  The dark soliton solution of the Gross-Pitaevskii equation in one dimension
has two parameters that do not change the energy of the solution: the global
phase of the condensate wave function and the position of the soliton. These
degeneracies appear in the Bogoliubov theory as Bogoliubov modes with zero
frequencies and zero norms. These ``zero modes'' cannot be quantized as the
usual Bogoliubov quasiparticle harmonic oscillators. They must be treated in a
non-perturbative way. In this paper I develop non-perturbative theory of zero
modes. This theory provides non-perturbative description of quantum phase
diffusion and quantum diffusion of soliton position. An initially well
localized wave packet for soliton position is predicted to disperse beyond the
width of the soliton.
