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. Time reversal symmetry violation, which says that a representation of T does not exist, can then be viewed as evidence for an arrow of time.3 It is in particular a weak arrow, in the language just introduced. Let me first illustrate the weak and strong arrows using a simple physical exam- ple, before diving into relativistic quantum field theory. Consider a bead constrained to a string, which we find always has positive velocity to the East (Figure 6.2). A representation of time translations maps each position-velocity pair x, x , always sat- ( €) isfying x > 0, to the pair ' x, x ⇤ x + xt, x . Such a description is obviously time € t( €) ( € €) asymmetric, and one can confirm this formally: since time reversal transforms states as T x, x ⇤ x, x , it follows that no time reversal operator exists in a state space with ( €) ( €) only positive velocities x > 0. Parity symmetry fails as well, since for the same reason, € the transformation P x, x ⇤ x, x does not exist either. ( €) ( €)

Figure 6.2: Time translations for the asymmetric bead-string system.

However, there is a non-standard representation of time reversal given by the combined PT reversal transformation, which I write, ⇡ x, x ⇤ x, x . This ⇡ cannot ( €) ( €) be written as a composition of P and T, neither of which exist. But it does define

1 a representation of time reversal symmetry, in that ⇡'t ⇡ ⇤ ' t, as can be easily checked.4 Therefore, PT is a dynamical symmetry of the system. This system has a weak arrow, but no strong arrow. But, since PT symmetry implies that t t is a symmetry of time translations, the advocate of temporalism 7! may conclude that time itself is symmetric. In contrast, the spacetime contextualist will resist this conclusion: this is only superficially a symmetry of time, more properly viewed as a symmetry of spacetime. And after all, if a system that always moves in

3Chapter 2 presented this picture of time reversal, Chapter 3 derived Wigner’s T as the unique representative of t t, Chapter 4 defined time reversal symmetry and T-violation, and Chapter 5 7! argued that its T-violation is associated with an arrow of time. 4 1 For all x, x we have, ⇡'t ⇡ x, x ⇤ ⇡'t x, x ⇤ ⇡ x + xt, x ⇤ x xt, x ⇤ ' t x, x . ( €) ( €) ( €) ( € €) ( € €) ( €)

156 one direction is not time symmetric, then what is? On this view, the PT transformation

⇡ does provide a representation of time reversal symmetry, but it is not the right one. Again, although I am sympathetic to spacetime contextualism, I do not have a strong view on this debate. What I would like to point out is that if this system fails to have a strong arrow of time, then it is hard to imagine one that cannot. This turns out to be because it is impossible — at least in quantum theory as we know it.

6.1.3 Generic reversibility and physical significance

In quantum theory, one standardly assumes that the time translations have the structure of R, + , the group of real numbers under addition. The state space as a projective ( ) Hilbert space, consisting of the equivalence classes of Hilbert space states related by a complex unit. A representation of time translations amongst the automorphisms of state space is then a strongly continuousf one parameter unitary representation t . 7! Ut A unitary representation by no means assures that the time reversal operator T is a symmetry, just as it is not assured in classical mechanics. For example, consider a unitary representation ⇤ e itH with, say, H ⇤ 1 P2 + PQ, interpreted in terms of Ut 2m position Q and momentum P. Then since time reversal changes the sign of P but not

1 1 Q, we find that THT , H, and consequently T tT , U t. But the time reversal U group element ⌧ by definition has the property that ⌧t⌧ 1 ⇤ t, and so ' ⇤ T is only a ⌧ 1 representation of it if T tT ⇤ t. This property is not satisfied here: thus, a system U U with this dynamics would be T-violating. However, here again the combination of parity and time reversal does turn out to be a symmetry.

Whether or not the time reversal operator T gives rise to a symmetry, it is always possible to construct some time reversing (antiunitary) operator that does, in that T 1 t ⇤ t. This might be PT, or it might be something else entirely. That fact is TU T U assured by the following.5

5This simple idea, that conjugation with respect to the unitary dynamics implements a time reversing symmetry, is similar in spirit to many proofs of the CPT theorem; see below, and for Swanson (2019) for a discussion in the context of algebraic proofs of CPT symmetry.

157 Proposition 6.1. Every strongly continuous unitary representation t of R, + extends 7! Ut ( ) to a representation of time reversal symmetry, in that there exists some antiunitary satisfying T 1 t ⇤ t. TU T U

Proof. Let H be the self-adjoint operator such that ⇤ e itH, assured to exist by Ut Stone’s theorem, with spectrum ⇤ R. Let H be its spectral representation on ✓ s L2 ⇤ , in that H x ⇤ x for all in its domain, and VHV 1 ⇤ H for some unitary ( ) s ( ) s V : L2 ⇤ (cf. Blank, Exner, and Havlíček 2008, §5.8). If K is the conjugation H! ( ) operator on L2 ⇤ , in that K ⇤ for all L2 ⇤ , then K, H ⇤ 0, since for all in ( ) ⇤ 2 ( ) [ s] the domain of H we have KH K 1 x ⇤ x x ⇤ H x . Thus, :⇤ V 1KV is the s s ( ) ( ) s ( ) T desired antiunitary operator, since our definitions imply that , H ⇤ 0, and hence [T ] 1 itH 1 it H 1 itH t ⇤ eT( )T ⇤ e T T ⇤ e ⇤ t. ⌅ TU T U

This provides a compelling argument against the possibility of a strong arrow of time: in any quantum theory with a unitary dynamics, including the entire standard model, some representation of time reversal symmetry is always possible, and so a strong arrow does not exist. The temporalist in particular will find there is no interesting arrow of time. The only way to have a truly strong arrow of time is to go beyond standard quantum theory, for example to contexts in which gravitational effects are incorporated. I will discuss this possibility briefly in Section 6.5.

However, if the standard time reversal operator T is known, then this generic antiunitary has the form ⇤ UT for some unitary U, whose physical significance T T is left uninterpreted.6 This leaves open a concern about whether it represents any transformation of physical significance. On the representation view, the meaning of a symmetry U arises as in a representation of some symmetry structure, like a group of spacetime symmetries. No such group is specified in the construction above, which might make this transformation unconvincing as a physical symmetry. At the very least, it would test the resilience of the temporalist’s commitment to accepting any time reversing symmetry whatsoever. A more convincing time reversing symmetry of quantum theory would be

6Just define U :⇤ T 1, and recall that the composition of two antiunitaries is always unitary. T

158 CPT symmetry. To make this case, we will need to say something about where this transformation gets its meaning, and what sort of operator it represents. This is the task of Section 6.3. But let me first review a charming alternative due to Feynman, that time reversal is ‘really’ the CT transformation in disguise.

6.2 The Feynman view on time reversal

6.2.1 Reversing time and charge

Richard Feynman argued that “positrons can be represented as electrons with proper time reversed relative to true time” (Feynman 1949, p.753). More generally, his view is that what we usually call ‘time reversal’ is, surprisingly, best understood as including charge conjugation. Feynman’s view made a scattering of remarkable appearances in the early 1950’s. It was advocated by Satosi Watanabe (1951), and then by Gerhart Lüders, who adopted it in his famous paper giving the first argument for a CPT theorem:

“In the following, two types of time reversal will appear: the time rever- sal ‘of the first kind’ which, loosely speaking, consists in a reversal of the ‘motion’ of all particles, and the time reversal ‘of the second kind’, a simul- taneous performance of a proper time reversal and a particle-antiparticle conjugation.” (Lüders 1954, p.4)

This interpretation was dropped in later sharpenings of the CPT theorem by Pauli (1955), Lüders (1957) and Jost (1957). However, Bell formulated a version of the CPT theorem that adopted a similar perspective:

“The reversal in time referred to above is that which arises naturally when the theory is approached in its field aspect; it is found to involve reversing the signs of electromagnetic charges. It is to be distinguished therefore from the ‘Wigner type’ time reversal, which perhaps arises more naturally

159 when the particle apsect is emphasized and which is defined as leaving the charges unaltered” (Bell 1955, p.479).

Feynman even appears to have smuggled his proposal into the modern theory of Feynman diagrams, which represent contributions to a scattering amplitude in a perturbative approximation. When the CT transformation exists, reflecting a Feynman diagram (Figure 6.3) about the vertical ‘time’ axis produces a CT transformation, which both reverses time and exchanges matter and antimatter!7

+ + µ µ e e

+ + e e µ µ

Figure 6.3: A Feynman diagram for an electron-positron decay (left), when vertically reflected, produces a description that both reverses time and exchanges matter and antimatter states (right).

The transformation that Feynman is proposing does have interesting physical significance. It also suggests that the appropriate transformation for understanding whether or not there is a weak arrow is what is commonly known as CT, insofar as turning around time itself ‘really’ involves charge conjugation too. What would bring one to say such a thing?

6.2.2 Justifying the Feynman view

Philosophers Arntzenius and Greaves have argued that the Feynman perspective is in fact necessary in order to explain a puzzle about the CPT theorem.8 Greaves (2008, 2010) formulates the puzzle in two parts:

“one can identify two positive sources of puzzlement:

7This was pointed out by Ramakrishnan (1967) and by Feynman (1985, Chapter 3) himself. 8See Greaves (2008), Arntzenius and Greaves (2009), Greaves (2010), Arntzenius (2011), and Arntze- nius (2012).

160 • How can it come about that one symmetry (e.g., Lorentz invariance) entails another (e.g., CPT) at all?

• How can there be such an intimate relationship between spatiotempo- ral symmetries (Lorentz invariance, parity reversal, time reversal) on the one hand, and charge conjugation, not obviously a spatiotemporal notion at all, on the other?”

(Greaves 2010, p.28)

Her solution to the second puzzle, echoed by Arntzenius (2011, 2012), is to adopt the Feynman view: the transformation normally called CT is what most naturally deserves the name ‘time reversal’. Her argument for this is that through a particular choice of quantisation, classical time reversal becomes what we normally call CT. As a result, she argues, “the transformation that is standardly called ‘CPT’ receives the name ‘PT’” (Greaves 2010, p.39). Since the non-spatiotemporal degrees of freedom are now renamed in terms of spatiotemporal ones, the second puzzle is found to be dissolved. Her solution to the first puzzle is then to argue that a restricted class of classical field theories are PT invariant, drawing on an argument of Bell (1955) that Greaves and Thomas (2014) have generalised. Not all authors would agree that the quantisation procedure Greaves advocates is the right one: after applying a standard quantisation procedure to symmetries

Wallace (2009) observes that the standard quantum time reversal operator T arises if one takes the classical time reversal transformation to be complex-antilinear, as it is standardly defined.9 In contrast, CT only arises if classical time reversal is taken to be complex-linear. The solution to the first puzzle, that classical field theory has a PT theorem, faces technical difficulties as well. For example, Swanson (2019, p.120) points out her PT theorem is too restrictive for the standard model, since it holds only for polynomial interactions of tensor fields, and thus does not apply to either non-

9 Baker and Halvorson (2010, p.118) raise a related concern: “We suspect that Feynman’s view arises from ignoring that, when the proper complex structure is applied to free particle systems, an antiparti- cle’s wave vector and its four-momentum have opposite temporal orientation. So, in the standard form of free QFT, ... both matter and antimatter systems always move ‘forward in time’ by virtue of meeting the spectrum condition.”

161 polynomial interactions or those involving spinorial tensors. Indeed, a significant part of the Greaves (2010) approach is to argue that in reasonable classical field theories, “there is no tensor field that represents temporal orientation and no more, in the context of a flat Lorentzian metric and a total orientation” (Greaves 2010, p.43). However, classical field theories with spinorial tensors have exactly the opposite character: a pair of (two component) spinorial vectors are co-oriented, and thereby determine a time orientation!

There are also two further conceptual issues facing this approach as a whole. In the first place, there is something strange about resolving the puzzle of ‘non- spatiotemporal’ degrees of freedom in this way. If feels confused about how the exchange of matter and antimatter are related to spacetime structure in the CPT the- orem, then that feeling is unlikely to go away when one renames ‘CPT’ as a ‘PT’ transformation. Whatever we call it, the transformation exchanges matter and anti- matter: Greaves’ proposal that its analogue in classical field theory reverses time alone does little to assuage the worry that in quantum field theory it does. Greaves’ argument might help explain how someone could arrive at this through quantisation. But, the explanation of how matter and antimatter are connected to spacetime symmetries still appears to be missing.

The second, the explanation Arntzenius and Greaves are proposing requires the existence of a CT operator. We begin with a classical time reversal operator, quantise in a way that produces a CT operator, and then compose it with P to produce a CPT theorem. The trouble is, a CT operator does not generally exist in the standard model, although CPT does. As I have pointed out above, symmetry violation does not just mean that the dynamics fail to be invariant under a symmetry operator: it means that no representation of that operator exists. This makes it hard to see how an explanation of this kind can get offthe ground. If CT is required to explain the CPT theorem, and no CT operator exists, it rather appears that the explanation making use of it does not exist either.

That said, Arntzenius and Greaves have proposed an extremely creative ap-

162 proach to understanding the direction of time, which does a lot to clarify the philo- sophical commitments of Feynman’s proposal. However, despite the charm of the Feynman view, it does become a little awkward in contexts where the CT transforma- tion does not exist. In such contexts, a more general strategy along these lines is to shift attention to the CPT transformation instead. One can formulate a very similar view in this context, as Wallace (2011b, p.4) suggests, on which time reversal is best viewed as reflecting space, reflecting time, and exchanging matter and antimatter. One might then take this to serve as the correct context for discussing the weak and strong arrows. However, the puzzles that Greaves has raised remain, and it would be nice to see some indication of how they can be resolved. In the next section, I will indicate an alternative road to their solution, which tracks more closely the more well-known proofs of the CPT theorem. This will also in turn help clarify the significance of the transformation for the arrow of time.

6.3 Representing C, P and T

On the face of it, there is indeed a puzzle about how matter-antimatter exchange could be relevant to the arrow of time. Why would this be the sort of thing that has significance for either the weak or strong arrows? The CPT theorem only appears to add to this sense of puzzlement, as Greaves (2010) has pointed out: how can the assumption of one group of symmetries, the restricted Poincaré group, imply that another symmetry like CPT holds too? And why should an apparently non- spatiotemporal quantity like the exchange of matter and antimatter have anything to do with it?

What makes these questions appear paradoxical is the implicit assumption that these two kinds of symmetry transformation are independent: continuous symmetries of spacetime on the one hand, and C, P and T on the other. In this section I argue that this implicit assumption is mistaken: C, P and T are each connected by their very definition to the continuous Poincaré transformations. Once this connection is

163 appreciated, the apparent paradox dissolves, and the close connection between CPT and the arrow of time becomes more apparent.

6.3.1 Relativistic spacetime symmetries

To explain the physical significance of CPT symmetry for the arrow of time, we must review where C, P and T get their meaning. Since the CPT theorem is formulated for quantum theories that are invariant under the continuous Poincaré transformations, we begin by discussing what that consists in. This will lead to an immediate explanation of Greaves’ first puzzle, about how symmetry under a continuous group can imply further symmetries under a discrete one. For readers who may wish to skip the technical details, here is the main idea: the discrete symmetries are in fact defined in terms of the continuous ones. We think of the continuous Poincaré transformations as being symmetries of Minkowski spacetime. But parity and time reversal are actually symmetries of those symmetries! In particular, parity reflects the spatial translations, while time reversal reflects the time translations. This makes it easy to imagine how a continuous symmetry could lead to reflection symmetries as well.

First, recall what it means to say that a transformation U on Hilbert space can be interpreted as a time translation, spatial translation, or boost. Hilbert space theory by itself is just an abstract state space. To give these transformations meaning, we must ‘tie it down’ to spacetime through a representation of a spacetime symmetry group. For local physics this is the Poincaré group, consisting of all the isometries of Minkowski spacetime. We build the Poincaré group up in steps, to see in particular where the discrete symmetries come from.10 First we have:

• The restricted Lorentz group L"+. This is the Lie group of Lorentz boosts and spatial rigid rotations about a spacetime point.

The arrow in L"+ reminds one that it preserves temporal orientation, while the plus indicates it preserves total orientation. We next extended it to:

10For further details, see Varadarajan (2007, §IX.2) and Landsman (1998, esp. §2.2 and Part IV).

164 • The restricted Poincaré group +". The elements of +" to consist in pairs a, ⇤ P P ( )2 4 R L"+. Since a boost or rotation ⇤ is an automorphism that changes the ⇥ orientation of a spacetime translations from a to ⇤ a , we must incorporate this ( ) into the group product:

a1, ⇤1 a2, ⇤2 :⇤ a1 + ⇤ a2 , ⇤1⇤2 . (6.1) ( )( ) ( ( ) )

11 4 This yields a semidirect product group denoted +" :⇤ R o L"+. P

Now comes the key step for this discussion: the discrete transformations are defined through a similar use of semidirect products. What is essential to note here is that this definition works because the discrete transformations are by definition (outer) automorphisms acting on the restricted Poincaré group. Parity p is the automorphism that reverses spatial translations, boosts and rotations; time reversal ⌧ is the one that reverses time translations, boosts and rotations; and p⌧ is their composition. This fact allows us to define:

• The complete Poincaré group . Let D ⇤ I,⌧,p, p⌧ be the group of space, time P { } and spacetime automorphisms of +", isomorphic to the Klein four-group. An P element of is a pair P, d +" D, with the group product defined in such a P ( )2P ⇥ way that the discrete transformations act appropriately on the continuous ones:

P1, d1 P2, d2 :⇤ P1 d1 P2 , d1d2 . (6.2) ( )( ) ( · ( ) )

The complete Poincaré group is thus the semidirect product group :⇤ +" o D. P P

We can now describe the symmetries of a relativistic quantum field theory, without appeal to any particular quantisation procedure. Successful quantisation will generally result in a representation of the restricted Poincaré group (cf. Landsman 1998, §3.3). So, we proceed to look at that representation directly. It is in particular a

11Semidirect products were introduced in Section 2.5.

165 strongly continuous homomorphism,

' : +" Aut , (6.3) P ! (H) from the restricted Poincaré group to the unitary operators on a Hilbert space. In a given reference frame, time translations take the form of a strongly continuous one parameter unitary representation ' t ⇤ of the time translations R, + . The ( ) Ut ( ) representation of a discrete symmetry, if it exists, is then just an extension of ' that includes some further elements of the complete Poincaré group . This might contain P the unitary parity operator P :⇤ ' p , the antiunitary time reversal operator T :⇤ ' ⌧ , ( ) ( ) or the antiunitary PT :⇤ ' p⌧ . ( )

6.3.2 How symmetries can give rise to more symmetries

Greaves’ first puzzle regarding the CPT theorem is: how can the assumption of one symmetry automatically give rise to a different one? The discussion above dissolves one aspect of the puzzle: the restricted Poincaré group naturally gives rise to further discrete symmetries, because the latter are not independent of the former. The discrete symmetries of parity, time reversal, and their composition are all symmetries of the Poincaré group itself: they are ‘higher order’ symmetries (see Section 4.3). So, one might well expect a theory that is invariant under the restricted Poincaré group to be invariant under its higher order discrete symmetries as well. In fact, the rather more pressing question appears to be: how could invariance under the discrete symmetries possibly fail? It is worth briefly addressing this puzzle as well.

Let me first make the puzzle even sharper. Let t be a time translation. Then by definition, the discrete symmetries transform this group element as,

1 1 1 ptp ⇤ t ⌧t⌧ ⇤ p⌧ t p⌧ ⇤ t. (6.4) ( ) ( )

Suppose we have a unitary or antiunitary representation of each of these group ele- ments, given by t , p P, ⌧ T and pt PT. Since a representation is a 7! Ut 7! 7! 7!

166 group homomorphism, these Hilbert space operators will satisfy the same relations:

1 1 1 P t P ⇤ t , T tT ⇤ PT t PT ⇤ U t , (6.5) U U U ( )U ( )

In other words: these operators are all symmetries12 of the dynamics! The very existence of a representation of parity, time reversal and PT guarantees that they are symmetries. How then can these symmetries ever fail? The answer is: a representation of the discrete transformations may not exist.

Given a quantum theory ‘tied down’ to spacetime by a representation of +", we next P face a technical question of whether it is possible to extend that representation to all of . What the symmetry violating experiments of the mid-twentieth century show is P that such an extension does not exist. I will discuss these experiments in more detail in Section 6.4. But first, notice the subtle conceptual point. Strictly speaking, parity violation is not the failure of in- variance under the parity operator P, and time reversal is not the failure of invariance under the time reversal operator. Rather, symmetry violation occurs if and only if a representation of these operators fails to exist at all.13 This has interesting philosophical consequences: since a representation both characterises the meaning of a transforma- tion, and also guarantees that it is a symmetry, symmetry violation implies not only that a symmetry has failed: the transformation has no meaning on state space at all. Understanding this subtlety will help us to see a curiosity in early proofs of the CPT theorem, which seem to have assumed the separate existence of transformations interpreted as C, P and T. But first, we must review what charge conjugation has to do with any of this.

6.3.3 Diverging perspectives on matter and antimatter

Greaves’ second puzzle about the CPT theorem is: how is invariance under a spacetime symmetry group connected to the exchange of matter and antimatter? If we do not

12For a discussion and definition of dynamical symmetries, see Chapter 4. 13In Chapter 4 I referred to this as the Symmetry-Existence Criterion.

167 adopt the Feynman perspective, then how can we understand this? Geroch has challenged the idea that there is one way to answer this question, proposing instead that charge conjugation contains a certain amount of conventional- ism:

“In general, there will be a number of different operators C which could be interpreted as effecting the replacement of particles by antiparticles. There is no obvious, unambiguous way of translating this physical notion into a mathematical operator.” (Geroch 1973, p.104)

A recent response by philosophers deserves mention here, though it is not the one I will focus on. This draws on the structure of local observables, as revealed by the DHR/BF superselection theory of Doplicher, Haag, and Roberts (DHR) and of Buchholz and Fredenhagen (BF), in order to derive the meaning of charge conjugation.14 Baker and Halvorson advocate DHR/BF theory as a foundation for the interpretation of matter and antimatter, in the spirit of the Representation View I set out above: “the physical property of a charge (or additive quantum number, or superselection sector) corresponds to a representation of the gauge group”, with matter-antimatter descriptions related by conjugate superselection sectors (Baker and Halvorson 2010, p.107). Swanson (2019) went on to give a substantial clarification of the relationship between spacetime symmetries and charge conjugation on this picture, in terms of the local expression of time translations (the ‘Lie product’).15 He summarises this approach:

“it explains why time-reversal and charge-conjugation are so closely linked. The spectrum condition entails that the only way to reverse the direction of time is to reverse the Lie product. But since the Lie product is also respon- sible for encoding the relationship between conjugate charges, antiunitary time reversal will also conjugate charge.” (Swanson 2019, p.121).

14These arose from a series of papers by Doplicher, Haag, and Roberts (1969a,b, 1971, 1974), later extended by Buchholz and Fredenhagen (1982). 15Given a C algebra with a one-parameter automorphism group defined by ↵ A ⇤ eitHAe itH for ⇤ t( ) all A, its differential form d ↵ ⇤ i H, A defines a Lie product. dt t t⇤0 [ ] 168 This deep perspective on charge conjugation does have some drawbacks: be- sides the known limitations of local algebraic quantum field theory,16 the extension of DHR/BF theory from global to local (variable phase) symmetries is a matter of ongoing research, as DHR themselves point out.17 I will set the details of this approach aside here, not because of any drawbacks, but to make a simpler point about the connection between spacetime symmetries and the matter-antimatter relationship, which I would not like to see lost in the details.

6.3.4 Covering group of the Lorentz group

There is a remarkably simple relationship between matter, antimatter and Minkowski spacetime, which arises out of the universal covering group SL 2, C of the Lorentz ( ) group: the universal cover is, in short, ‘two connected copies’ of the Lorentz group living in a simply connected space. This extra structure, which in a sense lives inside the Lorentz group already, turns out to be just what is needed to represent the exchange of matter and antimatter, in a sense I will explain shortly. On this perspective, matter and antimatter are not independent of spacetime, but rather baked into the structure of the Lorentz group through its universal cover. Let me briefly motivate this idea, by

first reviewing the structure of SL 2, C and its relationship to the Poincaré group. ( ) A covering group or cover over a Lie group G is a Lie group G⇤, together with a continuous homomorphism or ‘covering map’ from G⇤ onto G, such that every element of G has a neighbourhood that is evenly covered.18 The universal covering group is the unique simply connected covering group over G. It covers all other covering groups over G, and has the same local structure. So, if we are interested in the local structure of a group representation, then we can always lift up to representing the universal covering group instead, although there may be some degeneracy: multiple elements the covering group may get mapped to the same element in G under the covering map.

16Cf. the debate between Wallace (2006, 2011a) and Fraser (2009, 2011); for conciliatory replies see Swanson (2017) and Rédei (2020). 17Cf. Doplicher, Haag, and Roberts (1969a, p.4), referring to them as gauge groups “of the second kind”. See Doplicher (2010) for a discussion of future directions. 18A covering map is topologically a local homeomorphism or étale map; in the case of a Lie group, it induces an isomorphism of Lie algebras. See Hochschild (1965, Chapters IV and XII) for an introduction.

169 This means that, when we wish to encode the local structure of Minkowski spacetime in a state space using a representation, the Lorentz group L"+ is not really as ‘large as it can be’: more spacetime structure is hidden in the universal covering group. This turns out to be:

• Universal cover SL 2, C of the Lorentz group. The group of 2 2 complex matrices ( ) ⇥ with unit determinant, isomorphic to the universal covering group of L"+.

To see its relationship to the Lorentz group, we can construct an explicit representation of Minkowski spacetime ‘living inside’ SL 2, C , following the classic presentation of ( ) Naimark (1964, §3.9).

Let M SL 2, C be the subset of self-adjoint matrices A ⇤ A , with the ad- ⇢ ( ) ⇤ joint defined by the conjugate-transpose. Forgetting about matrix multiplication and just focusing on matrix addition and multiplication by real scalars, M forms a four- dimensional vector space under addition. A basis for this vector space is given by the Pauli matrices,

1 1 i 1 0 ⇤ 1 ⇤ 2 ⇤ 3 ⇤ . (6.6) 1 1 i 1 © ™ © ™ © ™ © ™ ≠ Æ ≠ Æ ≠ Æ ≠ Æ ⇤ ⇤ We define a´ symmetric¨ bilinear´ form by¨ setting i ´,j : ¨ ⌘ij, where ⌘´ij diag¨ 1, 1, 1, 1 h i ( ) is the Minkwoski metric. This makes M, , isomorphic to Minkowski spacetime, ( h i) viewed as a linear vector space sitting inside SL 2, C . Using the four basis elements, ( ) each element v M can be written in terms of some set of real numbers a such that, 2 i

3 a0 + a3 a1 ia2 v ⇤ ai i ⇤ . (6.7) i⇤0 a1 + ia2 a0 a3 ’ © ™ ≠ Æ We can also get a helpful expression of the´ inner product through¨ an explicit calculation of the determinant:

2 2 2 2 det v ⇤ a0 + a3 a0 a3 a1 ia2 a1 + ia2 ⇤ a a a a ⇤ v, v . (6.8) ( ) ( )( )( )( ) 0 1 2 3 h i

170 The Lorentz group L is the set of transformations ⇤ : M M that preserve the ! norm, Lv, Lv ⇤ v, v , which we can now write as equivalent to det Lv ⇤ det v . The h i h i ( ) ( ) restricted Lorentz group discussed above then consists in those elements that preserve temporal orientation ( v, ⇤v > 0) as well as total orientation (det ⇤ > 0). With h i ( ) the Lorentz group expressed in this way, one can construct an explicit covering map

S ⇤ , given by, 7! S ⇤Sv ⇤ SvS⇤ (6.9) for all v M. This map is straightforwardly shown19 to be a continuous, surjective 2 homomorphism ⇤ : SL 2, C L"+, which is a twofold covering in that ker ⇤ ⇤ I, I . ( )! { } This implies in particular that for all S SL 2, C , the pair of elements S, S get mapped 2 ( ) to the same Lorentz group element, ⇤S ⇤ ⇤ S L"+. 2 Although it is hard to visualise either of these groups, the subgroup SO 3 L"+ ( )⇢ of spatial rotations is easy to grasp, with its universal covering SU 2 . The relationship ( ) between these groups can be visualised as in Figure 6.4 in one dimension using a

Möbius strip: when an arrow is rotated through 2⇡, it is not transformed identically, but rather reverses: a second ‘copy’ of rotations through 2⇡ is needed to restore it to its original orientation.20

6.3.5 An account of charge conjugation

With these details in place, I can now give the basic perspective on the matter-antimatter relationship that I would like to defend. This view, which is in the same spirit as the Representation View above, can be summarised as follows.

• Charge conjugation, like parity and time reversal, is fundamentally a symmetry

19 ⇤S is in the Lorentz group because ⇤S v, ⇤S v ⇤ det SvS⇤ ⇤ det S det v det S ⇤ det v ⇤ v, v . Similar arguments show it preserves temporalh andi total( orientations.) ( ) It is( obviously) ( ) continuous,( ) h andi we can check that it is a homomorphism, ⇤ v ⇤ RS v RS ⇤ R SvS R ⇤ ⇤ ⇤ v . To see that it RS( ) ( ) ( )⇤ ( ⇤) ⇤ R S( ) is a twofold covering, let S ker ⇤, so that v ⇤ ⇤ v ⇤ SvS for all v. It follows by Schur’s lemma that 2 S ⇤ S ⇤ cI for some c C. We thus have that 1 ⇤ det S ⇤ c2, and hence that S ⇤ I, which is to say that 2 ( ) ± ker ⇤ ⇤ I . {± } 20You can experience SU 2 yourself by holding your palm upright, and noticing that by twisting your arm, it is possible to rotate( ) your hand 360 degrees all while keep your palm upright. By continuing to rotate it even further through another 360 degrees, always palm up, you can then untwist your arm to its original state!

171 Figure 6.4: On a Möbius strip, a rotation through 2⇡ reverses the arrow, which is returned to its initial state by another rotation through 2⇡.

of a group of continuous spacetime symmetries.

• But, whereas p and ⌧ are symmetries of the Lorentz group L"+, charge conju-

gation c is a symmetry of its covering SL 2, C , which acts identically on L"+, ( ) characterising the fact that it is an ‘internal’ symmetry.

• When charge conjugation symmetry exists, then just like parity and time reversal, it can be given a state space representation.

Geroch (1973, p.104-105) has proposed a related set of conditions to characterise charge conjugation in a Hilbert space representation: that C preserves the Hilbert space norm, that it commute with the action of the Poincaré group, and that it exchanges positive and negative frequency modes of a quantum field representation. All of these conditions are met automatically on the present account of charge conjugation: that it is a continuation of the ordinary representation theory of the Lorentz group, lifted to the universal covering SL 2, C . ( ) The Lie group SL 2, C has four non-trivial automorphisms, which are listed ( ) in Figure 6.5: two ordinary automorphisms (satisfying RS ⇤ R S ) and two ( ) ( ) ( ) anti-automorphisms (satisfying RS ⇤ S R ). One can immediately see from ( ) ( ) ( ) our definitions that one of them, the conjugate-transpose (adjoint) c : S S , acts 7! ⇤ invariantly on the restricted Lorentz group L"+ defined above. Namely, since ⇤ v ⇤ S( )

172 SvS⇤ and v⇤ ⇤ v, we find that the conjugate-transpose acts as,

c ⇤ v ⇤ SvS⇤ ⇤ Sv⇤S⇤ ⇤ SvS⇤ ⇤ ⇤ v. (6.10) ( S)( ) ( ) S

conjugate S S¯ automorphism 7! inverse-transpose S S 1 automorphism 7! ( )> conjugate-inverse S S¯ 1 antiautomorphism 7! conjugate-transpose S S :⇤ S¯ antiautomorphism 7! ⇤ ( )> Figure 6.5: The non-trivial automorphisms and antiautomorphisms of SL 2, C ( )

This is exactly as one would expect charge conjugation to behave: it is a ‘reversal’, in that applying it twice is the identity; it is a non-trivial symmetry of our ‘doubled’ spacetime structure SL 2, C ; and, it has no effect on spacetime or on the underlying ( ) Lorentz transformations L"+. This turns out to be the only automorphism with this

21 property, as the other three turn out to transform L"+ non-trivially. We thus adopt the following basic perspective: charge conjugation is the conjugate-transpose c : SL 2, C ( )! SL 2, C . Following the same procedure as above, we can now extend SL 2, C to ( ) ( ) include the charge conjugation:

• Universal cover with charge conjugation SL 2, C o I, c , where c : SL 2, C ( ) { } ( )! SL 2, C is the conjugate-transpose (charge conjugation), and the semidirect prod- ( ) 1 uct is as for the discrete elements of the Lorentz group above, so that cSc ⇤ S⇤.

Viewing the restricted Lorentz group L"+ as map on self-adjoint complex matrices v ⇤ v⇤ as above, it follows that the action of charge conjugation leaves the Lorentz transformations unchanged. The properties of the charge conjugation operator identified by Geroch (1973) can now be recovered automatically in a Hilbert space representation ' : SL 2, C ( )! Aut amongst the unitary operators on a Hilbert space . This representation might H H 21 Referring to the basis defined by 0, 1, 2, and 3, a straightforward calculation shows: (1) conjugation reverses the 2 axis (y); (2) inverse-transpose reverses the 1 and 3 axes (z and x); and (3) conjugate-inverse reverses the 3 axis (x).

173 have the property that ' S ⇤ US ⇤ U S ⇤ ' S , in which case it is isomorphic to the ( ) ( ) restricted Lorentz group, since as we have seen, L"+ is isomorphic to the quotient group SL 2, C I, I . However, this is not always the case: a ‘faithful’ representation of ( )/{ } SL 2, C , with one exactly element ' S ⇤ U for each S, is called a spinor representation. ( ) ( ) S These representations are well-studied, and it goes beyond my purposes to review them in detail.22 For this discussion, the crucial point is that in a spinorial representation, the conjugation map takes a vector A to its complex conjugate ¯ A0 up to a phase, where on a typical interpretation A is a positive-frequency solution to a wave equation and the ‘dotted’ vector ¯ A0 is a negative-frequency solution. Elementary considerations on the assumption that energy is bounded from below can be further used to show that this operator C cannot be antiunitary, and is therefore unitary (Geroch 1973, p.106). Although I have only sketched the details of this approach to matter and an- timatter, the view is straightforward: we can always replace a spacetime structure associated with the Lorentz group L"+ with the covering group SL 2, C . The meaning ( ) of charge conjugation in this context is fundamentally the map c S ⇤ S that takes ( ) ⇤ each element S of the covering group to its conjugate-transpose. On the representation view, the meaning of charge conjugation in a quantum theory is then given by a Hilbert space representation of SL 2, C o I, c . When a representation of SL 2, C is extended ( ) { } ( ) to include c as a group element, we find that the representative C ⇤ 'c of this element corresponds to precisely what we expect charge conjugation to be: it is a reversal op- erator that preserves elements of the Lorentz group; and, in a spinor interpretation of the underlying Hilbert space, this operator is an application of complex conjugation (up to phase) that reverses dotted and undotted indices.

6.3.6 Constructing CPT

Having understood C, P and T individually, we can now turn to the construction of these transformations together. The first step is to extend SL 2, C to a group that ( ) includes spacetime translations.

22A classic reference is Naimark (1964, §III.3.4); see also Geroch (1973, §18) and Wald (1984, Chapter 13).

174 Using a semidirect product in the same way as before, we can construct the universal covering group of the restricted Poincaré group, which I will denote:23

4 • S +" ⇤ R o SL 2, C : the elements a, L consisting in a translation a together an P ( ) ( ) element L SL 2, C of the covering group of the Lorentz group L"+. 2 ( )

Just as SL 2, C is a twofold covering of the restricted Lorentz group L"+, so this group ( ) is a twofold covering of the restricted Poincaré group +", with each pair of elements P P, P S +" mapped to the same element of +" by the covering map. Motivated in 2 P P this way, suppose we now describe a general spacetime structure consisting not just in the restricted Poincaré group +", but in its universal cover S +". P P We can extend the conjugate-transpose map on SL 2, C to a map c : S +" S +" ( ) P ! P by requiring that it acts invariantly on spacetime translations. This leads to a group element c that retains its essential character in describing the exchange of matter and antimatter: it is a non-trivial (anti)automorphism of our general spacetime structure

2 S +" satisfying c ⇤ I, but which leaves the underlying Poincaré group P+" unchanged. P We moreover use the strategy described above to describe the discrete transformations corresponding to parity p and time reversal ⌧. Our group of discrete symmetries now consists in eight transformations of S +": P

D⇤ ⇤ I, p,⌧,c, p⌧, pc, c⌧, cp⌧ . (6.11) { }

This collection forms a group isomorphic to Z2 Z2 Z2. Since each element is an ⇥ ⇥ automorphism of S +", we can use it just as before to describe the ‘complete’ spacetime P symmetry group, given by a semidirect production:

• S +" o D : the complete spacetime symmetry group, consisting of the universal P ⇤ covering group S +" of P+" , together with its discrete symmetries, including cp⌧. P The physical significance of these transformations is now again just a problem of representation theory: instead of defining a relativistic quantum field theory to be

23There is no standard symbol to denote this group. This is sometimes called the inhomogeneous SL 2, C group ISL 2, C (cf. Wald 1984, p.341). I avoid this notation to make clear that it is a semidirect product,( ) as one finds( e.g.) in Landsman (2017, p.272).

175 a representation of the restricted Poincaré group, we take it to be a representation of

S +". Its irreducible representations are labelled by the same Casimir operators as the P Poincaré group, m2 for 4-momentum and S2 for squared angular momentum, since the two have the same Lie algebra.24 The question of symmetry violation then becomes the question of whether a given representation of the continuous spacetime symmetries in

S +" can be extended to include some or all elements of the complete group S +" o D . P P ⇤ In particular, a CPT invariant quantum field theory is one in which the representation can be extended to include cp⌧. Viewed from this perspective, charge conjugation can hardly be called ‘non- spatiotemporal’: it is tied directly to the Poincaré group +" of spacetime symmetries, P being a non-trivial automorphism of its universal covering group that preserves +". P And, the very existence of a representation of cp⌧ is enough to establish that it is a dynamical symmetry, since this transformation has the effect of reversing time trans- lations. This solves the second of Greaves’ puzzles, in a way that is much closer to standard arguments for CPT symmetry stemming from the work of Jost (1957), which invariably draw on the structure of S +". P Thus, a richer representation theory dissolves the puzzles about what CPT has to do with spacetime, and how a relativistic quantum theory might be expected to be symmetric under it. In the next section, we finally turn to CPT symmetry in detail, in order to understand its significance for the arrow of time.

6.4 The significance of CPT symmetry

Viewed from the representation theory of the Poincaré group, there are a number of physically significant ways to ‘reverse time’, including T, CT, PT, and CPT. All of these transformations are closely connected to spacetime structure, through the universal covering of +". However, the significance of the CPT transformation is the one that P can be given physical meaning through a representation. To see how this is the case,

24Streater and Wightman (1964, §1-3) and Wald (1984, Chapter 13) give textbook treatments of the representations of S +" using the techniques developed by (Wigner 1939) and Bargmann (1964). P

176 we must take a little tour through the history of quantum field theory. Quantum field theory arguably began with the Klein-Gordon and Dirac equa- tions, with their were puzzling negative energy solutions. Todeal with this problem, we now standardly introduce a distinction between matter and antimatter, reinterpreting the negative-energy descriptions as positive-energy ones with matter and antimatter exchanged. The result is a theory that is well-behaved, in the sense of producing a hyperbolic wave equation, and fully relativistic. However, this seemingly great deal comes with a catch, astutely pointed out by Swanson (2014, 2019):

“This trick only works to restore Lorentz invariance and hyperbolicity, though, if there is an exact correspondence between particles and antipar- ticles: they must be indistinguishable except for their charge. The CPT theorem accounts for this.... It is only because the theory is CPT-invariant that we can reinterpret negative energy states as describing antiparticles in a manner consistent with the requirements of relativistic causality.” (Swan- son 2019, p.107)

Of course, the most straightforward way to assure this would be if it happened that quantum field theory is invariant under the exchange of matter and antimatter C. Unfortunately, that was shown to be experimentally false. Let me briefly discuss how this is known, and how remarkable it is that only CPT remains.

6.4.1 Violation of P,C, CT and PT

Before the 1950’s, quantum field theory was thought to be parity invariant. One might well wonder how it is possible to assert this, when the standard model was still decades from being invented. Without a dynamics, what does it mean to say that parity is a dynamical symmetry? Practitioners at the time drew on the following fact.25

Proposition 6.2. Let S be a unitary (scattering) matrix, and let P : be any unitary H!H bijection. Then the statement that R is a dynamical symmetry, in that PS ⇤ SP, is equivalent

25Proof: see Roberts (2015b,c). Interestingly, this proposition can be viewed as an instance of Curie’s Principle, that every symmetry of a cause is a symmetry of an effect.

177 to the statement that P in in if and only if P out ⇤ out. ! Even without knowing the dynamics, this fact allows one to predict on the basis of parity invariance that in-states and out-states must transform under parity in the same way. An application arose with the discovery of two-pion and three-pion outgoing states that transform in opposite ways under parity:

+ + P⇡ ⇡0 ⇤ ⇡ ⇡0 (6.12) + + + + P⇡ ⇡ ⇡ ⇤ ⇡ ⇡ ⇡.

Parity invariance implies that these two outgoing states cannot both arise from the same ingoing state, as an application of Proposition 6.2. So, it was assumed that these decay products must have come from different ingoing states, which were labelled

✓ and ⌧. However, strangely, the ingoing states appeared to have exactly the same mass and lifetime, which gave rise to the famous ‘✓-⌧ puzzle’: what could explain this remarkable coincidence? Lee and Yang (1956, p.254) controversially suggested that the problem could be solved by accepting parity violation. This proposal was dramatically confirmed when Chien-Shiung Wu et al. (1957) proved that the decay of the Cobalt-60 atom violates parity. This solved the ✓-⌧ puzzle: the two states can be identified as one and the same particle, now understood to be the states of a K meson or kaon. However, the solution also immediately introduced a more seriously difficulty: these same outgoing pion decays violate charge conjugation symmetry, and therefore the trick of assuming a correspondence between matter and antimatter states! This fact was confirmed directly in a follow-up experiment by Garwin, Lederman, and Weinrich (1957) soon after Wu’s. And, insofar as CPT symmetry holds, it follows from these two facts that both CT and PT symmetry are violated as well. In particular, if the Feynman (1949) view is taken to require that CT is a symmetry, then that view is incompatible with experiment. The response of most physicists was to postulate that the combined transforma- tion by parity and charge conjugation (interpreted as matter-antimatter exchange), or

178 CP, must remain a dynamical symmetry. A number of elegant CP invariant models were developed, including by Weinberg (1958). This of course led to another dramatic turn of events.

6.4.2 Violation of CP and T

Long-lived neutral kaon states are best known for their decay into three pions, K L ! 0 + ⇡ ⇡ ⇡. The neutral pion does not ionise in a spark chamber and so is invisible, but its trajectory can be calculated from the trajectories of the other two by conservation of momentum. This decay is compatible with CP invariance, by an application of

0 + Proposition 6.2: both the ingoing KL and the outgoing ⇡ ⇡ ⇡ reverse sign under CP, + and so respect CP symmetry. In contrast, a two-pion state ⇡ ⇡ is left unchanged by + CP, and so a decay from KL into ⇡ ⇡ would violate CP symmetry. In 1964, Cronin and Fitch26 set out to check whether they could show that, to a high degree of accuracy, no CP-violating two-pion decay events could be found. After a long analysis of all the photographs, they found on the contrary that the two-pion outgoing state occurs! After checking for possible confounding facts and discussing this result with colleagues at Brookhaven, it was agreed that the signal was unmistakable. After explaining it to their colleague Abraham Pais over coffee, Pais reported that, “[a]fter they left I had another coffee. I was shaken by the news” (Pais 1990). Thus, we again lost the argument of a correspondence between matter and antimatter! This correspondence is restored, as Swanson (2019) points out, by the presence of CPT symmetry. This is the last remaining way to make the correspondence: virtually all other discrete symmetry transformations fail to exist. This scarcity of options, together with our success rate above, might lead one to be concerned: can we really trust CPT symmetry after a history of events like this? The answer is not clear. However, unlike the earlier arguments, there are arguments for CPT symmetry that draw only on very general assumptions about the structure of relativistic quantum

26See Christenson et al. (1964) and Cronin and Greenwood (1982).

179 field theory. So, perhaps it stands a better chance. Given CPT symmetry, the discovery of CP violation implies the violation of time reversal symmetry as well. Thus, this discovery of 1964 also established the remarkable fact that T symmetry is violated. In the language of this chapter: the experiment shows that time has a weak arrow. This discovery has only been found to be more robust in the last decades: observations of time reversal invariance without appeal to CPT symmetry have also been achieved, through an application of the principle of detailed balance. This is known as ‘Kabir’s principle’ in this context, and is captured by the following fact.27

Proposition 6.3. Let S be a unitary (scattering matrix) and let T be an antiunitary bijection. Then time reversal symmetry TST 1 ⇤ S 1 is equivalent to the statement that in, S out , h i T out, ST in for all in, out. h i

Oscillating kaon-antikaon states turned out to be a fruitful place to test for T- violation using this technique: if the kaon-to-antikaon decay happens with different probability amplitude than the time reversed antikaon-to-kaon decay, then we would have a measurement of T-violation. This is exactly what was discovered by Angelopou- los et al. (1998). Related arguments later showed that T-violation occurs in the B meson sector as well (Lees et al. 2012). Some preliminary recent evidence even suggests a small amount of CP and T violation in the lepton sector through neutrino oscillations (T2K

Collaboration 2020). Thus, the experimental evidence for T-violation in the standard model, and thus the weak arrow of time, is becoming ever more secure. In summary, the experimental study of symmetry violation has found that most discrete transformations of S +", the universal covering group of +", are not P P symmetries of the standard model. But by their very definition, these transformations are automatically symmetries whenever a representation of them exists. As a result, one is forced to conclude that in the standard model, most of the discrete transformations do not exist. To be precise, experimental evidence shows that no representation of any

27See Roberts (2015a,c) and Ashtekar (2015).

180 of the following discrete transformations exists in the standard model:

C, P, T, PC, CT, and PT.

This leaves just one remaining possible discrete transformation in S +" that one might P be able to represent: the CPT transformation.

6.4.3 The great misnomer

An important lesson of the historical discussion above is that CPT is not really the composition of C, P and T. But when Lüders (1954) formulated his first version of the

CPT theorem, he did it as a composition of two transformations: CT, which he called time reversal “of the second kind”, and P, writing that one must “assume that the field theory in question is invariant under reflection in space; a formal reflection in time acts as an intermediate step in the proof” (Lüders 1954, p.4). What we have just seen is that this statement of the CPT theorem is an empty truth of the standard model: no spatial reflection transformation P exists, and so there is no non-trivial sense in which the proposition is true. This problem is often glossed in textbook treatments of the CPT theorem. How- ever, Sachs (1987, §11.2) points out that it is a serious problem:

“Before the discovery of parity violation in 1956, the definitions of the trans-

formations P, T, and C appeared much more straightforward because the invariance of all interactions was taken as a basic assumption. P violation by the weak interactions showed that this assumptino is untenable.... In such a situation there is a question whether it is meaningful to assert the existence of a kinematic (i.e., independent of the interaction) transformation associated with the violated symmetry.” (Sachs 1987, p.267)

Although Sachs proposes a strategy for attempting to avoid the problem, by ‘carrying over’ the representation of each of these operators from a context in which it is a symmetry to one in which it is not, the argument is a mighty struggle, which leads

181 Sachs himself to conclude that proofs of this kind are ultimately unsatisfactory. Fortunately, more careful proofs of the CPT theorem following Jost (1957, 1965) now proceed by direct construction of a representation of the abstract CPT transfor- mation cp⌧ S +" o D . A limitation of Jost’s own approach is its formulation in terms 2 P ⇤ of Wightman fields, which for all their precision have found limited fewer modelling applications than more standard formulations of quantum field theory.28 However, most modern proofs adopt his core technique of extending a representation of SL 2, C . ( ) Jost denoted his CPT operator ⇥, to avoid confusion with an operator that is necessarily the composition of three others.29 Referring to this transformation as CPT is thus a misnomer, though a convenient one, which I will continue in the discussion to follow.

6.4.4 Representations of CPT

Let me finally comment briefly on the construction of CPT itself. This invariably be- gins with a spinorial representation of S +", with field operators x having some P ( ) number of dotted and undotted spinorial indices. The representation has the partic- ular property that the spacetime translations (including time translations) satisfy Ua 4 a x a ⇤ x + a for all a R . One then assumes it satisfies some plausible U ( )U ( ) 2 physical conditions, usually involve constraining energy to the forward light cone, as well as postulating local relativistic causality. What the CPT construction then shows is that a representation of cp⌧ exists, and is given by a representative of this transfor- mation on a Hilbert space. I will not review the proof here,30 which has many steps.

28See the no-go results of Strocchi (2013, §7.3) for limitations of Wightman’s theory. Wightman fields appeared in a widely-circulated manuscript by Arthur Wightman written in 1951 (Wightman 1996, p.174). Rédei (2014b, 2020) and Swanson (2017) provide a philosophical appraisals. Streater reports his road to finding this influential paper: “I had asked [Abdus] Salam ‘what is a quantised field’, and received the answer ‘Good; I was afraid you would ask me something I did not know. A quantised field, x at the point x of space-time, is that operator assigned by the physicist using the correspondence ( ) principle, to the classical field at the point x’. I went away thinking about this; then I realised that what I needed was a statement of which operator is assigned by the physicist. I complained to P. K. Roy, who said that I should read Wightman’s paper” (Streater 2000). There is of course room for debate as to whether Salam or Roy gave Streater the better advice as to what a quantum field ‘really is’. 29More carefully, he writes: “This involution is in theories which are invariant under time inversion T, space reflection P and matter-antimatter conjugation C represented by the product TCP” (Jost 1965, p.100). 30See Haag (1996, §II.5) or Greenberg (2006) for a proof-sketch of Jost’s argument, Bain (2016, Chapter 1) for a review of a number of different arguments, and Swanson (2019) for an analysis of the algebraic

182 Instead, let me just note that in Jost-style proofs, if a spinor x has some number of ( ) m of dotted indices and n undotted spinor indices, then the CPT operator ⇥ is shown to satisfy the relation,

1 m F ⇥ x ⇥ ⇤ 1 i ⇤ x , (6.13) ( ) ( ) ( ) ( ) where F ⇤ 0 if x x ⇤ x x (boson) and F ⇤ 1 if x x ⇤ x x ⇤( ) ( ) ( ) ⇤( ) ⇤( ) ( ) ( ) ⇤( ) (fermion). In some reference frame, we now write for the a parameter group of Ut time translations in S +", which satisfies x ⇤ x + t for each field operator. This P Ut ( ) ( ) implies, through an application of Equation (6.13), that,

1 ⇥ t⇥ ⇤ t , (6.14) U U which is to say that the CPT operator ⇥ is a time reversing dynamical symmetry. There are many approaches to proving the CPT theorem, collected together by Bain (2016, p.28) in his book, The CPT theorem and the Spin-Statics Connection. Noting that the formal assumptions and conclusions of these proofs are often quite different, Bain argues that one has little reason to think the various CPT theorems are establishing the same thing. In response, Swanson (2018) points out a number of interesting ways in which these differences only superficial, such as the way the way Poincaré invariance is hidden in a technical assumption on the algebraic approach known as modular covariance. I would like to emphasise a simpler point about the CPT theorem: that in order for ⇥ to be reasonably interpreted as a CPT operator, it must consist in a representation of the cp⌧ transformation, defined as an automorphism of the universal covering of

31 a spacetime symmetry group, such as the covering group S +" over +". Any CPT P P theorem deserving of the name must at least be interpretable as doing this. Of course, the particular character of the representation will in general be different depending on the structure of our theory. But this is to be expected in quantum field theory, whose approach. 31This sort of construction can of course be carried out for the Galieli group as well, although a CPT theorem in this context does not appear to be available in 3 + 1 dimensions (Lévy-Leblond 1967).

183 precise mathematical foundation has not yet been completely settled.

6.5 CPT and the weak and strong arrows

Experimental evidence shows that there is a weak arrow of time, in that time reversal symmetry is violated. For a strong arrow to occur, it would have to be the case that no other time reversing symmetry is possible either. I have argued that in quantum theory, that is not the case: there is no strong arrow of this kind. However, a more moderate perspective on the strong arrow is to say that it requires no physically relevant time reversing symmetries are possible, where physical relevance means the transformation is an automorphism of the Poincaré group, or at least its universal cover. Then we get remarkably close to a strong arrow. On this definition, the physically relevant time reversing transformations are,

T, PT, CT, CPT.

And, it appears that only one of these is a symmetry in relativistic quantum field theory, the CPT transformation. However, relativistic quantum field theory is also an extremely limited context in which to study the arrow of time. After all, Minkowski is incredibly symmetric, in a way that global spacetime structure is not. It is absolutely remarkable that there is so much evidence for an arrow of time in this context, even in the weak sense. But, to establish whether or not there is a strong arrow of time, considerations of global spacetime structure become necessary. I reserve those considerations for Chapter 8.

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