Geometric Phase in Path Integral and Quantisation Rule

A description of geometric phase in terms of path integral formalism is presented. It is proved that this adiabatic phase can appear in the propagator or Green function of an adiabatic system. In the semiclassical approximation, following the Green function expression of the electronic density of states, the corresponding generalized Bohr-Sommerfeld quantisation rule can thus be obtained. It is shown that this rule has been corrected by the geometric phase.