Electromagnetic Potentials & Gauge Theory
Electromagnetic Potentials & Gauge Theory
The scalar and vector potentials \(\phi, \mathbf{A}\) reformulate Maxwell's equations in terms of fewer functions. Gauge freedom — the arbitrariness in choosing potentials — is a deep symmetry that extends to all of modern physics.
Definition
Define \(\mathbf{B} = \nabla\times\mathbf{A}\) and \(\mathbf{E} = -\nabla\phi - \partial_t\mathbf{A}\). A gauge transformation \(\mathbf{A}\to\mathbf{A}+\nabla\Lambda\), \(\phi\to\phi-\partial_t\Lambda\) leaves E and B unchanged.
Key Result
In Lorenz gauge (\(\nabla\cdot\mathbf{A} + (1/c^2)\partial_t\phi = 0\)), both potentials satisfy the wave equation driven by sources: \(\Box^2\phi = -\rho/\epsilon_0\).
Example 1
The Aharonov-Bohm effect: an electron traveling around a solenoid acquires a phase \(\Delta\phi = (e/\hbar)\oint\mathbf{A}\cdot d\mathbf{l}\) even where \(\mathbf{B}=0\), demonstrating that \(\mathbf{A}\) is physically observable.
Example 2
In Coulomb gauge (\(\nabla\cdot\mathbf{A}=0\)), the scalar potential satisfies Poisson's equation \(\nabla^2\phi = -\rho/\epsilon_0\), with instantaneous Coulomb interaction.
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Practice
- What is gauge invariance, and why is it important in physics?
- Write Maxwell's equations in terms of the 4-potential \(A^\mu\).
- Explain the Aharonov-Bohm effect in physical terms.
- How does gauge invariance relate to charge conservation via Noether's theorem?
Show Answer Key
1. Gauge invariance: $\mathbf{A}\to\mathbf{A}+\nabla\chi$, $\Phi\to\Phi-\partial\chi/\partial t$ leave $\mathbf{E}=-\nabla\Phi-\partial\mathbf{A}/\partial t$ and $\mathbf{B}=\nabla\times\mathbf{A}$ unchanged. It means the potentials are not unique — physics depends only on gauge-invariant quantities ($\mathbf{E},\mathbf{B}$, and observable phases). It is fundamental to quantum field theory (local gauge symmetry → force-carrying bosons).
2. $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$. Maxwell: $\partial_\mu F^{\mu\nu} = \mu_0 J^\nu$ becomes $\Box A^\nu - \partial^\nu(\partial_\mu A^\mu) = \mu_0 J^\nu$. In Lorenz gauge ($\partial_\mu A^\mu = 0$): $\Box A^\nu = \mu_0 J^\nu$ (wave equations for each component).
3. Even where $\mathbf{B}=0$, the vector potential $\mathbf{A}$ can affect quantum phases. An electron encircling a solenoid (with flux $\Phi$ inside, $\mathbf{B}=0$ outside) acquires phase $e\Phi/\hbar$, causing observable interference shifts. This shows $\mathbf{A}$ is physically meaningful in quantum mechanics, not just a mathematical convenience.
4. Noether's theorem: continuous symmetry → conservation law. Local $U(1)$ gauge symmetry of the Lagrangian $\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ (where $D_\mu = \partial_\mu + ieA_\mu$) gives $\partial_\mu J^\mu = 0$ (charge conservation). The conserved current is $J^\mu = e\bar{\psi}\gamma^\mu\psi$.