I start with the most basic concepts of Lie group and Lie algebra theory. 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 functions on -- not the simplest Hilbert space to start with. If one ignores these issues and concentrates solely on spin, the relevant Hilbert space is . (Feynman's Lectures on Physics, volume III, was the first textbook to take this approach in its first few chapters.) makes its entrance as a symmetry group on . I conclude with a few hand-waves on some loose ends.
© 2001 Michael Weiss