  If potential energy is the timelike component of a four-vector, then there
must be a corresponding spacelike part which would logically be called the
potential momentum. The potential four-momentum consisting of the potential
momentum and the potential energy taken together is just the gauge field of the
associated force times the charge associated with that force. The canonical
momentum is the sum of the ordinary and potential momenta. Refraction of matter
waves by a discontinuity in a gauge field can be used to explore the effects of
gauge fields at an elementary level. Using this tool it is possible to show how
the Lorentz force law of electromagnetism follows from gauge theory. The
resulting arguments are accessible to students at the level of the introductory
calculus-based physics course and tie together classical and quantum mechanics,
relativity, gauge theory, and electromagnetism. The resulting economy of
presentation makes it easier to include modern physics in the one year course
normally available for teaching introductory physics.
