  It is commonly assumed that quantum field theory arises by applying ordinary
quantum mechanics to the low energy effective degrees of freedom of a more
fundamental theory defined at ultra-high-energy/short-wavelength scales. We
shall argue here that, even for free quantum fields, there are holistic aspects
of quantum field theory that cannot be properly understood in this manner.
Specifically, the ``subtractions'' needed to define nonlinear polynomial
functions of a free quantum field in curved spacetime are quite simple and
natural from the quantum field theoretic point of view, but are at best
extremely ad hoc and unnatural if viewed as independent renormalizations of
individual modes of the field. We illustrate this point by contrasting the
analysis of the Casimir effect, the renormalization of the stress-energy tensor
in time-dependent spacetimes, and anomalies from the point of quantum field
theory and from the point of view of quantum mechanics applied to the
independent low energy modes of the field. Some implications for the
cosmological constant problem are discussed.
