Mathematical representation of higher dimensions

[Halitosis Crunch]

I'm rather new to physics in general, so bear with me in my potential ignorance.

Considering we have no idea of the absolute properties of higher dimensions, how is it that they're identified in equations? This especially perplexes me when thinking about the Kaluza-Klein theory, or even Superstring theory. How does one know how many dimensions necessary for the given forces to unify without falling on pure speculation?

Assuming it ultimately has to do with spatial limitations, and that in higher dimensions it naturally allows for unification, how do the mechanics of it actually work?

I suppose I enjoy the questions more than answers. Hopefully I'm making sense.

 

[selfAdjoint]

In string theory you start with any number of dimension (but it's all relativistic, so one of the dimensions is time). Then you work up your physics, and it turns out that the physics math will blow up unless you specify a certain number of dimensions. And that's how they get 26 dimensional spacetime for bosonic string theory and 10 dimensional for superstring theory, and 11-dimensional for M-theory. Different dimensions because somewhat different physics in each case.

In Kalusza-Klein theory they started with just one extra dimension past the four of normal spacetime. Then if you set up an Einstein-like action on this 5-dimensional manifold, the physics broke out so the 4-dimensional manifild had Einstein's GR and the extra dimension carried Maxwell's equations of electro magnetism.

 

[Halitosis Crunch]

I see, so the necessity for more dimensions is a direct result of the equations.

 

[selfAdjoint]

I see, so the necessity for more dimensions is a direct result of the equations.

Really of the physics described by the equations. The problem turns up in the math, but it's basically caused by the underlying physics.

 

[sol2]

I see, so the necessity for more dimensions is a direct result of the equations.

A good example, is the inclusion of GR. Although we have not satisfied the direct experiment verification of gravity waves, certain realizations of the Webber bar reveal something was happening, so they had to progress experimentally to LIGO?

 

[Halitosis Crunch]

In effect it's like expanding the box to make things fit in harmony, I suppose. I can see how the inclusion of GR in higher dimensions provides a remedy for the discrepancies between Relativity and Quantum Theory.