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

The quadratic path integral isIntroduce the positive spatial operatorCompleting the square in the Gaussian functional integral over $\delta \rho$ yieldsThe derivative expansion $A^{-1}=(2\lambda {)}^{-1}+O({\nabla }^2)$ therefore givesThis is the long-wavelength Goldstone boson action. Keeping the spatial derivative in $A$ gives ${\omega }^2=v^2{k}^2+k^4/(4m^2)$.