The following questions are germane to our understanding of gauge-(in)variant
quantities and physical possibility: how are gauge transformations and
spacetime diffeomorphisms understood as symmetries, in which ways are they
similar, and in which are they different? To what extent are we justified in
endorsing different attitudes — nowadays called sophistication, haecceitism,
and eliminativism — towards each? This is the first of four papers taking up
this question.
This first paper will discuss notions of symmetries and isomorphisms that
will be used in the remaining papers in the series. There are several such
notions in the literature and the question of how they mesh with empirical
discernibility is a delicate one; even the orthodox view that symmetries are
empirically unobservable (even in principle) has recently been challenged by
Belot. Focusing on local field theories, I will provide a precise definition of
dynamical symmetries in terms of the space of states of the theory at hand. I
will then apply the definition to Yang-Mills theories and general relativity
and show that these symmetries correspond to automorphisms of `natural’
geometric structures: the small diffeomorphisms of the spacetime manifold and
the small fiber-preserving diffeomorphisms of a fibered space. Finally, I will
show that these automorphisms can be given a passive gloss, since they
correspond 1-1 to the coordinate transformations of the underlying manifolds.
