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Formulation of String Theory

  1. Sep 29, 2014 #1
    This will, no doubt, sound like a silly question but I am just getting my feet wet in string theory. It seems from what I know so far, that string theory is very far removed from our observable universe. What I mean by that statement is that it seems we just say "well, maybe our universe is a brane within a larger space" or "maybe the extra-dimensions are just compactly curled up in very small radii."

    A few years ago, when I knew even less, I sort of intuited what string theory might consist of if I had to invent it myself. I wanted to actually start out with our observable 4-dim Minkowski space (curved or un-curved) and then attach at bundle structure to the manifold where the strings evolved. That way, what we see as a point evolving in the base-space (our universe) is actually a string evolving in the bundle space of some dimension. Furthermore, the image of the string evolution in the bundle will be the worldsheet I think.

    The one problem I see immediately is that in this formulation, closed strings would still project onto our universe which I don't think is a good thing.

    This seems to me to be a much more sensible starting point given that it makes explicit where the observable world is in the larger picture...but I'm not so arrogant as to think there's any sense to it! So why doesn't motivating the theory like this work? Thanks for any help!
     
  2. jcsd
  3. Sep 29, 2014 #2
    Superficially that doesn't sound too different to string theory with six dimensions compactified, with some Calabi-Yau space as the fiber of the bundle. I guess I'd say there are two main differences between what you describe, and compactified string theory.

    First, the fiber bundle approach suggests that there is some unique, objectively special division of the overall space into the fibers, whereas in a Kaluza-Klein theory like string theory, such a subdivision is just a matter of coordinates.

    Consider the common example (in popular science writing) of a garden hose. From a distance it looks one-dimensional, but close up you can see it has the topology of a cylinder. If we were building up a "garden-hose"-shaped space using your fiber-bundle method, we would start with a line, and then attach circular ring shapes all along the line, and attach them to each other to make the cylinder shape; and there would be a unique objective decomposition of the cylinder into the rings.

    But the actual garden hose can be broken down into a line of adjacent ring shapes in different ways. For example, you could slice it perpendicular to the cylinder, producing a stack of circles; or you could slice it with a slight "tilt", producing instead a stack of ellipses. The tilt of the slicing could even vary along the length of the hose, so that some slices were circular, others elliptical. What's fundamental about the garden hose is its two-dimensional shape, and not any particular decomposition of that shape into a stack of one-dimensional objects.

    The same goes for the product spaces appearing in Kaluza-Klein theories, like "four-dimensional space-time times some six-dimensional Calabi-Yau space". The overall space is fundamentally ten-dimensional on small scales, just like the surface of the garden hose is two-dimensional. An ant can crawl around on the garden hose and not care about one slicing into ellipses or another, they are just coordinate systems. Something analogous holds for the division of the ten-dimensional product space into an array of Calabi-Yaus - there are alternative "slicings" into six-dimensional "fibers".

    The other difference that stands out for me, is that the "fiber" in string theory is dynamical, because it's actually a slice of spatial geometry with its own metric. The Calabi-Yau can warp and wobble, as in general relativity. It has degrees of freedom - its moduli, in mathematical jargon - and their dynamics produces particular excitations in string theory, known as moduli fields.

    I'm sure it's possible mathematically to develop a theory along the lines you describe, with "strings" confined to fiber spaces. This may even arise sometimes in conventional string theory, as a sort of gauge fixing, and it might be educational to know what sorts of "fiber-space string theory" exists as a gauge-fixing of orthodox string theory, and which ones do not. I suspect this would in turn be related to the technical and philosophical differences between loop quantum gravity and string theory.
     
  4. Sep 29, 2014 #3
    Very helpful; thank you! So you mention that in the method I propose, there is an inherent notion of "which fiber is above which point" which should not be there? That's what you're getting at I think with your example of slicing the cylinder into circles, ellipses, etc. But is this not similar to gauge invariance in string theory as far as static gauge vs. light-cone gauge etc? Perhaps would a connection on the bundle help in any way?

    Finally, what do you think of the problem with closed strings that I mentioned? If closed string existed in a bundle, they'd still be projected into the space as a point particle...which is bad, correct?
     
  5. Sep 29, 2014 #4
    Yes, but now explain why. You obviously "believe" that you already hold the answer to your question, otherwise you would not have insisted that attention be paid to that particular element of your original post in this thread.

    So ... why would it be "bad?"
     
  6. Oct 3, 2014 #5
    Well what I meant was that, from what I know, we do not observe things like the graviton possibly because they don't have endpoints to attach to our universe. So it bothered me that a closed string in the bundle would still be seen as a "point" in the base space. I was asking if this is truly a problem or if I was overlooking something. If it is a problem, is there a common way around it?
     
  7. Oct 8, 2014 #6
    Something else that I didn't mention, is that the idea that the string is inside a single fiber, is at odds with how strings can behave in string theory. If we think of the background geometry as M4 x CY6, then this amounts to saying that each string is always inside a specific copy of the CY6. But the strings can also be extended in the macroscopic, M4 directions - cutting across the bundle of CY6 fibers.

    So what you propose might be relevant to a particular limit of orthodox string theory, in which the strings are effectively confined to one CY6 at a time, and excitations in the M4 directions are weak. There might be a "string-in-the-fiber" approximation for such a limit.

    Your graviton worry doesn't seem right. Maybe you're thinking of a braneworld, where the gravitons are closed strings in the space off the brane, and the other particles are open strings ending on the brane. But in the heterotic models, there are no braneworlds, it's just Kaluza-Klein and all the strings are closed. Also, there isn't inherently a problem with having a point-localized graviton. Yes, a gravity QFT based on that conception runs into problems, but the real theory proposed here is strings-in-fibers.
     
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