**Authors:** Felix M. Lev

We consider a system of two free bodies in de Sitter invariant quantum mechanics.
De Sitter invariance is understood such that representation operators satisfy commutation
relations of the de Sitter algebra. Our approach does not involve quantum
field theory, de Sitter space and its geometry (metric and connection). At very large
distances the standard relative distance operator describes a well known cosmological
acceleration. In particular, the cosmological constant problem does not exist and
there is no need to involve dark energy or other fields for solving this problem. At
the same time, for systems of macroscopic bodies this operator does not have correct
properties at smaller distances and should be modified. We propose a modification
which has correct properties, reproduces Newton's gravity, the gravitational redshift
of light and the precession of Mercury's perihelion if the width of the de Sitter momentum
distribution δ for a macroscopic body is inversely proportional to its mass m.
We argue that fundamental quantum theory should be based on a Galois field with a
large characteristic p which is a fundamental constant characterizing laws of physics
in our Universe. Then one can give a natural explanation that δ = constR/(mG)
where R is the radius of the Universe (such that λ = 3/R^{2} is the cosmological constant)
and G is a quantity defining Newton's gravity. A very rough estimation gives
G ~ R/(m_{N}lnp) where mN is the nucleon mass. If R is of order 10^{26}m then lnp is of
order 10^{80} and therefore p is of order exp(10^{80}). In the formal limit p → ∞ gravity
disappears, i.e. in our approach gravity is a consequence of finiteness of nature.

**Comments:** 94 pages, 2 figures. Chapter 3 (considering observable gravitational effects) has been considerably revised.

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[v1] 24 Apr 2011

[v2] 6 Jun 2011

[v3] 28 Sep 2011

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