I just finished a series of blog posts about doing quantum theory using the real numbers, the complex numbers and the quaternions… and how Nature seems to use all three. Mathematically, they fit together in a structure that Freeman Dyson called The Three-Fold Way.
You read all those blog posts here:
• State-observable duality – Part 1: a review of normed division algebras.
• State-observable duality – Part 2: the theorem by Jordan, von Neumann and Wigner classifying ‘finite-dimensional formally real Jordan algebras’.
• State-observable duality – Part 3: the Koecher–Vinberg classification of self-dual homogeneous convex cones, and its relation to state-observable duality.
• Solèr’s Theorem: Maria Pia Solèr’s amazing theorem from 1995, which characterizes Hilbert spaces over the real numbers, complex numbers and quaternions.
• The Three-Fold Way – Part 1: two problems with real and quaternionic quantum mechanics.
• The Three-Fold Way – Part 2: why irreducible unitary representations on complex Hilbert spaces come in three flavors: real, complex and quaternionic.
• The Three-Fold Way – Part 3: why the “q” in “qubit” stands for “quaternion”.
• The Three-Fold Way – Part 4: how turning a particle around 180 degrees is related to making it go backwards in time, and what all this has to do with real numbers and quaternions.
• The Three-Fold Way – Part 5 – a triangle of functors relating the categories of real, complex and quaternionic Hilbert spaces.
• The Three-Fold Way – Part 6 — how the three-fold way solves two problems with real and quaternionic quantum mechanics.
All these blog posts are based on the following paper… but they’ve got a lot more jokes, digressions and silly pictures thrown in, so personally I recommend the blog posts:
And if you’re into normed division algebras and physics, you might like this talk I gave on my work with John Huerta, which also brings the octonions into the game:
Finally, around May, John and I will come out with a Scientific American article explaining the same stuff in a less technical way. It’ll be called “The strangest numbers in string theory”.
Whew! I think that’s enough division algebras for now. I’ve long been on a quest to save the quaternions and octonions from obscurity and show the world just how great they are. It’s time to declare victory and quit. There’s a more difficult quest ahead: the search for green mathematics, whatever that might be.