EM Connection GR Style

[Quetzalcoatl]

Hi All,

 

Sorry for being so lengthy, I was trying to be as precise as possible, and efficiently bunch up all of my questions on this topic in this one post. Most of this post is just a review. I am not a physicist, so if there are errors, please point them out so I can learn from them. My questions follow.

 

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[latex]\tilde{A}[/latex] is the EM potential one-form such that [latex]F_{\mu\nu}=(d\tilde{A})_{\mu\nu}[/latex] using the exterior derivative.

[latex]\vec{\Psi}[/latex] is a wave-function (wave-vector). It is a function of space-time coordinates, with its "vectorness" values being in fiber coordinates.

This distinction in coordinates means that [latex]\mu[/latex] and [latex]\nu[/latex] below are somewhat different.

 

In QED, this is the equation for the EM connection:

[latex](\nabla_\mu \vec{\Psi})^\nu=(\partial_\mu-ieA_\mu)\Psi^\nu[/latex]

 

Via some algebraic manipulations, we get:

[latex](\nabla_\mu \vec{\Psi})^\nu=\partial_\mu\Psi^\nu+\frac{eA_\mu}{i}\Psi^\nu=\partial_\mu\Psi^\nu+(\frac{e\tilde{A}}{i})_\mu\delta^\nu_\xi\Psi^\xi[/latex]

 

Now trying to put this into the form of a connection, in a General Relativitic fashion:

[latex](\nabla_\mu \vec{\Psi})^\nu=\partial_\mu\Psi^\nu+\Gamma^\nu_{\mu\xi}\Psi^\xi[/latex]

Where [latex]\Gamma[/latex] is the EM connection in this context and not gravity (i.e. not Levi-Civita).

 

Equating the two formulas, we get:

[latex]\Gamma^\nu_{\mu\xi}=(\frac{e\tilde{A}}{i})_\mu\delta^\nu_\xi[/latex]

 

This is different from the Levi-Civita connection, as it is not symmetric in space-time index [latex]\mu[/latex] and fiber index [latex]\xi[/latex].

 

My questions:

• Does the EM connection not being symmetric mean that this connection has torsion?
• If there is torsion, is that why we have [latex]\frac{1}{4}F_{\mu\nu}F^{\mu\nu}[/latex] in the QED Lagrangian?
• Does [latex]\Gamma^\nu_{\mu\xi}=(\frac{\partial{\vec{e}_\mu}}{\partial{x^\xi}})^\nu[/latex] still hold, for some vector/fiber field frame [latex]\vec{e}_\mu[/latex] analogously to how the Levi-Civita connection is defined in GR for gravity?
• Are these [latex]\vec{e}_\mu[/latex] some of the extra dimensions in M-Theory?
• Is there a "greater connection" that includes both the gravity connection and this EM connection? How would it handle the difference between the fiber dimensions and the space-time dimensions? Is this usually called [latex]\omega[/latex] by any chance?

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It is well known that [latex]\tilde{A}[/latex] has gauge freedom of [latex]\tilde{A}+df[/latex] where d is the exterior derivative and f is a scalar field. Defining [latex]f=\frac{1}{e}\beta[/latex], a change of gauge would look like:

[latex]\vec{\Psi}'=e^{i\beta}\vec{\Psi}[/latex]

[latex]\tilde{A}'=\tilde{A}+\frac{1}{e}d\beta[/latex]

Plugging [latex]\tilde{A}[/latex] into the EM covariant derivative [latex]\tilde{\nabla}[/latex] we get:

[latex]\tilde{\nabla}'=\tilde{\nabla}-id\beta[/latex]

 

This finally gives us (after some algebra):

[latex]\nabla'_\mu\vec{\Psi}'=...=e^{i\beta}\nabla_\mu\vec{\Psi}=(\nabla_\mu\vec{\Psi})'[/latex]

 

As I understand it, [latex]\beta[/latex] would be the [latex]U(1)[/latex] coordinate parameter for the EM gauge group.

• Is the fiber of the Levi-Civita connection (i.e. gravity) just the tangent vector space which is the 4D space-time? This would be analogous to the fiber of the EM connection being the Lie algebra [latex]\mathfrak{u}(1)[/latex], with the [latex]\beta[/latex] coordinate for the corresponding Lie group, [latex]U(1)[/latex], right?
• Do theories exist which treat space-time dimensions as being finite cyclic dimensions, like [latex]U(1)[/latex], [latex]SU(2)[/latex], etc? This way all dimensions, space-time dimensions (x, y, z, t) and QFT gauge dimensions (such as EM's [latex]\beta[/latex]), work the same way in principle.
• Is there a [latex]``\Gamma'_{Levi-Civita}=\Gamma_{Levi-Civita}+dG"[/latex] gauge transformation for the Levi-Civita connection? It seems like this should somehow be related to gauge freedom of the Lorentz group on space-time and space-time tensors.
• If true, would this mean that the space-time dimensions (x, y, z, t) are merely the gravity potential's (i.e. the Levi-Civita connection's) gauge freedoms, in a way analogous to the EM potential's ([latex]\tilde{A}[/latex]) gauge freedom, the [latex]\beta[/latex] coordinate?

If you read thus far, thanks!

[Mordred]

You've got so many questions that it would be difficult to answer directly.

 

An article that may help is here as it deals with fiber bundles with the Levi ceveti connection and U(1)

 

http://www.google.ca/url?sa=t&source=web&cd=5&rct=j&q=fiber%20bundles%20and%20quantum%20theory&ved=0ahUKEwjB2sv9i5LKAhUE6GMKHVRWAw4QFggzMAQ&url=http%3A%2F%2Farxiv.org%2Fpdf%2Fphysics%2F0005051&usg=AFQjCNFXLchCy1CAWURhJ-kdCxG9FabipA&sig2=b7whyKZ3vSmAZIbYSo9qyA

 

Some of the notation above seems off but I can't put my finger on it.

[ajb]

Does the EM connection not being symmetric mean that this connection has torsion?

You cannot define torsion for connections on a general fibre bundle, so no.

 

You have a meaningful notion of torsion for tangent bundle connections and connections on Lie algebroids and some generalisations thereof.

 

Does [latex]\Gamma^\nu_{\mu\xi}=(\frac{\partial{\vec{e}_\mu}}{\partial{x^\xi}})^\nu[/latex] still hold, for some vector/fiber field frame [latex]\vec{e}_\mu[/latex] analogously to how the Levi-Civita connection is defined in GR for gravity?

You have used a 'frame field' (as you put it) and these are specifically tied to the tangent bundle. You are basically picking a non-coordinate basis. You should look up the Cartan formalism.

 

You can do something similar on a Lie algebroid or any anchored vector bundle.

 

 

Are these [latex]\vec{e}_\mu[/latex] some of the extra dimensions in M-Theory?

No, but they are needed to define spinors on a manifold and so do play an important role in, for example, supergravity.

 

 

Is there a "greater connection" that includes both the gravity connection and this EM connection? How would it handle the difference between the fiber dimensions and the space-time dimensions? Is this usually called [latex]\omega[/latex] by any chance?

Look up KK theory. This does not work properly, but it gives you the idea that it may be possible to do what you have said.

 

 

Is the fiber of the Levi-Civita connection (i.e. gravity) just the tangent vector space which is the 4D space-time? This would be analogous to the fiber of the EM connection being the Lie algebra [latex]\mathfrak{u}(1)[/latex], with the [latex]\beta[/latex] coordinate for the corresponding Lie group, [latex]U(1)[/latex], right?

Levi-Civita connections are always to do with the tangent bundle (or sometimes more exotically a Lie algebroid).

 

Is there a [latex]``\Gamma'_{Levi-Civita}=\Gamma_{Levi-Civita}+dG"[/latex] gauge transformation for the Levi-Civita connection? It seems like this should somehow be related to gauge freedom of the Lorentz group on space-time and space-time tensors.

There are ways of thinking about 'frame fields' and so on in a language more like gauge theory. You can look at gauge transformations of the frame fields on the tangent bundle for example.

 

 

I hope I have pointed you in the right direction with some of your questions. They seem more advanced that we usually see here.

[Quetzalcoatl]

Mordred,
Thanks for the link. I'm reading it now, and it looks promising!

ajb,
Thank you so much for the quick answers. I've looked at KK in the past, but not in any detail. I'll also review Cartan formalism as you suggest.

I hope I have pointed you in the right direction with some of your questions. They seem more advanced that we usually see here.

 

How much time does it take physicists to learn all this stuff in academia???

[Klaynos]

How much time does it take physicists to learn all this stuff in academia???

Around 4 to 10 years of post 18 education. Although you could also argue that we carry on trying to understand the ideas in our own areas of expertise so a lifetime?

[ajb]

How much time does it take physicists to learn all this stuff in academia???

Like Klaynos says, 3-4 years undergraduate study, 1 year Masters and 3-4 years PhD and then a lifetime to understand it properly!

 

I think with some hard reading, you could grasp the basic of differential geometry in gauge theory within a year. This depends on where you are starting of course.