Quantum theory without the Axiom of choice, and Lefschetz Quantum Physics

In this conceptual paper, we discuss quantum formalisms which do not use the famous Axiom of Choice. We also consider the fundamental problem which addresses the (in)correctness of having the complex numbers as the base field for Hilbert spaces in the K{\o}benhavn 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 fundamental open questions in these programs which come along the way.