Example of a one-time many-particle relativistic wave equation

An example is given of a relativistic wave equation of a system of $N$ particles with interaction potential containing only $N$-particle forces. The equation is formulated in the variables $t$, $g_1$, $g_2,\dots$, where $g_i$ are the three-dimensional parts of the four-velocities, and the equation is a direct generalization of the Sehrödinger equation in the $p$-representation. The transformations of the wave function allowed by the equation form a group that is isomorphic to the Poincare group and, when the interaction is switched off, they form a group that is isomorphic to the direct product of Poincare groups. An analog of the configuration $x$-space is constructed and it is shown that the equation is consistent with classical relativistic mechanics of many bodies.