Representations of Maxwell's equations: Part II: relativistic geometric algebra (STA)

title: Representations of Maxwell's equations: Part II: relativistic geometric algebra (STA)

This is the 2nd video in a series about representations of Maxwell's equations.

In the previous video, we started with the vector form of Maxwell's equations (for linear isotropic media), and then:

• adjusted the equations for dimensional consistency
• merged them into the single geometric algebra form of Maxwell's equation (singular), using the geometric algebra for R^3 (Euclidean 3D space.)

In this video, we derive the STA (space time algebra) form of Maxwell's equation (from Maxwell's equation for Euclidean space.) STA is the geometric algebra that uses the Dirac basis (what is known as gamma matrices in QFT.) Unlike QFT, we won't care about which matrix representation is used, and in fact won't even care that there is a matrix representation. Instead we treat this as an abstract basis, subject to a specific metric (i.e. specification of the dot products of the unit vectors in the basis.)

In videos to follow, we will

• derive the tensor form of Maxwell's equations (using the STA representation.)
• derive the (relativistic) differential forms representation of Maxwell's equations. This is very similar to the tensor form, and also requires two equations.

The viewer may also be interested in the following math.se answer:

https://math.stackexchange.com/a/4409471/359

which provides an overview of the material that we will be covering for this video series.

The playlist for this Maxwell's equation video series will also include one STA background video, where proper time, velocity, momentum, and invariance are discussed briefly. In particular, that video tries to demonstrate why we represent the Euclidean spatial basis as space-time bivectors in STA.