Understanding Abelian Groups, QLG, and Fiber Bundles in String Theory

[Phred101.2]

Can anyone tell me what type of string theory QLG is? Or explain
fiber bundles and (non)abelian groups? This isn't homework, or anything btw.

 

[Coin]

I don't know what "QLG" is. If you're asking about "LQG" there's several threads trying to describe it in the "beyond the standard model" forum.

"Nonabelian groups" just means a group where the commutative property does not hold. So if you have a nonabelian group, and it has members A and B, it might be the case that A*B is not equal to B*A.

I personally can't describe a "Fiber bundle" very well, but generally the idea is that you take two spaces, and you "paste" one of them onto every point of the other. For example, if you have a circle, and you also have a line, you could "paste" the line onto the circle in such a way that the start of the line appears at every point of the circle. The result would be a cylinder. The line, the circle, and the method by which you decided to "paste" the one to the other (all of which together describe this cylinder) would together make up a "fiber bundle". (I hope I got that right.)

There are a lot of good descriptions of this kind of thing at wikipedia.org and http://mathworld.wolfram.com.

 

[Phred101.2]

Pasting is otherwise called mapping, right? And that was a typo, mate.

 

[Jim Kata]

Well, loop quantum gravity is not a string theory. There's a huge difference between the two. Namely, string works with a fixed background and LQG is a dynamic background theory. In laymans talk that means that the background has been quantized in LQC. In doing such gravity can be treated as a fundamental and not an effective theory.

 

[Jim Kata]

I must apologize to physics forums I drink a lot

I'll try to answer your question about fiber bundles and non-abelian gauge theories in one fatal swoop, in a physics way, not to mathy.

Ok, you have a manifold, $$M$$. At each point on the manifold you can create a vector space of all vectors tangent to the manifold at that point. This is called the tangent vector space at that point, $$TM_x$$. Now you can think of a disjoint union of all the tangent spaces at every point on the manifold and this is called the tangent bundle $$TM = \coprod\limits_{x \in M} {TM_x }$$. Similarly, the cotangent bundle is the disjoint union of all the, orthogonal vectors, one forms to each point on the manifold $$T^* M = \coprod\limits_{x \in M} {T^* M_x }$$. The important part to a physicist is that the directional derivative of tensors changes from point to point on a manifold. The connection $$A$$ or in relativity $$\Gamma _{\beta \gamma }^\alpha$$ tells you how the directional derivative changes from point to point on a manifold. This is where the concept of a bundle comes in. So what you get is an exterior derivative. For Yang Mills, non abelian gauge theories, this exterior derivative is $$D_\mu = \partial _\mu - iA_\mu ^\beta (x)t_\beta$$ where $$t_\beta$$ are the generators of semi simple lie algebras. Now, looking at the holonomies, parallel transports, of the exterior derivative you get $$P\exp \left({i\oint\limits_C{d{\mathbf{x}}{\mathbf{A}}(x)} } \right)$$. It should be noted that this is very similar to looking at the fundamental groups in topology. Except in that case, you let your loops get contracted to the base point. Now expanding $$P\exp \left({i\oint\limits_C{d{\mathbf{x}}{\mathbf{A}}(x)} } \right)$$ you get $$P\exp \left( {i\oint\limits_C {d{\mathbf{x}}{\mathbf{A}}(x)} } \right) = e + \frac{1} {2}\iint {dx^\mu } \wedge dx^\tau \left( {\partial _\mu {\mathbf{A}}_\tau - \partial _\tau {\mathbf{A}}_\mu - [{\mathbf{A}}_\mu ,{\mathbf{{\rm A}}}_\tau ]} \right) + \cdots$$
. Now the curvature term is $${\mathbf{F}}_{\mu \tau } = \partial _\mu {\mathbf{A}}_\tau - \partial _\tau {\mathbf{A}}_\mu - [{\mathbf{A}}_\mu ,{\mathbf{A}}_\tau ]$$. What they mean by non-abelian is that $$[{\mathbf{A}}_\mu ,{\mathbf{A}}_\tau ] \ne 0$$

 

[Phred101.2]

Thanks I'll try to get this.