# String Theory: 6 extra dimensions

## Main Question or Discussion Point

I'm confused about the 6 extra curled up dimensions in that they are always portrayed like this:
http://people.cs.uchicago.edu/~mbw/astro18200/calabi-yau-grid-small.jpg"

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:
http://people.cs.uchicago.edu/~mbw/astro18200/calabi-yau-space3.jpg"

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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.

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?

The 6 extra dimensions being the 3d spatial in the past and 3d spatial of the future.

apeiron
Gold Member
it doesn't mean that when you move from one point of the grid to another you also necessarily move in the other dimensions.
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?

tom.stoer
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.

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.

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apeiron
Gold Member
.

So coming back to your question: there is a local CY 6-space in each point in space.
Thanks, that's a useful simplification which makes the issues clear.

tom.stoer
... 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.
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.

Nevertheless:
- they are local copies
- they are always perpendicular to the ordinarey 3-space
- the whole 10(11)-spacetime is (locally) Lorentzian

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?

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

So the space-time in ST is just a 11 dimensional vector space with a Lorentzian metric?
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?)

I don't know what you mean by local copies, unless you're referring to the tangent space?
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).

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).

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.
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. tom.stoer
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.

apeiron
Gold Member
A way to think about this is that those curled up dimensions are believed to cause particles to have the characteristics we observe.
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.

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

tom.stoer
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".

tom.stoer
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.
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.

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apeiron
Gold Member
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.

tom.stoer
Are there pictures of this in any papers or books you can reference?
No, unfortunately not. Simply think about a stack of sheets of paper and form a cylinder out of it - done.

Are the compact circular CY dimensions to be thought of as a bunch of space-like orthogonal directions branching off from each 3D point?
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.

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?
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.

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.
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.

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 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?

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.
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.

apeiron
Gold Member
Do you know the treatment of the q.m. harmonic oscillator?
Yes, I understand the general principle. So the answer is that string theory's concern is with the objects in the compactified space - the strings. But I asking also about the "space" around these strings. If such a thing exists. Does it have any properties of its own? Or is it regarded as properly empty in a way the vacuum was once supposed to be?

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.

I get it this time. Head down the cylinder and you are moving in flat 3-space. Go round the cylinder and you are moving through CY space.

And would you have a model of a hyperspheric 4D realm if the cylinder is turned into a donut?

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.
The question arises because my main interest is in models of causality and construction and constraint are complementary ways usually of arriving at the same outcome. You can construct something from the bottom-up (like bricks make a house) or constrain things from the top-down (like a magnetic field organises the dipoles in a bar magnet).

But I can also see constraints apply to compactification scenarios. Constraint would attempt to limit dimensionality towards zero, but constraint would run out of steam at the planck scale due to QM uncertainty. So constraint could not eliminate compactified dimensions.

This line of speculation comes more from thinking about what makes just three dimensions a natural minima. There could be an infinity of dimensions, yet we live in a world reduced to just three spatial ones. There are arguments as to why just three is indeed the end point for a constraints based causality. Where compactified dimensions then fit into this scenario then becomes the next question.

I'm asking really as it is interesting to see how people in different fields are conceptualising dimensionality.

A vortex is another good example of a dimensionality created via top-down constraint. A smooth flow can break into turbulence when the flow gets too viscous to dissipate its energy efficiently. Prevented from moving faster, it instead goes into gurgling spirals and loses the surplus energy in a different direction - orthogonally, so to speak, as heat and sound and scouring of the riverbed.

tom.stoer
Regarding your first question if spacetime has its own properties in string theory: I think it is a major weakness of string theory (some people in this forum - perhaps better experts on this subject than me - will not agree) that spacetime itself has no real physical properties. It acts as a passive stage on which physics happens (on which strings move and vibrate); but spacetime itself is a non-dynamical background. This is called background-dependence; in a theory of full quantum gravity one expects that the spacetime itself becomes dynamical.

I do not understand exactly what you mean be the 4D hypersphere and the donut. In our picture this would mean that all spatial dimensions are compactified and none is left over to form the well-known 3-space.

Regarding dimensions: have you ever thought about a model where spacetime is not the fundamental building block but "emerges" from some underlying basic structure?

LQG is one theory where the fundamental structure is not a 4d manifold. Instead it's a graph that carries an additional algebraic structure (so-called spin networks). Besides some other interesting things the graph itself has no "dimension"; this is something that emerges together with the smooth spacetime manifold as a kind of long-distance limit. LQG and other theories of quantum gravity suggest that the "number of dimensions" of spacetime depend on the resolution you are using to study it. In the long-distance limit spacetime is four-dimensional, but at very short distances the spacetime becomes effectively two-dimensional.

Causal Dynamical triangulation uses "4d Minkowskian simplices" as fundamental building blocks of spacetime. An interesting result is that again the dimensionality of spacetime depends on the resolution. There are some results from which one may derive that there is no fundamenal length scale, but that spacetime becomes a fractal without a shortest distance.

All these theories are not yet completed, but they are already indicating that we should not limit ourselves to a smooth spacetime manifold but should consider spacetime as a long-distance limit, similar to the surface of water; with a better resolution the surface disappears and the fundamental structure (molecules) becomes visible ... In all these cases the meaning of dimensionality disappears together with the manifold itself, just like wavelengths of water shorter than the typical distance of molecules are meaningless.

apeiron
Gold Member
Regarding dimensions: have you ever thought about a model where spacetime is not the fundamental building block but "emerges" from some underlying basic structure?
.
These would indeed be the kind of constraint satisfaction approaches I would expect to work. Spacetime emerges as a sum over histories - an equilibrium solution - that minimises energy, tension, curvature, whatever.

And the rivalry between strings and loops - between background dependent and background independent approaches - seems to boil down to these opposing views of causality. One is based on construction and one based on constraint.

Construction assumes local atoms - substance is fundamental in other words. And strings make a better kind of substantial atom (but demands then the pre-existence of a global canvas)

Constraint assumes global organisation - and so form is fundamental in this view. And spin networks or other loopy structures that must satisfy global constraints can then generate global states of order, in the continuum limit.

Having said that, loop approaches do not seem pure top down constraint-based models. Instead, they atomise the form, so to speak. You start with a causal triangle or a spinor or some other such "atom" which has some random orientation. Then by allowing free interaction of these parts, global order - via emergent global constraints - emerges.

So it would seem that loop approaches sneak in a form of background dependence as well. Each element of the network is like a fragment of dimensionality and so already presumes the existence of dimensionality. The dimensionality is just crumbled up into smallest planckscale bits. The background dependence is just shrunk from a global scale stage as in strings to a local scale - one the size of each loop (or whatever is chosen as the models "atom of form")

That is one of the reason I ask about people's notions of dimensionality. Is it based on constraint or construction? Is it a local or global property of their models?

Dimensionality seems to be taken too easily as a given, a known. But what is a dimension - beyond a direction, a degree of freedom, a symmetry, an inertia, etc?

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?
Point particles are not what is represented by string theory. A string has spacial extent and is NOT a point particle.

Regarding your first question if spacetime has its own properties in string theory: I think it is a major weakness of string theory (some people in this forum - perhaps better experts on this subject than me - will not agree) that spacetime itself has no real physical properties.
Ah but it is not necessarily a problem: think about the Kaluza/Klein addition of another (curled up) dimension to Einstein's general relativity. Simply formulating a four plus one dimensional representation of Einstein's GR causes Maxwell's equations to pop out!!!...so obviously an additional dimension has SOME sort of property favoring electromagnetic fields....(Or think of it as a purely mathematical result if you wish) and all this is simply a classical approach, nothing like fancy string theory. Somehow dimensions affect observables. (Of course adding a single extra dimension ran into limitations and eventually proved unsatisfactory)

String theory demands 10 plus one dimensions to replicate this universe..otherwise it has variations from observed conditions. And string theory even requires specified shapes: six dimensional calabi-Yau compatified dimensions. (Klein's compactified circles were not sufficient.)
Seems like with Planck sized strings vibrating with substantial energy and planck sized compactified dimensions vibrating with quantum jitters strings would have a tough time navigating around the curled up compactified dimensions...before they got very far they'd be back where they started...so it seems within reason strings might coast along with consistent vibrational patterns. When these extra dimensional shapes and sizes are altered, string vibrational patterns change and hence particles properties as well.

For those interested, Brian Greene offers extensive qualitative descriptions in Fabric of the Cosmos, 2004 edition, pages 345-375. And of course he makes it clear all this is a work in progress.

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tom.stoer