  This comment on the Phys. Rev. A paper "Nonlinear quantum effects in
electromagnetic radiation of a vortex electron" by Karlovets and
Pupasov-Maximov [Phys. Rev. A 103, 12214 (2021)] addresses their criticism of
the combined experimental and theoretical study "Observing the quantum wave
nature of free electrons through spontaneous emission" by Remez et al,
published in Phys. Rev. Lett. [Phys. Rev. Lett. 123, 060401 (2019)]. We show,
by means of simple optical arguments as well as numerical simulations, that the
criticism raised by Karlovets and Pupasov-Maximov regarding the experimental
regime reported by Remez et al is false. Further, we discuss a necessary
clarification for the theoretical derivations presented by Karlovets and
Pupasov-Maximov, as they only hold for a certain experimental situation where
the final state of the emitting electron is observed in coincidence with the
emitted photon - which is not the common scenario in cathodoluminescence. Upon
lifting the concerns regarding the experimental regime reported by Remez et al,
and explicitly clarifying the electron post-selection, we believe that the
paper by Karlovets and Pupasov-Maximov may constitute a valuable contribution
to the problem of spontaneous emission by shaped electron wavefunctions, as it
presents new expressions for the emission rates beyond the ubiquitous paraxial
approximation.
