Invariants and Green’s functions of a relativistic charged particle in electromagnetic fields

Abstract

In the papers (1.4) the exact expressions for the Green's functions of the Klein-Gordon and Dirae equations in some special types of external electromagnetic fields (namely, various combinations of uniform electric and magnetic fields and the fields of plane waves) were obtained by different and sometimes rather complicated methods. The possibility to obtain exact solutions in all these cases is explained by the fact that the corresponding relativistic problems can be reduced to solving the nonrelativistic SchrSdinger equations with quadratic Hamiltonians. Nonrelativistic systems with arbitrary quadratic N-dimensional Hamiltonians were recently studied in detail in the papers (5,e). Therefore in the present article we want to apply the general methods of ref. (5,e) to the relativistic equations in the external fields of the types described above and to show that these methods enable one to obtain exact Green's functions in all mentioned cases by means of avery simple procedure. Moreover, we shall obtain also exact results in some new interesting cases. For simplicity and brevity we consider only the Klein-Gordon equation. The (second order) Dirae equation can be considered analogously, but the formulae in this case are more cumbersome. We use the usual notations goo = -- gll ~ -- g22 = -- ga3 = l, t~ = c = 1, Pu = i~/~r~, # = 0, 1, 2, 3, where the eontravariant four-vector r is (t, x, y, z). The causal Green's function of the Klein-Gordon equation can be represented, according to (L2), as the integral over the proper time: