Electromagnetic energy-momentum equation without tensors: a geometric algebra approach

TL;DR

This paper introduces a geometric algebra approach to derive electromagnetic energy-momentum equations without tensors, providing a new algebraic formalism that simplifies the understanding of energy and momentum conservation in electromagnetism.

Contribution

It presents a tensor-free, geometric algebra-based derivation of electromagnetic energy-momentum equations, connecting conservation laws and wave equations within the Clifford algebra framework.

Findings

Derived energy-momentum conservation laws from complex electromagnetic fields.

Established a wave equation for the energy-momentum density.

Linked the formalism to Dirac-Pauli-Hestenes algebra in Clifford algebra.

Abstract

In this paper, we define energy-momentum density as a product of the complex vector electromagnetic field and its complex conjugate. We derive an equation for the spacetime derivative of the energy-momentum density. We show that the scalar and vector parts of this equation are the differential conservation laws for energy and momentum, and the imaginary vector part is a relation for the curl of the Poynting vector. We can show that the spacetime derivative of this energy-momentum equation is a wave equation. Our formalism is Dirac-Pauli-Hestenes algebra in the framework of Clifford (Geometric) algebra Cl_{4,0}.