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String Theory: 6 extra dimensions

  1. Jun 21, 2007 #1
    I'm confused about the 6 extra curled up dimensions in that they are always portrayed like this:

    But according to that picture it seems there are multiple "extra" dimensions, and that if you move to a different location in 3D space you are in a "new" Calabi-Yau dimension. Well, the problem is that it was always my understanding that dimensions were independant of motion in one another, so wouldn't there just be ONE and ONLY one like this:
    Last edited by a moderator: Apr 22, 2017
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  3. Jun 22, 2007 #2
    It's just a matter of graphical representation :

    example think of normal euclidean 3d space, suppose you want to represent it on a sheet of paper, so you draw a 2 dimensional grid representing (x,y) and at each point of the grid you place a vertical line representing the 3rd dimension (z). So any point in space (x0, y0, z0) can be placed picking one point of the grid (x0,y0) and a point on the vertical line (z0) which crosses at that same point.

    Of course, you can do better than that if you represent 3d on a plane with a perspective drawing where you put z as a diagonal line and we are all used to that, you don't need to put a vertical line on each point.
    But if you want to add instead of 1 additional dimension (z), many others, that becomes tricky doesn't it...

    that's why the graphic that you have in the first pictrure is just the same, it doesn't mean that when you move from one point of the grid to another you also necessarily move in the other dimensions.
  4. Jul 27, 2009 #3
    Are the dimensions possibly unknown and the result or inference to calculate the math?

    Either way, are the 10 dimensions possibly:

    3d space of the present 3d space of the past 3d space of the future + time in general = 10 dimensions?
  5. Jul 27, 2009 #4
    The 6 extra dimensions being the 3d spatial in the past and 3d spatial of the future.
  6. Jul 27, 2009 #5


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    This raises a basic issue I've not seen clearly answered (maybe the answer is just so obvious?).

    Does every point in 4-space have its own local CY 6-space? Or does every point of the extended world share the one compactified space?

    If the latter, that would of course make it pretty crowded if it is the scene for "stringy action".

    And also, moving from one 4D locale to another as a string would seem to involve moving "through" 6-space in some sense. A particle would have to carry its dimensionality with it.

    So perhaps taking an example like shifting an electron 2 cm to the right. It has just moved through extended 4-space. But did it also a) move through 6-space, b) move IN 6-space, or c) move its 6-space with it?
  7. Jul 28, 2009 #6


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    We should try to discuss this in three dimensions.

    Reduce the ordinary 3 spatial dimensions to 2 and reduce the 6 compactified dimensions to 1. Let's assume for a moment that they are not compactified.

    So in each point in space (labelled by two coordinates) you have a 1-dim space that extends into the 3rd dimensions. w/o compactification is't just a line, so in total it looks like familiar 3-space.

    So you have a line in each point in ordinary space. You can move along the line w/o moving in ordinary space, you can move through space w/o moving through the 3rd dimension, but in every new location in ordinary space you "see" a new line pointing into the third dimension. All these lines look the same, but they are not identical. Of course you con move through ordinary space AND through the third dimension if you wish.

    Now if you compactify the third dimension to a circle all this remains valid, even if it's harder to imagine how it looks like. If the circle of the third dimension is large enough (or if you are small enough) you would not even feel the difference between the circle and the straight line.

    So coming back to your question: there is a local CY 6-space in each point in space.
  8. Jul 28, 2009 #7
    I don't think anyone knows how it's supposed to work out. Some suggest that the extra dimensions are small, others seem to suggest we're living on a surface (brane) in a 11 dimensional ambient (Lorentzian?) space. There's probably people who think it's a mix as well.

    If someone can clarify this it'd be great.
    Last edited: Jul 28, 2009
  9. Jul 28, 2009 #8


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    Thanks, that's a useful simplification which makes the issues clear.
  10. Jul 29, 2009 #9


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    I don't know either, but that doesn't matter in this context. The only differences would be whether the local copies of the additional 6(7) dimensions are big or small and if they are circles, lines, CY, G2, ... or something else.

    - they are local copies
    - they are always perpendicular to the ordinarey 3-space
    - the whole 10(11)-spacetime is (locally) Lorentzian
  11. Jul 31, 2009 #10
    So the space-time in ST is just a 11 dimensional vector space with a Lorentzian metric?

    I don't know what you mean by local copies, unless you're referring to the tangent space?
  12. Jul 31, 2009 #11


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    Sorry for the confusion. I'll try to clarify that.

    First of all the spacetime in string theory (M-theory) has 10 (11) dimensions. The signature is (+---...) so it's Lorentzian.

    That does not necessarily mean that the metric is flat. As far as I know one can study ST on more complicated backgrounds, but the metric must not be too complicated, otherwise no formulation of the theory is know. The metric must be a solution of the Einstein equations; this consistency condition follows as a result of the ST equations (I only know this statement, I haven't seen the proof in a paper or a textbook; does anybody know a reference?)

    There was the question if there is only one CY or if there is one CY attached to each point in 4d spacetime. I was only saying that there is a "local CY" for each point in 4d spacetime. That's what I mean by local copy. If you look at a one-dimensional space (a line) and construct a two-dimensional space (a plane) you are simply adding one perpendicular direction (another line) to each point of the original line. So you have "local copies" of the second line - one copy for each point in the 1d case.

    If you are referring to the shape of the new dimension it's always the same, so the shapes of the "local copies" are identical - they are always 1d lines. That's why one often talks about "the CY". It means that you attach the SAME CY to every point in spacetime.

    What would happen if you allow different CYs? Look at the 1d example. Now construct a 2d space by attaching local copies of 1d lines. At a certain point of the original line you change from a line to a circle! The resutling manifold is a combination of a half-plane and a half-cylinder with a rather strange singularity. So I think that's the reason why one is studying a fixed CY geometry (I know that topology changes of the CYs have been discussed, but we should exclude this topic here).
  13. Jul 31, 2009 #12
    I could not access the links provided in post 1 by phoenix...anybody have others?

    My understanding is consistent with Tom's..there is a local CY 6-space in each point in space.

    A way to think about this is that those curled up dimensions are believed to cause particles to have the characteristics we observe...different strings are constrained in different ways via their vibrational energy and pattern....and as those move thru spacetime we don't expect a proton to convert to a neutron nor an electron as space changes...(that's likely a wildly exaggerated example, just trying to make the point).
  14. Aug 1, 2009 #13
    Just to think of a one-dimensional space or a line takes time, but I don't think you can look at a one dimensional space and see a line, the best You could see of a one dimensional motion is a point. To see a line you have to think of at least three dimensions, or motion in two dimensions. :wink:
  15. Aug 1, 2009 #14


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    It depends what you mean by "see". I do not want want to talk about motion; in this context it's confusing. The line "is" there, it is not generated by a moving point.
  16. Aug 1, 2009 #15


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    Which is where my difficulties with visualising this comes in. If an electron is taken to be a point particle with part of its existence represented in a compact and circular 6-space - a local gauge symmetry resonance - then how does this part of itself move smoothly with it?

    A wiggly string can be imagined to move through space. Where a point particle traces out a world line, the string carves out a fat tube. We've seen the illustrations of particle interactions as tubes coming together and splitting.

    So do we just extrapolate this mental image of a 2D tube to a 6D "tube" which moves a writhing CY space, a tangle of wriggles? This would seem to follow from Tom's description.
  17. Aug 2, 2009 #16
    Sorry, you are the one that said "If you look at a one-dimensional space (a line)" I can think of the space I just can't look at it as a line. sorry
  18. Aug 2, 2009 #17


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    Hello petm1,

    all what I wanted to do is to simplify things as much as possible. That means that we neglect dynamics, motation and time, and talk about a purely geometrical / topological approach.

    So I make a new proposal:

    "If you look at a one-dimensional manifold and construct a two-dimensional manifold from it, you are simply adding one (perpendicular) direction (= a new dimension) to each point of the original manifold. An example is to construct R2 by attaching a line R1 to each point in an original R1; mathematically this reads R2 = R1 * R1".
  19. Aug 2, 2009 #18


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    No, hopefully not!

    Let's discuss the geometrical ideas in more detail:

    We start by reducing the 10d spacetime of ST to 3d in order to have a chance to visualize things; that means we have a 2d space and 1d for time. Geometrically the string is (in STs with only closed strings) a closed loop with a coordinate s labelling the loop and a coordinate t labelling time; for each t you have a loop = a "deformed" circle, as t changes the loop itself changes, e.g. it can vibrate or twist or something like that.

    The string is a field Xm(s,t), where m is now labelling the spacetime dimension m=0..2. Attention: these coordinates labelled by m are different from s and t; s and t are not labelled by m. Instead these coordinates are labelling the spacetime the string X lives in.

    In our low-dimensional example we can start with a flat, euclidean space R2 and a time coordinate. The entire 3d spacetime is now a stack of R2 spaces on top of each other, one R2 space for each time X0. In this picture we identify the time coordinate t of the string with the time coordinate X0 of the 3d spacetime. This is slightly misleading if you want to start calculations in string theory, but for our purpose it's OK.

    Now you can draw a loop in each R2 space of the whole stack. So the string lives in an R2 space and it moves in the third dimension as time evolves. For each time t you have a new R2. The coordinate s is a coordinate along the loop, the coordinates X1 and X2 are simply the coordinates within one R2.

    If you now look at the whole R3, the spacetime, your string will draw a world-tube as it moves through t=X0.

    Now we change to a picture where spacetime is poartially compactified. Instead of R2 we use a cylinder R1 * S1. There is one compactified dimension S1, which is topologically a circle.

    Now you can draw a loop on a cyclinder which is again the string seen at one instance of time t=X0; you are still identifying these to time coordinates. As time evolves you have to draw a stack of cylinders. You can do that by wrapping one cylinder at time X0 by a new cylinder at time X0'. If you would do that by a sheet of paper your cylinders (more exactly the dimension labelled by X2) grew, but that is only due to the limitation of using sheets of paper. Physically this dimension does not grow (in the R2 case the shape was constant).

    Looking at this stack of cylinders is harder to visualize, but still you have a 2d world-tube generated by the string as it moves through this stack of cylinders. Using the sheets of paper wrapping each other you have one closed loop on each sheet.

    So it should be clear that changing the shape of one dimension does not chnage the topology of the world tube. If you replace the 1d spatial dimension of the cylinder by a 3d space and if you replace the 1d circular dimension by a CY, the visualization of the wrapped cylinders breaks down, but the 2d world-tube survives mathematically.

    The last difficulty is that the length of the string (the loop) can be (much) larger that the length of the circular dimension S1. That means that the string can wind around the cylinder. You can compare this to a rubber band that fixes a poster. You can wind the rubber band once, twice, ... around the poster. So the rubber band can vibrate (like a violine string) and it can wind around the cylinder. The difference between types of elementary particles is partially due to different windings.

    Again this difficulty does not change the topology of the 2d world-tube.
    Last edited: Aug 2, 2009
  20. Aug 2, 2009 #19


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    Thanks again Tom. Wish I could say I could visualise your argument when it gets to the wrapped cylinder bit here. Are there pictures of this in any papers or books you can reference?

    It would also be interesting to hear more about how you conceive dimensionality itself.

    Are the compact circular CY dimensions to be thought of as a bunch of space-like orthogonal directions branching off from each 3D point? Or are they somehow not spatial?

    I suppose they would lack vacuum-like qualities for a start - virtual particles, dark energy, etc. So would they be make of something, filled with something in the ether/condensate way the spacetime void is?

    Another question is could a circular compact dimension actually be orthogonal? Seems like it would have to be tangental to the flatness of 4-space.

    It is also interesting that you explicitly *construct* dimensionality by gluing new dimensions at right angle to the existing ones. I can see this is the normal mathematical way to think about it (building up from simple spaces to complex spaces by additive steps), but does anyone instead take a *constraints* based approach to the creation of dimensions as far as you know?

    This would be going at it the other way. For example, in regular geometry you add together an infinity of 0D points to construct a 1D line. But you could instead imagine constraining a 2D plane until it is so hemmed in from either side that its only action is a 1D line.
  21. Aug 2, 2009 #20


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    No, unfortunately not. Simply think about a stack of sheets of paper and form a cylinder out of it - done.

    Yes. Again look at the cylinder: it is two-dimensional; the non-compact dimension of the cylinder is our well-known 3d space; the compact circle is the compactified CY. They are orthogonal.

    Do you know the treatment of the q.m. harmonic oscillator? It's the same with strings (in the simplest case): you quantize the vibrational modes of the string; for each mode you get a vacuum or zero point energy. The problem with realistic vacua is much more involved, as you do you have more complicated objects, not only strings - let's postpone this question as it has not so much to do with the basic geometrical discussion.

    That because the drawings you knwo from popular books are misleading. Think again about the cylinder: The two dimensions are orthogonal; one dimension is the 3d space, one dimension is the CY, the time dimension comes in when you draw the cylinder for different times.

    This explicit construction of new dimensions is exactly what I am doing here. You must not think that string theory invented new types of dimensions here; it's all standard. The only difficulty is that it's harder to visualize because of the complex topology of the CY.

    What do you mean by constraints?

    First of all why would you start with an infinite set of 0d points?

    OK, this is what you mean be constraints. But how could this help within ST? You have a 10d spacetime with a 6d CY. Would you like to restrict the string to move only in 4d by constraining the other 6d to 0?

    I have never though about that but all what I know is that there are very good reasons to have 10d spacetime with 6d compacified and not eliminated. If you would confine the string by adding some kind of confining potential acting in the 6d direction you would be in big trouble: you push the vacum energy to infinity, and you break a lot of symmetries. So a confinement to 4d should be a dynamical results, not something you use as input.

    The ST guys expect the CY to be something like a dynamical result, even if they are not able to prove this. Perhaps other "confinemrent" mechanism are possible, but as far as I know nobody has ever started to think about that.
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