Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 337 / 1 / b

Table of contents
- Solution b

Solution

Stability requires

λ > 0

. The classical potential

r ρ + λ ρ^{2}

has a symmetry-breaking minimum when

r < 0

, at

\overset{ρ}{ˉ} = - \frac{r}{2 λ} > 0.

ρ = \overset{ρ}{ˉ} + δ ρ

and discarding constants and total derivatives gives the quadratic Lagrangian

L_{2} = δ ρ \dot{θ} - \frac{( \nabla δ ρ ) ^{2}}{8 m ρ ˉ} - λ (δ ρ)^{2} - \frac{ρ ˉ}{2 m} (\nabla θ)^{2} .

The density fluctuation is a gapped amplitude mode, while the phase is the prospective Goldstone boson.

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