Representations of Maxwell's equations: Part III: four-potentials, gauge freedom, and tensor formulation

This is the third video (four including one supplementary video) in a series about representations of Maxwell's equations.

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.)
• Derived the STA (space time algebra) form of Maxwell's equation. 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 this video we continue briefly with the STA form of Maxwell's equation, where we

• show that the trivector term of Maxwell's equation implies that there is a four-potential representation of the electromagnetic field
• show that the Lorentz gauge choice leads to four independent equations, one for each component of the four potential.
• show how to make a gauge transformation to satisfy the Lorentz gauge choice
• show that the gauge choice is automatically filtered out of the field, by virtue of the curl operation

then we

• show how the four-potential representation of the field can be expanded in coordinates (i.e. tensor formulation starts to kick in)
• expand the trivector term of Maxwell's equations in coordinates to obtain the source free term of Maxwell's equations in the tensor representations
• do the same for the vector term of Maxwell's equation, to obtain the source dependent Maxwell's tensor equation
• finally, write down the relationships between the original vector electric and magnetic field components and the Maxwell tensor field

After this, we will do one more video, where the differential forms representation of Maxwell's equations is obtained, also from the STA representation. 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.