A Deflationary Account of Quantum Theory and its Implications for the Complex Numbers

TL;DR

The paper addresses why quantum theory relies on complex numbers and whether the Hilbert-space formalism is fundamental or a Markovian embedding of deeper non-Markovian dynamics. It develops a Markovian-embedding framework, reviews the Strocchi-Heslot classical oscillator reformulation, and proposes an indivisible stochastic-process interpretation to relate quantum systems to underlying non-Markovian processes, with complex numbers enabling the Markovian embedding via the Schrödinger dynamics $i\partial_t|\Psi(t)\rangle=H|\Psi(t)\rangle$. It argues that the Hilbert-space formalism can be understood as such an embedding and that the complex algebra is indispensable for this correspondence, illustrating a stochastic-quantum mapping. The work reframes wavefunctions and their ontological status, clarifies the role of $i$ and complex structure in quantum theory, and outlines directions for extending non-Markovian-to-Markovian reductions in quantum contexts.

Abstract

Why does quantum theory need the complex numbers? With a view toward answering this question, this paper argues that the usual Hilbert-space formalism is a special case of the general method of Markovian embeddings. This paper then describes the indivisible interpretation of quantum theory, according to which a quantum system can be regarded as an indivisible stochastic process unfolding in an old-fashioned configuration space, with wave functions and other exotic Hilbert-space ingredients demoted from having an ontological status. The complex numbers end up being necessary to ensure that the Hilbert-space formalism is indeed a Markovian embedding.