Nonlinear equations for fields radiated by particles
This paper is devoted to the determination of the Poincaré invariant second order partial differential equations satisfied by the field radiated by a classical relativistic particle, described by its world line in n dimensional Minkowski space time \(M_ n\). The field is assumed to be retarded, namely, for each \(x\in M_ n\), to depend only on the unique point of the world line which is time like separated from x and in the past of x, and on a finite number of derivatives of the position of the particle with respect to its proper time, taken at that point. The equation is assumed in addition to be independent of the world line outside of the latter. The problem is solved completely for scalar fields, yielding (up to a change of function) the Liouville equation for \(n=2\), and the conformal invariant equation for scalar fields for \(n\geq 3\) (in particular the wave equation for \(n=4)\). In addition, the equation has a source term for \(n=4\). The problem is also solved for vector fields in some special cases (for instance under the Lorentz condition). The paper ends with some comments on the possibility of describing the Poincaré covariant evolution of a set of relativistic particles by considering them as the set of singularities of a field satisfying an invariant equation.
Poincaré invariant second order partial differential equations
scientific article; zbMATH DE number 3867840