Re: [math-fun] misc + (Riemann surfaces & string theory)
1. Rich writes: << One of my favorite word puzzles is finding "consecutive" words, where the last letter is incremented to make a new word. CONVEX CONVEY ...
Just wondering in what sense you mean "consecutive" (since e.g. CONVEXITY would appear between these words alphabetically). ------------------------------------------------------------- 2. The NY Times had a nice short article by Dennis Overbye in Sunday's Week in Review section (June 29) that is a paean to the joys of math. (It's nice to have at least one math fan writing for the public.) This is freely available through next Saturday on the web; sorry I can't dig up the URL right now (but just go to www.nytimes.com and it should be easy to find, esp. with advanced search). Overbye mentions that string theory has made the subject of Riemann surfaces a hot topic. Can anyone please explain this remark to me? (Please bear in mind that I have only a vague metaphoric idea of what string theory is about.) (First guess: String theory as I understand it posits little simple closed curves in physics space (11 dimensional?) a what underlie elementary particles. Maybe now someone has suggested replacing the s.c.c.'s with surfaces?) --Dan
On Wed, Jul 02, 2003 at 08:05:00PM -0400, asimovd@aol.com wrote:
Overbye mentions that string theory has made the subject of Riemann surfaces a hot topic. Can anyone please explain this remark to me? (Please bear in mind that I have only a vague metaphoric idea of what string theory is about.)
(First guess: String theory as I understand it posits little simple closed curves in physics space (11 dimensional?) a what underlie elementary particles. Maybe now someone has suggested replacing the s.c.c.'s with surfaces?)
Not quite. String theory posits replacing particles with little curves, but then when the curves propogate you add an extra time dimension, getting surfaces. The resulting theory is nearly conformally invariant[1], so you end up studying Riemann surfaces. Usually physicists are happy doing calculations with (perforated) spheres and tori, but for higher-order corrections you need to understand the moduli of Riemann surfaces better. Peace, Dylan [1] One might well ask why the metric on the surface is Euclidean rather than Lorentzian. I think you do some sort of analytic continuation.
I have two questions. 1. Riemann surfaces are complex manifolds. Why does the string surface have the extra structure of C^1 rather than being simply R^2 ? 2. I can understand trying out the idea that particles are little curves. But why can't they propagate in ordinary 4-dimensional spacetime; why do we need 11 dimensions? --- Dylan Thurston <dpt@math.harvard.edu> wrote:
On Wed, Jul 02, 2003 at 08:05:00PM -0400, asimovd@aol.com wrote:
Overbye mentions that string theory has made the subject of Riemann surfaces a hot topic. Can anyone please explain this remark to me? (Please bear in mind that I have only a vague metaphoric idea of what string theory is about.)
(First guess: String theory as I understand it posits little simple closed curves in physics space (11 dimensional?) a what underlie elementary particles. Maybe now someone has suggested replacing the s.c.c.'s with surfaces?)
Not quite. String theory posits replacing particles with little curves, but then when the curves propogate you add an extra time dimension, getting surfaces. The resulting theory is nearly conformally invariant[1], so you end up studying Riemann surfaces.
Usually physicists are happy doing calculations with (perforated) spheres and tori, but for higher-order corrections you need to understand the moduli of Riemann surfaces better.
Peace, Dylan
[1] One might well ask why the metric on the surface is Euclidean rather than Lorentzian. I think you do some sort of analytic continuation.
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I asked question #2 of my neighbor, Leonard Susskind, one time, and what I understood of the answer was this: There's some sort of calculation of ground state energies in various string theories that only work out the way a person wants them if it's assumed that the dimensionality of the system changes. In some string theories/calculations, this number of space-time dimensions seems to want to be 26, in others, 11, in others, 5, etc. Susskind also recommended the book "D-Branes" by Clifford V Johnson (Cambridge 2003) as having a good exposition versus other string theory books. I looked at it in the bookstore and it's not a casual read. You may now return to your studies of general relativity and quantum field theory. Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm ----- Original Message ----- From: "Eugene Salamin" <gene_salamin@yahoo.com> To: <dpt@math.harvard.edu>; "math-fun" <math-fun@mailman.xmission.com> Sent: Tuesday, July 08, 2003 7:49 AM Subject: Re: [math-fun] misc + (Riemann surfaces & string theory)
I have two questions.
1. Riemann surfaces are complex manifolds. Why does the string surface have the extra structure of C^1 rather than being simply R^2 ?
2. I can understand trying out the idea that particles are little curves. But why can't they propagate in ordinary 4-dimensional spacetime; why do we need 11 dimensions?
--- Dylan Thurston <dpt@math.harvard.edu> wrote:
On Wed, Jul 02, 2003 at 08:05:00PM -0400, asimovd@aol.com wrote:
Overbye mentions that string theory has made the subject of Riemann surfaces a hot topic. Can anyone please explain this remark to me? (Please bear in mind that I have only a vague metaphoric idea of what string theory is about.)
(First guess: String theory as I understand it posits little simple closed curves in physics space (11 dimensional?) a what underlie elementary particles. Maybe now someone has suggested replacing the s.c.c.'s with surfaces?)
Not quite. String theory posits replacing particles with little curves, but then when the curves propogate you add an extra time dimension, getting surfaces. The resulting theory is nearly conformally invariant[1], so you end up studying Riemann surfaces.
Usually physicists are happy doing calculations with (perforated) spheres and tori, but for higher-order corrections you need to understand the moduli of Riemann surfaces better.
Peace, Dylan
[1] One might well ask why the metric on the surface is Euclidean rather than Lorentzian. I think you do some sort of analytic continuation.
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On Tue, Jul 08, 2003 at 07:49:25AM -0700, Eugene Salamin wrote:
1. Riemann surfaces are complex manifolds. Why does the string surface have the extra structure of C^1 rather than being simply R^2 ?
Don't forget, this is physics, so most spaces come with metrics. In this instance, the way these surfaces arise is really as conformal manifolds. The classical action only depends on the conformal class of the metric. (Hence the name of the related field, "Conformal Field Theory".)
2. I can understand trying out the idea that particles are little curves. But why can't they propagate in ordinary 4-dimensional spacetime; why do we need 11 dimensions?
On a quantum-mechanical level, the classical conformal symmetry is a little broken unless the space-time is the critical dimension. Exactly what the critical dimension is depends on the precise theory. (Yes, I know that's very vague.) Peace, Dylan
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Thane Plambeck