Quantum systems as indivisible stochastic processes

According to the stochastic-quantum correspondence, a quantum system can be understood as a stochastic process unfolding in an old-fashioned configuration space based on ordinary notions of probability and 'indivisible' stochastic laws, which are a non-Markovian generalization of the laws that describe a textbook stochastic process. The Hilbert spaces of quantum theory and their ingredients, including wave functions, can then be relegated to secondary roles as convenient mathematical appurtenances. In addition to providing an arguably more transparent way to understand and modify quantum theory, this indivisible-stochastic formulation may lead to new possible applications of the theory. This paper initiates a deeper investigation into the conceptual foundations and structure of the stochastic-quantum correspondence, with a particular focus on novel forms of gauge invariance, dynamical symmetries, and Hilbert-space dilations.