Xc. Gao et al., INVARIANT FORMULATION AND EXACT-SOLUTIONS FOR THE RELATIVISTIC CHARGED KLEIN-GORDON FIELD IN A TIME-DEPENDENT SPATIALLY HOMOGENEOUS ELECTRIC-FIELD, Physical review. A, 57(2), 1998, pp. 753-761
On the basis of the invariant formulation, we find the quantum and cla
ssical exact solutions and corresponding total phase for the relativis
tic charged Klein-Gordon (KG) field in a time-dependent spatially homo
geneous electric field. The total phase includes both the dynamical an
d geometric phases (Aharonov-Anandan phases). The connection between t
he quantum and classical solutions is then obtained. From this connect
ion we discuss the condition under which the geometric phase for the K
G held can be defined.