Philosophical Interpretations of Gauge Theory
Nicholas J Teh
PhD Candidate
Dept. of Philosophy
University of Pittsburgh
Gauge theories are the cornerstone of modern physics; they crop up in classical field theory, QFTs and string theory. In addition, they have been wildly successful in describing and predicting fundamental interactions between particles. This descriptive framework is known as the Standard Model (SU(3)C×SU(2)L×U(1)Y ). As such, gauge theories provide a stage of very good experimental and theoretical pedigree for philosophical activity. There is, however, a certain amount of technical language one needs to develop in order to discuss gauge theories in a contemporary setting: the apparatus of differential topology. It seems to me that the best way to proceed is to explain the language before engaging in discussion; I do this in the next section. I ask the reader to bear with me if he should think the treatment trivial on the one hand, or somewhat abstract1 on the other. At least, it will have the merit of fixing notational conventions.
Historically, the treatment of electromagnetism as a U(1) Abelian gauge theory helped reveal its nonlocal characteristics through the prediction and subsequent detection of the Aharonov-Bohm (AB) effect. This leads to several questions of philosophical interest: (a) What interpretative schemes are there for the AB effect? and (b) What are the implications of these schemes for determinism, nonlocality, and more specialized doctrines like Lewis' (1986) Humean supervenience?
I discuss these issues in section 3 while largely focusing on Nounou's (2003) topological interpretation, on the grounds that it has been hitherto undiscussed in the literature. Although I disagree with Nounou on most points, I think there are several interesting morals to be drawn from her interpretation. This discussion will bring us to consider several other general philosophical issues, such as the role of causal explanation in a theory, the equivalence of mathematical structure vis-a-vis the equivalence of physical structure, and the manner in which surplus structure seems to make its way into the realm of physical structure (the ontology!). These will be recurring themes throughout the rest of the paper.
In section 4, I discuss the case against connection realism and bundle substantivalism (cf. Healey (2001)). By and large, I agree with Healey about the bundle substantivalist's woes, but do not readily concede the superiority of the loop interpretation. In particular, I resist his proposal that the gauge potential be excluded from the realm of physical structure, i.e. regarded as not physically real; I think at the very least, it is too early to tell. I will then offer a variant of the connection interpretation in which the gauge potential might be counted as physical structure.
I have two reasons for thinking this might be possible: a heuristic reason and a non-heuristic reason. However, the full force of the non-heuristic reason will only be evident in section 5. There, I describe general gauge theories and supply evidence (both theoretical and experimental) in support of my views. These come from the Gribov ambiguity, quark confinement and anomalies.
Section 6 poses the question "Tu Quoque?" to the loop realist, for his position is plagued by its own set of problems. I bring against him a general charge of underdetermination that holds at both the classical and quantum levels, followed by a specific charge of underdetermination that arises for holonomy interpretations of pure Yang-Mills theory and quantum gauge field theory.
Finally in section 7, I review my arguments and indicate future research directions.