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.