Nonlinear Schrödinger dynamics of matrix D-branes

We formulate an effective Schrödinger wave equation describing the quantum dynamics of a system of D0-branes by applying the Wilson renormalization group equation to the world sheet partition function of a deformed σ-model describing the system, which includes the quantum recoil due to the exchange of string states between the individual D-particles. We arrive at an effective Fokker-Planck equation for the probability density with diffusion coefficient determined by the total kinetic energy of the recoiling system. We use Galilean invariance of the system to show that there are three possible solutions of the associated nonlinear Schrödinger equation depending on the strength of the open string interactions among the D-particles. When the open string energies are small compared to the total kinetic energy of the system, the solutions are governed by freely-propagating solitary waves. When the string coupling constant reaches a dynamically determined critical value, the system is described by minimal uncertainty wavepackets which describe the smearing of the D-particle coordinates due to the distortion of the surrounding space-time from the string interactions. For strong string interactions, bound state solutions exist with effective mass determined by an energy-dependent shift of the static BPS mass of the D0-branes.