  Collective coordinates are frequently employed in path integrals to manage
divergences caused by fluctuations around saddle points that align with
classical symmetries. These coordinates parameterize a manifold of zero modes
and more broadly provide judicious coordinates on the space of fields. However,
changing from local coordinates around a saddle point to more global collective
coordinates is remarkably subtle. The main complication is that the mapping
from local coordinates to collective coordinates is generically multi-valued.
Consequently one is forced to either restrict the domain of path integral in a
delicate way, or otherwise correct for the multi-valuedness by dividing the
path integral by certain intersection numbers. We provide a careful treatment
of how to fix collective coordinates while accounting for these intersection
numbers, and then demonstrate the importance of the fix for free theories. We
also provide a detailed study of the fix for interacting theories and show that
the contributions of higher intersections to the path integral can be
non-perturbatively suppressed. Using a variety of examples ranging from
single-particle quantum mechanics to quantum field theory, we explain and
resolve various pitfalls in the implementation of collective coordinates.
