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

The conserved number operator isSubstitution of $\Psi =\sqrt{\rho }{e}^{-i\theta }$ into the Landau-Ginzburg theory givesThe imaginary term is a total derivative, soThus $\rho$ is the number density and is the canonical momentum conjugate to $\theta$. The canonical commutation relation is $[\theta (\mathbf{x}),\rho (\mathbf{y})]=i{\delta }^{(d)}(\mathbf{x}-\mathbf{y})$. Consequently, for ${\theta }_0=\int d^{d}x\theta$ in volume $V$,Equivalently, the averaged phase $\bar{\theta }={\theta }_0/V$ obeys $\Delta \bar{\theta }\Delta N\ge 1/2$. This number-phase conjugacy means that a state of sharp $N$ has no sharp phase, whereas a phase-selected state exhibiting spontaneous symmetry breaking must superpose different number sectors. Such sectors become effectively degenerate in the thermodynamic limit.