  High-energy partons ($E \gg T$) traveling through a quark-gluon plasma lose
energy by splitting via bremsstrahlung and pair production. Regardless of
whether or not the quark-gluon plasma itself is strongly coupled, an important
question lying at the heart of philosophically different approaches to energy
loss is whether the high-energy partons of an in-medium shower can be thought
of as a collection of individual particles, or whether their coupling to each
other is also so strong that a description as high-energy `particles' is
inappropriate. We discuss some possible theorists' tests of this question for
simple situations (e.g. an infinite, non-expanding plasma) using thought
experiments and first-principles quantum field theory calculations (with some
simplifying approximations). The physics of in-medium showers is substantially
affected by the Landau-Pomeranchuk-Midgal (LPM) effect, and our proposed tests
require use of what might be called `next-to-leading order' LPM results, which
account for quantum interference between consecutive splittings. The complete
set of such results is not yet available for QCD but is already available for
the theory of large-$N_f$ QED. We therefore use large-$N_f$ QED as an example,
presenting numerical results as a function of $N_f\alpha$, where $\alpha$ is
the strength of the coupling at the relevant high-energy scale characterizing
splittings of the high-energy particles.
