The Bones of Sophus Lie

TL;DR

This paper constructs the gauge covariant derivative for wave functions using Lagrangian mechanics and covariance principles, elucidating its relation to gauge theory, Lie algebras, and curved spacetime.

Contribution

It introduces a novel derivation of the gauge covariant derivative based on covariance principles rather than group theory, applicable to curved spacetime.

Findings

Derived divergence equations for charged current and energy-momentum.

Unified framework for gauge covariant derivatives in flat and curved spacetime.

Clarified the role of Lie algebra in gauge theories.

Abstract

The gauge covariant derivative of a wave function is ubiquitous in gauge theory, and with associated gauge transformations it defines charged currents interacting with external fields, such as the Lorentz force exerted by an electromagnetic field. It is the gauge covariant derivative which defines how an external field acts upon the wave function. This paper constructs the gauge covariant derivative, then uses the elegant framework of Lagrangian mechanics to derive two ``divergence equations'' from a general Lagrangian, one applying to the charged current, the other to energy-momentum. The student will appreciate the construction of the gauge covariant derivative of a classical wave function using only matrices, linear transformations, external fields, and partial derivatives. More unusual is using the principle of covariance, rather than group theory as guidance in the construction, but with exactly the same result. Advantage is taken of the close analogy with coordinate covariance of tensors. The details of deriving these two divergences provides motivation and a path to understanding the gauge covariant derivative, the underlying non-abelian Lie algebra, its application to building Lagrangians, and the resulting definitions of charged current and energy-momentum. All results are generalized to curved space-time.