User talk:String4d

Welcome

Sorry for my cold welcome to you in Wikipedia, I hope you have a nice time and lots of contributions. Dan Gluck 10:04, 11 August 2007 (UTC)

Non-critical string theory: Lorentz invariance

Hi, I answered you in my talk page.Dan Gluck 18:51, 10 August 2007 (UTC)

Liouville's equation

Hello,

I noticed you added a link to Liouville equations. Note this is an equation for Gaussian curvature in differential geometry. Hope there was no misunderstanding. Katzmik 08:25, 12 August 2007 (UTC)

Thanks for your comment on the book by Barbashov and Nesterenko. Apparently you are referring to "Introduction to the relativistic string theory", translated into English in 1990. None of our university libraries have it, unfortunately (neither math nor physics). I see that in mathscinet, only one review refers to the book, and only 2 additional articles contain it in their bibliography. It does not seem to be a well-known book. Are you referring to some sort of Lagrangian versus Hamiltonian duality between the two forms of the equation? Liouville's theorem in Arnold's classic book "mathematical methods of classical mechanics" is formulated as the invariance of the volume element under phase flow. A generalisation in symplectic geometry is that the symplectic form is invariant under the flow. Thus, Liouville's theorem seems to be a pre-metric result, and its relation to Gaussian curvature, if any, needs to be understood. I am very curious about what you wrote and hope you can provide additional details. Katzmik 08:06, 14 August 2007 (UTC)

P.S. Are you saying the Liouville's theorem in physics is the same as Liouville's theorem in complex analysis, or that Liouville's equation in physics is the same is Liouville's equation in differential geometry? Katzmik 08:20, 14 August 2007 (UTC)

The connection with Polyakov's Lagrangian is very interesting. What you seem to be saying is that an equation identical to Liouville's curvature equation arises in the context of studying the Lagrangian. However, from the point of view of classical differential geometry, curvature is not a variable that can be changed, to the variable μ in Polyakov's case. This is at least the naive viewpoint. Another point that is bothering me is that the symplectic/Lagrangian framework is, as I mentioned already, pre-metric, i.e. "flabby" in Gromov's sense. More specifically, symplectic geometry has no local invariants, by the classical Darboux theorem, whereas Gaussian curvature is most decidedly a local Riemannian invariant. Which Polyakov paper are you referring to?

I imagine all this is very naive for a physicist, and I am certainly an ignoramus as far as field theory is concerned. I would like to understand all this better, so if you have the patience please go ahead. However, an argument can be made that a wikipedia reader should be able to understand Liouville's classical curvature equation without understanding field theory first. Katzmik 07:41, 15 August 2007 (UTC)