Path Integrals in finite dimensions

[Hymne]

Hello fellow physicists!
Last meeting with my supervisor I had just recovered from disease so all I have left are some equations for the math behind path integrals that don't make to much sense..
I was wondering if, maybe someone can help and clarify what he was trying to get at. It would be really appricieated! The change of basis and diagonalization is no trouble, but need some help with comments on the rest.
The equations are in the following pdf:

http://www.2shared.com/document/Y-JaDQoN/mall-okommenterad.html
(The math writing on this page did not work to well for me. )

All the best!
// Hymne

 

[Hymne]

I'm sorry for the crapy english and bad formulation - I wrote quite much for to long so when I tried to post every thing disappeared.
Anyway I think there might be some mistakes done with subscript and superscript.
If you can just write what is done in "equation #" and maybe a summary of the whole connection with path integrals, I would be really happy.

It feels like the more I write for the moment, the worse it gets.