Time-dependent symmetries: the link between gauge symmetries and indeterminism

Abstract

Mathematically, gauge theories are extraordinarily rich — so rich, in fact, that it can become all too easy to lose track of the connections between results, and become lost in a mass of beautiful theorems and properties: indeterminism, constraints, Noether identities, local and global symmetries, and so on. One purpose of this short article is to provide some sort of a guide through the mathematics, to the conceptual core of what is actually going on. Its focus is on the Lagrangian, variational-problem description of classical mechanics, from which the link between gauge symmetry and the apparent violation of determinism is easy to understand; only towards the end will the Hamiltonian description be considered. The other purpose is to warn against adopting too unified a perspective on gauge theories. It will be argued that the meaning of the gauge freedom in a theory like general relativity is (at least from the Lagrangian viewpoint) significantly different from its meaning in theories like electromagnetism. The Hamiltonian framework blurs this distinction, and orthodox methods of quantization obliterate it; this may, in fact, be genuine progress, but it is dangerous to be guided by mathematics into conflating two conceptually distinct notions without appreciating the physical consequences. The price paid by this article for abandoning the mathematics of gauge theory as far as possible is an inevitable loss of rigour. Virtually nothing will be ‘proved’ below; at most, the shape of proofs will be gestured at and strong plausibility-arguments advanced. For a more detailed understanding of the mathematics, the natural place to start is Earman’s contribution to this volume (to which my own article can be seen as a commentary). Further details can be found in many standard texts on general relativity or quantum field theory (Peskin and Schroeder (1995) is particularly clear; for a really in-depth mathematical analysis, consult Henneaux and Teitelboim (1992)). A note on terminology: the word ‘gauge’, used extensively in this introduction, will not often appear again. Its meaning is now thoroughly ambiguous (as Earman notes) and I felt it simpler to resort to the marginally more cumbersome, but clearly definable, notion of a ‘theory with a local symmetry group’. As will become clear below, there are genuine dynamical differences between