[Updated 7/22/25: Fixed some typos and added some clarifications.]
This is the first in a series of notes that explore the origin of half-integer spins. As an initial step, we need to extract the identity component of the Poincare group, as well as understand precisely how the components are glued together. As usual, this evolved from a simple calculation into an extensive set of notes replete with numerous digressions.
If you read these notes and come across any typos, points that could benefit from clarification, misconceptions, or downright errors, please let me know. This will help me improve my own understanding, and I will also make an effort to amend the notes. It is my intention to keep these notes as clean and accurate as possible, so any such feedback would be a great help.
In keeping with my usual approach, I’ve included a set of background mathematics that reflects points that can be confusing or minor gaps in my own background or things I found required some piecing together. The rough goal is to make the notes somewhat self-contained for a reader with a basic knowledge of undergraduate mathematics.
In this piece, the background digressions include:
- Quotient topologies
- Connectedness, path-connectedness, and components
- Semidirect products and exact sequences (a brief summary relative to my detailed notes on the subject, but also an appendix with some practical aspects of semidirect products that don’t appear in those notes).
- Pointed sets
- Homotopy to the extent needed to define homotopy groups
- Topological groups
- Zero-dimensional manifolds
The meat is in sections 3 and 4 on the zeroth homotopy group and the examples of its construction for various Lie groups that we care about.
I’ve provided 4 versions: large and small, bw and color.
A Place of Sand 