[Paper Review] Potential Momentum, Gauge Theory, and Electromagnetism in Introductory Physics

David J. Raymond|ArXiv.org|Mar 17, 1998

Experimental and Theoretical Physics Studies5 references3 citations

TL;DR

This paper introduces the concept of 'potential momentum'—the spatial component of a four-vector gauge field—as a pedagogical tool to unify classical and quantum mechanics, relativity, and electromagnetism in introductory physics. By modeling matter wave refraction at gauge field discontinuities, it derives the Lorentz force law from gauge theory, enabling a unified, accessible presentation of modern physics in a single-semester course.

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.

Motivation & Objective

• To bridge the gap between classical mechanics, quantum mechanics, and modern physics in a single-semester introductory course.
• To introduce the concept of potential momentum as the spatial component of a gauge four-vector, analogous to potential energy.
• To demonstrate how the Lorentz force law emerges naturally from gauge theory using elementary wave mechanics.
• To provide a unified, conceptually coherent framework that connects relativity, electromagnetism, and quantum mechanics at an accessible level.
• To simplify the teaching of modern physics by reducing conceptual and mathematical redundancy in standard curricula.

Proposed method

• Introduces the potential four-momentum as the combination of potential momentum (spatial part) and potential energy (timelike part), derived from the gauge field multiplied by charge.
• Uses the analogy of matter wave refraction at a discontinuity in a gauge field to model particle dynamics in electromagnetic fields.
• Applies the principle of least action and wavefront continuity to derive the deflection of de Broglie waves across a gauge discontinuity.
• Demonstrates that the resulting wavefront bending reproduces the Lorentz force law in the classical limit.
• Employs spacetime geometry and four-vector formalism to unify classical and quantum descriptions of momentum and force.
• Uses LaTeX-based figures and wavefront diagrams to illustrate refraction effects and gauge field interactions.

Experimental results

Research questions

• RQ1 How can the concept of potential momentum be introduced in a way that connects classical mechanics with quantum mechanics and relativity?
• RQ2 Can matter wave refraction at a gauge field discontinuity be used to derive the Lorentz force law in an introductory physics context?
• RQ3 What is the role of the gauge field in unifying the description of electromagnetic forces across classical and quantum frameworks?
• RQ4 How can the canonical momentum be understood as the sum of ordinary and potential momentum in a gauge-theoretic framework?
• RQ5 To what extent can modern physics concepts like gauge invariance and four-vectors be taught in a one-semester calculus-based physics course?

Key findings

• The potential momentum, as the spatial component of the gauge four-potential scaled by charge, provides a consistent and intuitive extension of momentum in gauge theories.
• Matter wave refraction at a discontinuous gauge field reproduces the classical Lorentz force law, validating the gauge-theoretic derivation of electromagnetic forces.
• The canonical momentum is rigorously shown to be the sum of mechanical momentum and potential momentum, aligning with the standard definition in Lagrangian mechanics.
• The derivation using wavefront continuity and Fermat’s principle in a gauge context offers a physically intuitive path to the Lorentz force, accessible to introductory students.
• The framework allows for a unified presentation of relativity, electromagnetism, and quantum mechanics without requiring advanced mathematical formalism.
• The method reduces conceptual and pedagogical fragmentation in introductory physics, enabling a more coherent and economical curriculum.