Potential Momentum, Gauge Theory, and Electromagnetism in Introductory Physics

TL;DR

This paper introduces the concept of potential momentum as part of gauge theory, linking classical and quantum physics, and demonstrates how electromagnetism's Lorentz force law can be derived at an introductory level.

Contribution

It presents a novel way to incorporate gauge fields and potential momentum into introductory physics, simplifying the connection between classical and modern physics.

Findings

Potential momentum is identified as the spacelike component of a four-vector.

Refraction of matter waves can illustrate gauge field effects.

Lorentz force law can be derived from gauge theory at an introductory level.

Abstract

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.