  To formulate the zero modes in a finite-size system with spontaneous
breakdown of symmetries in quantum field theory is not trivial, for in the
naive Bogoliubov theory, one encounters difficulties such as phase diffusion,
the absence of a definite criterion for determining the ground state, and
infrared divergences. A new interacting zero mode formulation that has been
proposed for systems with a single zero mode to avoid these difficulties is
extended to general systems with multiple zero modes. It naturally and
definitely gives the interactions among the quantized zero modes, the
consequences of which can be observed experimentally. In this paper, as a
typical example, we consider an atomic Bose-Einstein condensed system with a
dark soliton that contains two zero modes corresponding to spontaneous
breakdown of the U(1) gauge and translational symmetries. Then we evaluate the
standard deviations of the zero mode operators and see how the mutual
interaction between the two zero modes affects them.
