Making Symmetry Explicit: The Limits of Sophistication

TL;DR

This paper examines when local symmetries in physics should be explicitly handled versus left implicit, proposing a criterion to distinguish these cases and analyzing implications for different theoretical frameworks.

Contribution

It introduces the background-relative sophistication (BRS) criterion to determine when symmetry must be explicit, and applies it to general relativity and gauge theories.

Findings

BRS effectively predicts when symmetry handling is necessary

Explicit symmetry handling arises in specific representational frameworks

Quantum and subsystem contexts challenge the BRS criterion

Abstract

Symmetry is often treated in philosophy of physics as an interpretive problem. A particularly lively dispute concerns local symmetries: do they indicate surplus structure that ought to be expunged, or are they merely a harmless redundancy? One influential response favours the second option for certain theories -- those dubbed internally sophisticated. And indeed, in much of physics practice, local symmetries are left implicit: one simply works "up to isomorphism'' without pausing over invariance. But not always. In some settings, local symmetry and invariance become pressing practical concerns for physicists. Yet philosophical discussions of sophistication have paid little sustained attention to when, and why, this happens. Surveying textbook general relativity (GR) and gauge theory, I identify the settings in which diffeomorphism invariance or gauge invariance must be handled explicitly. (Here a setting is a choice of representational framework or background assumptions within which one formulates and uses the theory -- for instance, linearisation, an initial-value formulation, or a Hamiltonian $3+1$ formalism.) I propose an operational criterion -- background-relative sophistication (BRS) -- and argue that it accounts well for the pattern: it marks just where symmetry can stay implicit and where it must be made explicit. Quantum and subsystem settings raise a further difficulty: there, certain tasks (superposition and gluing) force symmetry into view even for theories that are BRS.