= Particle-number phase symmetry
{title2=$\psi\mapsto e^{i\chi}\psi$}
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.
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