This is the personal website and blog of Adam Freese. I'm a theoretical physicist and amateur microtonal musician. I will be using this website very infrequently to post musings about random physics-, music- or math-related things as they strike my fancy.
An ulterior motive for posting what and when I do will be to have an easily-accessible resource to double-check sign and factor \(i\) conventions.
Previews of the latest pages can be found below.
This post shows how to derive the Lorentz algebra from the Clifford algebra
\(\mathrm{Cl}_{1,3}\).
I assume Lorentz transformations take the form
\(x \mapsto L x L^{-1}\)
within the Clifford algebra,
and require \(L\) to map vectors to vectors.
By looking at \(L\) close to identity,
and imposing this vectors-to-vectors requirement,
you can show that the generators must be bivectors.
The Lorentz algebra then just follows from the Clifford algebra's
multiplication rules.
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May 10, 2025
This post explores some basic theory about 3D rotations:
the Lie algebra,
the fact that
\(\mathrm{SU}(2,\mathbb{C})\) is the double cover,
the origin of spinors,
and how quaternions and rotors fit into all this.
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