Can Compact Dimensions be Explained in Terms of Rudimentary Field Theory?

[Sojourner01]

I'm an undergrad just getting into lagrangian fields and symmetry, having my first advanced particle physics lecture tomorrow. I've read around the subject of high energy physics to some degree, but I'm very limited in what I can understand, largely due to the mathematical particulars that I haven't encountered before. I'd like to be able to visualise - or at least explain in a more common-sense way - what compact dimensions represent, why they're required, how they're supposed to come about, and so on. As mentioned, I don't have the mathematical background to understand the jargon - I can get by with being shown an overview of the mathematics involved, however. I find the common pop-science explanation of string theory and ephemera largely useless since they don't give any indication of how one might solve problems.

So - is there a reasonable ontological explanation of compact dimensions in terms of what one might be expected to know of rudimentary field theory?

 

[yanniru]

I may be able to help you visualize the compactified dimensions. In my opinion, the most important aspect of compactified space is that it still exists as a discrete subspace throughout all space. Depending on how the original dimension broke up into pieces of space, that subspace could be a uniform array of CY manifolds, or if the symmetry break was random, the array spacing could be random.

I like to think of the 9 space dimensions before symmetry broke as 3 dimensions aligned in each direction of 3D space. Then for one of the three to inflate, the other two had to deflate or compactify. I then imagine that in each direction, one dimension curls up say from east to west whereas the other curls up from west to east. So if they end up like tiny loops with spin, then the two loops have a spin and an equal and opposite anti-spin.

I presume that at any discrete point in the array, the dimensions curl up in orthagonol planes resulting in a spin vector in all 6 directions of 3d space, but the net spin at each point would be zero. However, it appears that any spin would be allowed at each point provided it is balanced by the anti-spin.

I am not learned in string theory and I wonder if this picture makes any sense to the learned ones on this forum?

 

[Sojourner01]

Well thanks for the effort, but the problem remains: what are these dimensions "curled up" into?

In any case, I'm not so much into visualisation - I can kind of do that as much as is possible for a person to extend their comprehension into more dimensions than we perceive directly. This, however, still doesn't help me solve any problems. Some sort of statement along the lines of "this so and so must have n degrees of freedom to be consistent and thus can't fit into three/four-dimensional space" would be satisfactory - some sort of logical proof that so many dimensions are required, from the point of view of mathematics I can understand.