Towards a probabilistic foundation for non-relativistic and relativistic quantum theory

The controversy on the interpretation and conceptual foundations of quantum mechanics has been accompanying the theory since its inception in the 1920s. The question of which probability theory ought to be used to statistically describe quantum phenomena is an aspect of this controversy which has so far only received comparatively little attention in the literature. The commonly held view is that von Neumann correctly identified this probability theory with his mathematical axiomatization of quantum mechanics. Historically, however, the mathematical foundations of quantum mechanics were established before Kolmogorov published his axiomatization of modern probability theory in 1933. In the contemporary literature this conflict is usually resolved by depicting Kolmogorov's "classical" theory as the special case of "quantum probability theory" for which the random variables commute, thus implying that von Neumann discovered the generalization of Kolmogorov's theory before the latter even published its axioms. This dissertation explores the alternative hypothesis that von Neumann, lacking a formal foundation of probability theory, axiomatized quantum mechanics using an expedient theory of probability. While at first this may seem to be an unfounded proposition, I show that there does indeed exist evidence within the Schrödinger theory of quantum mechanics that an axiomatization on the basis of Kolmogorov's theory of probability can adequately account for the empirical data. Due to the Born rule, this approach naturally identifies the position probability density for the quantum system as the basic quantity in non-relativistic quantum theory. Dynamical equations that reformulate Schrödinger's equation in terms of this density were discovered by Madelung in 1927. Even though the Madelung equations are central to many "classical" approaches to the foundations of quantum mechanics such as Bohmian and stochastic mechanics, no coherent mathematical theory has been developed so far for this system of partial differential equations. Wallstrom prominently raised objections against the Madelung equations, aiming to show that no such theory exists in which the system is well-posed and in which the Schrödinger equation is recovered without the imposition of an additional "ad hoc quantization condition". With a view on the existing literature, in this dissertation I clarify in which sense Wallstrom's objections are justified and in which sense they are not. It is found that it may be possible to construct a mathematical theory of the Madelung equations which is satisfactory in the aforementioned sense, though more mathematical research is required. If it was indeed possible to recast the Schrödinger theory on the basis of the Madelung equations and Kolmogorov's theory of probability, this would constitute a viable competitor to the respective quantum-mechanical theory. The second part of this dissertation considers the consequences of Kolmogorov's axiomatization for relativistic quantum theory. As opposed to quantum mechanics, however, quantum field theory still lacks a generally agreed upon mathematical foundation---despite the fact that mathematical physicists have been working on its construction since the 1950s. Based on the above statistical view on the non-relativistic theory, a new approach to this foundational problem is examined: The generalization of the Born rule to the general-relativistic setting. The dissertation discusses the (smooth) one-body generalization, which is essentially the theory of the general relativistic continuity equation. The subject matter has not been investigated this comprehensively and under such general conditions in the literature before. With the aim of an eventual generalization to the case of a varying number of bodies, this work also considers a differential-geometric generalization of the Reynolds Transport theorem. Though the resulting identities have been known for some time, the theorems shown overcome various "boundedness conditions" in the literature which are cumbersome to satisfy in common modeling situations. I conclude with a discussion on where we stand with regards to the implementation of Kolmogorov's axioms into non-relativistic and relativistic quantum theory as well as appropriate paths for future research.