On the applicability of Kolmogorov's theory of probability to the description of quantum phenomena. Part I

It is a common view that von Neumann laid the foundations of a "non-commutative probability theory" with his axiomatization of quantum mechanics (QM). As such, it is regarded a generalization of the "classical probability theory" due to Kolmogorov. Outside of quantum physics, however, Kolmogorov's axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann's axiomatization of QM was contingent on the absence of a general theory of probability in the 1920s. In this work I argue in favor of the latter position. In particular, I show that for non-relativistic N-body quantum systems subject to a time-independent scalar potential, it is possible to construct a mathematically rigorous theory based on Kolmogorov's axioms and physically natural random variables, which reproduces central predictions of QM. The respective theories are distinct, so that an empirical comparison may be possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series will address the projection postulate and the question of measurement in this approach.