  We study a way of $q$-deformation of the bi-local system, the two particle
system bounded by a relativistic harmonic oscillator type of potential, from
both points of view of mass spectra and the behavior of scattering amplitudes.
In our formulation, the deformation is done so that $P^2$, the square of center
of mass momentum, enters into the deformation parameters of relative
coordinates. As a result, the wave equation of the bi-local system becomes
nonlinear with respect to $P^2$; then, the propagator of the bi-local system
suffers significant change so as to get a convergent self energy to the second
order. The study is also made on the covariant $q$-deformation in four
dimensional spacetime.
