  A simple argument is presented which, based on the minimal coupling
Lagrangian for a many-body system, keeps the gauge invariance of the many-body
Schr\"odinger equation with explicit Coulomb potential. The elimination of
longitudinal electric field does not necessarily lead to the breakdown of gauge
invariance. The total time derivative term in the matter-EM field interaction
in the Lagrangian is canceled out by the choice of Coulomb gauge. The remaining
interaction is described by transverse vector potential and the longitudinal
electric field which is the homogeneous solution of Gauss law. This leads
directly to the gauge invariant forms of the linear and nonlinear constitutive
equations. It is discussed how to reconcile this result with the cases of an
isolated matter interacting with external charges.
