Algebraic solution of the Klein-Gordon equation

An algebraic approach to the solution of the Klein-Gordon equation is described for the case of a charged particle in the presence of plane-wave electromagnetic radiation. From an examination of the commutation relations between Pmu =-i( delta / delta xmu ) and Anu , P.A A.A, etc. one finds a new set of 'translation' operators Pi mu which commute with the total 'Hamiltonian'. The authors then construct a representation of the Poincare group out of the Pi mu and their canonically conjugate 'coordinates' Qnu . The solutions are shown to correspond to the spin zero mass m representation of the restricted Poincare group. Applications of the technique to other quantum-mechanical problems are also briefly discussed.