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Let's consider a 2D electromagnetic field defined in a square domain $[0,\Lambda]^2$, with periodic boundary conditions, with a constant charge distribution, uniform all over the aforementioned domain:

$$ \mathscr{L}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + j_\mu A^\mu$$ $$ j_\mu=(0,k,k) \hspace{26pt} k \in \mathbb{R} $$

I have to calculate the contribution to the energy of the system given by the charge distribution, in the limit of small $k$.

I can't figure out how to write down the energy for the charge distribution in terms of $k$. I always end up with formulas involving the stress-energy tensor, which in turns depends upon $F_{\mu \nu}$ which in turn depends upon the actual dynamics of the system. My intuition, along with analogies with some similar problems tells me that a simple (maybe approximate) elegant answer should be given in terms of $k^2$.

I have tried writing down the partition function for the system in the path integral formalism, performing the gaussian integral and evaluating the term $j_\mu (K^{-1})^{\mu \nu} j_\nu$, but that does not seem to help.

For more fun one can also consider the case where the photon has mass: $\mathscr{L}'=\mathscr{L}+m A_\mu A^\mu$ in the limit of $m \approx \Lambda$, so that Yukawa potential of the interaction for the gapful photon is still quite similar to the Coulomb one.

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OK, so what are you actually asking? E.g, what concept did you get stuck on? (Also, is this a homework or self-study problem? It reads like one... not that that's a problem, it's just that perhaps we should put the homework tag on it) –  David Z Apr 15 '12 at 22:31
    
David: well, my problem is that I can't write down the energy for the charge distribution in terms of $k$. I always end up with formulas involving the stress-energy tensor, which in turns depends upon $F_{\mu \nu}$ which in turn depends upon the actual dynamics of the system. My intuition, along with analogies with some similar problems tells me that a simple (maybe approximate) elegant answer should be given in terms of $k^2$. –  zakk Apr 15 '12 at 23:10
    
(That's not homework strictly speaking... I came up with this problem analyzing some other almost uncorrelated topic, I don't know about the etiquette here, anyway I have no problem tagging is as such!) –  zakk Apr 15 '12 at 23:13
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(2 comments up) OK, well, the content of that comment would be great to incorporate into your question! The best questions explicitly state what they're asking, like "How can I write down the energy in terms of $k$?" You could also put something like that in the question title. (1 comment up) Are you doing this problem because you need the answer, or to learn the method? In the latter case, the homework tag would be appropriate, even if it's not an actual homework assignment. It's not really a big deal; your question just sounded like it might be a HW question when I read it. –  David Z Apr 15 '12 at 23:48
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Great, that helps! I made one small change afterwards, hopefully you don't mind. –  David Z Apr 16 '12 at 0:01
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