Constraining the Physical State by Symmetries

TL;DR

This paper explores how symmetries and gauge invariance in generally covariant theories constrain the definition of physical states, highlighting differences between compact and non-compact spaces.

Contribution

It clarifies the conditions under which physical states are uniquely defined or have freedom, especially contrasting compact and non-compact spatial manifolds.

Findings

In gauge covariant theories with compact space, physical states are uniquely defined up to gauge transformations.

In non-compact spaces, there are additional options for defining physical equivalence classes.

The results relate to the hole argument and the nature of determinism in covariant theories.

Abstract

After reviewing the hole argument and its relations with initial value problem and general covariance, we shall discuss how much freedom one has to define the physical state of a system in a generally covariant (or gauge covariant) field theory. We shall show that in gauge covariant theories (and generally covariant theories with a a compact space) one has no freedom and one is forced to declare as physically equivalent two configurations which differ by a gauge transformation (or by a global spacetime diffeomorphism), as it is usually prescribed. On the contrary, when space is not compact, the result proven for the compact case does not hold true and one may have different options to define physically equivalent configurations, still preserving determinism.