Uniqueness of Maxwell's equations

[michael879]

Hi all,

I'm trying to derive for myself the uniqueness proof for Maxwell's equations, but I'm a little stuck at the end. I've managed to prove the following:
$$\dfrac{A^\mu}{\partial{t}}\nabla{A^\mu}|_S = \dfrac{A^\mu}{\partial{t}}|_{t_0} = \nabla{A^\mu}|_{t_0} =0 \Rightarrow \forall_{\vec{r}\in V, t > t_0} F^{\mu\nu} = 0$$
Where S is the surface bounding volume V, t₀ is some initial time, and:
$$A^\mu \equiv A_1^\mu - A_2^\mu, F^{\mu\nu} \equiv F_1^{\mu\nu} - F_2^{\mu\nu}$$
are the difference fields between the two solutions satisfying the same boundary conditions (at surface S, time t₀, and with the same charge distributions)

My problem is that I can't turn the left hand side of the first equation into any meaningful condition on the fields $F^{\mu\nu}$. As far as I can tell you need to know the potentials at the boundaries in order for uniqueness to hold, rather than just the field values... What I'm looking for is some expression of $F^{\mu\nu}$ on the boundaries that implies the potential conditions on the left hand side of the equation

 

[michael879]

Anyone? I've been googling this and nobody seems to provide a formal, complete proof for the uniqueness of solutions to Maxwell's equations. I'm trying to prove the uniqueness of an SU(N) gauge field theory, so the difficulty involved in the U(1) case is very discouraging...

 

[dauto]

The solution is not unique. That's what's meant by gauge freedom. It is the freedom to chose different solutions for A^μ which correspond to the same physical situation

 

[michael879]

Gauge freedom is on the potentials not the fields... As I showed in the OP the fields ARE unique as long as the derivatives of the potentials are known on the boundaries. My problem is that I can't turn the uniqueness conditions into expressions of the fields alone

 

[Rena Cray]

Do you have a formal, or clear statement of the uniqueness theorem?

 

[michael879]

Do you have a formal, or clear statement of the uniqueness theorem?

Yes, exactly what I stated in the OP... However the requirements for uniqueness involve knowledge of derivatives of the potentials, rather than of the field values and are not gauge-independent.

Let there exist 2 solutions to Maxwell's equations for identical charge distributions, with identical boundary conditions for some (not yet defined) space-time boundaries (spatial boundaries or initial conditions). Label these solutions 1 and 2, and define the difference between them as:
$$A^\mu \equiv A_1^\mu - A_2^\mu, F^{\mu\nu} \equiv F_1^{\mu\nu} - F_2^{\mu\nu}$$

These fields satisfy the empty space Maxwell's equations, and evaluate to 0 on any space-time boundary for which solutions 1 and 2 are constrained. The following statement can be derived for any volume V with boundary surface S, and some fixed time t₀:
$$\dfrac{\partial A^\mu}{\partial{t}}\nabla{A^\mu}|_S = \dfrac{\partial A^\mu}{\partial{t}}|_{t_0} = \nabla{A^\mu}|_{t_0} =0 \Rightarrow \forall_{\vec{r}\in V, t > t_0} F^{\mu\nu} = 0$$

Therefore, with sufficient boundary conditions solutions 1 and 2 are identical (the fields not the potentials, obviously). The condition:
$$\dfrac{\partial A^\mu}{\partial{t}}\nabla{A^\mu}|_S = \dfrac{\partial A^\mu}{\partial{t}}|_{t_0} = \nabla{A^\mu}|_{t_0} =0$$
is not gauge-independent though, so I am looking for some expression of boundary conditions on the FIELDS that leads to this. The conditions are purely derivatives of the potentials, which is good, but I don't see an obvious transformation to the field variables

*EDIT* sorry, I forgot to specify this but I used the Lorenz gauge in the derivation of the above equation so $\partial_\mu A^\mu = 0$

 

[UltrafastPED]

Does this help: http://faculty.uml.edu/cbaird/95.657(2013)/Maxwell_Uniqueness.pdf

Or this: http://www.thp.uni-koeln.de/alexal/pdf/electrodynamics.pdf

 

[michael879]

Does this help: http://faculty.uml.edu/cbaird/95.657(2013)/Maxwell_Uniqueness.pdf

No, I found this page while I was searching for an answer. The whole thing is pretty "hand-wavey" (e.g. 14 equations + 14 unknowns = unique solution), very informal, and under closer scrutiny suffers the exact same problems as me: the boundary conditions necessary for uniqueness are on the gauge-dependent potentials rather than the gauge independent fields.

Or this: http://www.thp.uni-koeln.de/alexal/pdf/electrodynamics.pdf

This link has plenty of uniqueness proofs, but I couldn't find a general one for an arbitrary system. It is very long though, so I may have just missed it... However, the ones I did see all simplified Maxwell's equations in some way, which makes the problem much simpler.

 

[marmoset]

I think it is possible to rewrite Maxwell's equation in a form (a symmetric, hyperbolic form) which let's you apply a general theorem on existence and uniqueness for such partial differential equations. I read about this in Geroch's paper "Partial Differential Equations of Physics", available here http://arxiv.org/abs/gr-qc/9602055.

 

[michael879]

I think it is possible to rewrite Maxwell's equation in a form (a symmetric, hyperbolic form) which let's you apply a general theorem on existence and uniqueness for such partial differential equations. I read about this in Geroch's paper "Partial Differential Equations of Physics", available here http://arxiv.org/abs/gr-qc/9602055.

I'm sure you're right, but the thing is I've already proven uniqueness! (I don't really care about existence at the moment). The issue isn't whether or not the solutions are unique, its what conditions are necessary for uniqueness.