Relativistic Generalization and Extension to the Non-Abelian Gauge Theory of Feynman's Proof of the Maxwell Equations

TL;DR

This paper extends Feynman's original proof of Maxwell's equations to relativistic and non-Abelian gauge theories, revealing the limitations on fields acting on quantum particles and broadening the theoretical framework.

Contribution

It formulates relativistic versions of Feynman's derivation and extends the scheme to non-Abelian gauge theories, highlighting the types of fields compatible with quantum particles.

Findings

Only scalar, gauge, and gravitational fields can act on quantum particles consistently.

Relativistic versions of Feynman's derivation are successfully formulated.

Extension to non-Abelian gauge theories is achieved in the special relativistic context.

Abstract

R.P. Feynman showed F.J. Dyson a proof of the Lorentz force law and the homogeneous Maxwell equations, which he obtained starting from Newton's law of motion and the commutation relations between position and velocity for a single nonrelativistic particle. We formulate both a special relativistic and a general relativistic versions of Feynman's derivation. Especially in the general relativistic version we prove that the only possible fields that can consistently act on a quantum mechanical particle are scalar, gauge and gravitational fields. We also extend Feynman's scheme to the case of non-Abelian gauge theory in the special relativistic context.