Review of Richard Healey's 2008 book. To appear in MIND

Baker, David John

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Review of Richard Healey, Gauging What’s Real: TheConceptual Foundations of Gauge TheoriesDavid John BakerDepartment of Philosophy, University of Michigandjbaker@umich.eduSeptember 11, 2009I loved this book. It’s not an easy read, but it’s deeply rewarding – an important mono-graph and a useful reference, all in one. Its 2008 Lakatos Prize was well-earned. GaugingWhat’s Real is also a book that metaphysicians should take notice of, because it overlapswith – and often challenges – much of the literature on natural properties and Humeansupervenience.But noticing Healey’s book is one thing, and comprehending its depth is quite another.The learning curve is very steep – steeper than necessary, perhaps. It would be outrageousto recommend that a non-specialist pick up Gauging What’s Real without a “prep course”in the philosophy of gauge theory. So let me first provide, insofar as space allows, a prepcourse and a list of introductory readings.The place to begin is Lange (2002), which discusses the metaphysics of classical fieldtheories. By the end of Chapter 2, you will have learned that electromagnetism employs avector quantity called the potential (one component is measured in the familiar unit of volts).The curious thing about the potential is that its value is of no predictive significance. Onlycertain derivatives of this quantity matter in the calculation of experimental predictions. AsLange notes, the “classic” response to this fact has been to deny that the potential is phys-ically real, appealing instead to the electric and magnetic fields as the theory’s fundamentalontology.If you don’t finish Lange’s book (I recommend that you do), make sure to skip aheadto page 285 of the final chapter, where he briefly explains the Aharonov-Bohm effect. This1experimentally-measured phenomenon casts considerable doubt on the “classic” ontology ofelectric and magnetic fields. It turns out that electromagnetism can influence a measurablefeature of quantum particles (their complex phase) even in regions with zero electromagneticfield. This motivates Healey’s main project: the search for a new ontology, beyond just thefield, for electromagnetism.With Lange under your belt, you’ll be prepared for the first six pages of Healey, beforehe starts talking about fiber bundles.This is a bit of a problem with the book, pedagogically – Healey breaks out the mathemat-ical big guns before he needs to. Much of what needs explaining, including Aharonov-Bohm,would benefit from an informal first-pass look which he doesn’t provide. Fortunately, theredoes exist an accessible introduction to the math Healey employs: Chapter 3 of Maudlin(2007). Healey would surely ask that you take Maudlin’s philosophical claims with a grainof salt, but I imagine he’d agree that this chapter provides an excellent introduction to fiberbundles, pitched specifically at the “core” philosopher.That’s the last of the assigned reading. Now on to Healey.His first four chapters form a self-contained study of classical (that is, non-quantum)gauge theories. Since the Aharonov-Bohm experiment depends crucially on the interaction ofelectromagnetism with a quantum particle, this may seem like an artificial distinction. But Iurge the reader not to be tempted by such thoughts. The metaphysical upshot of Aharonov-Bohm rests on the fact that the local values of the electric and magnetic fields do notdetermine the results of the experiment. Similar results could be achieved in purely classicalphysics, if there were a classical complex field that interacted with electromagnetism, andwhose complex phase differences were measurable. In our world, as it turns out, the only such“field” is the wavefunction of a quantum particle – but that seems to be a metaphysicallycontingent feature of our world. So the metaphysical implications of Aharonov-Bohm extendto at least some classical worlds, and do not essentially depend on quantum theory.What are those metaphysical implications? Healey argues that the worlds described byclassical gauge theories are not made up of fundamental fields localized at points, but insteadexhibit a sort of holism. There are three lines of response available to the Aharonov-Bohmexample. One is to retain the classic ontology of fields, but allow them to act instantaneouslyat a distance. This may sound unpalatable at face. Moreover, as Healey notes it doesn’tmake much sense (for technical reasons) to respond in this way to the Aharonov-Bohm2effect in gauge theories more complicated than electromagnetism, such as the theory ofchromodynamics that describes the strong nuclear force.One response that does work – technically, at least – is to accept the unobservablepotential as a real and fundamental physical quantity. This view breaks down into a numberof versions, but Healey argues that they all fall prey to the same problem: they entail thatelectromagnetism is indeterministic. The problem arises as follows. Suppose I hand youa specification of the electromagnetic potential everywhere at some time t. To figure outthe potential that should result at some t′ (either before or later), you’d naturally seeka solution of the theory that obeys Maxwell’s equations, which codify the known laws ofelectromagnetism. Then you’d check what value the potential takes on at t′. The problemis that there are infinitely many distinct solutions to the equations, each with a differentpotential at t′. These solutions are all empirically equivalent, so usually we ignore thedifferences between them. But if the potential is a fundamental quantity, we’d better stopignoring such differences – and we are left with a bizarre sort of unobservable indeterminism.This argument has been around for some time, but Healey backs it up with a new,entirely original argument. He shows that if one accepts some plausible assumptions ofDavid Lewis’s account of how theoretical terms refer, it is impossible for us to constructterms that refer to values of the unobservable potential. Thus, even if the interpretationsHealey opposes are correct, the fundamental physical facts according to these interpretationscannot be expressed in any of our theories.Healey thinks we can do better, by taking the third of the three options I mentioned. Heoffers an ontology for electromagnetism and other gauge theories that he calls the “holonomyinterpretation.” The distinguishing feature of this interpretation is its holism: rather thanbeing localized at points, the fundamental physical quantities inhere in closed loop-shapedregions of spacetime. These properties do not generally supervene on the features of thepoints making up the loops.All this has so far been restricted to classical gauge theories. Healey goes on in the secondhalf of his book to discuss the technical difficulties and ontological prospects of quantumgauge theories. I’m afraid these four chapters will resist comprehension by non-specialists,but for the specialist community they will prove invaluable. (The rest of my review willprobably elude the non-specialist, too, for which I apologize.)That said, the question of how to quantize gauge theories is so deeply vexed that, in-3evitably, a few important approaches are ignored. Where possible, Healey is careful towork within the framework of algebraic quantum field theory, a decision I applaud given theunique mathematical precision of this approach compared with other forms of quantum fieldtheory. But he is not always clear about the tension that exists between this approach andsome of the quantization schemes he discusses. For instance, he begins by discussing quan-tization that involves gauge-fixing (that is, imposing a deterministic equation of motion onthe potential). This otherwise-excellent discussion neglects the fact that research by FrancoStrocchi has shown that these forms of quantization, at least in the standard textbook formwhich Healey presents, are incompatible with the algebraic approach. Gauge-fixing methodsare widely employed, and I don’t think they should either be omitted or quickly dismissedfrom a discussion like this. But Healey’s book would have benefited from some discussion ofthe alternatives to the algebraic approach (e.g., effective field theories), which are currentlyneeded to make sense of methods like gauge-fixing.Readers interested in alternatives to the quantization schemes discussed by Healey shouldlook in two places. The major approach he leaves out (perhaps because it involves a discretepicture of spacetime) is lattice gauge theory. Thorough discussions can be found in anynumber of textbooks. The other, far less well known scheme is one due to Thirring andNarnhofer (1992). These authors suggest a way around the no-go theorems of Strocchiby proposing an algebraic gauge quantum field theory that employs non-separable Hilbertspaces. This approach has the interesting consequence that the potential is not even definedas a physical quantity on the resulting quantum field theory, laying to rest (if correct) thedispute about whether the potential is physically real. But it is far from clear that Thirringand Narnhofer’s approach will bear fruit, any more than the many alternatives discussed byHealey. I’m mainly thankful that our author treats quantum gauge theory the way it oughtto be treated – as an open question. He can be forgiven for overlooking a couple of the manypossible answers.ReferencesLange, Marc (2002), An Introduction to the Philosophy of Physics, Oxford: Blackwell.Maudlin, Tim (2007), The Metaphysics Within Physics, Oxford: Oxford UP.4Thirring, W. and N. Narnhofer (1992), “Covariant QEDWithout Indefinite Metric,” Reviewsin Mathematical Physics 4:Special Issue 197–211.5

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