Gauge-invariant Effective Action for the Dynamics of Bose-Einstein   condensates with a fixed number of atoms
  In this paper we present a particle-number-conserving (PNC) functional formalism to describe the dynamics of a cold bosonic gas. Treating the total number of particles as a constraint, whereby the phase invariance of the theory becomes local in time, we study this U(1) gauge theory using DeWitt's "gauge invariant effective action" techniques. Our functional formulation and earlier PNC proposals are shown to yield equivalent results to next-to-leading order in an expansion in the inverse powers of the total number of particles. In this more general framework we also show that earlier PNC proposals can be seen as different gauge (and gauge fixing condition) choices within the same physical theory. 
Esteban Calzetta and Bei-Lok Hu