Sunday, 24 January 2016

electromagnetism - Derivation of Maxwell's equations from field tensor lagrangian


I've started reading Peskin and Schroeder on my own time, and I'm a bit confused about how to obtain Maxwell's equations from the (source-free) lagrangian density L=14FμνFμν (where Fμν=μAννAμ is the field tensor).


Substituting in for the definition of the field tensor yields L=12[(μAν)(μAν)(μAν)(νAμ)]. I know I should be using Aμ as the dynamical variable in the Euler-Lagrange equations, which become LAμμL(μAν)=μL(μAν), but I'm confused about how to proceed from here.


I know I should end up with μFμν=0, but I don't quite see why. Since μ and ν are dummy indices, I should be able to change them: how do the indices in the lagrangian relate to the indices in the derivatives in the Euler-Lagrange equations?



Answer



Well, you are almost there. Use the fact that (μAν)(ρAσ)=δρμδσν

which is valid because μAν are d2 independent components.


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