  Adiabatic time evolution of degenerate eigenstates of a quantum system
provides a means for controlling electronic states since mixing between
degenerate levels generates a matrix Berry phase. In the presence of spin-orbit
coupling in n-type semiconductor quantum dots the electron Hamiltonian is
invariant under time reversal operation and the many body groundstate may be
doubly degenerate. This double degeneracy can generate non-Abelian vector
potentials when odd number of electrons are present. We find that the
antisymmetry of many electron wavefunction has no effect on the matrix Berry
phase. We have derived equations that allow one to investigate the effect of
electron correlations by expressing the non-Abelian vector potentials for many
electron system in terms of single electron non-Abelian vector potentials.
