Grand-canonical Hamiltonian of a Bose field ($K=H-\mu N$)

For a repulsive contact-interacting Bose-Einstein condensate, the classical Hamiltonian density generating evolution in a frame rotating at the chemical potential is $\mathcal{K}={\hslash }^2|\nabla \psi {|}^2/(2M)-\mu |\psi {|}^2+g|\psi {|}^4/2$. Its integral is the grand canonical ensemble generator $H-\mu N$, not the physical energy $H$ alone. For $g,\mu >0$ the uniform mean-field theory minima satisfy $|{\psi }_0{|}^2=\mu /g$, with a freely chosen constant phase. In the physical frame, the corresponding field has time factor $e^{-i\mu t/\hslash }$.