Thursday, February 2, 2017

electromagnetism - Lorenz gauge fixing


Is it always possible to define function ψ satisfying the Lorenz gauge equation μμψ+μAμ=0?



Answer



Yes. If you define f=μAμ then you can write the equation in the form μμψ=f

This is the Klein-Gordon equation with a nonzero source (f) and can be solved via Green's function methods. Once you have the Klein-Gordon propagator* G(x) (this is derived in any e.g. quantum field theory textbook) appropriate to the boundary conditions the solution can be written as ψ(x)=d4xG(xx)f(x)
since Green's functions by definition satisfy μμG(xx)=δ(xx)
where we take all differentiations to be with respect to x.


*You need the propagator in the position space representation to write this down. It is usually more convenient to write it in momentum space; you can go back and forth using (inverse) Fourier transforms.


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