1. Vector Potential & Gauge Transformations in Geometric Algebra.
Setup: Consider a uniform magnetic field \(\vec{B} = B_0 \mathbf{e}_z\text{.}\) In Geometric Algebra (GA), we represent this as the bivector:
\begin{equation*}
\bivec{B} = B_0 (\mathbf{e}_x \wedge \mathbf{e}_y) \, .
\end{equation*}
Note that in 3D, the traditional vector \(\vec{B}\) is the dual of this bivector via \(\vec{B} = -I \bivec{B}\text{,}\) where \(I = \mathbf{e}_x \mathbf{e}_y \mathbf{e}_z\text{.}\)
(a)
Find the vector potential \(\vec{A}\) such that \(\bivec{B} = \vec{\nabla} \wedge \vec{A}\text{.}\) Specifically, verify that the following two gauges satisfy this condition:
-
Symmetry Gauge: \(\vec{A}_{\text{sym}} = \frac{B_0}{2}(x\mathbf{e}_y - y\mathbf{e}_x)\)
-
Landau Gauge: \(\vec{A}_{\text{Landau}} = -B_0 y \mathbf{e}_x\)
Hint.
\(\vec{\nabla} \wedge \vec{A}\)\((\vec{\nabla} \wedge \vec{A})_{ij} = \partial_i A_j - \partial_j A_i\text{.}\)\(xy\)\(B_0\text{.}\)
(b)
Define a gauge transformation as \(\vec{A}' = \vec{A} + \vec{\nabla}\chi\text{.}\) What is the physical significance of the scalar function \(\chi\text{?}\)
(c)
Determine the specific scalar function \(\chi(x,y)\) that transforms the Landau gauge potential into the Symmetry gauge potential:
\begin{equation*}
\vec{A}_{\text{sym}} = \vec{A}_{\text{Landau}} + \vec{\nabla}\chi
\end{equation*}
Hint.
\(\vec{A}_{\text{sym}} - \vec{A}_{\text{Landau}}\)
(d)
Prove that the magnetic bivector is invariant under this transformation:
\begin{equation*}
\bivec{B}' = \vec{\nabla} \wedge \vec{A}' = \bivec{B}
\end{equation*}
(e)
Why does adding the gradient term \(\vec{\nabla}\chi\) not change the bivector field \(\bivec{B}\text{?}\) Explain this in terms of the properties of the wedge product and second derivatives.
