A nonrelativistic <Landau-Ginzburg theory> for a complex <order parameter> with global $U(1)$ symmetry is
$$
\mathcal L=\frac i2(\psi^*\dot\psi-\dot\psi^*\psi)
-\frac1{2m}|\nabla\psi|^2-r|\psi|^2-\frac u2|\psi|^4+cdots,
\qquad u>0.
$$
In the broken phase $r<0$, write $\psi=\sqrt n e^{i\theta}$. The first term is
$$
\frac i2(\psi^*\dot\psi-\dot\psi^*\psi)=-n\dot\theta,
$$
up to the sign convention for the phase. Thus the momentum conjugate to $\theta$ is $-n$: this is <number-phase conjugacy>. Canonical quantization gives the corresponding local commutator, while the spatial integral relates the global phase to total particle number.

Consequently a state with a sharply selected phase, as used to describe <spontaneous symmetry breaking>, cannot be an eigenstate of particle number. It is a coherent superposition of charge sectors with number fluctuations. Conversely, an exact finite-volume number eigenstate has no definite phase; the phase-selected broken-symmetry states emerge in the thermodynamic limit.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 337 / 1 / a / Solution

Number-phase conjugacy is the canonical relation between a conserved $U(1)$ charge and the phase of its order parameter. A state of sharp number cannot also have a sharply selected broken-symmetry phase.


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 Incoming links: Number-phase conjugacy