Constraint relativistic quantum dynamics

Abstract

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 Schr\odinger involving $N$ invariant time parameters ${\ensuremath{\tau}}_{a} (a=1,\dots{},N)$. Since the interaction functions (relativistic potential energies) can have a complicated momentum dependence, these equations do not become second-order equations in the representation ${p}_{a\ensuremath{\mu}}=\ensuremath{-}\frac{i\ensuremath{\partial}}{\ensuremath{\partial}{q}_{a}^{\ensuremath{\mu}}}$. The integrability condition ensures the existence of a unitary operator that propagates the system from one point to another in $N$-dimensional $\ensuremath{\tau}$ space independent of the path. M\o{}ller operators and the scattering operator are defined and the limits ${\ensuremath{\tau}}_{a}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ are studied. It is demonstrated how the separability of the interaction functions leads to a factorization of the $S$ matrix (cluster decomposition).