Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 305 / 1 / ii / Solution

Varying $A_{\nu }$ gives the Maxwell equationsOn the vacuum manifold, choose unitary gauge so that $\varphi =v$. Then $\mathbf{J}=-2e^2{v}^2\mathbf{A}$ and the London equation isFor a static configuration, $\nabla \times \mathbf{B}=\mathbf{J}$ and $\nabla \cdot \mathbf{B}=0$, soThe magnetic field therefore decays exponentially inside the condensate. This is the Meissner effect, with penetration depth equal to the inverse gauge-boson mass.