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Journal of Physical Studies 22(2), Article 2001 [7 pages] (2018)
DOI: https://doi.org/10.30970/jps.22.2001 FOUR-MOMENTUM AND ANGULAR FOUR-MOMENTUM OF THE ELECTROMAGNETIC FIELD OF A SYSTEM OF RELATIVISTIC CHARGED PARTICLES IN A WEAK INTERACTION APPROXIMATIONYuri Krynytskyi
Department for Theoretical Physics, Ivan Franko National University of Lviv, |
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We consider a system of two charged particles in a weak interaction approximation. In the approximation, the charges of the particles are assumed to be arbitrary small. This allows us to neglect the dependence of the Lorentz force on the acceleration of particles and make a correct reduction to the common moment of time. Also, in the approximation, radiation and radiation friction force can be neglected. In other words, the problem becomes purely mechanical. At the same time, we do not take into consideration restrictions on the velocity of the particles; therefore, the system is strictly relativistic. One can imagine the physical picture of interaction in such a system in the following way: the particles move towards each other from infinity, then interact, and then fly away to infinity in almost the same directions from which they initially came. The particles exchange 4-momentum and the angular 4-momentum through the electromagnetic field. The 4-momentum and the angular 4-momentum of the field together with the 4-momentum and the angular 4-momentum of the particles form the corresponding integrals of motion.
We obtain exact expressions for the 4-momentum and angular 4-momentum of the field in a weak interaction approximation as the functions of coordinates and the velocities of the particles. The expressions are generalized to the case of many particles. It is shown that in this case the 4-momentum and the angular 4-momentum of the field depend on the sums of the pairwise contributions. The expressions are analyzed for the particles flying away to infinity. We obtain the result that, in contrast to the 4-momentum, the angular 4-momentum of the field does not tend to zero. The law of 4-momentum conservation and the law of angular 4-momentum conservation are analyzed in the non-relativistic limit.
PACS number(s): 03.30.+p, 03.30.-z, 03.50.De