  Infrared divergences have long been heralded to cancel in sufficiently
inclusive cross-sections, according to the famous Kinoshita-Lee-Nauenberg
theorem which mandates an initial and final state sum. While well-motivated,
this theorem is much weaker than necessary: for finiteness, one need only sum
over initial $\textit{or}$ final states. Moreover, the cancellation generically
requires the inclusion of the forward scattering process. We provide a number
of examples showing the importance of this revised understanding: in $e^+e^-
\to Z$ at next-to-leading order, one can sum over certain initial and final
states with an arbitrary number of extra photons, or only over final states
with a finite number of photons, if forward scattering is included. For Compton
scattering, infrared finiteness requires the indistinguishability of hard
forward-scattered electrons and photons. This implies that in addition to
experimental limits on the energy and angular resolution, there must also be an
experimental limit on the momentum at which electric charge can be observed.
Similar considerations are required to explain why the rate for $\gamma \gamma$
to scatter into photons alone is infrared divergent but the rate for $\gamma
\gamma$ to scatter into photons or charged particles is finite. This new
understanding sheds light on the importance of including degenerate initial
states in physical predictions, the relevance of disconnected Feynman diagrams,
the importance of dressing initial or final-state charged particles, and the
quest to properly define the $S$ matrix.
