Thursday, July 14, 2011

Motl Math

Lubos has an interesting post entitled: Why is the sum of integers equal to -1/12

This idea, first explored by Leonard Euler, turns out to be related not only to the original series Euler was looking at but to numerous regularization schemes in QFT and String Theory - as Lumo shows.

Of course the integers don't actually have a finite sum - at least not under the usual rules of arithmetic, but the math that leads to Lumo's result is amusing and instructive - see his post for some of the details.  (And Wikipedia on the Riemann Zeta Function)

It's well known that Dirac and several other famous figures in quantum theory were deeply distrustful of the whole idea of renormalization.  Maybe the Euler-Lumo result is a hint that they were right after all.

Of course that would still leave the problem of explaining why it works.