How can a dimension be "curled up" and have a finite extent?

[Cody Richeson]

TL;DR Summary

The first three spatial dimensions are described as straight lines extending infinitely at perpendicular angles, but additional spatial dimensions are described as being microscopic. Why would there be a limit to their extent?

As I understand it, dimension is a way of describing direction, with the first three spatial dimensions being straight lines which extend infinitely in one direction, perpendicular to each other. In string theories, several additional dimensions are required, sometimes up to nine or 10, I believe. These dimensions are sometimes described as being "curled up" into microscopic spaces. I find this a bit confusing, because if the first three are straight lines extending forever in a given direction, why would there be dimensions which can only exist in very small spaces? I know that I'm probably misinterpreting things here, but I would like to understand whether these directions are also infinite straight lines. If not, what are they, and what does it mean that they might be confined to atomic scale spaces?

 

[phinds]

So far, string theory is an elegant mathematical construct that is not known to bear any relationship to reality.

 

[Demystifier]

I find this a bit confusing, because if the first three are straight lines extending forever in a given direction, why would there be dimensions which can only exist in very small spaces?

In principle, any of the dimensions could be straight or curled, string theory contains solutions of all kinds. But the ultimate goal is to find a solution that describes our own universe, which clearly has precisely 3 straight spatial dimensions.

 

[AdirianSoan]

"Straight" isn't quite accurate; curvature. And singularities can be considered as local cases of the dimensions curled in on themselves.

And 3 isn't quite accurate, either, as the SM has five dimensions; three spacial, one time, and one closed dimension (Kaluza-Klein).

You don't strictly need either time or Kaluza-Klein, but if your goal is minimizing the number of dimensions, you don't really need any of them. The important thing isn't really the number of dimensions, or their shapes, but how simple the model is to understand and work with. The important takeaway from the holographic principle is that the number of dimensions in a model doesn't, on it's own, affect whether or not that model is mathematically isomorphic with another model.

 

[Stephen Tashi]

As I understand it, dimension is a way of describing direction

If you want to think of it that way, how many dimensions are needed to describe the directions associated with an airplane? One three dimensional vector can describe the velocity of the center of mass at time t. However that doesn't answer the question of the orientation of the body of the plane. If you describe the orientation of the wings with respect to the velocity, you can do this using angles. An angle need not be represented by a coordinate that ranges over the entire number line.

 

[ohwilleke]

the SM has five dimensions; three spacial, one time, and one closed dimension (Kaluza-Klein).

The notion that the SM has a closed Kaluza-Klein dimension is an assertion I have never seen made before. Do you have any sources for that assertion?

 

[AdirianSoan]

The notion that the SM has a close Kaluza-Klein dimension is an assertion I have never seen made before. Do you have any sources for that assertion?

I don't. An expert correspondent made the assertion a while ago and I didn't find it surprising enough to check, so I just assumed they were correct.

In retrospect, thinking about it, I guess I should have found it surprising.

I withdraw the assertion; it isn't particularly noteworthy from my perspective, and the comment after mine makes a better point with regard to Kaluza-Klein anyways.

 

[arivero]

Not the SM, only the SM gauge group. And not with only one extra dimension, as the original KK model with U(1) group. In fact U(1) is misleading too because in this case the group and the extra dimension are the same, while in general case the extra dimensional manifold is not the group, but any manifold where the group can operate via isometries. For example, with a sphere of dimension n, the group will be SO(n+1).

To get the SM gauge group, one needs at least seven extra dimensions, and in fact there is not a unique way to do it, the trick can be seen as quotienting down from eight extra dimensions, the product of a 5-sphere times a 3-sphere. Such manifold has isometry group SO(6) times SO(4), which at the lie algebra level is same that su(4) + su(2)+su(2), so it is in some way logical that breaking down one dimension the group is the one of the SM. The source for this is a paper of Witten in 1981, "Realistic Kaluza Klein Theories". It sparked some interest for a few years, until people become convinced that it was not possible to find the fermions with the SM representations and the SM values of hypercharge. Not sure if they looked under all the corners, but at least there was a lot of important people looking at it.

 

[bhobba]

As I understand it, dimension is a way of describing direction

Not quite. I think you would benefit from studying linear algebra then tensor calculus and its application to curved manifolds.

Thanks
Bill