- #1
- 1,306
- 0
I'm wondering if it is true that any surface can be equated to a weighted sum of a basis of surfaces differring only by genus? I think this is asking whether the path integral formulation for strings is more general.
Thanks.
Thanks.
weighted sum?? how do you "add" surfaces?Originally posted by Mike2
I'm wondering if it is true that any surface can be equated to a weighted sum of a basis of surfaces differring only by genus?
more general than what?I think this is asking whether the path integral formulation for strings is more general.
So what you are saying is that a surface can only be conformally equivalent to a surface of one genus type, either it has a hole, or it does not, right?Originally posted by lethe
weighted sum?? how do you "add" surfaces?
as far as i know, there are no sensible algebraic operations defined on surfaces
yesOriginally posted by Mike2
So what you are saying is that a surface can only be conformally equivalent to a surface of one genus type, either it has a hole, or it does not, right?
i don t think that such a thing exists.I was wondering if there isn't a similar principle for surfaces?
where are higher genus surfaces used in physics other than in string theory? how can i compare their use in string theory?
More general than its use in quantum physics.
So what you are saying is that there is no higher dimensional application of the Taylor series expansion, where, for example a surface in 3D being describe by f(x,y)=z is not equivalent to a sum of other functions of (x,y)? I don't remember seeing any examples of this either, but I've not seen anything that rules it out. It seems to me one could easily expand the f(x,yc) with y a constant and then expand f(xc,y) with x constant separately, and then combine these separate expansions into various surfaces. The original surface would then be the result of a sum of surfaces.Originally posted by lethe
yes
i don t think that such a thing exists.
there certainly is Taylor series for functions in higher dimensions.Originally posted by Mike2
So what you are saying is that there is no higher dimensional application of the Taylor series expansion, where, for example a surface in 3D being describe by f(x,y)=z is not equivalent to a sum of other functions of (x,y)? I don't remember seeing any examples of this either, but I've not seen anything that rules it out. It seems to me one could easily expand the f(x,yc) with y a constant and then expand f(xc,y) with x constant separately, and then combine these separate expansions into various surfaces. The original surface would then be the result of a sum of surfaces.
Are the series expansions of a surface only for open surfaces as functions of (x,y)? If not, then are the expansion of surfaces into the sum of basis surfaces only for genus 0 surfaces?Originally posted by lethe
there certainly is Taylor series for functions in higher dimensions.
i do not see what that has to do with your idea about adding topological spaces
i cannot answer this, because there is no such thing as a series expansion of a surface.Originally posted by Mike2
Are the series expansions of a surface only for open surfaces as functions of (x,y)?
basis surfaces?? i cannot imagine what that means, since surfaces are not algebraic objects, and i cannot add themIf not, then are the expansion of surfaces into the sum of basis surfaces only for genus 0 surfaces?
Aren't surfaces described by functions? Don't functions have expansions? Can't functions, these expansion functions, perscribe surfaces? What?Originally posted by lethe
i cannot answer this, because there is no such thing as a series expansion of a surface.
taylor expansion is only defined for real valued functions, not for surfaces.
Technically, no; they're described by equations.Aren't surfaces described by functions?
Generally, no. And even when they do, they generally only on a small piece of the function.Don't functions have expansions?
Using a suitable interpretation of a function prescribing a surface1...Can't functions, these expansion functions, perscribe surfaces?
The point is that perturbation is done on functions, not surfaces.What?
I'm still not quite sure what's vibrating. Are the various points along the string oscillating in space with time, or is it some function on the string that is changing at various points on the string?Originally posted by selfAdjoint
Trying to get an equation for a worldsheet is the wrong way to go. String world sheets are Riemann surfaces, and their topology is determined, Riemann fashion, by the singularities of the meromorphic functions they describe. And the singularities are determined from the vertices, places where strings enter or leave the interaction. A beautiful theorem lets physicists calculate their operator expansions from mathematical functions defined around these vertices. It's these mathematical things that are quantized by the path integrals.
Polchinski's book is rather complete. no graduate math is needed as a prerequisite, as this book is intended for physicists. maths you would need to know would include tensor index notations, complex analysis, and it is beneficial to know as much differential geometry and topology as possible, although not a prerequisite, a little group theory, and some Lie theory. i dunno.... basically, it assumes you know a bit more math than your average physics grad student, but not too much more...Originally posted by Mike2
Also, do you consider Polaski's book complete? Or is there some graduate math needed as a prerequisite to this book? Also, how many chapters must you read through before you get to Superstring theory and not just practicing on a faulty theory?
Do you think Hatfield book, "Quantum Field Theory of Point Particles and Strings" would be sufficient to read Polaski?Originally posted by selfAdjoint
Someone who hasn't had complex analysis might be snowed by all the Conformal stuff, though I suppose Laurent series could be taken as introduced. You are very right about the physics, though. I think the requirement is to be mature in your understanding of field theory. Meaning not only do you know the facts, but you are right there with their interelationships. Down in the middle of Polchinski is no time to be having mysteries about how that integral transformed or where that bracket came from in the operator product expansion.
yes. if you can read Hatfield, then you will be able to read Polchinski.Originally posted by Mike2
Do you think Hatfield book, "Quantum Field Theory of Point Particles and Strings" would be sufficient to read Polaski?