Understanding the Classical Euclidean Action in Quantum Field Theory

[spaghetti3451]

This is the Euclidean classical action $S_{cl}[\phi]=\int d^{4}x\ (\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi))$.

It would be nice if somebody could explain the structure of the potential.

I don't understand why $\phi$ is used instead of a position vector $\textbf{r}$. Also, how can $(\frac{1}{2}(\partial_{\mu}\phi)^{2}$ be interpreted as the kinetic energy of the particle? I have integrated the Lagrangian over three spatial coordinates before, but why can the temporal coordinate be integrated over in this expression?

 

[PeterDonis]

This is the Euclidean classical action

For what? We need some more context to understand what you are trying to do here.

It would be nice if somebody could explain the structure of the potential.

To do that we would need more information than just $U(\phi)$. $U$ could be anything.

 

[Orodruin]

I don't understand why $\phi$ is used instead of a position vector $\textbf{r}$. Also, how can $(\frac{1}{2}(\partial_{\mu}\phi)^{2}$ be interpreted as the kinetic energy of the particle? I have integrated the Lagrangian over three spatial coordinates before, but why can the temporal coordinate be integrated over in this expression?

Apart from what Peter said, in what you have written down, $\phi$ is a scalar field and $S$ is the action of that field. The field takes a value in each point in space so this is the basics of field theory. If you instead had a single classical particle moving, you would have a different action and the space integral would not be there. You would instead have a time integral only and some function of the particle coordinates and velocity.

 

[spaghetti3451]

I am new to this kind of integral and the so-called Euclidean classical action. I was wondering what branch of physics I should learn about to become familiar with this concept and any textbooks or online resources (lecture notes, videos, etc.) you might suggest for that purpose.

 

[PeterDonis]

I am new to this kind of integral and the so-called Euclidean classical action.

Where did you encounter it? It's the sort of thing I would expect to see in quantum field theory.

 

[spaghetti3451]

I read it in a paper, actually.

I have seen classical field theories being covered in standard quantum field theory courses, so I guess I'll have to learn that topic to become familiar with what's being discussed.

 

[PeterDonis]

I read it in a paper, actually.

Can you give a link?

I have seen classical field theories being covered in standard quantum field theory courses

Yes, that is usually done, since classical field theories provide the Lagrangians for most quantum field theories.

 

[spaghetti3451]

Here's the link: http://arxiv.org/abs/hep-th/0511156

It would be really helpful if you could provide some reading materials for me to fully understand the Section I of the paper.

My background is that I am a fourth-year undergraduate student, and I have done courses only in Quantum Mechanics (Griffiths), Classical Mechanics (Marion and Thornton), Statistical Mechanics (Blundell). I have not studied Classical Mechanics (Goldstein) or Quantum Field Theory (Peskin and Schroeder).

 

[PeterDonis]

I have not studied ... Quantum Field Theory (Peskin and Schroeder).

Then I would definitely study it; it looks like that's the main background you need.