Relativistic Motion in a Constant Electromagnetic Field

TL;DR

This paper clarifies that the complex structure observed in the velocity of a relativistic charged particle in a constant electromagnetic field arises from the Lorentz group's algebraic properties, not from complex formalism, providing a new operator-based solution.

Contribution

It demonstrates that the complex electromagnetic field in relativistic motion results from Lorentz algebra, offering an operator evolution approach to solve equations of motion without intermediate fields.

Findings

The Lorentz algebra is equivalent to two complex su(2) algebras.

The complex structure naturally emerges from Lorentz group generators.

The solution provides direct velocity evolution without intermediate fields.

Abstract

For a relativistic charged particle moving in a constant electromagnetic field, its velocity 4-vector has been well studied. However, despite the fact that both the electromagnetic field and the equations of motion are purely real, the resulting 4-velocity is seemingly due to a complex electromagnetic field. This work shows that this is not due to some complex formalism used (such as Clifford algebra) but is intrinsically due to the fact that the $o(3,1)$ Lie algebra of the Lorentz group is equivalent to two commuting complex $su(2)$ algebras. Expressing the complex $su(2)$ generators in terms of the boost and rotation operators then naturally introduces a complex electromagnetic field. This work solves the equation of motion not as a matrix equation, but as an operator evolution equation in terms of the generators of the Lorentz group. The factorization of the real evolution operator into two commuting complex evolution operators then directly gives the time evolution of the velocity 4-vector without any reference to an intermediate field.