Exact Wavefunction in a Noncommutative Dipole Field Theory

The nonrelativistic case of noncommutative scalar dipole field theory with quartic interaction on a two-dimensional spacetime is analyzed. As there are two parameters in the general quartic interaction we try a way to find their relation. To do this we first investigate the formulation of the quantum mechanics for a particle carrying the noncommutative dipole. We point out a problem therein and propose a possible method to solve it. We use this prescription to determined the quartic interaction term in the field theory once the noncommutative dipole is turned on. The two particle Schr\"odinger equation is then solved and the exact wavefunction of the bound state and associated spectrum are found. It is seen that the wavefunction has three center positions in the relative coordinates and the separation of the center is equal to the dipole length $L$ of the dipole field, exhibiting the nature of the noncommutative dipole field.