Claim 2.1.1.
The wedge product captures oriented area and is analogous to angular momentum or flux in physical systems.
Warning 2.1.2.
The above diagram is not complete; just ignore it for the time being. Should be updated soon :)
Section 2.1 Geometric Algebra and Physics
The Eigenscribe © methodology system utilizes Geometric Algebra (Clifford Algebra) as a unifying language to reformulate classical electromagnetism and quantum mechanics, replacing fragmented concepts from vector calculus with a coherent geometric framework.
Subsection 2.1.1 Physical Interpretations of the Wedge Product
202604110004 | geometric algebra classical mechanics vector calculus | PreFigure Demo: 🔗 Interactive
Subsubsection 2.1.1.1 Wedge Product and Angular Momentum
🔗 Linked Notes:
🔖 References: [Hamiltonian Mechanics with Geometric Calculus, biblio]
Subsection 2.1.2 Vector Potential for a Uniform Magnetic Field
Definition 2.1.3. Vector Potential.
For a uniform magnetic field \(\vec{B}=B_0\mathbf{e}_z\text{,}\) represented as a bivector \(\bivec{B}=B_0(\mathbf{e}_x\wedge\mathbf{e}_y)\text{,}\) the vector potential \(\vec{A}\) must satisfy
\begin{equation*}
\bivec{B}=\vec{\nabla}\wedge\vec{A} \, .
\end{equation*}
Example 2.1.4. Common Gauge Choices (Uniform Magnetic Field).
- Symmetry Gauge
-
Preserves rotational symmetry.\begin{equation*} \vec{A}_\text{sym}=\frac{B_0}{2}\left(x\mathbf{e}_x+y\mathbf{e}_y\right) \end{equation*}
- Landau Gauge
-
Preserves translational symmetry along the \(x\)-axis.\begin{equation*} \vec{A}_\text{landau}=-B_0 y \mathbf{e}_x \end{equation*}
Lemma 2.1.5. Vector Potential for a Uniform Magnetic Field.
Consider two gauges of a uniform magnetic field \(\vec{B}=B_0\mathbf{e}_z\) with respective potentials \(\vec{A}_1\) and \(\vec{A}_2\text{.}\) For such gauges, there is always some scalar function $\chi$ that can be used to relate the respective vector potentials as follows:
\begin{equation*}
\vec{A}_1= \vec{A}_2 + \vec{\nabla}\chi \, .
\end{equation*}
