Charged Scalar Field Lagrangian Problem: Hawking, Ellis

[Nikos]

Hi,there! Here is the lagrangian for a charged scalar field http://www.photodump.com/direct/Bbking22/P-meson-Lagrangian.jpg 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 , while the result of the book is http://www.photodump.com/direct/Bbking22/P-meson-AmVar.jpg . 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 .

 

[George Jones]

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

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

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

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

$$\begin{align*} 0 & =\frac{\partial L}{\partial A_{f}}-\partial_{n}\frac{\partial L}{\partial\left( \partial_{n}A_{f}\right) }\\ & =-\frac{1}{2}ie\psi\left( \partial^{f}\overline{\psi}-ieA^{f}\overline {\psi}\right) +\frac{1}{2}\left( \partial^{f}\psi+ieA^{f}\psi\right) ie\overline{\psi}+\frac{1}{4\pi}\partial_{n}F^{nf} \end{align*}$$

 

[Nikos]

Thank you very much George!

 

[dextercioby]

There's no 1/2 in the lagrangian in the first place. For SED in flat spacetime one has the action

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

Daniel.