Abstract
Quantum manifolds can be used to describe strong quantum fluctuations of geometry which eliminate the ultraviolet divergences of the fields at microscopic level. It is shown that the numbers of scalar wave functions on quantum spheres in 2, 3 and 4 dimensions are finite when the deformation parameter is a root of unity. The deformation parameter is determined by the radius of the universe and the scale of the ultraviolet cutoff. Moreover, it is pointed out that the non-commutative nature of the wave functions depends on their scales. The wave functions and their energies are almost classical below the intermediate scale. If we set the scale far beyond our scale, the effect is almost invisible for our scale. A possible scenario for the quantum evolution of the universe is also discussed
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