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Axial gauge

  1. Feb 21, 2017 #1
    V. Rubakov: Classical Theory of Gauge Fields, Problem 4: Find the residual gauge transformations and the general solution of the Maxwell equations in the axial gauge ([itex]\vec{\textbf{n}} \cdot \vec{\textbf{A}}=0[/itex]), where [itex]\vec{\textbf{n}}[/itex] is some fixed unit three-vector, which is constant in spacetime.

    I am using Rubakov notation, so gauge transformation is [itex]A'_{\mu}(x)=A_{\mu}(x)+ \partial_{\mu} \alpha(x)[/itex] and [itex]\eta_{\mu \nu}=diag(1,-1,-1,-1).[/itex] If [itex]\vec{\textbf{n}} \cdot \vec{\textbf{A}}=0 \Longrightarrow \vec{\textbf{n}} \cdot \vec{\textbf{A}'}=0[/itex], so:
    [tex]\vec{\textbf{n}}(\vec{\textbf{A}}-\vec{\nabla} \cdot \alpha)=0\\
    \vec{\textbf{n}}(\vec\nabla \cdot \alpha)=0.[/tex]
    I assume that form this relation, [itex]\vec{\textbf{n}}[/itex] must be orthogonal to [itex]\vec\nabla \cdot \alpha[/itex], but is there any further or deeper explanation? Do I have to choose particular [itex]\vec{\textbf{n}}[/itex], for example [itex](0,0,0,1)[/itex]? In this case:
    [tex]A'_z=0 \\
    A_z-\partial_z \alpha = 0 \\
    \Longrightarrow \alpha = \int A_z dz+f(x,y,t)[/tex]
    I still don't know how to proceed to Maxwell equations.
     
  2. jcsd
  3. Feb 26, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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