Comments on: Physics is hard http://hwasungmars.wordpress.com/2007/12/21/physics-is-hard/ reviews, squash, and technology Sat, 14 Nov 2009 16:26:38 +0000 http://wordpress.com/ hourly 1 By: hwasungmars http://hwasungmars.wordpress.com/2007/12/21/physics-is-hard/#comment-9705 hwasungmars Wed, 16 Jul 2008 09:02:46 +0000 http://hwasungmars.wordpress.com/2007/12/21/physics-is-hard/#comment-9705 A.J, actually that is a very good point. Thanks for sharing your insight. A.J, actually that is a very good point. Thanks for sharing your insight.

]]> By: A.J. Tolland http://hwasungmars.wordpress.com/2007/12/21/physics-is-hard/#comment-9704 A.J. Tolland Tue, 15 Jul 2008 18:23:00 +0000 http://hwasungmars.wordpress.com/2007/12/21/physics-is-hard/#comment-9704 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 $latex \langle \phi(x) \rangle = \int_F d\phi e^{i S(\phi)} \phi(x)$ he's going to be confused. Too many $latex \phi$'s. But if you tell him that there's a function $latex ev_x$ on the space of classical fields, which sends $latex f$ to $latex f(x)$, and that the expectation value of the operator $latex \phi(x)$ can be computed as $latex \langle \phi(x) \rangle = \int d\mu(f) exp(i S(f)) ev_x(f)$ for some measure $latex d\mu$, your mathematician will understand perfectly well what you mean. 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 $latex d\mu$ is obtained by regularization and renormalization, and does not assign equal weight to arbitrarily jittery fields. 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
\langle \phi(x) \rangle = \int_F d\phi e^{i S(\phi)} \phi(x)
he’s going to be confused. Too many \phi’s. But if you tell him that there’s a function ev_x on the space of classical fields, which sends f to f(x), and that the expectation value of the operator \phi(x) can be computed as
\langle \phi(x) \rangle = \int d\mu(f) exp(i S(f)) ev_x(f)
for some measure d\mu, your mathematician will understand perfectly well what you mean.

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 d\mu is obtained by regularization and renormalization, and does not assign equal weight to arbitrarily jittery fields.

]]> By: Review: Srednicki’s QFT « The Art of Equations http://hwasungmars.wordpress.com/2007/12/21/physics-is-hard/#comment-9629 Review: Srednicki’s QFT « The Art of Equations Tue, 11 Mar 2008 21:07:09 +0000 http://hwasungmars.wordpress.com/2007/12/21/physics-is-hard/#comment-9629 [...] 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! [...] [...] 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! [...]

]]> By: Lecture Notes of 2007 « The Art of Equations http://hwasungmars.wordpress.com/2007/12/21/physics-is-hard/#comment-9523 Lecture Notes of 2007 « The Art of Equations Mon, 31 Dec 2007 16:30:12 +0000 http://hwasungmars.wordpress.com/2007/12/21/physics-is-hard/#comment-9523 [...] These lecture notes are an excellent introduction for people who want to learn the difficult QFT. [...] [...] These lecture notes are an excellent introduction for people who want to learn the difficult QFT. [...]

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