Riemannian structure of space-time as a consequence of quantum mechanics

Different axiomatic approaches to general relativity which use light rays and classical test particles as primitive concepts remain incomplete because they end with a Weylian instead of a Riemannian structure of space-time. It is shown that the final step to a Riemann space can be obtained as a necessary consequence if quantum mechanics, as the theory of matter, is made part of the total scheme. Quantum mechanics must contain classical particle mechanics as a limiting case. The self-consistency requirement that in Weyl space this limiting case should agree with the axiomatically introduced classical-test-particle behavior implies the conclusion that the Weyl geometry of space-time must be restricted to the special case of a Riemann geometry. This is shown in detail for massive spin-½ particles after a general discussion of the theory of unquantized tensor fields and two-spinor fields in Weyl space. The result is independent of the Weyl type chosen for the orthotetrad (Lorentz basis). The same conclusion is obtained from massive Klein-Gordon theory in Weyl space in demanding that the physically reasonable current should be divergence-free.