Path ordered integral over simplices?

[ianhoolihan]

Hi all,

I am trying to follow the derivation by Carroll of the parallel propagator in http://preposterousuniverse.com/grnotes/grnotes-three.pdf notes, beginning page 66 or so.

My question is with the integral in equation 3.40.

1) why is it that this is an integral over a simplice, and not an n-cube (which the limits of integration seem to suggest)?

2) why is it that $\eta_n \geq \eta_{n-1}\geq \ldots \geq \eta_1$ so that path ordering must take place? (See between equations 3.40 and 3.41 for discussion.)

Cheers,

Ianhoolihan

 

[chiro]

Hey ianhoolihan.

I can't answer your question specifically, but one thing that you may want to think about is the actual final map that your products of your individual A maps would do to your integral for each simplex.

My guess is that the orientation would be screwed up completely for one and this would affect the integral in a bad way and not give the right result.

Just out of curiosity, what does your linear map A represent?

 

[ianhoolihan]

I can't answer your question specifically, but one thing that you may want to think about is the actual final map that your products of your individual A maps would do to your integral for each simplex.

My guess is that the orientation would be screwed up completely for one and this would affect the integral in a bad way and not give the right result.

Just out of curiosity, what does your linear map A represent?

Hmmm, the A maps are (I'm guessing here) an infinitesimal "parallel propagation" in the direction tangent to the curve $\gamma$. As for the product of the maps, I'm not sure.

I wonder if the reason for $\eta_n \geq \eta_{n-1}\geq \ldots \geq \eta_1$ is to do with the iteration process...actually, I think that might be it. For those not looking at the pdf, the solution for the parallel propagator is

$${P^\mu}_\rho (\lambda,\lambda_0) = {\delta^\mu}_\rho + \int^\lambda_{\lambda_0} {A^\mu}_\sigma(\eta) {P^\sigma}_\rho(\eta,\lambda_0) d\eta$$

So solving by iteration,

$${P^\mu}_\rho (\lambda,\lambda_0) = {\delta^\mu}_\rho +\int^\lambda_{\lambda_0} {A^\mu}_\rho(\eta)d \eta+ \int^\lambda_{\lambda_0} \int^\eta_{\lambda_0} {A^\mu}_\sigma(\eta){A^\sigma}_\rho(\eta')d\eta d\eta' + \ldots$$

The point being that, in the first equation it is ${P^\sigma}_\rho(\eta,\lambda_0)$, so the next substitution must only range from $\eta$ to $\lambda_0$, if you understand the abuse of language.

Yup, I think that works.

Now, for the simplices...oh, maybe it follows from $\eta_n \geq \eta_{n-1}\geq \ldots \geq \eta_1$ quite obviously. For example, in the second term, the integral is over both $\eta,\eta'$ such that $\eta' \leq \eta$. Hence, simplices.

Thank you.

 

[chiro]

I don't think I did much to warrant a thank-you, but I'm glad you got it in the end.

 

[blue_raver22]

Hello Ianhoolihan,

Thank you for your question. The path ordered integral over simplices is a mathematical tool used in the study of differential geometry and general relativity. It is a way of integrating a function over a path in a curved space, taking into account the curvature of the space.

To answer your first question, a simplex is a geometric shape that is higher dimensional than a line segment, but lower dimensional than a cube. It is essentially a higher dimensional triangle, and is often used in the study of differential geometry. In this context, the integral in equation 3.40 is over a simplex because the path being integrated over is a curved path in a higher dimensional space.

To address your second question, the \eta_n terms represent the tangent vectors to the path being integrated over. In order for the integral to be well-defined, the tangent vectors must be ordered in a specific way, with each successive vector being less than or equal in magnitude to the previous one. This is why the path ordering is necessary, as it ensures that the integral is taken in a consistent and well-defined manner.

I hope this helps to clarify the concepts behind the path ordered integral over simplices. If you have any further questions, please feel free to ask. Good luck with your studies!

Best regards,
(Scientist)