![Sample our Humanities journals, sign in here to start your FREE access for 14 days]({"type":"image" , "src" : "/sda/5939/TOC-Subject-Sample-2021-Humanities.jpg"})
Abstract
I discuss the idea of relativistic causality, i.e., the requirement that causal processes or signals can propagate only within the light‐cone. After briefly locating this requirement in the philosophy of causation, my main aim is to draw philosophers’ attention to the fact that it is subtle, indeed problematic, in relativistic quantum physics: there are scenarios in which it seems to fail.
I set aside two such scenarios, which are familiar to philosophers of physics: the pilot‐wave approach, and the Newton–Wigner representation. I instead stress two unfamiliar scenarios: the Drummond–Hathrell and Scharnhorst effects. These effects also illustrate a general moral in the philosophy of geometry: that the mathematical structures, especially the metric tensor, that represent geometry get their geometric significance by dint of detailed physical arguments.
Acknowledgements
For comments, conversation and correspondence, I am very grateful to audiences in Dubrovnik, Oxford and Cambridge; two referees, and Bill Demopoulos, Ian Drummond, Gordon Fleming, John Norton, Brian Pitts, Graham Shore; and especially to Steve Adler, Harvey Brown, Dennis Lehmkuhl, Bob Wald and Steven Weinstein.
Notes
[1] I confess at the outset that I set aside a recent line of argument which ‘goes in the opposite direction’ to what follows, i.e., which uses relativistic causality at high energies (mostly in the form of an analytic S‐matrix) to select low‐energy effective theories—thereby concluding from effects like those I shall discuss that QED is not embeddable in any causal high‐energy theory. Thanks to Hugh Osborn for this, and for referring me to Adams, Arkani‐Hamed et al. (Citation2006). That paper also discusses other superluminal effects, including one from a brane model—and how it might be detected by tiny deviations in the moon’s orbit! Shore (Citation2007) includes a response to this line of argument, drawing on his work on the Drummon–Hathrell effect, details of which are in section 5.2 below. So far as I know, this line of argument awaits philosophers’ attention.
[2] Though these effects are unfamiliar, Brown is not alone in pointing to their philosophical importance. Weinstein mentions the Drummond–Hathrell effect as one of several examples, in his recent judicious defence of superluminal signalling (Citation2006, 390, 393); and Weinstein (Citation1996) urges a moral similar to Brown’s. What follows is much indebted to them both.
[3] The remarks under (2) apply equally to the two examples, the pilot‐wave theory and the Newton–Wigner representation, which I have set aside. Thus for the pilot‐wave theory, in the non‐relativistic version, the guidance equation is strictly speaking a mathematical statement of how the velocity of a particle depends on the position of distant particles. But it is usually, and I think rightly, taken to indicate action‐at‐a‐distance—and not joint effects of a common cause. A similar remark applies to the quantum potential; and there are similar equations, again indicative of action‐at‐a‐distance, in relativistic versions. But I admit that this interpretation is not mandatory. Indeed, Dickson (Citation1998, sec. 9.4, 196–208) denies it. (His overall position is reminiscent of Norton’s causal anti‐fundamentalism. He is wary of localized facts or events as causal relata and suggests that under determinism, the most we can say is that the total present state is the effect of earlier states.)
[4] Given just his suspicion of causality, Norton could even rejoice at my examples violating relativistic causality: he could see them as providing another nail in the coffin of (the modern representative of) the principle of causality. But as noted in the next paragraph, Norton joins most other philosophers of physics in being gung‐ho about relativistic causality.
[5] In this paragraph, I am indebted to Klaas Landsman and Fred Muller for correspondence and references.
[6] I learnt this argument, apparently common in the folklore of AQFT, from K. Fredenhagen and S. Summers, in conversation—to whom my thanks.
[7] For an approach to quantum field theory on non‐globally hyperbolic spacetimes, cf. Kay (Citation1992, especially sec. 6).
[8] I am very grateful to Bob Wald for teaching me what follows. One caveat: to date, the framework is secured for scalar fields only. Wald tells me he is very sure it can be extended to Dirac fields, and pretty sure for QED, a happy prospect, also for this paper’s topic, since it is what we need for fully understanding section 5’s effects.
[9] Though the words are mine, this section is due to Brown. He urges this moral, and related ones, in various passages of his (Citation2005) article; cf. also his associated papers, especially Brown and Pooley (Citation2001, Citation2006). Since drafting this paper, I have read Weinstein’s brief but rich (Citation1996) article. This article (i) floats a moral like Brown’s, for general relativity and kindred theories, and lists some adjacent issues; (ii) illustrates the moral with several examples of non‐minimal coupling, not just two as in our section 5; and (iii) discusses coupling in terms of action principles. By the way, though the moral is controverted by philosophers, my impression in discussion with physicists is that they endorse it—at least the way I say it!
[10] For Brown’s discussion, cf. (Citation2005), 25–26 and 161–163. Cf. also Norton (Citation1985) and Ghins and Budden (Citation2001).
[11] In both these theories, there is a distinction between timelike and spacelike curves, and so the theory asserts the particle to travel along a timelike geodesic. The main difference is that in special relativity, the connection is uniquely determined by (the requirement of compatibility with) the metric. We need not consider now the issue whether this assertion needs a dynamical or ‘constructive’ explanation of the kind Brown favours. But I will discuss this issue in section 4.2.3.1.
[12] Partial derivatives are usually represented by a comma; and covariant derivatives, by a semi‐colon. So the semi‐colon abbreviates the correction terms, and Minimal Coupling is sometimes called the ‘comma‐to‐semi‐colon’ rule.
[13] Aficionados know several such: thanks to Steve Adler for mentioning the conformal massless Klein–Gordon field.
[14] For details, cf., e.g., Misner, Thorne, and Wheeler (Citation1973, 471–480), Wald (Citation1984, 73), and Geroch and Jang (Citation1975). Brown notes various subtleties about these theorems. In particular, although the form of the field equations determines gij to be a (0,2) symmetric tensor, nothing in the equations dictates that gij should have Lorentzian signature.
[15] A detailed study of photon dispersion and birefringence (polarization‐dependent phenomena) is Adler (Citation1971). And in recent years, a framework for unifying these results has begun to emerge:, e.g., Dittrich and Gies (Citation1998).
[16] Brown’s own discussion is in (Citation2005), 165–172. Among other references, he cites Shore (Citation2003b) and Liberati et al. (Citation2002). These and Shore (Citation2003a, Citation2007), and some of their references, have been my sources for what follows. I again stress, as in sections 1 and 3.3, that our present understanding of these effects is undoubtedly incomplete—there are plenty of open questions hereabouts, for both physicists and philosophers.
[17] Incidentally, the emergence in relativistic pilot‐wave theories of Lorentz‐invariance at the observable quantum level from the non‐Lorentz‐invariant sub‐quantum level is an example of such ‘implicit occurrence’.
[18] Section 5.2.1 will give a numerical estimate for a solar mass black hole.
[19] My main sources are Shore (Citation2003a, Citationb); cf. note 16.
[20] I presume his idea is that only with this can we be sure that there can be a process from B to C which is like that from A to B. But the threatened zig‐zag might exploit different processes in its two legs. But never mind, we will see, now and in section 5.3.2, stronger reasons to doubt that there can be such zig‐zags.
[21] My main source is Liberati et al. (Citation2002); cf. note 16.
[22] For the history and philosophy of this effect, cf. Rugh, Zinkernagel and Cao (Citation1999).
[23] Drummond and Hathrell also make this point (Citation1980, 353). One might add: consider such propagation, or even instantaneous action‐at‐a‐distance, in a Newtonian spacetime.