NON-RELATIVISTIC QUANTUM MECHANICS MICHAEL DICKSON Abstract. This chapter is a discussion of the philosophical and foundational is- sues that arise in non-relativistic quantum theory. After introducing the formalism of the theory, I consider: characterizations of the quantum formalism, empirical content, uncertainty, the measurement problem, and non-locality. In each case, the main point is to give the reader some introductory understanding of some of the major issues and recent ideas. Keywords: quantum theory, quantum mechanics, measurement problem, uncer- tainty, nonlocality Contents 1. The Theory 3 1.1. The Thought Behind Starting with Formalism 4 1.2. The Standard Formalism 4 1 1.3. Simple Example: A Spin- 2 Particle 19 1.4. Dirac Notation 25 1.5. Transformations 27 1.6. Preview of Philosophical Issues 37 2. Whence the Kinematical Formalism? 39 2.1. From Propositions to Hilbert Space 40 2.2. From States to Hilbert Space 42 2.3. Hardy’s Axioms 51 3. Empirical Content 53 3.1. Measurement 53 3.2. The Issue of Empirical Content in Terms of POVMs 59 3.3. Symmetries 60 3.4. Reference Frames 66 3.5. A Group-Theoretic Characterization of Empirical Content 70 Author’s address: Department of Philosophy, University of South Carolina, Columbia, SC 29208, USA. Thanks to Jeremy Butterfield and John Earman for their comments and suggestions. Thanks also to the participants of a workshop at the University of Pittsburgh in November, 2004, for helpful comments. 2 4. Uncertainty 73 4.1. Canonical Commutation Relations 73 4.2. The Uncertainty Relations 75 4.3. Interpretation of the Uncertainty Relations 79 4.4. The Einstein-Podolsky-Rosen Argument 82 5. The ‘Measurement Problem’ 89 5.1. The Basic Problem 89 5.2. Measurement 90 5.3. Generality of the Problem 91 5.4. Non-Solutions 94 5.5. Interpretations 103 6. Non-locality 118 6.1. No-Go Theorems 119 6.2. Reactions to the Theorems 127 7. Mathematical Appendix 136 7.1. Hilbert Spaces 136 7.2. Operators 139 7.3. The Hilbert Space C2 141 7.4. Posets and Lattices 142 7.5. Topology and Measure 143 7.6. Groups 144 References 147 This article is an introduction to some of the most important philosophical and foun- dational issues that arise from or concern non-relativistic quantum theory. The chap- ter has six main sections. The first introduces the theory, including some of the important mathematical results required to formulate and address many of the philo- sophical and foundational issues. This section is the longest, and most important, for it will begin to give the careful reader the background needed to understand and evaluate much of the vast literature on non-relativistic quantum theory. And that literature is indeed vast—there is no way that it can even be summarized in a chapter of this length. Instead, in the five subsequent sections, I will consider some of the more important: foundational characterizations of the formalism of quantum theory, empirical content, quantum uncertainty, the measurement problem, and non-locality. There are many other issues one could discuss, and some recent movements that merit 3 consideration. Alas, we will not have time for them. A careful reading of the material here is a start, however, towards understanding these other issues. Of these five issues, the first two are somewhat less discussed, especially in the Anglo-American philosophical literature. Those sections are therefore longer, relative to the final three, than some readers might expect. This fact is not meant to imply anything about the relative importance of the issues, but is an attempt to redress a relative lack of coverage in certain circles. Much of the material presented here—especially from §4,§5, and §6—is largely my review of standard material that can be found in many places. I have therefore chosen not to provide extensive bibliographic information. Indeed, I have kept bibliographic references to a minimum. This article is thus not intended to be a compendium of work in the field, much less an extensive annotated bibliography. The reader is encouraged to seek additional resources to fill out the brief accounts given here. Such resources are numerous. The final section is a brief mathematical appendix, reviewing essential definitions and results, mostly from the theory of Hilbert spaces and groups. It may serve one of two purposes, depending on the reader: a brief reminder of concepts learned elsewhere; or a prompt to learn the concepts elsewhere. It is unlikely that a reader who is completely unfamiliar with these concepts will absorb them just from what is said here. Reference is made to the relevant subsections of this appendix at the appropriate places in the text. While at some points I have made some effort at rigor, for the most part, the discussion here is only partly rigorous, with the occasional attempt made to point towards what further would be required for complete rigor. The reader is, again, encouraged to consult the literature for mathematical details, and in any case is encouraged to bear in mind that much of the discussion here is not intended to be entirely mathematically rigorous, while, I hope, also not being misleading. 1. The Theory This section is an introduction to the formalism of quantum theory. After a brief justification of this approach (§1.1), I will introduce the major elements of the formal- ism (§1.2), followed by a simple, but important, example (§1.3). I will then introduce the commonly used ‘Dirac’ notation (§1.4), and conclude by considering the role of transformations (groups) in the theory (§1.5), including dynamical transformations (equations of motion) and finally (§1.6) a brief preview of the philosophical issues to 4 come. More than subsequent sections, this section will rely heavily on the material from the mathematical appendix (§7), with references where appropriate. 1.1. The Thought Behind Starting with Formalism. Why begin an account of a physical theory with its formalism? Why not begin, instead, with its basic physical insights, or fundamental physical principles? One problem with such an approach here is that, in the case of quantum theory, there is not much significant agreement about what the basic physical insights, or fundamental physical principles, are. Some argue that the collapse postulate (to be discussed later) is at the heart of the theory. Others argue that it must be excised from the theory. Some argue that the theory is fundamentally indeterministic, while others argue that we can make sense of it only in terms of an underlying determinism. Some argue that the familiar notion of a ‘particle’ with a definite location is a casualty of the theory, while others argue that the theory makes sense only if one takes such a notion as fundamental. Now, advocates of these different views tend also, it is true, to advocate different formulations of the theory, but they will not suggest that formulations other than their preferred one are wrong, only that they, perhaps, emphasize the wrong points. (Indeed, there is no disputing that the standard formalism—the one presented here— is empirically successful; advocates of different views will ultimately have to account for that success in their own terms.) Hence, while the choice of a single formalism at the start of our discussion might slant our point of view somewhat, it will, unlike the choice of basic physical insights or fundamental physical principles, not prejudge the central issues. 1.2. The Standard Formalism. I begin with a very brief sketch of a common un- derstanding of the formalism, which I shall flesh out and generalize subsequently. (The reader is not expected to have a deep understanding of any aspect of the formalism merely as a result of reading this subsection.) 1.2.1. Hilbert Space. The formalism of quantum mechanics is normally understood in terms of the theory of Hilbert spaces (§7.1). A Hilbert space is a vector space (§7.1.1) with an inner product (§7.1.3) that is also complete with respect to the norm (§7.1.4) defined by this inner product. A standard example is the space, `2, of (modulus)- square-summable sequences of complex numbers. In this space, the inner product of P∞ ∗ two vectors, (x1, x2,...) and (y1, y2,...) is n=1 xnyn. Another standard example is the space, L2(RN ), of (modulus)-square-integrable, Lebesque-measurable, complex- valued functions on RN , where we identify two functions (i.e., they represent the same vector) if (and only if) they differ only on a set of Lebesque measure (§7.5.4) zero. 5 Here the inner product of two vectors, f(x) and g(x), is R f ∗(x)g(x)dx (where f(x) and g(x) are arbitrary representatives from their respective equivalence classes).1 1.2.2. Observables. The ‘observables’ of the theory—the physical quantities, or prop- erties, whose value or presence one can, in principle at least, measure, or ‘observe’— are normally taken to be represented by the self-adjoint operators (§7.2.1, §7.2.3) on the Hilbert space. (The nature of the representation—that is, which operators represent which observables—can depend on the physical situation being described.) Via the spectral theorem (discussed below), one can identify each observable with a spectral family of projection operators, the observable being given, essentially, by a map from Borel sets (§7.5.5) of possible values of the observable to elements in the spectral family. This subsection reviews these ideas briefly. 1.2.2.1. Positive Operator Valued Measures. It is often useful to adopt a broader notion of an observable, as a ‘positive-operator-valued measure’ (POVM).