Intrinsic Geometrical Structure and Fiber Bundles
Department of Philosophy
Abstract: One mathematical object that plays a central role in gauge theories is the fiber bundle. Fiber bundles associate abstract spaces (a fiber) with each point in a base space, tying the whole into an integrated object by means of both topological structure (in defining the fiber bundle) and a connection. If one interprets these fibers as representing internal degrees of physical freedom at each point, then a natural understanding of fiber bundles entails rejection of the classical picture of universals. Mathematically, a metaphysics of universals would be represented by a product space rather than a fiber bundle: the physical state at each point would be drawn from a single, fixed set of possibilities, held in common by all the points in space-time. I call this understanding of the significance of fiber bundles “hyperlocality”, indicating a metaphysics more local than that of universals. Richard Healey has argued that the right moral to draw from gauge theories is rather a fundamental non-separability in nature: a metaphysics less local than classical theory. These nonseparable aspects of nature are, he says, directly represented not by connections on fiber bundles but by holonomies. His objections to hyperlocality arise from worries about counting: he claims that the same set of holonomies are consistent with many different fiber-bundle-cum-connections. This is supposed to lead to epistemic or semantic difficulties. The main burden of this talk is to show that the counting worries are misplaced. There are not multiple fiber bundles or connections that are underdetermined by the holonomies or by the empirical data. I will discuss the fundamental definitions of a fiber bundle and a connection, and provide a somewhat simplified account of the Aharonov ‐ Bohm effect from this point of view.