  Dynamical locality is a condition on a locally covariant physical theory,
asserting that kinematic and dynamical notions of local physics agree. This
condition was introduced in [arXiv:1106.4785], where it was shown to be closely
related to the question of what it means for a theory to describe the same
physics on different spacetimes. In this paper, we consider in detail the
example of the free minimally coupled Klein--Gordon field, both as a classical
and quantum theory (using both the Weyl algebra and a smeared field approach).
It is shown that the massive theory obeys dynamical locality, both classically
and in quantum field theory, in all spacetime dimensions $n\ge 2$ and allowing
for spacetimes with finitely many connected components. In contrast, the
massless theory is shown to violate dynamical locality in any spacetime
dimension, in both classical and quantum theory, owing to a rigid gauge
symmetry. Taking this into account (equivalently, working with the massless
current) dynamical locality is restored in all dimensions $n\ge 2$ on connected
spacetimes, and in all dimensions $n\ge 3$ if disconnected spacetimes are
permitted. The results on the quantized theories are obtained using general
results giving conditions under which dynamically local classical symplectic
theories have dynamically local quantizations.
