  We establish that many fundamental concepts and techniques in quantum field
theory and collider physics can be naturally understood and unified through a
simple new geometric language. The idea is to equip the space of collider
events with a metric, from which other geometric objects can be rigorously
defined. Our analysis is based on the energy mover's distance, which quantifies
the "work" required to rearrange one event into another. This metric, which
operates purely at the level of observable energy flow information, allows for
a clarified definition of infrared and collinear safety and related concepts. A
number of well-known collider observables can be exactly cast as the minimum
distance between an event and various manifolds in this space. Jet definitions,
such as exclusive cone and sequential recombination algorithms, can be directly
derived by finding the closest few-particle approximation to the event. Several
area- and constituent-based pileup mitigation strategies are naturally
expressed in this formalism as well. Finally, we lift our reasoning to develop
a precise distance between theories, which are treated as collections of events
weighted by cross sections. In all of these various cases, a better
understanding of existing methods in our geometric language suggests
interesting new ideas and generalizations.
