Generalized uncertainty principle in quantum cosmology for the maximally symmetric space

The new uncertainty relation is derived in the context of the canonical quantum theory with gravity for the case of the maximally symmetric space. This relation establishes a connection between fluctuations of the quantities which determine the intrinsic and extrinsic curvatures of the spacelike hypersurface in spacetime and introduces the uncertainty principle for quantum gravitational systems. The generalized time-energy uncertainty relation, which takes into account gravity, is proposed. It is shown that known Unruh's uncertainty relation follows, as a particular case, from the new uncertainty relation. As an example, the sizes of fluctuations of the scale factor and its conjugate momentum are calculated within an exactly solvable model. All known modifications of the uncertainty principle deduced previously from different approaches in the theory of gravity and string theory are obtained as particular cases of the proposed general expression.