Axial gauge

  • #1
63
0
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.
 

Answers and Replies

Related Threads on Axial gauge

Replies
9
Views
870
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
8
Views
4K
  • Last Post
Replies
10
Views
844
  • Last Post
Replies
5
Views
5K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
0
Views
12K
  • Last Post
Replies
1
Views
624
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
688
Top