I've been studying the application of Clifford Algebra in quantum mechanics, more specifically in spin, but I'm stuck in a basic analogy that is part of the basics before starting to do things using Clifford Algebra. The point is that we have a spinor in spherical coordinates and then say that the equivalent representation of it in the way made below, but it looks like something taken out of nowhere, or simply established in a way that things could work. I am following three references about them:
- Geometric Multiplication of Vectors An Introduction to Geometric Algebra in Physics (Compact Textbooks in Mathematics) by Miroslav Josipović;
- Geometric Algebra for Physicists by Chris Doran, Anthony Lasenby;
- William E. Baylis - Clifford (Geometric) Algebras With Applications to Physics, Mathematics, and Engineering - Birkhäuser Basel (1996)
I would like some intuition or explanation about why we do this identification $$\psi = a^0 + a^k \sigma_k$$ where $I=\sigma_1 \sigma_2 \sigma_3$ but I'd like more than because it works in that way, if possible. I searched a lot but I couldn't find anything.
I really appreciate your attention and help.