Hilary Greaves

Time reversal in classical electromagnetism

Richard Feynman has claimed that anti-particles are nothing but particles ‘propagating backwards in time’; that time reversing a particle state always turns it into the corresponding anti-particle state. According to standard quantum field theory textbooks this is not so: time reversal does not turn particles into anti-particles. Feynmans view is interesting because, in particular, it suggests a non-standard, and possibly illuminating, interpretation of the CPT theorem. This paper explores a classical analog of Feynmans view, in the context of the recent debate between David Albert and David Malament over time reversal in classical electromagnetism. 1. Introduction2. Time Reversal and the Direction of Time3. Classical Electromagnetism: The Story So Far 3.1. The standard textbook view3.2. Alberts proposal3.3. Malaments proposal3.4. Albert revisited4. The ‘Feynman’ Proposal5. Structuralism: A Third Way? 5.1. Structures: the debate recast5.2. Relational structures5.3. Malament and Feynman structures as conventional representors of a relational reality6. Conclusions and Open Questions Introduction Time Reversal and the Direction of Time Classical Electromagnetism: The Story So Far 3.1. The standard textbook view3.2. Alberts proposal3.3. Malaments proposal3.4. Albert revisited The standard textbook view Alberts proposal Malaments proposal Albert revisited The ‘Feynman’ Proposal Structuralism: A Third Way? 5.1. Structures: the debate recast5.2. Relational structures5.3. Malament and Feynman structures as conventional representors of a relational reality Structures: the debate recast Relational structures Malament and Feynman structures as conventional representors of a relational reality Conclusions and Open Questions

Empirical Consequences of Symmetries

It is widely recognized that ‘global’ symmetries, such as the boost invariance of classical mechanics and special relativity, can give rise to direct empirical counterparts such as the Galileo-ship phenomenon. However, conventional wisdom holds that ‘local’ symmetries, such as the diffeomorphism invariance of general relativity and the gauge invariance of classical electromagnetism, have no such direct empirical counterparts. We argue against this conventional wisdom. We develop a framework for analysing the relationship between Galileo-ship empirical phenomena on the one hand, and physical theories that model such phenomena on the other, that renders the relationship between theoretical and empirical symmetries transparent, and from which it follows that both global and local symmetries can give rise to Galileo-ship phenomena. In particular, we use this framework to exhibit an analogue of Galileo’s ship for the local gauge invariance of electromagnetism. 1 Introduction 2 Analogues of Galileo’s Ship? Faraday’s Cage and t’Hooft’s Beam-Splitter   2.1 Faraday’s cage   2.2 t’Hooft’s beam-splitter 3 A Framework for Symmetries I: Systems and Subsystems 4 An Example: Coulombic Electrostatics 5 A Framework for Symmetries II: The Relationship between Theoretical and Empirical Symmetries 6 Newtonian Gravity 7 Local Symmetries that Are Not Boundary-Preserving: Classical Electromagnetism and Faraday’s Cage 8 Local Boundary-Preserving Symmetries: Klein-Gordon-Maxwell Gauge Theory and t’Hooft’s Beam-Splitter 9 Summary 10 Conclusions 1 Introduction 2 Analogues of Galileo’s Ship? Faraday’s Cage and t’Hooft’s Beam-Splitter   2.1 Faraday’s cage   2.2 t’Hooft’s beam-splitter   2.1 Faraday’s cage   2.2 t’Hooft’s beam-splitter 3 A Framework for Symmetries I: Systems and Subsystems 4 An Example: Coulombic Electrostatics 5 A Framework for Symmetries II: The Relationship between Theoretical and Empirical Symmetries 6 Newtonian Gravity 7 Local Symmetries that Are Not Boundary-Preserving: Classical Electromagnetism and Faraday’s Cage 8 Local Boundary-Preserving Symmetries: Klein-Gordon-Maxwell Gauge Theory and t’Hooft’s Beam-Splitter 9 Summary 10 Conclusions

Towards a Geometrical Understanding of the CPT Theorem

The CPT theorem of quantum field theory states that any relativistic (Lorentz-invariant) quantum field theory must also be invariant under CPT, the composition of charge conjugation, parity reversal and time reversal. This paper sketches a puzzle that seems to arise when one puts the existence of this sort of theorem alongside a standard way of thinking about symmetries, according to which spacetime symmetries (at any rate) are associated with features of the spacetime structure. The puzzle is, roughly, that the existence of a CPT theorem seems to show that it is not possible for a well-formulated theory that does not make use of a preferred frame or foliation to make use of a temporal orientation. Since a manifold with only a Lorentzian metric can be temporally orientable—capable of admitting a temporal orientation—this seems to be an odd sort of necessary connection between distinct existences. The paper then suggests a solution to the puzzle: it is suggested that the CPT theorem arises because temporal orientation is unlike other pieces of spacetime structure, in that one cannot represent it by a tensor field. To avoid irrelevant technical details, the discussion is carried out in the setting of classical field theory, using a little-known classical analog of the CPT theorem. 1. Introduction2. The Connection between Dynamical Symmetries and Spacetime Structure3. A Puzzle about the CPT Theorem4. A Classical PT Theorem 4.1. Bells theorem4.2. Auxiliary constraints5. Resolution of the Puzzle6. Galilean-Invariant Field Theories 6.1. Temporal orientation in Galilean spacetime6.2. Counterexample to the Galilean PT hypothesis7. Conclusions Introduction The Connection between Dynamical Symmetries and Spacetime Structure A Puzzle about the CPT Theorem A Classical PT Theorem 4.1. Bells theorem4.2. Auxiliary constraints Bells theorem Auxiliary constraints Resolution of the Puzzle Galilean-Invariant Field Theories 6.1. Temporal orientation in Galilean spacetime6.2. Counterexample to the Galilean PT hypothesis Temporal orientation in Galilean spacetime Counterexample to the Galilean PT hypothesis Conclusions

On the CPT theorem

Abstract We provide a careful development and rigorous proof of the CPT theorem within the framework of mainstream (Lagrangian) quantum field theory. This is in contrast to the usual rigorous proofs in purely axiomatic frameworks, and non-rigorous proof-sketches in the mainstream approach. We construct the CPT transformation for a general field directly, without appealing to the enumerative classification of representations, and in a manner that is clearly related to the requirements of our proof. Our approach applies equally in Minkowski spacetimes of any dimension at least three, and is in principle neutral between classical and quantum field theories: the quantum CPT theorem has a natural classical analogue. The key mathematical tool is that of complexification; this tool is central to the existing axiomatic proofs, but plays no overt role in the usual mainstream approaches to CPT.

A Reconsideration of the Harsanyi–Sen–Weymark Debate on Utilitarianism

Harsanyi claimed that his Aggregation and Impartial Observer Theorems provide a justification for utilitarianism. This claim has been strongly resisted, notably by Sen and Weymark, who argue that while Harsanyi has perhaps shown that overall good is a linear sum of individuals’ von Neumann–Morgenstern utilities, he has done nothing to establish any connection between the notion of von Neumann–Morgenstern utility and that of well-being, and hence that utilitarianism does not follow. The present article defends Harsanyi against the Sen–Weymark critique. I argue that, far from being a term with precise and independent quantitative content whose relationship to von Neumann–Morgenstern utility is then a substantive question, terms such as ‘well-being’ suffer (or suffered) from indeterminacy regarding precisely which quantity they refer to. If so, then (on the issue that this article focuses on) Harsanyi has gone as far towards defending ‘utilitarianism in the original sense’ as could coherently be asked.

Discounting for public policy: A survey

This article surveys the debate over the social discount rate. The focus is on the economics rather than the philosophy literature, but the survey emphasizes foundations in ethical theory rather than highly technical details. I begin by locating the standard approach to discounting within the overall landscape of ethical theory. The article then covers the Ramsey equation and its relationship to observed interest rates, arguments for and against a positive rate of pure time preference, the consumption elasticity of utility, and the effect of various sorts of uncertainty on the discount rate. Climate change is discussed as an application.

SPACETIME SYMMETRIES AND THE CPT THEOREM

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 nonspatio-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,