We carry out a quantization of a classical relativistic particle dynamics, that is, a theory of N spinless point masses in mutual interaction. It is of Hamiltonian form, manifestly covariant, and involves N first-class constraints. In the resultant relativistic quantum dynamics these constraints are N invariant simultaneous "Schrödinger equations" involving N invariant time parameters (a=1,N). Since the interaction functions (relativistic "potential energies") can have a complicated momentum dependence, these equations do not become second-order equations in the representation pa=-iqa. The integrability condition ensures the existence of a unitary operator that "propagates" the system from one point to another in N-dimensional space independent of the path. Møller operators and the scattering operator are defined and the limits a'± are studied. It is demonstrated how the separability of the interaction functions leads to a factorization of the S matrix (cluster decomposition).