Physics is hard
I was talking with a post-doc who is doing research in algebraic topology, and somehow we ended up talking about quantum field theory. I was amazed by his knowledge of it, usually you don’t expect mathematicians, especially pure mathematicians, to know what quantum field theory is.
I asked him where he learned all his quantum field theory. He said originally he was doing physics as a undergraduate, but he switched to mathematics after he couldn’t understand quantum field theory. He had taken three lectures, including two that he took when he was doing his Ph.D.
Quantum field theory was too hard for me. It felt like hitting a brick wall. It was impossible to learn it step by step. [...] It was like learning a language. The understanding comes years later you start using it.
- Paraphrased by myself.
I agree with his view. Physics is hard, and it is harder than mathematics in a different sense. You can find a similar view in Cox and Katz’s Mirror Symmetry and Algebraic Geometry book.
Mirror symmetry has made some surprising predictions in algebraic geometry, [...]. Yet to understand where these predictions come from, the algebraic geometer must plunge into the language of physics, which is unfamiliar and sometimes frustratingly nonrigorous.
While physicists complain that mathematics is hard because it is technical and rigorous, mathematicians complain that physics is hard because it is nonrigorous. I guess different fields has different approaches to a similar material. But what made the nonrigorous approach so successful?
I think the most powerful tool of physicists is experiments (in a broadest sense, also including Gedankenexperiments, and computer simulations). For example, even while mathematicians were complaining that path integrals are not well-defined, physicists just used them and nature (experiments) backed them up. I think this is the reason why physicists were able to jump to the “post-rigorous stage”, even without having to spend a lot of time in the rigorous stage (in comparison to mathematicians).
But, I think I am ending up like the post-doc who I was talking to. I am uncomfortable with jumping into the post-rigorous stage, without spending lots of time in the rigorous stage. There is a lot to learn in mathematics, but I feel that I am drifting off towards it.
[...] These lecture notes are an excellent introduction for people who want to learn the difficult QFT. [...]
Lecture Notes of 2007 « The Art of Equations said this on December 31, 2007 at 4:30 pm |
[...] QFT. My favorite book and the book that I always consult first is Srednicki. I think if my friend had encountered this book before, he wouldn’t have ended up being an algebraic topologist! [...]
Review: Srednicki’s QFT « The Art of Equations said this on March 11, 2008 at 9:07 pm |
So… I spend a lot of time talking with mathematicians about QFT, and I think many physicists misunderstand what makes physics difficult for mathematics. The lack of rigor isn’t really a problem. No one cares that the naive path integral measure doesn’t actually exist. The problem is that physicists don’t usually define their terms clearly enough for mathematicians to follow the story.
Consider, for example, the path integral definition of the expectation value of a scalar field. If you give a mathematician the expression
’s. But if you tell him that there’s a function 
on the space of classical fields, which sends 
to 
, and that the expectation value of the operator 
can be computed as
, your mathematician will understand perfectly well what you mean.
he’s going to be confused. Too many
for some measure
Of course, your mathematician might get curious about what exactly $d\mu$ is. If you tell him it’s Lebesgue measure on the space of functions, you’re lying, even as far as the physics goes. The correct definition of
is obtained by regularization and renormalization, and does not assign equal weight to arbitrarily jittery fields.
A.J, actually that is a very good point. Thanks for sharing your insight.