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In this conceptual paper, we discuss quantum formalisms which do not use thefamous Axiom of Choice. We also consider the fundamental problem whichaddresses the (in)correctness of having the complex numbers as the base fieldfor Hilbert spaces in the K{\o}benhavn interpretation of quantum theory, andpropose a new approach to this problem (based on the Lefschetz principle).Rather than a Theorem--Proof--paper, this paper describes two new researchprograms on the foundational level, and focuses on fundamental open questionsin these programs which come along the way.

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