Extra dimensions and fibre bundles

[Micha]

Hi, this is my first post. Congratulations to a very interesting forum.

The issue, I want to bring up, has probably been discussed already, but hopefully at least not in exactly the same way. I have read the book "Road to Reality" from Roger Penrose, which by the way I found quite interesting, although a little straining , because it was sometimes too hard and sometimes too easy for me. The view of Penrose seems to be, that everything in physics is geometry, which may or may not be true.
Anyway coming to my point, Penrose argues against extra dimensions, that they would be unstable on a classical level and he puts forward reasons, why the classical level and not the quantum level applies to this problem. In other parts of the book, Penrose introduces the concept of fibre bundles and explains its success in the context of gauge theory. Now my question is, how is a fibre bundle exactly different from a uniform and stable extra dimension? Isn't a fibre bundle just an extra dimension, which is uniform and stable by definition? I would guess, that the difference is, that a fibre bundle would be regarded as a purely mathematical concept and not as a physical reality like an extra dimension. But can you already draw a clear line between the two concepts on a purely mathematical level?

 

[garrett]

Yes, you are right, a fiber bundle is an extra-dimensional geometry. The compactified manifold at each spacetime point in the Kaluza-Klein framework is the fiber in the fiber bundle framework.

The difference is in the Lagrangian. In KK the Lagrangian is taken to be the curvature scalar, while for a fiber bundle the Lagrangian is quadratic in the curvature. The metric plays different roles in each framework. But we can play with mixing them up.

The KK framework adds a piece to the metric, or rather to the frame, which turns out to act like a fiber bundle gauge field. Similarly, you can add a piece to your fiber bundle gauge field that acts like a frame, making it a Cartan geometry.

I've been playing around with a lot of this stuff on my wiki. You might find it interesting to poke around. Here's a good place to start:

http://deferentialgeometry.org/#[[Ehresmann principal bundle connection]]

 

[Micha]

Thank you for your answer.

I have once seen the KK derivation, where starting from the curvature as the Langrangian in five dimensions (4 space + 1 time) and assuming, that one dimension is curled up, general relativitiy in four dimensions plus electromagnetism were derived. I find it tempting to believe, that this "means" something, but it seems, many very smart people including Einstein could not solve this riddle so far.

I am not sure, however, how a quadratic Langrangian in the fibre bundle case would look like. Is the Lagrangian somehow split from the beginning into two terms, on for the base manifold and one for the fibre? (Just taking the square of the curvature as the Langrangian should give the same solution I think.)

Your wiki is a little over my head at the moment. But maybe it will be useful for me later, thanks.

 

[garrett]

I have once seen the KK derivation, where starting from the curvature as the Langrangian in five dimensions (4 space + 1 time) and assuming, that one dimension is curled up, general relativitiy in four dimensions plus electromagnetism were derived. I find it tempting to believe, that this "means" something, but it seems, many very smart people including Einstein could not solve this riddle so far.

The mathematics of KK is pretty straightforward. Let's see, I wrote up an intro a while ago... here:

http://sifter.org/~aglisi/Physics/KKnewnew.pdf

And it works for all gauge fields, including the electro-weak and strong, by including more compact dimensions with the right shape. The two problems with KK are:
1) The compact dimensions produce harmonics -- giving infinite "tower"s of massive particles, which we don't see.
2) It's hard to get chiral symmetry breaking spinors to work correctly.

I am not sure, however, how a quadratic Langrangian in the fibre bundle case would look like. Is the Lagrangian somehow split from the beginning into two terms, on for the base manifold and one for the fibre? (Just taking the square of the curvature as the Langrangian should give the same solution I think.)

Sort of. Traditionally one writes the gauge field Lagrangian over the base manifold as
$$L = \frac{1}{4} tr( F_{i j} F^{ij} )$$
with $F_{ij}$ the gauge field curvature -- a Lie algebra valued 2-form. This Lagrangian is called "quadratic" because it's (heuristically) $F^2$. But, if you work in the total space of the fiber bundle, the Ehresmann connection and curvature are well defined over this space, and pull back along sections to give the familiar A and F over the base. A different choice of section produces a gauge transformation. Working in the total space like this is just like doing KK.

Your wiki is a little over my head at the moment. But maybe it will be useful for me later, thanks.

Sure -- the wiki is more for me than anyone else. Just a way to organize my understanding, and see how everything interconnects. I consider Penrose's excellent book to be a minimum prerequisite to understanding anything in it -- but I've gone off on quite a few tangents that would be hard to grok without other background.

 

[Micha]

Very interesting.
Your reply made me read the relevant passages in Penrose's book again
and I noticed, I should have known some of the stuff. :-)
I found there the formula you gave for the Lagrangian of Electromagnetism and also the your statement, that the field strenght tensor Fij can be regarded as the curvature tensor of the gauge field. In the mathematical part of the book, Penrose says, that this gauge curvature tensor has really four indices, but two go into the fibre direction and are usually suppressed. This makes the analogy to the Riemann curvature tensor of general relativity very close, I suppose.
It is intersting to speculate then, what the gauge concept means then. I just read "The life of the cosmos" by Lee Smolin, where he says, that the gauge principle, means, that different particles are just different aspects of the same particle, because their properties can be rotated into each other via gauge transformations. But it also seems from what has been said here, that the gauge principle has a deep connection to geometry.

 

[Micha]

Does anybody know, how loop quantum gravity deals with gauge invariance? Does the gauge principle have any geometrical meaning in this theory (or any other intuitive meaning)?