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Lagrangian problem

  • Thread starter Nikos
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  • #1
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Hi,there! Here is the lagrangian for a charged scalar field http://www.photodump.com/direct/Bbking22/P-meson-Lagrangian.jpg [Broken] as it can be found in “the large scale structure of space-time” Hawking, Ellis on page 68. It seem’s that I have problem varying Aa on the lagrangian because I get http://www.photodump.com/direct/Bbking22/P-meson-Myresult.jpg [Broken] , while the result of the book is http://www.photodump.com/direct/Bbking22/P-meson-AmVar.jpg [Broken] . Does anyone have the same problem? Does anyone gets the book’s result? Thanks for the help. Here are all the results for this example http://www.photodump.com/direct/Bbking22/Example3.jpg [Broken] .
 
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Answers and Replies

  • #2
George Jones
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Science Advisor
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I seem to get the same result as you. Below are my calculations, which I have done in flat so that I could use familiar notation. I haven't checked my calculations very closely, so I could easily have made a mistake.

Have you looked in a quantum field theory book? Almost all books should include a flat space version of this. I don't have any physics books with me right now, so I can't check.

Regards,
George

[tex]
L=-\frac{1}{2}\left( \partial_{a}\psi+ieA_{a}\psi\right) g^{ab}\left(
\partial_{b}\overline{\psi}-ieA_{b}\overline{\psi}\right) -\frac{1}{2}
\frac{m^{2}}{\hbar^{2}}\psi\overline{\psi}-\frac{1}{16\pi}F_{ab}F_{cd}
g^{ac}g^{bd}
[/tex]

[tex]
\begin{align*}
\frac{\partial L}{\partial A_{f}} & =-\frac{1}{2}ie\delta_{a}^{f}\psi
g^{ab}\left( \partial_{b}\overline{\psi}-ieA_{b}\overline{\psi}\right)
+\frac{1}{2}\left( \partial_{a}\psi+ieA_{a}\psi\right) g^{ab}ie\delta
_{b}^{f}\overline{\psi}\\
& =-\frac{1}{2}ie\psi g^{fb}\left( \partial_{b}\overline{\psi}-ieA_{b}
\overline{\psi}\right) +\frac{1}{2}\left( \partial_{a}\psi+ieA_{a}
\psi\right) g^{af}ie\overline{\psi}
\end{align*}
[/tex]

[tex]
\begin{align*}
\frac{\partial L}{\partial\left( \partial_{n}A_{f}\right) } & =-\frac
{1}{16\pi}\left[ \frac{\partial F_{ab}}{\partial\left( \partial_{n}
A_{f}\right) }F_{cd}+F_{ab}\frac{\partial F_{cd}}{\partial\left(
\partial_{n}A_{f}\right) }\right] g^{ac}g^{bd}\\
& =-\frac{1}{16\pi}\left[ \frac{\partial}{\partial\left( \partial_{n}
A_{f}\right) }\left( \partial_{a}A_{b}-\partial_{b}A_{a}\right)
F_{cd}+F_{ab}\frac{\partial}{\partial\left( \partial_{n}A_{f}\right)
}\left( \partial_{c}A_{d}-\partial_{d}A_{c}\right) \right] g^{ac}g^{bd}\\
& =-\frac{1}{16\pi}\left[ \left( \delta_{a}^{n}\delta_{b}^{f}-\delta_{b}
^{n}\delta_{a}^{f}\right) F_{cd}+F_{ab}\left( \delta_{c}^{n}\delta_{d}
^{f}-\delta_{d}^{n}\delta_{c}^{f}\right) \right] g^{ac}g^{bd}\\
& =-\frac{1}{16\pi}\left[ \left( g^{nc}g^{fd}-g^{fc}g^{nd}\right)
F_{cd}+F_{ab}\left( g^{an}g^{bf}-g^{af}g^{bn}\right) \right] \\
& =-\frac{1}{16\pi}\left[ F^{nf}-F^{fn}+F^{nf}-F^{fn}\right] \\
& =-\frac{1}{4\pi}F^{nf}
\end{align*}
[/tex]

[tex]
\begin{align*}
0 & =\frac{\partial L}{\partial A_{f}}-\partial_{n}\frac{\partial L}{\partial\left(
\partial_{n}A_{f}\right) }\\
& =-\frac{1}{2}ie\psi\left( \partial^{f}\overline{\psi}-ieA^{f}\overline
{\psi}\right) +\frac{1}{2}\left( \partial^{f}\psi+ieA^{f}\psi\right)
ie\overline{\psi}+\frac{1}{4\pi}\partial_{n}F^{nf}
\end{align*}
[/tex]
 
  • #3
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Thank you very much George!!
 
  • #4
dextercioby
Science Advisor
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Insights Author
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There's no 1/2 in the lagrangian in the first place. For SED in flat spacetime one has the action

[tex] S^{SED}\left[A_{\mu},\phi,\phi^{*}\right]= \int \ d^{4}x \ \left[\left(D^{\mu}\phi\right)\left(D_{\mu}\phi\right)^{*} -\frac{1}{4} F^{\mu\nu}F_{\mu\nu}-\mu^{2}\phi\phi^{*}\right] [/tex]

Daniel.
 

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