Constraint relativistic quantum dynamics

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 ${\tau }_a (a=1,\dots ,N)$. Since the interaction functions (relativistic "potential energies") can have a complicated momentum dependence, these equations do not become second-order equations in the representation $p_{a\mu }=-\frac{i\partial }{\partial q_a^{\mu }}$. The integrability condition ensures the existence of a unitary operator that "propagates" the system from one point to another in $N$-dimensional $\tau$ space independent of the path. Møller operators and the scattering operator are defined and the limits ${\tau }_a\to \pm \infty$ are studied. It is demonstrated how the separability of the interaction functions leads to a factorization of the $S$ matrix (cluster decomposition).