QUANTUM COSMOLOGY AND EDDINGTONS LARGE COSMIC NUMBERS

A theory that contains three fundamental constants from which one can build length, time, and mass (or force) etalons satisfies with that a necessary criterion of a ''universal unified field theory.'' In order to interpret such a theory physically, one has to translate it into th e Galilei-Newtonian language. This leads to classical ''pictures'' who se compatibility is ensured by introducing appropriate measurement-the oretical principles which imply corresponding uncertainty relations. I n this paper we compare different (mainly gravitational) theories from the paint of view of the fundamental constants underlying each case, and of the respective uncertainty relations. Assuming Eddington's hypo thesis of large cosmic numbers, it is argued in particular that in qua ntized general relativity one arrives at less stringent limitations on cosmology than in other conceivable approaches satisfying the princip le of minimal coupling. These limitations, however, are in all cases s trong enough to bar the way to quantum cosmology. Instead one is led t o a Diophantean concept.