Path integrals and the sum of surfaces, is this general?

weighted sum?? how do you "add" surfaces?

as far as i know, there are no sensible algebraic operations defined on surfaces

more general than what?

 

Re: Re: Re: Path integrals and the sum of surfaces, is this general?

yes

i don t think that such a thing exists.

where are higher genus surfaces used in physics other than in string theory? how can i compare their use in string theory?

 

Re: Re: Re: Re: Re: Path integrals and the sum of surfaces, is this general?

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

 

Re: Re: Re: Re: Re: Re: Re: Path integrals and the sum of surfaces, is this general?

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.

basis surfaces?? i cannot imagine what that means, since surfaces are not algebraic objects, and i cannot add them

 

Technically, no; they're described by equations.

If you meant that a surface (in [tex]\mathbf{R}^3[/tex]) can be represented in the form [tex]z=f(x, y)[/tex], then the answer is, in general, definitely no.

Generally, no. And even when they do, they generally only on a small piece of the function.

Using a suitable interpretation of a function prescribing a surface¹...

The expansion (if it exists) will prescribe a surface, a piece of which is the same as a piece of the surface prescribed by the original function; it is possible that they may only have a single point in common!

Each partial sum of the expansion will prescribe a surface. The surfaces prescribed by the partial sums may or may not eventually look like the surface prescribed by the expansion.

Each term of the expansion will prescribe a surface. However, because we cannot add surfaces, this does not prescribe a perturbative expansion of the surface.

The point is that perturbation is done on functions, not surfaces.


Footnotes:

¹: An example of a suitable definition of a function prescribing a surface is:

Definition: The function [tex]f(\vec{x})[/tex] is said to prescribe the surface [tex]S[/tex] if and only if the points of [tex]S[/tex] coincide with the solution set of [tex]f(\vec{x})=0[/tex].

 

Hurkyl has explained this very nicely, and captured the essence of my point.

let me say it one more way: a function z=f(x,y) is a one-to-one valued map, which means that it can never "double back" on itself. yet, i can conceive of many surfaces which do "double back", like the circle, or the sphere. these manifolds can never be represented in their entirety by a function, and i can ascribe no meaningful content to the notion of a perturbation expansion of these surfaces.

 

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.

 

The string is osciliating and moving. Also it is splitting off strings and joining with other strings in interactions. The interactions are what are interesting to the physicists. The motion of the string in spacetime is represented by potentials on the worldsheet. These are universally designated the X_i, and the number of them is one less than the number of spacelike dimensions of the background. One less, because one of the spacelike dimensions is taken up on the worldsheet by the length of the string.

Each of the X_i is a real valued potential on the worldsheet - the coordinate in that dimension which that point of the string occupies at that moment. As the string moves the coordinates, and hence the potentials on the worldsheet, change. Oh the worldsheet is a busy, complicated place. It takes Polchinski several chapters to introduce all the math and physics.

 

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...

on the other hand, it does assume a lot of graduate level physics prerequisites. if you haven't studied quantum field theory for quite a while, you may find polchinski's book illegible.

as for how many chapters you spend on a toy model before you get to the superstring, well the book is in two volumes, and all of volume I is on the nonsupersymmetric string. so volume I is 9 chapters, although you don t necessarily have to finish all of those before moving to volume II.

but umm.... i don t know... what do you have against spending some time with a toy model? it can be quite useful...

 

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.

and please note: his name is Polchinski, not Polaski.