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Few weeks ago I came upon the following problem:

Let B be a vector field derivable from a vector potential A (on a simply connected topological space, smooth enough and everything well established so that mathematicians do not have to care about), i.e. [tex]\vec B=rot \vec A=\vec\nabla\times\vec A[/tex].

Now, we know from Elecrodynamics that we could alter the vector potential A by some gradient field (guage transformation).

Further, assume [tex]\Delta\vec B\neq 0[/tex], i.e. B itself does not satisfy Laplace's equation.

The question is now, whether a scalar field [tex]\phi[/tex] exists, such that [tex]\vec A'= A+grad\phi[/tex][tex]\Delta\vec B=0[/tex]?and

The question is reasonable, since:

[tex](\Delta\vec B)_i=\partial^2_l B_i=\partial^2_l\varepsilon_{ijk}\partial_j(\vec A+grad\phi)_k=\partial^2_l\varepsilon_{ijk}\partial_j(A_k+\partial_k\phi)=\varepsilon_{ijk}\partial_j\partial^2_l(A_k+\partial_k\phi)[/tex]

although we know [tex]\partial^2_l A_k\neq 0[/tex], why shouldn't some [tex]\phi[/tex] exist such that [tex]\partial^2_l(A_k+\partial_k\phi)=0[/tex]?!

(I tend to think that this is impossible, for the following reason: Let B be the magnetic field and A be the respective vector potential. We know that an EM-wave is a wave, where both the Electric and the Magnetic field obey the wave equation. Now if the former were true, I could always pick up that gauge for the A potential and make the magnetic wave equation trivial, which would be somekind of embarrassing, since I'm talking on my cell phone every day making use of the electroMAGNETIC waves :) - of course this is by no way a rigorous mathemacial disproof)

so I would be glad to here what you think of this :)

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# Is the following gauge transformation possible?

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