#dirac basis

A multivector Lagrangian for Maxwell’s equation: A summary of previous exploration.

This summarizes the significant parts of the last 8 blog posts. [Click here for a PDF version of this post] STA form of Maxwell’s equation. Maxwell’s equations, with electric and fictional magnetic sources (useful for antenna theory and other engineering applications), are \begin{equation}\label{eqn:maxwellLagrangian:220} \begin{aligned} \spacegrad \cdot \BE &= \frac{\rho}{\epsilon} \\ \spacegrad…

Progressing towards coordinate free form of the Euler-Lagrange equations for Maxwell’s equation

This is the 6th part of a series on finding Maxwell’s equations (including the fictitious magnetic sources that are useful in engineering) from a multivector Lagrangian representation. [Click here for a PDF version of this series of posts, up to and including this one.]  The first, second, third, fourth, and fifth parts are also available here on this blog. We managed to find Maxwell’s equation…

Gauge freedom and four-potentials in the STA form of Maxwell’s equation.

[If mathjax doesn’t display properly for you, click here for a PDF of this post] Motivation. In a recent video on the tensor structure of Maxwell’s equation, I made a little side trip down the road of potential solutions and gauge transformations. I thought that was worth writing up in text form. The initial point of that side trip was just to point out that the Faraday tensor can be expressed in…

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Motivation.

The notation I prefer for relativistic geometric algebra uses Hestenes’ space time algebra (STA) [2], where the basis is a four dimensional space \( \setlr{ \gamma_\mu } \), subject to Dirac matrix like relations \( \gamma_\mu \cdot \gamma_\nu = \eta_{\mu \nu} \).

In this formalism a four vector is just the sum of the products…