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Hello, so I was asked a question in two parts (Peskin & Schroeder problem 2.1). The first part asked me to derive the source-free Maxwell's equations from the action:
S=\int{d^4 x \frac{-1}{4}F_{\mu\nu}F^{\mu\nu}}
Given that the vector potential itself is the dynamical variable.
I derived (source-free) Gauss's law and Ampere's law from that action by setting the variation to zero. Is it possible to derive Faraday's law and Gauss's law for magnetism from that action? I was under the impression that those two laws just came from our definition of the E and B field. Since I carried through the whole process and got the 4 equations of motion:
\partial_\nu F^{\mu\nu}=0
I don't see how I could extract the other two Maxwell's equations from this action. Certainly I can't extract them from this equation of motion.
The second part of the problem asked me to find the energy-momentum-stress tensor of E&M. I started by using Noether's theorem, using the translational (in 4 directions) invariance of the action. I think I was going somewhere until I hit a roadblock.
In a nutshell, I used the transformations:
x^\mu \rightarrow x^\mu+\epsilon^\mu
As my symmetry, and then as per the usual formulation I made \epsilon a function of the space-time coordinates instead of a constant vector. I reached a point where:
F_{\mu\nu} \rightarrow F_{\mu\nu} + \epsilon^\rho(\partial_\rho F_{\mu\nu})+(\partial_\rho A_\nu)(\partial_\mu \epsilon^\rho)-(\partial_\rho A_\mu)(\partial_\nu \epsilon^\rho)
I think if I can just take care of the last 2 terms, it should be ok, but I can't think of a way to manipulate them in such a way to combine them into one term, or factor it or something like that. Any ideas? Thanks!
EDIT
If I'm not providing enough info, just tell me, but some help would be appreciated, thanks!
S=\int{d^4 x \frac{-1}{4}F_{\mu\nu}F^{\mu\nu}}
Given that the vector potential itself is the dynamical variable.
I derived (source-free) Gauss's law and Ampere's law from that action by setting the variation to zero. Is it possible to derive Faraday's law and Gauss's law for magnetism from that action? I was under the impression that those two laws just came from our definition of the E and B field. Since I carried through the whole process and got the 4 equations of motion:
\partial_\nu F^{\mu\nu}=0
I don't see how I could extract the other two Maxwell's equations from this action. Certainly I can't extract them from this equation of motion.
The second part of the problem asked me to find the energy-momentum-stress tensor of E&M. I started by using Noether's theorem, using the translational (in 4 directions) invariance of the action. I think I was going somewhere until I hit a roadblock.
In a nutshell, I used the transformations:
x^\mu \rightarrow x^\mu+\epsilon^\mu
As my symmetry, and then as per the usual formulation I made \epsilon a function of the space-time coordinates instead of a constant vector. I reached a point where:
F_{\mu\nu} \rightarrow F_{\mu\nu} + \epsilon^\rho(\partial_\rho F_{\mu\nu})+(\partial_\rho A_\nu)(\partial_\mu \epsilon^\rho)-(\partial_\rho A_\mu)(\partial_\nu \epsilon^\rho)
I think if I can just take care of the last 2 terms, it should be ok, but I can't think of a way to manipulate them in such a way to combine them into one term, or factor it or something like that. Any ideas? Thanks!
EDIT
If I'm not providing enough info, just tell me, but some help would be appreciated, thanks!
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