  A new method based on Synge's world function is developed for determining
within the WKB approximation the gravitationally induced quantum phase shift of
a particle propagating in a stationary spacetime. This method avoids any
calculation of geodesics. A detailed treatment is given for relativistic
particles within the weak field, linear approximation of any metric theory. The
method is applied to the calculation of the oscillation terms governing the
interference of neutrinos considered as a superposition of two eigenstates
having different masses. It is shown that the neutrino oscillations are not
sensitive to the gravitomagnetic components of the metric as long as the spin
contributions can be ignored. Explicit calculations are performed when the
source of the field is a spherical, homogeneous body. A comparison is made with
previous results obtained in Schwarzschild spacetime.
