Gauge-invariance and the empirical significance of symmetries

This paper explicates the direct empirical significance (DES) of symmetries. Given a physical system composed of subsystems, such significance is to be awarded to physical differences about the composite system that can be attributed to symmetries acting solely on its subsystems. The debate is: can DES be associated to the local gauge symmetries, acting solely on subsystems, in gauge theory? In gauge theories, any quantity with physical significance must be a gauge-invariant quantity. Using this defining feature, we can recast the existence of DES as a question of holism: if a larger system is composed of (sufficiently) isolated subsystems, are the individual gauge-invariant states of the subsystems sufficient to determine the gauge-invariant state of the larger system? Or is the relation between the subsystems underdetermined by their physical states, and does the underdetermination carry both empirical significance and a relation to the subsystem symmetries? To attack the question of DES from this gauge-invariant angle, the straightforward method is gauge-fixing; for the values of gauge-invariant quantities are entirely determined by gauge-fixed representations of the system and subsystem states. However, gauge-fixings are subtle for field theories \textit{in the presence of boundaries}. There are two qualitatively different types of boundary: an internal boundary, dividing the system into subsystems, and an external one, standing outside the entire composite system (`the whole Universe'). We find: (i) for internal boundaries, DES cannot be associated to local gauge symmetries, only to the global components of these symmetries; (ii) here there is no DES-symmetry in vacuum in a simply-connected manifold; but (iii) we do recover the recent literature's standard notions of DES for point-particle mechanics, as exemplified by the thought-experiment known as `Galileo's ship'; finally (iv) we recover previous construals of DES [Greaves Wallace, 2014, Wallace 2019], but only for external boundaries and for sufficiently inhomogeneous configurations of non-Abelian gauge theories