The Three-Fold Way

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:

Quantum theory and division algebras.

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:

Higher gauge theory, division algebras and superstrings.

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.

4 Responses to The Three-Fold Way

  1. Tim van Beek says:

    John wrote:

    Maria Pia Solèr’s amazing theorem from 1995, which characterizes Hilbert spaces over the real numbers, complex numbers and quaternions.

    I’d put it the other way around: Pick a field k and say that you would like to play quantum mechanics in an infinite dimensional Hilbert space over k, fixing a minimal set of axioms to make sense of that already fixes k to be the real, complex or quarternion numbers.

    (BTW: The link to the n-cafe post about Soler’s theorem does not seem to work, and why does the history over at the n-cafe end with April 2010?).

    • John Baez says:

      I fixed the link. I don’t know why the n-Café’s monthly archives end in April 2010 — maybe something broke then and nobody noticed. I noticed it while writing this blog post! Perhaps more useful than the month-by-month archives is the entry called the whole enchilada.

  2. The Cosmist says:

    Do you really think the mathematical tools of reductionist physics are going to be very useful in “green mathematics”? Doesn’t analysis of the complex systems encountered in biology, ecology etc. require more “green computing”, statistics, data mining and pure empiricism than anything else? In my limited experience, mathematical equations are of little use in modelling really complex real world systems. This is unfortunate, since reductionist math models were comprehensible and aesthetically pleasing, while “green mathematics” is probably going to be an ugly and incomprehensible mess — one best understood not by human minds, but by increasingly intelligent machines!

    • John Baez says:

      The Cosmist wrote:

      Do you really think the mathematical tools of reductionist physics are going to be very useful in “green mathematics”?

      Great question — I’ll tell you at the end of this century!

      Doesn’t analysis of the complex systems encountered in biology, ecology etc. require more “green computing”, statistics, data mining and pure empiricism than anything else?

      It will certain require a lot of those things. It will also require a lot of good new ideas. And those ideas, if they’re formulated with sufficient logical clarity and generality, may turn out to be mathematics.

      Two examples:

      My friend Christopher Lee heads the Center for Computational Biology at UCLA. He does heavy-duty data mining for genomics: for example, check out his analysis of HIV evolution and drug resistance from sequencing of 50,000 clinical AIDS patient samples. But his work involves lots of beautiful new math. His new paper on potential information metrics is a great example:

      It is common to measure the information value of a model as its average prediction power for some observable variable of interest. Then the absolute goodness-of-fit of a statistical model to a set of observations can be formulated as the total remaining information obtainable by the set of all models that we have not yet computed (one of which might fit the observations much better than our current model). In a paper just published in Information, I define this metric as the potential information, and show that it can estimated directly from the observations, without actually computing any of the remaining models.

      This addresses a simple question in Bayesian inference: how do we know when we’re done?

      Another: check out Steven N. Evans‘ work on metagenomics and metrics on spaces of probability distributions.

      In my limited experience, mathematical equations are of little use in modelling really complex real world systems…

      I don’t think of math as being primarily about equations. While equations are certainly a big part of it, I’d say math is really about noticing very general patterns, inventing definitions that help formalize these patterns, and reasoning with these definitions. I’ve published plenty of math papers where equations are not very important. And sometimes the equations look like this:

      (The left side equals the right; click for details—and thanks to Scott Carter for allowing me to use this picture.)

      So math is very flexible, and I think we’ll continue to see it grow.

      This is unfortunate, since reductionist math models were comprehensible and aesthetically pleasing, while “green mathematics” is probably going to be an ugly and incomprehensible mess — one best understood not by human minds, but by increasingly intelligent machines!

      The world is too complex for us to understand all of it, but that’s okay: math is always about finding the beautiful pieces that we can understand. There is something beautiful about this leaf, for example:

      And while we can’t understand each individual vein, we may be able to understand the overall pattern. See Qinglan Xia’s paper The formation of a tree leaf, for a start.

      So green mathematics may require careful judgement to separate the ugly and incomprehensible from the beautiful and comprehensible, and let humans focus on the latter while machines deal with the former.

      Of course if the machines get intelligent enough, they may want to work on the beautiful part.

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