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

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- Solution ii

Solution

Varying

A_{ν}

gives the Maxwell equations

\partial_{μ} F^{μν} = J^{ν}, J^{ν} = i e [(D^{ν} ϕ)^{†} ϕ - ϕ^{†} D^{ν} ϕ] .

On the vacuum manifold, choose unitary gauge so that

ϕ = v

. Then

J = - 2 e^{2} v^{2} A

and the London equation is

\nabla \times J = - 2 e^{2} v^{2} B .

For a static configuration,

\nabla \times B = J

and

\nabla \cdot B = 0

, so

\nabla^{2} B = 2 e^{2} v^{2} B = \frac{1}{ξ ^{2}} B, ξ = \frac{1}{2 e v} = \frac{1}{m _{A}} .

The magnetic field therefore decays exponentially inside the condensate. This is the Meissner effect, with penetration depth equal to the inverse gauge-boson mass.

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