SPACETIME SYMMETRIES AND THE CPT THEOREM BY HILARY GREAVES A dissertation submitted to the Graduate School|New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Philosophy Written under the direction of Frank Arntzenius and approved by New Brunswick, New Jersey May, 2008 ABSTRACT OF THE DISSERTATION Spacetime symmetries and the CPT theorem by Hilary Greaves Dissertation Director: Frank Arntzenius This dissertation explores several issues related to the CPT theorem. Chapter 2 explores the meaning of spacetime symmetries in general and time reversal in particular. It is proposed that a third conception of time reversal, `geometric time reversal', is more appropriate for certain theoretical purposes than the existing `active' and `passive' conceptions. It is argued that, in the case of classical electromagnetism, a particular nonstandard time reversal operation is at least as defensible as the standard view. This unorthodox time reversal operation is of interest because it is the classical counterpart of a view according to which the so-called `CPT theorem' of quantum field theory is better called `PT theorem'; on this view, a puzzle about how an operation as apparently non- spatio-temporal as charge conjugation can be linked to spacetime symmetries in as intimate a way as a CPT theorem would seem to suggest dissolves. In chapter 3, we turn to the question of whether the CPT theorem is an essentially quantum-theoretic result. We state and prove a classical analogue of the CPT theorem for systems of tensor fields. This classical analogue, however, ii appears not to extend to systems of spinor fields. The intriguing answer to our question thus appears to be that the CPT theorem for spinors is essentially quantum-theoretic, but that the CPT theorem for tensor fields applies equally to the classical and quantum cases. Chapter 4 explores a puzzle that arises when one puts the CPT theorem alongside a standard way of understanding spacetime symmetries, according to which (latter) spacetime symmetries are to be understood in terms of background spacetime structure. The puzzle is that a `PT theorem' amounts to a statement that the theory may not make essential use of a preferred direction of time, and this seems odd. We propose a solution to that puzzle for the case of tensor field theories. iii Acknowledgements Each of the main chapters in this dissertation is, or is a very slightly adapted version of, a research article prepared for submission to a professional journal. The articles in chapters 2 and 4 are, at the time of writing, being considered for publication by The British Journal for the Philosophy of Science. That in chapter 3 has not yet been submitted. Chapter 2 was written jointly by Frank Arntzenius and myself. A draft con- taining ancestors of much of sections 2.2, 2.3 and 2.4 was produced by Arntzenius, at that time working alone, in July 2005. These have been significantly rewrit- ten and added to (by both of us) following discussion between the two of us. Section 2.5 (on structuralism) was written by me. We estimate that our overall contributions to this paper are equal. Chapters 3 and 4 are single-authored. I would like to thank Robert Geroch for extensive correspondence on the `classical PT theorem' proved in chapter 3, and an anonymous referee for The British Journal for the Philosophy of Science for extremely helpful feedback on the paper reproduced here as chapter 4. iv Table of Contents Abstract :::::::::::::::::::::::::::::::::::: ii Acknowledgements ::::::::::::::::::::::::::::: iv 1. Introduction :::::::::::::::::::::::::::::::: 1 2. Time reversal in classical electromagnetism :::::::::::: 5 2.1. Introduction . 6 2.2. Time reversal and the direction of time . 7 2.3. Classical electromagnetism: the story so far . 11 2.3.1. The standard textbook view . 11 2.3.2. Albert's proposal . 13 2.3.3. The triviality problem . 14 2.3.4. Justifying time reversal operations: the pragmatic program 17 2.3.5. Justifying time reversal operations: the ontological program 19 Spacetime structure from time-reversal invariance: a first pass. 19 Time-reversal invariance of coordinate-dependent formula- tions. 20 2.3.6. Malament's proposal . 23 Electric and magnetic fields. 27 2.3.7. Albert revisited . 29 2.4. The `Feynman' proposal . 31 Fundamental ontology. 31 v Time reversal. 32 More on the `Feynman' proposal. 32 2.5. Structuralism: A Third Way? . 35 2.5.1. Structures: the debate recast . 36 2.5.2. Relational structures . 43 2.5.3. Malament and Feynman structures as conventional repre- sentors of a relational reality . 44 2.6. Conclusions and open questions . 48 Summary of conclusions from this paper. 48 Theory choice. 49 Conventionality of spacetime structure? . 51 Field theory. 52 3. A classical PT theorem ::::::::::::::::::::::::: 53 3.1. Introduction . 54 3.2. Definitions and lemmas to be invoked . 56 3.2.1. Definitions . 56 Tensor fields. 56 Lorentz group. 57 Tensor representations of the Lorentz group. 57 3.2.2. Lemmas to be invoked . 58 3.3. Classical PT theorem for tensor field theories . 60 3.4. Limitations of the classical PT theorem and projects suggested . 64 Restrictions required on the dynamical equations. 64 Relationship between the classical PT theorem proved here and the QFT CPT theorem. 66 3.5. Spinor field theories . 66 vi 3.6. Conclusions . 69 4. Towards a geometrical understanding of the CPT theorem :: 71 4.1. Introduction . 72 4.2. The connection between dynamical symmetries and spacetime struc- ture . 75 Spacetime theories. 75 Symmetries. 75 Spacetime symmetries. 76 The connection between symmetries and spacetime structure. 79 Non-generally covariant formulations of spacetime theories. 81 Example. 82 4.3. A puzzle about the CPT theorem. 84 4.4. A classical PT theorem . 87 4.4.1. Bell's theorem . 89 4.4.2. Auxiliary constraints . 90 4.5. Resolution of the puzzle . 92 4.6. Galilean-invariant field theories . 96 4.6.1. Temporal orientation in Galilean spacetime . 97 4.6.2. Counterexample to the Galilean PT hypothesis . 99 4.7. Conclusions . 100 References ::::::::::::::::::::::::::::::::::: 102 Vita ::::::::::::::::::::::::::::::::::::::: 103 Colleges attended . 103 Positions held . 103 Publications . 103 vii 1 Chapter 1 Introduction 2 This dissertation consists of three papers in the philosophy of physics. All three began as attempts to understand the CPT theorem of quantum field theory. This theorem states that any relativistic (that is, Lorentz-invariant) quantum field theory must also be invariant under CPT, the composition of charge conjugation, parity reversal and time reversal. Two things seem initially puzzling about such a theorem: • How can there be such an intimate relationship between spatiotemporal symmetries (Lorentz invariance, parity reversal, time reversal) on the one hand, and charge conjugation, not obviously a spatiotemporal notion at all, on the other? • How can it come about that one symmetry (e.g. Lorentz invariance) entails another (e.g. CPT) at all? These two types of puzzlement are quite distinct. The first paper, `Time re- versal in classical electromagnetism' (chapter 2), began as an attempt to solve the first puzzle. The underlying thought here is that, while the mathematics that goes into proving that such-and-such a transformation on quantum fields is (for present purposes) beyond doubt, one calls the theorem in question a `CPT theo- rem' only on the basis of certain assumptions about which transformations on sets of fields are to be interpreted as implementing time reversal, charge conjugation and parity reversal, and that these latter assumptions are not beyond question. In fact, they have occasionally been questioned. John S Bell, for example, notes that while the transformation for which we prove a theorem is naturally inter- preted as `CPT' under a particle interpretation of quantum field theory, it is more naturally viewed as a PT transformation from the point of view of field theory (Bell, 1955, passim); and Richard Feynman's slogan `antiparticles are particles travelling backwards in time' is fairly naturally interpreted as suggesting the same 3 idea. If this is correct, it dissolves the first puzzle: there is no such connection between Lorentz invariance and a genuinely `non-spatio-temporal' symmetry. The arguments in favour of interpreting the key transformation as CPT rather than as PT, however, can seem very compelling. One of the main aims of the paper of chapter 2 is to remedy this by exploring, in the context of classical elec- tromagnetism, a `time reversal' transformation that is the classical counterpart of the quantum transformation usually called `TC', and arguing that it is no less plausible to regard this as the implementation of time reversal than it is to regard the transformation that is usually called `time reversal' as the implementation of time reversal in classical electromagnetism. The project requires a thorough re-examination of the meaning of time reversal; the second main aim of this paper is to articulate and advocate a novel conception of time reversal (distinct from the usual notions of `active' and `passive' time reversal) that we think is implicit in recent work of David Malament (2004). For the purposes of the remainder of the dissertation, I presuppose that the transformation usually called `CPT' is, indeed, properly interpreted as a PT transformation. The papers of chapters 3 and 4 also pursue the approach of attempting to illu- minate the PT theorem by exploring the analogous issues in the classical context. Chapter 3 focusses on the issue of whether or not an analogue of the quantum- theoretic PT theorem can be proved in the context of classical field theory. The (surprising) answer is that it can for classical tensor field theories, but not for classical spinor field theories. The core of this paper is mathematical: it states and proves the classical PT theorem for tensor field theories.