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6. L CPT Reversal 6. CPT reversal l Précis. On the representation view, there may be weak and strong arrows of time. A weak arrow exists. A strong arrow might too, but not in current physics due to CPT symmetry. There are many ways to reverse time besides the time reversal operator T. Writing P for spatial reflection or ‘parity’, and C for matter-antimatter exchange or ‘charge conjugation’, we find that PT, CT, and CPT are all time reversing transformations, when they exist. More generally, for any non-time-reversing (unitary) transformation U, we find that UT is time reversing. Are any of these transformations relevant for the arrow of time? Feynman (1949) captured many of our imaginations with the proposal that to understand the direction of time, we must actually consider the exchange of matter and antimatter as well.1 Writing about it years later, he said: “A backwards-moving electron when viewed with time moving forwards appears the same as an ordinary electron, except it’s attracted to normal electrons — we say it has positive charge. For this reason it’s called a ‘positron’. The positron is a sister to the electron, and it is an example of an ‘anti-particle’. This phenomenon is quite general. Every particle in Nature has an amplitude to move backwards in time, and therefore has an anti-particle.”(Feynman 1985, p.98) Philosophers Arntzenius and Greaves (2009, p.584) have defended Feynman’s proposal, 1In his Nobel prize speech, Feynman (1972, pg.163) attributes this idea to Wheeler in the development of their absorber theory (Wheeler and Feynman 1945). Stueckelberg (1942) had arrived earlier at a similar perspective. 151 interpreting it as the claim that “the operation that ought to be called ‘time reversal’ — in the sense that it bears the right relation to spatiotemporal structure to deserve that name — is the operation that is usually called TC”. However, a more common physics adage today is perhaps the one expressed by Wallace, that, “in quantum field theory, it is the transformation called CPT, and not the one usually called T, that deserves the name”. (Wallace 2011b, p.4) The intuition behind this is that in the standard model of particle physics, only CPT is guaranteed to be a symmetry. CPT symmetry is moreover deeply connected to what it means to be a relativistic quantum field theory of any kind, by a collection of results known as the CPT Theorem. It is also connected to the spin statistics connection, that half-integer spin systems obey Fermi-Dirac statistics, while integer spin systems obey Bose-Einstein statistics.2 This chapter will evaluate these ‘alternative’ arrows of time, and argue in par- ticular that CPT symmetry does have implications for time’s arrow. How important it is depends on how one interprets two different perspective on time’s arrow. First: experimental evidence shows that T symmetry is violated, which establishes one sense in which time is asymmetric. I call this the weak arrow of time. However, there is stronger sense of time asymmetry, which is to say that no time reversing symmetry is available at all: neither T, nor any non-time-reversing transformation composed with it. I call this a strong arrow of time. One’s philosophical commitments might bring one to prefer one sense of an arrow over the other; however, I will argue that consideration of both provides insight into the structure of time. In particular, I will argue for a perspective on matter and antimatter according to which their exchange is just a representation of a particular symmetry of a spacetime structure, the universal covering group of the Poincaré group. This provides a clearer physical sense of how CPT can be interpreted as essentially a spacetime transformation. I begin Section 6.1 with a simple pictorial discussion of these arrows and their philosophical underpinnings. I then give a formal argument that a strong arrow is 2A classic reference is Streater and Wightman (1964); see Bain (2016) for a philosophical review. 152 incompatible with the ordinary unitary dynamics of quantum theory: some ‘time reversing’ operator is always a symmetry. This argument stops short of giving a complete physical interpretation of the operator. Thus, I next turn to considering which time reversing symmetries might have physical significance. Section 6.2 reviews Feynman’s proposal, that time reversal also exchanges matter and antimatter, and its defence by Arntzenius and Greaves. Unfortunately, experimental evidence for discrete symmetry violation makes trouble for this view, but that it can be saved through consideration of the CPT theorem. Section 6.3 then gives an account of the physical significance of CPT symmetry, as an automorphism of a spacetime structure on the representation view. I claim this proposal solves some of the puzzles regarding the CPT theorem highlighted by Greaves (2010). Section 6.4 then briefly reviews the statement of the CPT theorem, correcting in particular the great misnomer that it is a ‘composition’ of three operators, and arguing that it prohibits a strong arrow. Section 6.5 finally discusses the possibility of how one might yet establish a strong arrow of time, in theories that go beyond the standard model. 6.1 Weak and strong arrows 6.1.1 Philosophical underpinning Our main aim in this chapter is to clarify status of the following two notions of an arrow of time: 1. a weak arrow occurs when no time reversing symmetry exists that can be inter- preted as reversing ‘time alone’; 2. a strong arrow occurs when no time reversing symmetry exists at all. To illustrate, consider the 2 2 chequered grid of Figure 6.1, with a vertical t-axis ⇥ representing time, and a horizontal x-axis representing space. This description is invariant under PT symmetry, but not T symmetry. In particular, the reversal of ‘time alone’ t t is not a symmetry: so, there is a weak arrow. However, the combined 7! − 153 reversal of both time and space t, x t, x is a symmetry: so, there is no strong ( )7! (− − ) arrow. t t t − − x x − Figure 6.1: Time reversal is not a symmetry, but space-time reversal is. What could motivate interest in the weak arrow of time, as opposed to the strong? The answer depends on one’s philosophical commitments regarding the role of time within the broader spacetime structure of the world. Here are two perspectives that arrive at different results. • Temporalism: any symmetry that transforms time translations as t t establishes 7! − a symmetry of time. The temporalist holds that in order to understand the sym- metries of time, only time is needed, without regard for any other degrees of freedom. So, since PT symmetry reverses time translations in the sense that PT : t t, it establishes a time symmetry, nevermind that it transforms space 7! − as well. • Spacetime Contextualism: only a symmetry of time translations t t that does 7! − not affect any time-independent degrees of freedom establishes a symmetry of time. The spacetime contextualist is concerned with time translations in their broader con- text as a spacetime symmetries. So, although PT symmetry may be an important spacetime symmetry, it does not really belong to the class of temporal symmetries, because it also transforms space. On first glance, one might be tempted by temporalism: we are after all ultimately interested in the symmetries of time translations. What does it matter if this symmetry is established by T,PT or CPT? Once we know t t is a symmetry of time translations, 7! − 154 we may conclude that time does not have an arrow. For the temporalist, only the failure of all time reversing symmetries would be good evidence for an arrow of time: the only interesting arrow is a strong arrow. However, the spacetime contextualist has a response that I find convincing. The temporalist view is motivated by a ‘purist’ attitude about time, that only the time translations matter. However, transformations like PT and CPT cannot be defined at all without appeal to non-temporal degrees of freedom. So, by helping one’s self to the availability of PT, CT, or CPT, one has already accepted some amount of spacetime contextualism, by accepting that other degrees of freedom are relevant to time’s struc- ture. In that case, it seems disingenuous to ignore that other degrees of freedom are transformed. The very availability of transformations like PT, CT, and CPT is reason to think that neither transformation can be appropriately described as ‘time reversal’. As a result, existing evidence of T-violation establishes an interesting arrow, though a weak one. Although temporalism and spacetime contextualism are helpful for framing this discussion, I do not see that a great deal rides on the distinction. A middle ground is simply to accept that there are two interesting notions of an arrow, weak and strong. I will soon show a sense in which a strong arrow is incompatible with the dynamics of quantum theory. For this we will need to introduce language that is a little more precise. 6.1.2 Representations of weak and strong arrows Let me begin by taking stock of the previous chapters. One of the main themes of this book is that time reversal is not a transformation of spacetime coordinates, but of time translations t t. These can in general be implemented by a time reversal 7! − transformation ⌧ with the property that ⌧t⌧ 1 ⇤ t. Now, on what I have called − − the Representation View, a dynamical theory is just a representation of spacetime transformations amongst the automorphisms of a state space. And, when we look at the representatives of time translations in quantum theory, we find a set of unitary 155 operators generating solutions to the Schrödinger equation, while time reversal Ut ⌧ is uniquely represented by Wigner’s (antiunitary) time reversal operator T.
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