Particle-number phase symmetry ($\psi \mapsto e^{i\chi }\psi$)

Conservation of particle number corresponds to global circle group phase rotations of a nonrelativistic Bose field. A positive-density uniform mean-field saddle chooses a phase, with degenerate saddle manifold a circle. The associated Goldstone boson is the long-wavelength phase mode. This is a thermodynamic symmetry-selection statement; an exact finite-volume number eigenstate does not itself choose a phase.