  In the presence of spin-orbit coupling and inversion symmetry of the lateral
confinement potential a single electron does not exhibit matrix Berry phases in
quasi-two-dimensional semiconductor quantum dots. In such a system we
investigate whether many-body correlation effects can lead to finite matrix
Berry phases. We find that the transformation properties of many-electron
wavefunctions under two-dimensional inversion operation do not allow finite
matrix Berry phases. This effect is exact and is independent of the form of
electron-electron interactions. On the other hand, quasi-two-dimensional
semiconductor quantum dots with lateral confinement potential without inversion
symmetry can have finite matrix Berry phases. We find that many-body quantum
fluctuations can change matrix Berry phases significantly in such systems.
