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In units $\hslash =1$, polar fields rewrite the coherent-state action asafter removing the periodic total density derivative. The potential has positive-density minimum ${\rho }_0=\mu /g$ for $\mu >0$. Density curvature gives a finite static amplitude cost, while phase changes cost only derivatives. Number-phase conjugacy couples their dynamics; the two fields do not automatically represent two independent excitation branches.

The density-phase action of a Bose gas follows directly from $\psi =\sqrt{\rho }{e}^{i\varphi }$:The total derivative integrates to zero for periodic density. HenceA uniform density fluctuation has nonzero quadratic cost $g(\delta \rho {)}^2/2$, whereas a uniform phase change has no energy cost. Thus the radial fluctuation is massive in the static quadratic-action sense, while the phase is massless. The number-phase conjugacy term $i\delta \rho {\partial }_{\tau }\varphi$ couples their dynamics: this does not imply an additional independent gapped quasiparticle branch in the nonrelativistic Bose gas.

The compact phase winding term $i{\rho }_0\int {\partial }_{\tau }\varphi$ vanishes for the smooth zero-winding phonon sector. Globally the phase can wind by $2\pi$ while the complex field remains periodic, so that term should not be discarded across all topological sectors without a further argument.

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.

A nonrelativistic Landau-Ginzburg theory for a complex order parameter with global $U(1)$ symmetry isIn the broken phase $r<0$, write $\psi =\sqrt{n}{e}^{i\theta }$. The first term isup 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.