the old quantum theory, 1916–1918

I discuss in detail the contents of the adiabatic hypothesis, formulated by Ehrenfest in 1916. I focus especially on the paper he published in 1916 and 1917 in three different journals. I briefly review its precedents and thoroughly analyze its reception until 1918, including Burgers’s developments and Bohr’s assimilation of them into his own theory. I show that until 1918 the adiabatic hypothesis did not play an important role in the development of quantum theory. An erratum to this article can be found at     

Quantum metaphysical indeterminacy and the ontological foundations of orthodoxy

This paper develops quantum state individualism, a fundamental ontology for what is usually known as ‘orthodox quantum mechanics.’ The central import of this ontology is that allows for a systematic evaluation of some of the main conclusions of the recent literature on quantum metaphysical indeterminacy. In particular, quantum state individualism supports the ‘gappy’ version of Jessica Wilson's determinable-based account of metaphysical indeterminacy; it implies that fundamental reality is perfectly precise; and third, it provides a non-disjunctive definition of determinables and thereby shields Wilson's account against the charge that it requires either a departure from classical logic or a revision of the quantum formalism.     

Einstein and the Kaluza–Klein particle

In his search for a unified field theory that could undercut quantum mechanics, Einstein considered five-dimensional classical Kaluza–Klein theory. He studied this theory most intensively during the years 1938–1943. One of his primary objectives was finding a non-singular particle solution. In the full theory this search got frustrated, and in the x⁵-independent theory Einstein, together with Pauli, argued it would be impossible to find these structures.     

Proposition 10, Book 2, in the <Emphasis Type="Italic">Principia</Emphasis>, revisited

In Proposition 10, Book 2 of the Principia, Newton applied his geometrical calculus and power series expansion to calculate motion in a resistive medium under the action of gravity. In the first edition of the Principia, however, he made an error in his treatment which lead to a faulty solution that was noticed by Johann Bernoulli and communicated to him while the second edition was already at the printer. This episode has been discussed in the past, and the source of Newton’s initial error, which Bernoulli was unable to find, has been clarified by Lagrange and is reviewed here. But there are also problems in Newton’s corrected version in the second edition of the Principia that have been ignored in the past, which are discussed in detail here.     

Quantum jumps,superpositions, and the continuous evolution of quantum states

The apparent dichotomy between quantum jumps on the one hand, and continuous time evolution according to wave equations on the other hand, provided a challenge to Bohr's proposal of quantum jumps in atoms. Furthermore, Schrödinger's time-dependent equation also seemed to require a modification of the explanation for the origin of line spectra due to the apparent possibility of superpositions of energy eigenstates for different energy levels. Indeed, Schrödinger himself proposed a quantum beat mechanism for the generation of discrete line spectra from superpositions of eigenstates with different energies.However, these issues between old quantum theory and Schrödinger's wave mechanics were correctly resolved only after the development and full implementation of photon quantization. The second quantized scattering matrix formalism reconciles quantum jumps with continuous time evolution through the identification of quantum jumps with transitions between different sectors of Fock space. The continuous evolution of quantum states is then recognized as a sum over continually evolving jump amplitudes between different sectors in Fock space.In today's terminology, this suggests that linear combinations of scattering matrix elements are epistemic sums over ontic states. Insights from the resolution of the dichotomy between quantum jumps and continuous time evolution therefore hold important lessons for modern research both on interpretations of quantum mechanics and on the foundations of quantum computing. They demonstrate that discussions of interpretations of quantum theory necessarily need to take into account field quantization. They also demonstrate the limitations of the role of wave equations in quantum theory, and caution us that superpositions of quantum states for the formation of qubits may be more limited than usually expected.     

Quantum propensities

This paper reviews four attempts throughout the history of quantum mechanics to explicitly employ dispositional notions in order to solve the quantum paradoxes, namely: Margenau's latencies, Heisenberg's potentialities, Maxwell's propensitons, and the recent selective propensities interpretation of quantum mechanics. Difficulties and challenges are raised for all of them, and it is concluded that the selective propensities approach nicely encompasses the virtues of its predecessors. Finally, some strategies are discussed for reading similar dispositional notions into two other well-known interpretations of quantum mechanics, namely the GRW interpretation and Bohmian mechanics.