Geometric Description of Free EM Field

[dextercioby]

I'm not a geometer, so I beg for indulgence on the below:

In a modern geometrical description of electromagnetism (either in flat or in curved space-time*), I see at least 3 (or 4) (fiber) bundles over the 4D space-time taken to be the base space:

* 1 the cotangent bundle and the bundle of p-forms over space time (de Rham complex). Differential operator: d Here the EM field appears as: the potential is a 1 form field, the field strength is a 2 form field. F=dA.

* 2 the (principal?)$SO_{\uparrow}(1,3)$ bundle of tensors over space time. Here if the group is replaced by SL(2,C), we get the spinor bundle over space time, a concept described in the 13th chapter of Wald's book - in the context of a curved space time. Here I imagine A and F as covariant and double covariant tensors. Differential operator = $\partial_{\mu}$ or $\nabla_{\mu}$.

* 3 the so-called U(1) gauge bundle over space time (principal/associated bundle ?) , with F and A having particular descriptions. Here we have another differential operator d-bar, such as F = d-bar A.

Questions:

a) How are 1,2,3 exactly related ?
b) Is d-bar at 3 related to d at 1 ? I suspect yes. How can it be proven ? The operator at 2, how's it related to d-bar at 3 or to d at 1 ?

Thank you!

 

[fzero]

2 is the frame bundle of 1. As the discussion there describes, the bundle of forms is an example of an vector bundle associated to a principal G-bundle. As for 3, any G-bundle has associated vector bundles associated with the representations of G, see here. We could more accurately claim that the gauge fields $A_\mu$ is a section of a bundle which is itself a product of a principal $U(1)$ bundle with $T^*(X)$. While it's a bit transparent here, we can say that $T^*(M)$ is the associated vector bundle for the trivial $U(1)$ representation.

 

[dextercioby]

Hi Fzero, thank you for the answer. One more question, though. Wiki is fine for definitions (as any encyclopedia), but can't be used as a textbook. So which sources (as in books) did you use (in college or in research/job after college) to acquire in your mind a valid geometric description of field theory ?

And d-bar of mine above is the same as d in 1, thus the differentiation is unique ?

 

[fzero]

Hi Fzero, thank you for the answer. One more question, though. Wiki is fine for definitions (as any encyclopedia), but can't be used as a textbook. So which sources (as in books) did you use (in college or in research/job after college) to acquire in your mind a valid geometric description of field theory ?

Nakahara, Geometry, Topology, and Physics is decent. If you want something a bit more rigorous, Choquet-Bruhat and DeWitt-Morette, Analysis, Manifolds and Physics is a 2-vol set, though I think the basics are covered in the first volume. AMP has quite a bit more functional analysis than a typical geometry book, since part of their focus was to put various physics methods in a rigorous mathematical framework. If you want a reference tailored to the mathematics alone, Kobayashi and Nomizu, Foundations of Differential Geometry is pretty accessible, if I recall correctly.

And d-bar of mine above is the same as d in 1, thus the differentiation is unique ?

We can be more specific. If $G$ were non-Abelian, then the gauge fields are $\mathrm{ad}~G$ valued 1-forms, so we can say that they live in $\omega^1(X, \mathrm{ad}~G)$. There is a covariant exterior derivative $d_A$ on this space that extends from the ordinary exterior derivative on $\omega^1(X)$. In your case the $U(1)$ bundle has a trivial connection so we just have the ordinary derivative.

 

[dextercioby]

Thank you for the reccomendations, I'll get hold of them. One more question. There's a 4th bundle in classical field theory: the jet bundle. Where do they come in and how are they related to 1->3 ?

Thanks!

 

[fzero]

Thank you for the reccomendations, I'll get hold of them. One more question. There's a 4th bundle in classical field theory: the jet bundle. Where do they come in and how are they related to 1->3 ?

Thanks!

I've heard the term (probably in the context of geometric quantization), but I'm not really familiar with jet bundles. What I can gather from the wiki description is that, given a bundle $E\rightarrow X$ with some sections $\sigma$, we can construct jet bundles in such a way that the sections of the jet bundle are the vectors $(\sigma, \partial^1 \sigma, \ldots, \partial^r \sigma)$, where $\partial^i \sigma$ is the ith derivative with respect to the coordinates on $X$.

There's an example on the wiki where they explain that the 1st jet bundle is diffeomorphic to $\mathbb{R} \times T^*(X)$. The first term corresponds to the function, while the 1-form corresponds to its derivative. I would say that the function should more properly be thought of as a section of an $\mathbb{R}$ bundle over $X$, but since this is trivial, I suppose that the claimed product structure is correct.

It seems that this structure is useful to some people because the data is precisely what you want to define a Lagrangian functional. I've just never had a reason to learn about any of this.