A Rough Road-map

The basic plan of attack: show how elements of $SU(2)$ correspond to rotations; then apply this to the spin of the electron.

I start with the most basic concepts of Lie group and Lie algebra theory. $SO(3)$ is the ideal illustrative example: readily pictured, yet complicated enough to be interesting. The main goal is the double covering result. I do not take the most direct path to this goal, attempting to make it appear ``naturally''. Once we do have it, the urge to explore some of the related topology is irresistible.

Next comes physics. Usually introductory quantum mechanics starts off with things like wave/particle duality, the Heisenberg uncertainty principle, and so forth. Technically these are associated with the Hilbert space of complex-valued $L^2$ functions on ${\bf R}^3$-- not the simplest Hilbert space to start with. If one ignores these issues and concentrates solely on spin, the relevant Hilbert space is ${\bf C}^2$. (Feynman's Lectures on Physics, volume III, was the first textbook to take this approach in its first few chapters.) $SU(2)$ makes its entrance as a symmetry group on ${\bf C}^2$. I conclude with a few hand-waves on some loose ends.

© 2001 Michael Weiss

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