  The normalization in the path integral approach to quantum field theory, in
contrast with statistical field theory, can contain physical information. The
main claim of this paper is that the inner product on the space of field
configurations, one of the fundamental pieces of data required to be added to
quantize a classical field theory, determines the normalization of the path
integral. In fact, dimensional analysis shows that the introduction of this
structure necessarily introduces a scale that is left unfixed by the classical
theory. We study the dependence of the theory on this scale. This allows us to
explore mechanisms that can be used to fix the normalization based on cutting
and gluing different integrals. "Self-normalizing" path integrals, those
independent of the scale, play an important role in this process. Furthermore,
we show that the scale dependence encodes other important physical data: we use
it to give a conceptually clear derivation of the chiral anomaly. Several
explicit examples, including the scalar and compact bosons in different
geometries, supplement our discussion.
