Space-Time-Matter: Finite Projective Geometry as a Quantum World with Elementary Particles

A unified theory for space-time and matter might be based on finite

projective geometries instead of differentiable manifolds and gauge

groups. Each point is equipped with a quadratic form over a finite Galois field which define neighbors in the finite set of points.

Due to the projective equivalence of all quadratic forms this world

is necessarily a 4-dimensional Lorentz-invariant space-time with a

gauge symmetry G(3)xG(2)xG(1) for internal points which represent elementary particle degrees of freedom. Matter appears as a

geometric distortion by an inhomogeneous field of quadrics and all

physical properties (spins, charges) of the standard model seem to

follow from its geometric structure in a continuum limit. The finiteness inevitably induces a fermionic quantization of all matter fields

and a bosonic for gauge fields. This unity of space-time and matter

was already sought 1918 by Hermann Weyl in a gauge theory as an

extension of Einstein’s general theory of relativity, but not found -

probably because of the assumption of a continuous geometry.