From Isham to Klebanov

Recently Professor Isham from Imperial College, restarted my interest in an old passion of mine. What is the nature of Information?

He proposes the study of topoi. These are mathematical constructs that generalize the concept of sets. In his view classical mechanics is described by Set Theory, and Quantum Mechanics by a different Category. Is like if Quantum Mechanics had its own mathematics different from Classical Mechanics. I will say that Information is different in the classical world as it is in the quantum world.

On the other hand Professor Klebanov from Princeton University has recently rekindled my interest into another deep mathematical problem, the Riemann Hypothesis. Here I put these ideas from mathematics and physics into context.

Information is different than matter and energy, but no less real. Loosely speaking matter is the stuff we are made of, energy is its motion, and information is how it moves. Recently Professor Joe Mazur from Marlboro College wrote the book The Motion Paradox, that revisits the thousands of years old idea of motion. Any one of these points I write about here is a deep part of our mental wiring, here I only touch them at the surface, and try to glimpse at some connections among them.

Let us start with motion. Mechanical motion is the change of position in time, but then, what is position and what is time? Time is necessary so everything does not happen at once. In this same line of thinking, space is necessary so that everything is not in the same place. Abusing language I will say that Information is necessary so not everything happens the same way. Just like matter and energy are related through the most famous formula in Physics, E = mc², I expect that Information and Energy are related through something similar; I = f(E).

Isham has a series of papers on the use of Category Theory to build Physics Theories. This will better represent how a quantum object can be in several places at once, and a classical object cannot. You can read the last one at the Los Alamos Archive here.

In my view what Isham is saying is that the how things move of classical physics is different from the how they do in the microscopic world. On the other hand I believe that classical physics is an approximation of quantum physics, and therefore I will be inclined to consider the category of classical physics as contained in the quantum physics one. My point is that Information is related to Energy but is not Energy, unless the whole shebang is just a bit; "It from Bit", as John Archibald Wheeler said.

At the other side of the Atlantic Ocean from Isham we have Igor Klebanov, a young theoretical physicists at Princeton University. He found out that String Theory and Supersymmetric Yang-Mills theories are mathematically related; of course within the context of Set Theory as is used in Quantum Mechanics. Klebanov is not working on the foundations of Quantum Mechanics as Isham is, he directed a group of young scientists in a very precise numerical calculation to support a conjecture, strong and weak coupling expansions are related. This is a breakthrough, because it has proven very hard to calculate properties of the matter we are made of, like Carbon nuclei, with high energy physics theories. Now there seems to be a path through Klebanov et al.'s results.

The Riemann Hypothesis states that Bernhard Riemann's Zeta function only has zeros on the complex plane along a known line. This line happens to be related to the number 1/2 that appears prominently in the Princeton group's calculations. There is a type of singularity, a cut singularity, that is prominent in both the numerical calculation and the hypothesis.

Are these results related?

I do not know how to use Isham ideas. I never studied Category Theory, it is not part of the theoretical physics curriculum, you know; but in the quagmire that String Theory had turned out to be nowadays, it looks like a way out, maybe one can decide now what is the right theory, or at least what is the role of all the great variety of known string theories. On the other hand it seems more likely to me that the young turks at Princeton now know how to proceed, and maybe they will even be able to prove the more than a hundred year old Riemann Conjecture, just after Perelman proved Poincaré's Conjecture recently.