Geometric field theory and weak Euler–Lagrange equation for classical relativistic particle-field systems
Peifeng Fan1,2, Hong Qin3,4,5(), Jian Liu3, Nong Xiang1,4, Zhi Yu1,4
1. Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, China 2. Science Island Branch of Graduate School, University of Science and Technology of China, Hefei 230031, China 3. Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China 4. Center for Magnetic Fusion Theory, Chinese Academy of Sciences, Hefei 230031, China 5. Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, USA
A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is developed. The connection between the space-time symmetry and energy-momentum conservation laws of the system is established geometrically without splitting the space and time coordinates; i.e., spacetime is treated as one entity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that the particles and the field reside on different manifolds. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of the electromagnetic fields and also a functional of the particles’ world lines. The other difficulty associated with the geometric setting results from the mass-shell constraint. The standard Euler–Lagrange (EL) equation for a particle is generalized into the geometric EL equation when the mass-shell constraint is imposed. For the particle-field system, the geometric EL equation is further generalized into a weak geometric EL equation for particles. With the EL equation for the field and the geometric weak EL equation for particles, the symmetries and conservation laws can be established geometrically. A geometric expression for the particle energy-momentum tensor is derived for the first time, which recovers the non-geometric form in the literature for a chosen coordinate system.
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