From: baez@galaxy.ucr.edu (John Baez) Subject: mathematical physics Date: 27 Oct 2000 23:02:01 GMT Newsgroups: sci.physics.research Summary: [missing] In article , A.J. Tolland wrote: >On 24 Oct 2000, Toby Bartels wrote: >> Right, and then mathematical physicists (like John Baez) come along >> and attempt to find a rigorous framework to put the ad hoc rules into. >I'm not sure that the mathematical physicists are really doing >much along the lines of the latter these days. They seem to have become >somewhat distinct from mathematicians trying to study physical models >rigorously (as did for instance I. Segal or B. Simon). From what I can >tell, their job description now reads: "Use mathematical weaponry/magic to >obtain physical insights." This, if you like, is the Witten school of >mathematical physics. > If anyone knows more about the history of this terminology, I'd >love to hear about it. When I went to college in 1979, "mathematical physics" had a rather special meaning - namely, the art of using analysis to make rigorous various calculations that physicists had already done in nonrigorous way. I studied this kind of thing from people like Elliot Lieb, who is famous for rigorously proving the stability of matter starting from Schrodinger's equations. This had already been nonrigorously demonstrated by Freeman Dyson, and it's exactly the sort of thing that ordinary physicists have trouble getting excited about: a rigorous proof of something that "everyone knows". The bible of this sort of mathematical physics is Reed and Simon's "Methods of Modern Mathematical Physics", and I studied it assiduously. In fact, I started after Kochen refused to work with me on a junior project, on the grounds that I didn't know the spectral theorem for self-adjoint operators. This pissed me off, so I spent the summer learning about the spectral theorem from Reed and Simon's book. When I finished college in 1982, I knew I wanted to work on quantum gravity. But I didn't know who worked on that - especially in a math department. So I wound up going to grad school at MIT, in part because I liked C*-algebras and Irving Segal, the inventor of C*-algebras, taught there. I wound up doing my thesis on constructive quantum field theory and 4d conformal geometry - two of Segal's obsessions. As a postdoc, I worked on scattering theory for nonlinear wave equations - another of Segal's interests. All of this was good old-fashioned mathematical physics in the above sense: lots of analysis, a little bit of geometry and algebra, with an emphasis on making existing stuff more rigorous. But in the meantime, the more modern style of mathematical physics was taking over, thanks to the boom in string theory and the influence of people like Michael Atiyah and Isadore Singer (who was also at MIT, and in fact a student of Segal). This more modern style emphasized geometry and topology, with analysis taking the back seat (although for Singer it was still very important). Instead of cleaning up old messes and making things rigorous, it was more about discovering new connections between different branches of math and physics. There is also an endless amount of work to be done in proving all the conjectures that Witten and other string theorists generate. This could also be considered a form of "cleaning up messes", but they are new rather than old messes. The old messes are often harder to clean up! So we now have lots of people thinking about the math of string theory, while nobody even knows if quantum field theory (much less string theory) makes sense - except perturbatively. It's a different ballgame. >Prof. Baez, are you mainly interested in putting physical models on >mathematically firm footing, or are you more interested in the physical >implications of your mathematical constructions? Or does this question >create a false distinction? After a while of learning both the old and new styles of mathematical physics, I decided to work on quantum gravity, but in a mathematically rigorous way. So I'm trying to do new physics, but I don't want to drift *too* far into talking about stuff that I can't make precise. I publish lots of papers in math journals, and prove honest theorems. You see, I worry that doing theoretical physics with neither experimental evidence nor mathematical rigor to guide me would amount to playing tennis with the net down. I feel much more relaxed knowing that even if my physics theories are on the wrong track, I've at least proved some solid theorems that may be of use to people. Without this, I would be nervous and unhappy. Of course, I *also* want my physics to be correct! But for this, only time and experiment will tell.