  Effects of the quantum gravity under Minkowski space-time background are
considered. It is shown that despite the absence of the complete theory of
quantum gravity, some concrete predictions could be made for the influence of
the quantum gravitational fluctuations on the propagation of the massive
particles. We demonstrate that although the gravitational potential
fluctuations do not produce particle scattering, they cause decoherence of the
matter waves due to off-shell effects. For point-like massive particles of the
Planck mass order, the effect is considerable. However, this type of
decoherence is beyond the measurable possibility for the real particles of the
finite size.
