[FWN Special] Journey to Dirac Spinors and the Dirac Equation

Yesterday (8 / 8 / 2022) was the 120th birthday of P.A.M. Dirac. It’s been a long time, “enough time to fit a lot of progress in there”.

In his honour, I proceeded to dust off my QFT and related maths notes from long ago and will attempt to present a modern journey to the Dirac equation and the description of relativistic spin-1/2 particles, from the mathematical formalisms.

Notes will be gradually posted during the week. Full references and additional work will be updated on the final page of this blog post.

Intro Update 2 (17 Aug 2022)

This essay / note will be arranged in a pedagogical order, without much historical burden of the order in which things were discovered. I take this inspiration from schools such as MIT who famously teach quantum mechanics to undergraduates in a similar fashion.

Section 0 will be a mathematical toolbox of the relevant results we will use, such as some group theory basics and Clifford algebra (needed to properly understand Dirac gamma matrices). In Section 1 we will outline the definition and properties of the Lorentz Group, and reach the Klein-Gordon equation for a scalar field. In Section 2 we consider the effects of spin, and work towards the Dirac equation and Dirac Lagrangian, all from first principles.

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