Path Integrals in finite dimensions

• Hymne
In summary, a physicist recovering from an illness is seeking clarification on the math behind path integrals from their supervisor. They are struggling with understanding the equations provided in a PDF and are asking for help with the comments and any mistakes made with subscripts and superscripts. They also express difficulty in explaining their thoughts clearly due to their current state.

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:

(The math writing on this page did not work to well for me. )

All the best!
// 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.

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1. What are path integrals in finite dimensions?

Path integrals in finite dimensions are mathematical tools used to calculate the probability of a particle or system moving from one position to another in a fixed amount of time. It takes into account all possible paths the particle or system could take, rather than just the most likely one.

2. How are path integrals used in physics?

Path integrals are used in physics to study quantum systems, such as particles and fields. They allow for a more intuitive and visual understanding of these systems by considering all possible paths instead of just the most probable one. They are also used in statistical mechanics to calculate thermodynamic properties of a system.

3. What is the main difference between path integrals in finite dimensions and infinite dimensions?

The main difference between path integrals in finite dimensions and infinite dimensions is the number of possible paths that are considered. In finite dimensions, only a finite number of paths are considered, while in infinite dimensions, an infinite number of paths are taken into account.

4. Can path integrals be used in classical mechanics?

Yes, path integrals can be used in classical mechanics to calculate the most probable path of a particle or system moving from one position to another. However, they are more commonly used in quantum mechanics due to the probabilistic nature of quantum systems.

5. Are path integrals difficult to calculate?

The complexity of calculating path integrals depends on the specific problem at hand. In some cases, they can be relatively straightforward to calculate, while in others, they may require advanced mathematical techniques. However, with practice and a good understanding of the underlying principles, path integrals can be calculated effectively.