# The logic of the 11 dimensions in M theory

## Main Question or Discussion Point

According to M-theory as I understand, the 7 additional spatial dimensions to our familiar 3 are curled up all around us but they are too small for us to see. However I have difficulty in understanding how this constitutes a "dimension" because dimensions allow additional degrees of freedom.

I understand Rob Bryanton's view of it which explains how (and this is my brief interpretation) each extra dimension is reached by folding the current one. Ie fold a '2D' piece of paper into the '3D' dimension and you get a '3D' object. So in a sense each dimension is fundamentally "extra" and allows instantaneous movement from one point to another in the dimension below.

But this version of extra dimensions seems to differ entirely in that if I were to fold 4D spacetime into the 5th dimension, and entered that dimension, I should be able to move instanteously to any other point in the 4th dimension, so thereby having free access to go to any point in my current timeline. Whereas M theory's extra dimensions doesn't seem to make sense.

Put another way, if you pretend a straw is one of these curled up dimensions, the fact that I am able to trace a path around the curved surface doesn't mean I am suddenly moving in a different dimension from 3 space. And if the straw kept getting smaller and smaller, it would be still be occupying 3 space... would it not...?

## Answers and Replies

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bapowell
That's correct. Curling a 2D sheet back on itself to form a cylinder still leaves you with a 2D surface. As a denizen of the sheet, you cannot access the higher (3rd) dimension that the surface is embedded in. This is not the way extra dimensions are interpreted in string theory. First, why does a dimension being curled up too small to see somehow make it not a an additional degree of freedom? Because it is not accessible to us? True -- at low energies, the additional degrees of freedom afforded by extra dimensions are frozen (they are integrated out in going from the high energy theory to the low energy effective theory). But to strings and other high energy phenomena that can probe the relevant length scales of the compactified dimensions, they are relevant degrees of freedom. The best way to conceive of these extra dimensions is to pin them onto every point in the non-compact 3D manifold that we call home. If these extra dimensions were also non-compact, then at any point in space you could access them just as we access (up,down), (left,right), (back, forward). But since they are tiny and curled up, as we move through our non-compact 3-space, we actually move through the extra dimensions as well, doing tiny laps through them. But again, the relevance of these extra dimensions (the effect of doing these laps) depends on the energy of the probe: strings move through the extra dimensions and explore a full 10D world, whereas we humans don't notice them because the size of the compactified space goes to zero in the low energy limit.

Thanks for your reply. I still have a problem understanding how a dimension, regardless of how compactified, still counts as an additional degree of freedom just because we can't enter it. Because it is still encompassed within in our 3D reality.

Not sure if you know the other view on dimensions I was referring to in the OP, but this is the video: .

The idea of dimensions as per that video seem completely unrelated to the way the dimensions are thought of in M theory, at least in my understanding -- or am I wrong and they are one and the same?

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I've seen some the videos of the person you linked to and while what he says is not all wrong, he seems to be a crackpot and his ideas are not correct. If I recall he made it seemed like it mattered to the 5th or 6th dimension that the 4th is weird and is time. It doesn't its just a label. My advice, dont trust him, he is misleading and many crackpots online do not understand what a dimension really is.

A dimension is simply a degree of freedom. Think about a sphere. Imagine you are an amoeba living on a sphere. It is all you know. At a given point you can move in two possible directions. Every other direction is a simple combination. To be more mathematical, if you zoom in on the two sphere (the two sphere is just a shell) it looks like the flat plane. Thats why we say its two dimensional. This is no different then on the surface of the earth which usually looks flat to us.

When we talk about smooth surfaces (or manifolds) the dimension of the space is simply what its tangent space, or what it looks like, very close up. If you zoom in enough, the circle looks like the the line, the circle is one dimensional. Along a circle I have one degree of freedom, I can go forward or backwards thats it.

Ok now imagine there are 4 non-compact spatial dimensions. That is we live on R4 and I'm ignoring any difficulty about time. You can go along the x,y,z, or w axes forever. Or at least thats what you think. But lets say after going along the w axis for a long enough time you realize things start to repeat. This extra dimension is compact but very very large so you can traverse it. The w-axis is a circle in the sense that if you go along far enough, everything repeats. Or you can think of a different case, the cylinder with a very large radius. It looks non compact in both directions, but its actually compact in one of them when you travel long enough. You'd agree that regardless of the compactness, at any point in time along the cylinder you have two degrees of freedom to move in.

Finally we consider the 11 dimensions of M theory. Now to you there are only 4 dimensions, 3 space, 1 time. You disregard the idea of any extra dimensions because to you there are no extra degrees of freedom. However, you're just too large to experience these extra dimensions. A string can travel through all dimensions and has much more degrees of freedom it can realize then you can as a macroscopic object.

These dimensions can not be thought of as existing within our 3 spatial dimensions. They are extra degrees of freedom that strings can move in. It would be like saying we can consider the flat plane as a subset of the real line or the cylinder as a subset of the real line. compactness doesn't matter, it doesnt make sense.

Surely even with a thing moving in a curled up 'extra' dimension, you could still describe its movement from point A to B with two sets of three dimensional co-ordinates?

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bapowell
Thanks for your reply. I still have a problem understanding how a dimension, regardless of how compactified, still counts as an additional degree of freedom just because we can't enter it.
Perhaps it would help if you spelled out precisely what you mean by "degree of freedom" here. My point is that the relevant number of degrees of freedom of a theory is generally dependent on the energy scale.
Not sure if you know the other view on dimensions I was referring to in the OP, but this is the video: .
Yes, I commented on this view in the very beginning of my post, and indicated that it is not the correct way to view extra-dimensional space in string theory. See my original reply.

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Surely even with a thing moving in a curled up 'extra' dimension, you could still describe its movement from point A to B with two sets of three dimensional co-ordinates?
In a way yes. If I have 4 coordinates (x,y,z,w), I can fix w and say that for each fixed value of w I have a set of three dimensional coordinates. And if I just cared about two points I would need two sets of three dimensional coordinates with a notion of distance between them, which depends on the fourth dimension (is it compact or not). However, if I have a set of three dimensional coordinates parametrized by a fourth coordinate with a notion of distance, then I have really four coordinates.

You can think about many coordinate systems like this, but its not very practical.

Ok so are the extra 'dimensions' in string theory really curled up surfaces/manifolds that are added to make describing the strings behavour easier.... ie: it could be done with just four dimensions, but the mathematics would be un-nessicerally long winded..... ie: its far more efficient to add manifolds?

bapowell
Ok so are the extra 'dimensions' in string theory really curled up surfaces/manifolds that are added to make describing the strings behavour easier.... ie: it could be done with just four dimensions, but the mathematics would be un-nessicerally long winded..... ie: its far more efficient to add manifolds?
No, the extra dimensions are required for internal consistency of the theory, not for convenience. For example, certain anomalies only cancel, and some important symmetries are only retained, in specific numbers of dimensions.

No, the extra dimensions are required for internal consistency of the theory, not for convenience. For example, certain anomalies only cancel, and some important symmetries are only retained, in specific numbers of dimensions.
Topological consistency, right? String theory only retains conformal (a little trick where a complex function can take more than one 'path' and still be equivalent) invariance if it's formulated in 11 or 26 dimensions, right?

The curled dimensions are just like the normal dimensions. They increase the number of degrees of freedom of any particle namely by another position coordinate and momentum.

However, because they are so small, the position becomes irrelevant because of the uncertainity principle. The position of a particle is not definite - in fact the position is "smeared" all over the length of the new dimension.

On the other hand, the momentum in the directions of the curled dimensions is quantized. This happens for the same reason why bound electron momentum is quantized in atoms. The wavefunction interferes with itself since the new dimension is curled.

The new dimensions add new symmetries to Lagrangian. It is possible to add any gauge symmetry to the Lagrangian by postulating new dimensions. The exact group depends on the topology. For instance, the U(1) symmetry of electromagnetism can be explained by just one compactified dimension.

That said, in my opinion the string theory went too far and moved much beyond neccessary. It is no longer a simplification. Another dimensions are a nice idea, but the string theory abused it.

haushofer
Topological consistency, right? String theory only retains conformal (a little trick where a complex function can take more than one 'path' and still be equivalent) invariance if it's formulated in 11 or 26 dimensions, right?
I'm not sure what topology has to do with it.

In string theory you start off with the so called Nambu-Goto (NG) action, which means that you say that the dynamics of the string is obtained by minimalizing the surface it traverses in spacetime. This is inspired by e.g. the point particle, where the length of its spacetime path is minimalized.

However, the NG action is hard to quantize. But you can write down an action which gives, classically, the same dynamics: the Polyakov action. However, this is only true because of a certain symmetry of this Polyakov action, called Conformal Symmetry (after gauge fixing and all the technical details). The reason is that in the rewriting you use a trick by introducing an "auxiliary field" not present in the NG formulation, which can only be written away using the symmetries of the Polyakov action. This also only works for strings, hence one good reason for choosing strings instead of higher dimensional objects.

So far, so good. Untill you want to quantize it. Quantization threatens to spoil the conformal symmetry, which goes under the name of "anomaly". That would be a disaster: you wouldn't be talking about the original theory anymore, because without conformal symmetry the Polyakov formulation does not equal the NG formulation. This, as it turns out, puts a constraint on the number of spacetime dimensions.

In ten dimensions the anomaly is circumvented, and this is the reason why people call (super)string theory consistent in ten dimensions. It's not the only way out; e.g. in three spacetime dimensions you also circumvent the anomaly. But we only know how to compactify "redundant" dimensions, not the other way around. So the choice of three dimensions is barely mentioned in textbooks, I guess.

haushofer
The curled dimensions are just like the normal dimensions. They increase the number of degrees of freedom of any particle namely by another position coordinate and momentum.

...

That said, in my opinion the string theory went too far and moved much beyond neccessary. It is no longer a simplification. Another dimensions are a nice idea, but the string theory abused it.
It does not abuse it, it requires it. As such it became much more than merely "a nice idea".

MathematicalPhysicist
Gold Member
@haushofer, you seem to be quite versed in string theory, may I ask what your formal background?

You can reply to me via PM if it's too personal.

haushofer
I'm a PhD student doing most of my research in GR and Supergravity related topics. But string theory has some of my attention, partly because my group is also doing research in it. But mostly from a curious point of view ;)

However I have difficulty in understanding how this constitutes a "dimension" because dimensions allow additional degrees of freedom.
What makes you think curled up dimensions fail to provide additional degrees of freedom? They do. Let's call them 'constraints'. But you are not alone...see the comment from David Gross, below....we have a lot to learn.

Michio Kaku:
As a string moves in space time it executes a complicated set of motions....which require a string to obey a large set of self consistency conditions....they place extraordinary restrictions on space-time.
David Gross, one of the founders of Heterotic string theory looks at geometry (spacetime) creating matter:

To build matter itself from geometry-that is in a sense what string theory does. It can be thought of that way, especially in a theory like heterotic string which is inherently a theory of gravity in which the particles of matter as well as the other forces of nature emerge in the same way that gravity emerges from geometry.....superstring theory ultimately comes from a geometric principle, whose precise nature is still unknown./QUOTE]

Lee Smolin says in the trouble with Physics,, 2007,