A curious multifoliate caustic in the magnetic Green function

Abstract

The classical analogue of the quantal time-dependent Green function for particles in a uniform magnetic field is the family of helical trajectories of particles emitted from a point source isotropically and with the same energy. These helices envelop a caustic surface consisting of infinitely many leaves meeting the symmetry line in conical cusps; the symmetry line is itself part of the caustic. An exact integral representation for the quantal Green function is constructed from the classical time-dependent action. Under semiclassical conditions, i.e. when many quantum flux units thread a circular magnetic orbit, the Green function is enhanced by a factor of order h(cross)-1/6 on the caustic leaves and by h(cross)-1/2 on the symmetry line. Quantisation into Landau levels is a consequence of constructive interference between the infinity of trajectories passing through a point.