## Introduction

There are many important questions that do not fall neatly into any one discipline; rather, their full investigation requires the integration of two or more distinct fields. This book is about just such a question – one that arises at the intersection of physics, philosophy, and history. The question can be simply stated as “What is the relation between classical and quantum mechanics?” The simplicity of the question, however, belies the complexity of the answer. Classical mechanics and quantum mechanics are two of the most successful scientific theories ever developed, and yet how these two very different theories can successfully describe one and the same world – the world we live in – is far from clear. One theory is deterministic, the other indeterministic; one theory describes a world in which chaotic behavior is pervasive, and the other a world in which it is almost entirely absent. Did quantum mechanics simply replace classical mechanics as the new universal theory? Do they each describe their own distinct domains of phenomena? Or is one theory really just a continuation of the other?

In the philosophy literature, this sort of issue is known as the problem of intertheoretic relations.^{1} Currently, there are two accepted philosophical frameworks for thinking about intertheoric relations: the first is reductionism,
and the second, pluralism. As we shall see, these labels each actually describe a family of related views. Reductionism is roughly the view that one theory can be derived from another, either by means of a logical deduction or the
mathematical limit of some parameter. Theoretical pluralism, by contrast, takes each scientific theory to have its own distinct domain of laws, entities, and concepts,
which cannot be reduced to those of any other theory. The central thesis of this book is that neither reductionism nor pluralism
adequately describes the relation between quantum and classical mechanics.

In searching for a new philosophical framework for thinking about intertheory relations, I turn to the history of science,
and examine the philosophical views of three of the founders of quantum theory: Werner Heisenberg, Paul Dirac, and Niels Bohr. Perhaps surprisingly, all three of these figures accorded to classical mechanics a role of continued *theoretical* importance; none of them took classical mechanics to be a discarded theory, rendered useless for all but “engineering” purposes. Moreover, I shall argue
that none of them took the relation between classical and quantum mechanics to be captured by the usual reductionist account in terms of the classical limit (*ħ*→0). Despite these two important similarities, however, all three of them held a very different view of the quantum–classical relation. As we shall see, Heisenberg’s account of classical and quantum mechanics as “closed theories” led him to adopt a version of theoretical pluralism; Dirac, by contrast, saw a deep analogy or structural continuity between these theories; and Bohr viewed quantum theory as a “rational generalization” of classical mechanics. I shall show that not only do these historical views have many interesting
parallels with contemporary debates in the philosophy of science, but they can also suggest new ways in which our present-day
debates might be moved forward.

The question of the relation between classical and quantum mechanics cannot be decided on purely historical or philosophical
grounds, but also requires delving into contemporary research in physics. I shall focus on an area of scientific research
known as semiclassical mechanics. Very roughly, semiclassical mechanics can be thought of as the study of “mesoscopic” systems that are in the overlap between the classically described macroworld and the quantum mechanically described
microworld. As such, it is a field ideally suited for exploring questions about the relationship between classical and quantum
mechanics. More specifically, I shall focus on a subfield known as quantum chaos. The name “quantum chaos” is something of a misnomer, since quantum systems cannot exhibit the sort of sensitive dependence on initial conditions characteristic of classically chaotic behavior.^{2} Instead, the field of quantum chaos is concerned with the study of quantum systems whose classical counterparts are chaotic.
As we shall see, these quantum-chaotic systems pose a number of unique challenges for an adequate characterization of the
quantum–classical relation. At the heart of this book is a summary of four areas of research in semiclassical mechanics that involve quantum systems
whose classical counterparts are chaotic. These are the semiclassical solution of the helium atom, diamagnetic Rydberg atoms, wavefunction scarring, and quantum dots. These case studies will function as the “data” against which the adequacy of the philosophical accounts of intertheory
relations will be tested.

I shall argue that there are three surprising lessons to draw from this examination of semiclassical research: First, there is a variety of *quantum* phenomena ranging from atomic physics to condensed-matter physics, for which semiclassical mechanics – not pure quantum mechanics
– provides the appropriate theoretical framework for investigating, calculating, and *explaining* these phenomena. Second, these semiclassical methods and explanations involve a thorough hybridization of classical and quantum
ideas. Far from being incommensurable theoretical concepts, they can be combined in both empirically adequate and conceptually fruitful ways. Finally,
the classical structures (such as periodic orbits) appealed to in semiclassical mechanics are not simply useful calculational
devices, but are actually manifesting themselves in surprising ways in quantum *experiments* (that is, in ways that are not simply the quantum behavior mimicking the classical behavior). This speaks to a much richer continuity of dynamical structure across classical and quantum mechanics than is usually recognized.

These features of semiclassical research pose two important challenges for contemporary philosophy of science. First, the semiclassical appeals to classical structures in explaining quantum phenomena do not fit easily with either of the current orthodox accounts of scientific explanation (that is, they are neither deductive–nomological nor causal explanations). I shall argue that a new philosophical account of scientific explanation is called for and outline what such an account might look like.

Second, this semiclassical research also poses a challenge to the adequacy of our current philosophical frameworks for thinking about intertheory relations. More specifically, it reveals that an adequate account of the relation between classical and quantum mechanics should not just be concerned with the narrow (though of course important) question of how to recover classical behavior from quantum mechanics, but rather should recognize the many structural correspondences and continuities between the two theories. Of the three historical views that I examine, I shall argue that Dirac’s “structural continuity” view provides the most adequate foundation for a new philosophical account of intertheory relations – one that can incorporate these insights from semiclassical research. This new view, which I call interstructuralism, takes from theoretical pluralism the insight that predecessor theories such as classical mechanics are still playing an important theoretical role in scientific research; that is, quantum mechanics – without classical mechanics – gives us an incomplete picture of our world. From reductionism, however, it takes the lesson that we cannot rest content with the view that each of these theories describes its own distinct domain of phenomena. We stand to miss out on many important scientific discoveries and insights if we do not try to bring our various theoretical descriptions of the world closer together.

^{1} In the philosophical literature, as in the remainder of this book, the terms ‘intertheoretic’ and ‘intertheory’ are used interchangeably.

^{2} For a review of the reasons why there cannot typically be chaotic behavior in quantum systems see Bokulich (2001, p. x).

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