  Intuition dictates that a very long, very thin cavity (e.g., a fiber optic
cable) could perhaps be modeled as an approximately one dimensional system. In
this paper we rigorously explore the validity of such intuition from the
perspective of a localized probe coupling to a quantum field inside a cavity
(e.g., an atom or an Unruh-DeWitt particle detector in a fiber optic cable). To
do so, we introduce the notion of subfield decomposition in which a $D+1$
dimensional quantum field in an axially-symmetric cavity can be reduced to an
infinite collection of uncoupled, massive $1+1$ dimensional fields. We show
that the ability to approximate a higher-dimensional scenario by a $1+1$
dimensional model is equivalent to making a certain change of the probe's shape
in the higher-dimensional space. The approximation is justified whenever this
change of shape is "small enough". In this light, we identify the dynamically
relevant norm by which the magnitude of these changes in probe shape ought to
be judged. Finally, we explore this approximation in particular setups
corresponding to quantum optics and superconducting circuits.
