High-order field theory and a weak Euler–Lagrange–Barut equation for classical relativistic particle-field systems

In both quantum and classical field systems, conservation laws such as the conservation of energy and momentum are widely regarded as fundamental properties. A broadly accepted approach to deriving conservation laws is built using Noether's method. However, this procedure is still unclear for relativistic particle-field systems where particles are regarded as classical world lines. In the present study, we establish a general manifestly covariant or geometric field theory for classical relativistic particle-field systems. In contrast to quantum systems, where particles are viewed as quantum fields, classical relativistic particle-field systems present specific challenges. These challenges arise from two sides. The first comes from the mass-shell constraint. To deal with the mass-shell constraint, the Euler–Lagrange–Barut (ELB) equation is used to determine the particle's world lines in the four-dimensional (4D) Minkowski space. Besides, the infinitesimal criterion, which is a differential equation in formal field theory, is reconstructed by an integro-differential form. The other difficulty is that fields and particles depend on heterogeneous manifolds. To overcome this challenge, we propose using a weak version of the ELB equation that allows us to connect local conservation laws and continuous symmetries in classical relativistic particle-field systems. By applying a weak ELB equation to classical relativistic particle-field systems, we can systematically derive local conservation laws by examining the underlying symmetries of the system. Our proposed approach provides a new perspective on understanding conservation laws in classical relativistic particle-field systems.