Conceptual Foundations of Field Theories in PhysicsAuthor(s): Andrew WayneReviewed work(s):Source: Philosophy of Science, Vol. 67, Supplement. Proceedings of the 1998 Biennial Meetingsof the Philosophy of Science Association. Part II: Symposia Papers (Sep., 2000), pp. S516-S522Published by: The University of Chicago Press on behalf of the Philosophy of Science AssociationStable URL: http://www.jstor.org/stable/188691 .Accessed: 30/07/2012 17:33

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Discussion: Conceptual Foundations of Field Theories in Physics

Andrew Waynett Concordia University

This discussion provides a brief commentary on each of the papers presented in the symposium on the conceptual foundations of field theories in physics. In Section 2 I suggest an alternative to Paul Teller's (1999) reading of the gauge argument that may help to solve, or dissolve, its puzzling aspects. In Section 3 I contend that Sunny Au- yang's (1999) arguments against substantivalism and for "objectivism" in the context of gauge field theories face serious worries. Finally, in Section 4 I claim that Gordon Fleming's (1999) proposal for hyperplane-dependent Newton-Wigner fields differs im- portantly from his previous arguments about hyperplane-dependent properties in quan- tum mechanics.

1. Introduction. One of the main aims of science is to develop theories which enable us to explain and predict observable phenomena by appeal to unobservable underlying mechanisms. Many of the most powerful and successful contemporary theories in physics fall under a field-theoretic par- adigm, in which fields (electromagnetic, gravitational, quantum, etc.) play crucial explanatory and ontological roles. Despite their paramount im- portance in contemporary physics, field theories have, until recently, been the subject of little interpretive work by philosophers of science. Early work by Mary Hesse (1962) and Howard Stein (1970) raised several im- portant interpretive questions about field theories, as has work on the structure and ontological status of spacetime (Friedman 1983, Earman 1989). In the last fifteen years there has been increasing interest in inter- pretive problems of quantum field theory (e.g., Redhead 1983, Brown and Harre 1988, Huggett and Weingard 1994, Teller 1995, Auyang 1995,

tSend requests for reprints to the author, Department of Philosophy, Concordia Uni- versity, 1455 de Maisonneuve Blvd. West, Montreal, H3G 1M8, Canada.

I1 would like to thank Sunny Auyang, Gordon Fleming, and Paul Teller for useful comments on earlier drafts of this paper.

Philosophy of Science, 67 (Proceedings) pp. S516-S522. 0031-8248/2000/67supp-0039$0.00 Copyright 2000 by the Philosophy of Science Association. All rights reserved.

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Fleming 1996). Nonetheless, interest in the conceptual foundations of field theories remains limited to a relatively small group of philosophers of physics and philosophically-minded physicists. The goal of the present symposium is to broaden this interest by presenting a sample of current work in the conceptual foundations of field theories. Paul Teller (1999) attempts to unravel the mystery of "the gauge argument" used to intro- duce gauge fields into quantum field theory. Sunny Auyang (1999) makes an argument for the existence of absolute (yet non-substantial) spacetime based on the fiber bundle formalism of gauge fields. Gordon Fleming (1999) defends a hyperplane-dependent formulation of quantum field the- ory using Newton-Wigner localized fields. This discussion provides a brief commentary on each paper.

2. Geometry, Conventionalism, and the Gauge Argument. The stated aim of Teller's paper (1999) is to lay out the gauge argument as clearly as possible. Teller's focus is the question: given the element of convention- ality in the requirement of invariance of the characterization of the state function under a local phase transformation, how can apparently sub- stantive conclusions about the possible existence and properties of gauge fields follow from such a requirement? As part of the explicatory project, Teller presents a particular account of what precisely this "element of conventionality" consists of in the case of the electromagnetic field. The fact that the local phase transformation must be made to all state functions and the fortuitous cancellation of the local phase term from the expres- sions for A(x) and xg(x) should have something to do with the conven- tionality of the theoretical formalism. On Teller's account, choosing a 0(x) in the gauge transformation of A(x) and the local phase transformation of xg(x) is purely a matter of notation; to change the local phase is simply to adopt new coordinative definitions (as Reichenbach called them) for the terms A(x) and xV(x). Invariance under a local phase transformation applied to all state functions is to be expected, simply because this trans- formation amounts to changing notation. So how can empirical facts about the possible existence and properties of gauge fields follow from these formal facts about notation? They cannot, according to Teller, and the gauge argument shouldn't be read as making any such claim. Teller rejects the reading of the gauge argument as providing an argument from a premise about local phase transformations to a conclusion about the possible existence of a gauge field. Rather, the argument plays an "episte- mic" role, one of alerting us to the assumptions in gauge theories that do underwrite the possible existence and properties of gauge fields.

What are these assumptions? How do they embody the possibility of a physical gauge field? In the final section of the paper Teller sketches a "geometrization" of gauge fields, where the local phase is interpreted as

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a sort of geometrical connection. Teller suggests that if the geometrization project were successful it would help answer these questions. In geometry, the connection F describes the curvature of space, in the sense that it describes the influence of geometry on vectors free from manifest local disturbance. In the geometrized electromagnetic gauge field, the "phase connection" A(x) describes another kind of curvature of space: it describes the influence of "another kind of geometry" (as Teller puts it) on phases free from manifest local disturbance. The gauge field is an interaction field, the interaction is produced by a curvature of a phase "connection," and the connection at a point can be understood as the limit of the change of phase around a loop as that loop is shrunk to a point. Hence we have substantive conclusions about the possible existence and properties of gauge fields based on understanding gauge fields as a kind of geometry.

Teller canvasses several worries about the geometrization program, and I would like to suggest one more. On Teller's approach to the gauge ar- gument, the element of conventionality is a purely notational one, limited to how we characterize the state function under a local phase transfor- mation. Consider taking a more robust conventionalist line about gauge fields, in the spirit of Poincare's conventionalism. On this approach, we have a number of alternative total theories of the world (with all the same experimental consequences, of course) that differ in the extent to which observable phenomena are produced by the gauge "geometry" and by non-geometrical forces. These theories are incompatible, but they are con- structed so that any evidence supporting one equally supports all the oth- ers. For the Poincare conventionalist, the geometrization of gauge fields is not an all-or-nothing affair; there are many ways of splitting up the account of gauge interactions between geometry and non-geometrical forces and no principled way to choose among them. We may go even further, with the Reichenbach empiricist, and deny that these various the- ories are genuine alternatives at all. The mystery of the- gauge argument is solved, or rather dissolved: properly understood, gauge field theory in- volves no claims (except perhaps conventional ones) about the possible existence or properties of gauge fields.

3. Desubstantializing Substantivalism. The moral of Auyang's paper (1999) is an oft-repeated but important one: we should not infer metaphysical claims naively from a scientific theory's formal characteristics. Auyang applies this moral to the case of a substantivalist interpretation of space- time in the context of contemporary gauge field theories, and she reaches two conclusions. First, the argument for substantivalism about spacetime is based on a naive interpretation of certain formal considerations in dif- ferential geometry that Auyang labels "the constructivist approach." Sec- ond, an alternative, "analytic" approach to gauge field theories naturally

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leads to an alternative interpretation on which spacetime is absolute and objective but not substantial.

The substantivalist might object that Auyang's account of the argument for substantivalism is incomplete. According to Auyang, the source of a substantival interpretation of spacetime can be traced to the constructivist approach to physical theory exemplified by many textbooks in differential geometry. On this approach, we begin by positing a differentiable mani- fold and then introduce various constructions on the manifold. One such construction is a fiber, a complex mathematical object of which a tangent space is a simple example. Fibers indexed by points in the manifold make up a fiber bundle, a powerful way to characterize the content of a gauge field theory. The manifold functions as a sort of mathematical substratum for geometry, and, absent any distinction between mathematical and physical fields, this naturally leads us to interpret the manifold as repre- senting a physical substratum for our physical theories (an interpretation of spacetime known as manifold substantivalism). The substantivalist would agree with Auyang that this cannot possibly constitute an argument for substantivalism. The main arguments for substantivalism are variants on Newton's argument: only the substantivalist can explain the distinction between particles in genuine inertial motion and those that are absolutely accelerated. Interestingly, the argument from inertial effects doesn't apply in any obvious way to systems composed exclusively of gauge fields. Au- yang contends that there is no good argument for any kind of substanti- valism in the context of physical systems containing gauge fields alone. The substantivalist might object, however, that no substantivalist has ever claimed that there is.

Auyang presents an alternative, "analytic" approach to gauge field the- ories that, she claims, supports an interpretation on which spacetime is absolute and objective but not substantial. On the analytic approach, we begin with facts about various properties at different points in a system and then analyze these in terms of individuals and the relations they bear. In the context of the fiber bundle approach to gauge field theories, we begin by postulating that the entities that make up a gauge field all share the same set of possible properties related by a gauge group G. We posit a set of gauge orbits, each comprised of an arbitrary quality and its equiv- alence class of possible qualities. So far, gauge orbits are all identical and lack any principle of individuation. The function of the spatiotemporal parameter x is one of individuation and identification of the gauge orbits into fibers Gx that make up the fiber bundle. As well, the parameter x describes a spacetime manifold. Here, the spacetime manifold is an objec- tive structure, indispensable and absolute, in precisely Newton's sense, but not a physical substance. Call this position manifold objectivism: when a gauge field is posited, an objective spacetime manifold is posited and con-

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versely, when no entity or local field is posited there is no independently- existing manifold.

What of Auyang's claim that the analytic approach provides support for manifold objectivism? Here Auyang would seem to be disregarding the main moral of her paper, for manifold objectivism can be no more read off the fiber bundle formulation of gauge field theories than can mani- fold substantivalism. For one thing, a relationalist reinterpretation of this talk of an absolute, objective spacetime manifold in terms of actual and possible spacetime relations is available and tempting. For another, the hole argument would seem to present as big a challenge to manifold ob- jectivism as it does to manifold substantivalism. Recall that for the hole argument to get off the ground all that is needed are objective spacetime locations in the manifold not any claim that spacetime is a material sub- stance-and these objective spacetime locations are equally present in the ontology of manifold objectivism.

4. A Hyperplane-Dependent Ontology for Quantum Field Theory? The Reeh-Schlieder theorem implies a surprising quantum entanglement on a large scale, recently dubbed "superentanglement" by Rob Clifton et al. (1998). Interestingly, Newton-Wigner fields-that is, quantum fields in the Newton-Wigner position basis-avoid the Reeh-Schlieder theorem and its counterintuitive consequences. Fleming (1999) shows that relative to a Newton-Wigner scalar field superentanglement disappears, and the vacuum state has the same structure as does the non-relativistic vacuum. As he points out, Michael Redhead recognized this fact several years ago but argued that the Newton-Wigner basis ought not count as a bona fide, physically significant position representation, and for two reasons (Red- head 1995). First, Newton-Wigner position is not an objective fact about a particle across reference frames. This is expressed by the fact that Newton-Wigner localized states are not covariant under any transforma- tion involving a Lorentz boost. Second, even within a reference frame, Newton-Wigner "localization" is not what we would want to admit as physical localization at all: a Newton-Wigner field "localized" at a point in space and time is not incompatible with the localization of the same field at a different time at a spacelike separate point!

Fleming defends the use of Newton-Wigner fields to avoid the Reeh- Schlieder theorem, and he does so by replying to the two charges in turn. The first step is to construct Newton-Wigner fields that are covariant. These fields turn out to be hyperplane-dependent: the Newton-Wigner field is indexed by a Newton-Wigner position and a hyperplane. This co- variant construction is a significant achievement, but Fleming recog- nizes from long experience explaining hyperplane-dependence to philos- ophers, no doubt that it is only a small part of the battle. Localization

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on a hyperplane is an extremely counterintuitive form of localization, for reasons well explained by Fleming, and it isn't clear that we are any farther ahead than we were with local fields, traditional Minkowski coordinates, and the Reeh-Schlieder theorem.

Fleming tries to resolve this dilemma by arguing that we have an inde- pendent reason to suppose that Newton-Wigner localization has physical significance. Fleming uses the term "physically significant" interchangeably with "physically meaningful," "not a mere formal construction," and "rep- resents a physical property." By these terms he means any quantity "that can play an important role in the description and analysis of the structure and behavior of physical systems." These physical properties are to be dis- tinguished from fundamental physical properties, and on a realist approach that distinction is made in the following way. Fundamental properties are what David Lewis (1986) calls natural properties, they carve nature at its joints. Fixing the fundamental physical properties of a system fixes all the properties of the system. All other physical properties supervene upon and reduce to the fundamental physical properties. Physically significant quan- tities, then, refer ultimately to facts about fundamental physical properties.

Hyperplane-dependent Newton-Wigner fields refer to non-fundamental physical properties. Fleming constructs an operator, the position-of- center-of-energy operator, whose eigenvectors are the Newton-Wigner field position eigenvectors. Newton-Wigner fields are thus fields which create and annihilate quanta and anti-quanta in states with specific, well- defined hyperplane-dependent center-of-energy values, and this, Fleming claims, is their physical significance. I am sympathetic to Fleming's pro- posal, although we clearly need both more philosophical and technical work before we can confidently conclude that the Minkowski coordinate for hyperplane-dependent Newton-Wigner fields is physically meaningful, in the sense of helping us describe and explain measurable properties of quantum fields.

What of the ontological status of these hyperplane-dependent fields? I contend that Fleming's present proposal differs importantly from his well- known work on hyperplane dependence in quantum mechanics (1965,1966, 1988, 1992, 1996). Fleming has argued in the past that photons in the singlet state in an EPR-Bell type experiment have a fundamental physical prop- erty-their spin state-which is hyperplane-dependent. Facts about the spin state of the photon pair are hyperplane-dependent and do not reduce to any non-hyperplane-dependent facts about the system. This claim opens up worries about the ontology of hyperplane-dependent properties which have so bothered philosophers of physics and are well articulated, for ex- ample, by Tim Maudlin (1994; cf. Wayne 1997). The present proposal en- genders no such worries because it is no part of Fleming's claim that the world contains, at bottom, any hyperplane-dependent physical prop-

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erties at all! The hyperplane-dependent center-of-energy position operator is analogous to the center of energy of a classical relativistic system. In some classical relativistic systems, the center of energy of the system is a hyper- plane-dependent quantity (recall the rotating sphere example). And the cen- ter of energy position 4-vector is a physically significant-although not fun- damental-hyperplane-dependent property of classical relativistic systems. The position of the center of energy of a classical relativistic system is defined in terms of, supervenes upon, and reduces entirely to, non-hyper- plane-dependent facts about the system. Analogously, the position-of- center-of-energy operator in quantum field theory reduces to non-hyper- plane-dependent physical facts about the quantum field.

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