Nonrelativistic Newton-Cartan Quantum Geometries

Abstract

The basic physical principles and ideas of geometro-stochastic (GS) quantum theory were described in Secs. 1.3–1.5, and their implementation within a general relativistic context will be carried out in the next and subsequent chapters. However, as a basic testing ground for these principles and ideas in general, and of the central concept of GS propagation in particular, we shall choose in this chapter the more familiar, as well as experimentally more extensively investigated, territory of nonrelativistic quantum theory. In this realm the concept of sharp localization gives rise to no foundational inconsistencies at the theoretical level, and the experimental confirmation of the conventional theory is indubitable and conclusive. Thus, we shall demonstrate that in the nonrelativistic context the proposed GS framework merges in the sharp-point limit (cf. Sec. 3.5) into a framework that is equivalent to conventional nonrelativistic quantum mechanics in the presence of an external Newtonian gravitational field. In this manner, we shall reach the assurance that the GS framework gives rise to no inconsistencies with that part of quantum theory that has already received experimental confirmation under the empirical limitations stipulated by the imposition of the nonrelativistic regime. Hence, the motivation for the investigations in this chapter is, in this last respect, analogous to the motivation for considering the linearized gravity approximation of the classical theory of general relativity (CGR): the existence of such a weak-gravity limit, in which classical general relativistic solutions merge into their Newtonian counterparts, has provided, at the inception1 of general relativity, the necessary assurance that the CGR framework does not give rise to any conflict with the wealth of observational data supporting Newtonian theory, despite the total dissimilarities in the mathematical structures of these two frameworks for describing gravitational phenomena.