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Classical Euclidean Action

  1. Mar 17, 2015 #1
    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?
     
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  3. Mar 17, 2015 #2

    PeterDonis

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    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.
     
  4. Mar 17, 2015 #3

    Orodruin

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    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.
     
  5. Mar 17, 2015 #4
    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.
     
  6. Mar 17, 2015 #5

    PeterDonis

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    Where did you encounter it? It's the sort of thing I would expect to see in quantum field theory.
     
  7. Mar 17, 2015 #6
    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.
     
  8. Mar 17, 2015 #7

    PeterDonis

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    Can you give a link?

    Yes, that is usually done, since classical field theories provide the Lagrangians for most quantum field theories.
     
  9. May 23, 2015 #8
    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).
     
  10. May 23, 2015 #9

    PeterDonis

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