A new path-integral representation for the many-particle Green function of the relativistic particles

Starting from the second quantizied functional integral representation for field path-integral representation for the total many-particle Green function for relativistic and nonrelativistic point-like charged Bose and Fermi particles in $(3+1)$ or in $(2+1)$ interacting via Maxwell or Chern-Simon fields is constructed and shown to be only an integral over the trajectories of the particles. The effective action depends on coordinates and velocities of the particles, and is nonlocal in time due to causal interactions between the particles. In static (nonrelativistic) approximation the action is local in time and leads to expressions for the Hamiltonian for Coulomb interaction in $(3+1)$, and for anyon interaction in $(2+1)$ dimensions. This path integral representation automatically includes the usual connection between spin and statistics for the cases of infinite flat space and trivial topology for the manifold of the charged fields. Our results are generalized in the presence of an external magnetic field. It is shown how to take into account the contribution of the vacuum polarization effects in the framework of the approach.