The Kustaanheimo prime

Paul Kustaanheimo
Paul Kustaanheimo (1924-1997) was a Finnish astronomer and mathematician. In celestial mechanics, his name is associated with the Kustaanheimo-Stiefel transform or shortly KS transform which allows to regularize the Kepler problem using Clifford algebras. In this elegant picture, the motion of the two bodies becomes a rotation in three dimensions rendering therefore the Kepler motion into a harmonic oscillator. The KS transform generalizes the Levi-Civita regularization, which was found already in 1920 by Tullio Levi-Civita. The Kustaaanheimo Stiefel transform is still studied today. See this article for example.

(Image Source: originally

I learned about his attempt to do physics on finite fields in a linear algebra lecture of Ernst Specker. By the way, the absorption of mathematical material was a treat, but seeing lectures and side remarks from teachers like for example the above remark about Kustaanheimo is something which I consider now the most priceless part of my own college years. Knowledge is important, but it is cheep as it can again be looked up. Insight, excitement and suggestions from masters in the field is much more valuable. By the way, the picture on the Wikipedia entry of Specker from 1982 was taken just at the time, when I took his linear algebra course for us freshmen mathematicians and physicists (who took then both the same identical lectures in the first two years).

The fact that Specker mentioned Kustaanheimo is not an accident. Stiefel (1909-1978) (the same Stiefel from the Stiefel-Whitney classes of vector bundle), was an academic brother of Ernst Specker as they both were students of Heinz Hopf. Eduard Stiefel also founded the institute for applied mathematics at ETH. Stiefel was from 1948 on professor at ETH. Specker got his PHD in 1949 and was professor at ETH since 1955 so that Stiefel and Specker certainly knew each other. Kustaanheimo stayed in Zuerich in the summer of 1964 to work (as acknowledged in their paper) with Stiefel on the “perturbation theory of the Kepler problem” so that also Kustaanheimo and Specker must have known each other (even so Specker did not tell us (the linear algebra class) when mentioning the Kustaanheimo work).

Kustaanheimo muses about doing physics over a finite field. The starting point is to approximate a geometric space by a finite geometry. He gives an upper bound of the order of a googoplex but estimates the actual prime to be much smaller. The Kustaanheimo paper is of a similar scope then my “particles and prime” paper as it deals with a rather clear cut elementary number theoretical problem, which it is motivated by a speculation that our physical space could be modeled by a finite field. Here is the article (thanks to the Harvard librarians for digging the paper up):