Ships, Lifts and Cages: A unified interpretation of symmetry in classical and quantum physics

Abstract

In this thesis, I explore the extent to which a focus on transformations and symmetry can highlight the conceptual unity of classical spacetime theories and quantum (gauge) theory, in particular general relativity (GR) and non-relativistic quantum- electromagnetism (Q-EM). In the literature, there is a general belief that because the transformations of spacetime theories act on external space, they have a different interpretation from the transformations of gauge theory, which act on internal space. By focusing both on the role of transformations within these different theories, and the possibility of interpreting the transformations actively, I show that, despite important differences, there is more conceptual unity than is often realised. Specifically, I aim to answer the two following questions. 1) What role do symmetry transformations play in GR and Q-EM? I show that the equivalence principle and the gauge principle play analogous roles in GR and Q-EM respectively. This leads to an analogue of the equivalence principle in Q-EM, and the formulation of the notion of an inertial gauge. The claim that “dynamical forces restore symmetry” is demystified. 2) Is there a fundamental difference between spacetime transformations and gauge transformations? It is commonly accepted that gauge transformations, unlike spacetime transformations, cannot be interpreted actively. I show why this is mistaken by considering the action of gauge transformations on both classical and quantum systems. In the latter case, I suggest that the particular nature of quantum systems is responsible for the differences that there appear to be between active gauge transformations and active spacetime transformations.