This is Entry #1 of this blog, the “Theoretical Atlas”, in which introductions are made. It is the first of a few “meta” entries, setting out where we’re going.
My name is Jeffrey Morton. I am a mathematician interested in physics – and life. I’m starting this blog as I start out on my first postdoctoral position after finishing my Ph.D with John Baez at UCR.
The starting point for my research to date was my interest in efforts to find a quantum theory of gravity. It started innocently enough, learning some differential geometry, which led into General Relativity; that, in turn, led into studying quantum field theory in curved spacetimes. There many limitations on what you can say about that: general relativity and quantum field theory are based in very different mathematical vocabularies (not to mention grammar). Relativity theory describes gravity in terms of the geometry of spacetime – which is quite definite. Quantum field theory, on the other hand, describe matter in a very different way, in which the observed values of physical quantities can be any of the eigenvalues of certain linear operators on some Hilbert space. In particular, it doesn’t always predict a definite, specific value for the concentration of mass in any given location. But it’s mass-energy, in particular places, which supposedly creates gravity. These theories have a hard time talking to each other unless you carefully limit how much they interact, or assume a great deal of symmetry.
I encountered this fact when I was studying the Einstein-Dirac-Maxwell equations for my M.Sc. and an interest in confronting it led me to UCR. What I learned from Dr. Baez once there pushed my interests in several other directions. As one might expect: if the problem is that two theories use very different language, a step to reconciling it is to develop a new language which can handle both of them. This is where category theory entered the story. Category theory is a very general mathematical language, which can be applied to many subjects within mathematics, and thence to their many and various applications. One way to state the essential idea is that it takes both “things” and “relations” between things (in a very general way) as fundamental concepts. So far, I’ve been thinking about various ideas regarding how this can show up in physics.
Now I have introduced myself and two of the the main conceptual characters, what about the blog?