# Classical Euclidean Action

1. Mar 17, 2015

### 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?

2. Mar 17, 2015

### Staff: Mentor

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

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

3. Mar 17, 2015

### Orodruin

Staff Emeritus
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.

4. Mar 17, 2015

### 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.

5. Mar 17, 2015

### Staff: Mentor

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

6. Mar 17, 2015

### 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.

7. Mar 17, 2015

### Staff: Mentor

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

8. May 23, 2015

### spaghetti3451

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).

9. May 23, 2015

### Staff: Mentor

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