Quantum Theory without the Axiom of Choice, and Lefschetz Quantum Physics

In this paper, we discuss quantum formalisms that do not use the axiom of choice. We also consider the fundamental problem that addresses the (in)correctness of having the complex numbers as the base field for Hilbert spaces in the København interpretation of quantum theory, and propose a new approach to this problem (based on the Lefschetz principle). Rather than a theorem–proof paper, this paper describes two new research programs on the foundational level, and focuses on basic open questions that arise in these programs.