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. 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.

Introduction

The philosophical importance of semiclassical appeals to classical structures in explaining quantum phenomena was first recognized by Robert Batterman (1993; 1995; 2002). More generally, Batterman has argued that when it comes to explaining phenomena in the asymptotic domain between two theories, the fundamental theory describing that domain is often explanatorily deficient, and an adequate explanation must make essential reference to the less fundamental theory. For example, in discussing the relation between classical and quantum mechanics, Batterman writes, “There are many aspects of the semiclassical limit of quantum mechanics that cannot be explained purely in quantum mechanical terms, though they are in some sense quantum mechanical … [T]hese quantum mechanical features require reference to classical properties for their full explanation” (Batterman 2002, pp. 109–10). He has extended these arguments to other theory pairs with singular limits as well, arguing that in optics, for example, one finds explanations of wave-theoretic phenomena that make an essential and ineliminable appeal to ray-theoretic structures such as caustics.

More recently, Batterman's arguments have been the subject of considerable criticism from figures such as Clifford Hooker (2004), Michael Redhead (2004), and Gordon Belot (2005). These criticisms have tended to cluster around the following two assumptions: First, in saying that a classical structure (such as a trajectory or caustic) explains, one is thereby committed to the claim that the structure exists.

Introduction

As we saw in the last chapter, the received account of the relation between classical and quantum mechanics is a form of reductionism, where classical mechanics is supposed to emerge from quantum mechanics in the limit of some parameter. When we examine more closely the views of three of the key founders of quantum theory, however, we see that none of them took the relation between these theories to be adequately captured by such a reductionist limit. Indeed as we shall see presently, Werner Heisenberg's account of the relation between classical and quantum mechanics is actually a strong form of theoretical pluralism, quite similar to Nancy Cartwright's metaphysical nomological pluralism introduced in Section 1.3. Rather than viewing quantum mechanics as the fundamental theory that replaced classical mechanics, Heisenberg argues that both theories are required, each having its own proper domain of applicability and each being a perfectly accurate and final description of that domain.

Throughout his career, Heisenberg held a highly original view in the philosophy of science, centered on his notion of a closed theory. Very briefly, a closed theory is a tightly knit system of axioms, definitions, and laws that provides a perfectly accurate and final description of a certain limited domain of phenomena. This notion has profound implications for Heisenberg's understanding of scientific methodology, theory change, intertheoretic relations, and realism.

Introduction

The issue of intertheoretic relations is concerned with how our various theoretical descriptions of the world are supposed to fit together. As the physicist Sir Michael Berry describes it, “Our scientific understanding of the world is a patchwork of vast scope; it covers the intricate chemistry of life, the sociology of animal communities, the gigantic wheeling galaxies, and the dances of elusive elementary particles. But it is a patchwork nevertheless, and the different areas do not fit well together” (Berry 2001, p. 41). This uncomfortable patchwork exists even if we restrict our attention to within the field of physics alone. Physics itself consists of many subtheories, such as quantum field theory, quantum mechanics, condensed-matter theory, thermodynamics, classical mechanics, and the special and general theories of relativity – just to name a few. Each of these theories is taken to be an accurate description of some domain of phenomena, and insofar as they are supposed to be describing one and the same world, it is important to ask how these very different – and in many cases prima facie mutually inconsistent – theories are supposed to fit together.

Hitherto, the philosophical frameworks available for thinking about intertheory relations have been rather limited. Traditionally, discussions of intertheoretic relations have been framed in terms of reductionism.

Open theories

When it comes to the issues of reductionism, scientific methodology, and theory change, the views of Werner Heisenberg and Paul Dirac diverge in fundamental and interesting ways. They revisited their disagreements over these philosophical issues many times throughout their careers, and their disagreements can be most succinctly described as a debate over whether physical theories are “open” or “closed.” As we saw in the last chapter, Heisenberg's belief that classical and quantum mechanics are closed leads him to view these theories as perfectly accurate within their domains, inalterable, and correct for all time. Although Dirac never uses the term, his own views on classical and quantum mechanics can be fruitfully understood as a rival account of “open theories.” Dirac argues that even the most well-established parts of quantum theory are open to future revision; indeed he takes no part of physics to be a permanent achievement, correct for all time. Instead of viewing classical mechanics as a theory that had been replaced, he sees it as a theory that should continue to be developed, modified, and extended.

Unlike Heisenberg, who views physics as a set of consistent axiomatic systems, Dirac sees physics as a discipline much closer to engineering.

Introduction

Semiclassical mechanics can be broadly understood as the theoretical and experimental study of the interconnections between classical and quantum mechanics. More narrowly, it is a field that uses classical quantities to investigate, calculate, and even explain quantum phenomena. Its methods involve an unorthodox blending of quantum and classical ideas, such as a classical trajectory with an associated quantum phase. For these reasons, semiclassical mechanics is often referred to as “putting quantum flesh on classical bones,” where classical mechanics provides the skeletal framework on which quantum quantities are constructed.

There are three primary motivations for semiclassical mechanics: First, in many systems of physical interest, a full quantum calculation is cumbersome or even unfeasible. Second, even when a full quantum calculation is within reach, semiclassical methods can often provide intuitive physical insight into a problem, when the quantum solutions are opaque. And, third, semiclassical investigations can lead to the discovery of new physical phenomena that have been overlooked by fully quantum-mechanical approaches. Semiclassical methods are ideally suited for studying physics in the so-called mesoscopic regime, which can roughly be understood as the domain between the classically described macro-world and the quantum mechanically described micro-world. An area in which semiclassical studies have proven to be particularly fruitful is in the subfield of quantum chaos.

SPACE AND TIME IN COSMOLOGY

The question about the nature of space and time is intimately linked with the question of cosmology: Did space and time have a beginning? Do they go on forever? Space and time form the framework for our picture of cosmology, while our large-scale view of the Universe puts the limits on what space and time are.

The nature of space and time underwent a radical change from Newton to Einstein. As Newton set out in his Principia Mathematica, space and time was an unchanging Aristotelian background to the unfolding play of particles and waves. But even this seemingly innocuous assumption caused Newton problems. Gravity acted instantaneously everywhere (action at a distance); a radical idea for the 1770s used to the idea that every effect had a direct cause. If the Universe was infinite in extent, the forces acting on any given point would depend instantaneously on the influence of all of the matter throughout the Universe. But because the volume of space increases rapidly with distance these forces would accumulate and increase without limit in an infinite Universe.

SPACETIME STRUCTURE

Einstein's general theory of relativity gives a mathematical description of space, time and gravitation which is extraordinarily concise, subtle and accurate. It has, however, the appearance of being concise only to those who are already familiar with the mathematical formalism of Riemannian geometry. To someone who is not familiar with that body of mathematical theory – a theory which, though remarkably elegant, is undoubtedly sophisticated, and usually becomes extremely complicated in detailed application – Einstein's General Relativity can seem inaccessible and bewildering in its elaborate structure. But the complication and sophistication lie only in the details of the formalism. Once that mathematics has been mastered, the precise formulation of Einstein's physical theory is, indeed, extremely compact and natural. Although a little of this formalism will be needed here, it will be given in a compact form only that should be reasonably accessible.

The mathematical theory of Riemannian geometry applies to smooth spaces of any (positive whole) number N of dimensions. Such a space M is referred to as an N-manifold, and to be a Riemannian manifold it must be assigned a metric, frequently denoted by g, which assigns a notion of ‘length’ to any smooth curve in M connecting any two points a, b. (See Figure 3.1.) For a strictly Riemannian manifold, this length function is what is called positive definite which means that the length of any such curve is a positive number, except in the degenerate situation when a = b and the curve shrinks to a point, for which the length would be zero.

INTRODUCTION

Whereas the previous chapter tells us about mysteries surrounding the physical structure of the Universe from a largely observational point of view, in this essay I will approach the problem of space and time from a theoretical point of view. This is about the conceptual structure of physics, why in fact our current concepts of space and time are fundamentally flawed and how they might be improved. I will explain in detail why I think that spacetime is fundamentally not a smooth continuum at the pre-subatomic level due to quantum-gravity effects and why a better although still not final picture is one where there are no points, where everything is done by algebra much as in quantum mechanics, what I therefore call ‘quantum spacetime’.

The idea of ‘moving around’ in space in this theory is replaced by ‘quantum symmetry’ and I shall need to explain this to the reader. Symmetry is the deepest of all notions in mathematics and what emerged in the last two decades is that this very concept is really part of something even more fundamental. Indeed, these quantum symmetries not only generalise our usual notion of symmetry but have a deep self-duality in their very definition in which the role of the composition of symmetry transformations and a new structure called a ‘coproduct’ is itself symmetric. For our purposes, quantum symmetries are needed in order to extend Einstein's theory of Special Relativity to quantum spacetimes.

Not many of us are perplexed about space. We can move around in it and its nature seems experientially obvious. Yet even in the case of spatial properties, the philosophically minded can deem the existence of reliable measuring rods, capable of metricating space, as not being as straightforward a matter as commonsense might suppose. Moreover, when physical cosmologists theorise about the Universe, they find that its vast spatial domains exhibit an intrinsic curvature, corresponding to General Relativity's account of the nature of gravity. There are certainly subtleties about the nature of space, which go beyond the expectations of everyday thought, but they are nothing like as perplexing as those we encounter when we attempt to think about the nature of time.

Time travel is not available to us and we have to take our experience of time ‘as it comes to us’, in the succession of those fleeting present moments which as soon as we experience them recede immediately into the inaccessible fixity of the past. Famously, St Augustine, meditating on temporality in the Confessions, said that as long as he did not think about it, he knew what time was, but as soon as he reflected on the nature of temporal flux, he began to be perplexed. To commonsense, the one thing that does seem clear is that time flows. Yet one of the central issues in the modern discussion of temporality is whether this is indeed the case, or whether our human sense of the flow of time is merely a trick of psychological perspective, and the fundamental reality of time is quite different.

INTRODUCTION

Physics and metaphysics are two distinct occupations of human beings, and not long ago a lot of effort was invested into keeping them strictly apart. Nowadays, however, the situation seems to be changing. A couple of years ago, an international conference was organised at Cambridge, UK, the proceedings of which bear the title ‘Physics Meets Philosophy at the Planck Scale’. What is special about the Planck Scale that physics and philosophy (of which metaphysics is an essential part) seem to have something to tell to each other?

In physicists’ jargon the ‘Planck Scale’ or the ‘Planck era’ means either the most fundamental level of the physical Universe, or an edge at which our present theories of physics break down (this is why the Planck era is also called the Planck threshold). Currently, these two meanings are almost synonymous since the fundamental theory of physics lies beyond the reach of our well-founded physical theories and models.

There are two directions along which we could approach the Planck era.We can either adopt the path followed by cosmologists, or that followed by elementary particle physicists. In cosmology, one tries to reconstruct the history of the Universe starting from our present era as far backward in time as possible. As one moves in this direction, the Universe contracts and becomes denser and denser, till one reaches a density of the order of 1095 g/cm3, and then one finds oneself at the Planck era.