A NEW PATH-INTEGRAL REPRESENTATION FOR THE MANY-PARTICLE GREEN-FUNCTION OF THE RELATIVISTIC-PARTICLES

Starting from the second quantized functional integral representation, the field path-integral representation for the total many-particle Gr een, function for relativistic and nonrelativistic point-like charged Bose and Fermi particles in (3 + 1) or in (2 + 1) interacting via Maxw ell or Chern-Simon fields is constructed and shown to be only an integ ral over the trajectories of the particles. The effective action depen ds on the coordinates and velocities of the particles, and is nonlocal in time due to causal interactions between the particles. In a static (nonrelativistic) approximation, the action is local in time and lead s to expressions for the Hamiltonian for Coulomb interaction in. (3 1), and for anyon interaction in (2 + 1) dimensions. This path integra l representation automatically includes the usual connection between s pin and statistics for the cases of an infinite flat space and trivial topology for the manifold of the charged fields. Our results are gene ralized in the presence of an external magnetic field. It is shown how to take into account the contribution of the vacuum polarization effe cts within the framework of the approach.