The Weyl-Cartan Space Problem in Purely Affine Theory

According to Poincaré, only the epistemological sum of geometry and physics is measurable. Of course, there are requirements of measurement to be imposed on geometry because otherwise the theory resting on this geometry cannot be physically interpreted. In particular, the Weyl-Cartan space problem must be solved, i.e., it must be guaranteed that the comparison of distances is compatible with the Levi-Civita transport. In the present paper, we discuss these requirements of measurement and show that in the (purely affine) Einstein-Schrödinger unified field theory the solution of the Weyl-Cartan space problem simultaneously determines the matter via Einstein's equations. Here the affine field ⁱ _kl represents Poincaré's sum, and the solution of the space problem means its splitting in a metrical space and in matter fields, where the latter are given by the torsion tensor ⁱ _kl].