  The spin of an electron trapped in a quantum dot is a promising candidate
implementation of a qubit for quantum information processing. We study the
central spin problem of the effect of the hyperfine interaction between such an
electron and a large number of nuclear moments. Using a spin coherent path
integral, we show that in this limit the electron spin evolution is well
described by classical dynamics of both the nuclear and electron spins. We then
introduce approximate yet systematic methods to analyze aspects of the
classical dynamics, and discuss the importance of the exact integrability of
the central spin Hamiltonian. This is compared with numerical simulation.
Finally, we obtain the asymptotic long time decay of the electron spin
polarization. We show that this is insensitive to integrability, and determined
instead by the transfer of angular momentum to very weakly coupled spins far
from the center of the quantum dot. The specific form of the decay is shown to
depend sensitively on the form of the electronic wavefunction.
