The next math department colloquium at Stanford will feature Lenny Susskind lecturing on p-adic numbers and cosmology, here’s the abstract:
The biggest conceptual problem of cosmology is called the measure problem. It has to do with the assignment of probabilities in an exponentially inflating universe, which falls apart into separate causally-disconnected regions. Neither I nor my friends had ever intended to learn about p-adic numbers until we realized how similar such a universe is to an endlessly growing tree-graph. The result has been some new insights from p-adic number theory into the measure problem and other puzzles of eternal inflation. Within the constraints of a one-hour lecture, I will explain as much of this as I can.
I’ve no idea what this is about, but I’m guessing that Susskind is somehow drawing inspiration from two facts:
It’s hard to believe that any of the special features of these mathematical structures will make the problems of eternal inflation go away, but who knows…
Coincidentally, I’ve spent a lot of time recently learning about the p-adics, with a very different motivation. The way these things come up in mathematics is that you can think of number theory as being about a space, the space of prime numbers. The p-adics appear naturally when you decide to ask what happens locally near one point (i.e. at one prime). P-adic integers correspond to power series expansions, p-adic numbers to Laurent series. Various people have thought about analogies between conformal field theories on a Riemann surface, where one also wants to focus on what happens at a point and use representation theory methods, and the Langlands program which does something similar in number theory. This is part of the geometric Langlands story, and has roots in a remarkable paper of Witten’s from 1988 entitled Quantum field theory, Grassmannians, and algebraic curves.
As I’ve mentioned before, this semester here at Columbia we have Harvard’s Dick Gross as Eilenberg lecturer, and he’s giving a wonderful series of lectures starting with local Langlands. I’m hoping at some point to put together what I’ve been learning about this and possible connections to QFT in some readable form, but at the moment things are still too speculative and hazy. In any case, no sign that these ideas are going to solve the problems of cosmology…
Update: The Susskind et al. paper on this topic is now out at the arXiv. A p-adic model is studied, but no reason is given to believe that it has anything to do with eternal inflation and cosmology.