# Chapter 4: Phase and Gauge

## Units

```{list-table}
:class: toc-table
:header-rows: 1
:widths: 5 60 15

* - Unit
  - Title
  - Textbook
* - 4.1
  - [Gauge Field](4-1-gauge-field)
  - §9.1
* - 4.2
  - [Berry Phase](4-2-berry-phase)
  - §9.4
* - 4.3
  - [Landau Level](4-3-landau-level)
  - §9.2
* - 4.4
  - [Spin and Monopole](4-4-spin-and-monopole)
  - §9.3, §9.5
```

## Review & Summary

:::{glossary}
**Gauge transformation**
  $\psi \to e^{i\chi(x)}\psi$, $A_\mu \to A_\mu - \partial_\mu\chi$. Leaves all observables unchanged.

**Covariant derivative**
  $D_\mu = \partial_\mu + iA_\mu$. Ensures the Schrödinger equation is gauge-covariant.

**Minimal coupling**
  $H = \frac{1}{2m}(\mathbf{p} - q\mathbf{A})^2 + q\Phi$.

**Aharonov-Bohm effect**
  Phase shift $\Delta\phi = \frac{q}{\hbar}\oint \mathbf{A}\cdot d\mathbf{l} = \frac{q\Phi_B}{\hbar}$ from enclosed magnetic flux, even where $\mathbf{B}=0$.

**Landau levels**
  $E_n = \hbar\omega_c(n+\tfrac{1}{2})$, with cyclotron frequency $\omega_c = eB/m$ and magnetic length $\ell_B = \sqrt{\hbar/eB}$.

**Dirac quantization**
  $eg = n\hbar c/2$. Magnetic monopoles imply electric charge quantization.

**Berry phase**
  $\gamma = i\oint \langle n(\mathbf{R})|\nabla_{\mathbf{R}} n(\mathbf{R})\rangle \cdot d\mathbf{R}$. Geometric phase from adiabatic cyclic evolution.
:::