The final version of a recent approach towards quantum foundation

TL;DR

This paper simplifies a foundational approach to quantum mechanics by removing the need for an inaccessible variable, deriving the Hilbert space formalism from the existence of two complementary accessible variables.

Contribution

It presents a simplified, purely mathematical basis for quantum foundations that derives the Hilbert space formalism from the existence of two complementary variables.

Findings

Hilbert space formalism can be derived from two maximal accessible variables.

The assumption of an inaccessible variable can be dropped.

The theory connects mathematical formalism with physical variables.

Abstract

In several articles, this author has advocated an alternative approach towards quantum foundation based upon a set of postulates, and based upon the notions of theoretical variables and of accessible theoretical variables. It is shown in this article that this basis can be considerably simplified. In particular, the assumption that there exists an inaccessible variable $\phi$ such that all the accessible ones can be seen as functions of $\phi$, can be dropped. This assumption has been difficult to motivate in the previous articles. From this, I get a simple basis for the main Theorems.The essential assumption is that there in the given context exist two different maximal accessible variables, what Niels Bohr would have called two complementary variables. From this, the whole Hilbert space formalism may be derived. It is also discussed in some detail how this Hilbert space can be chosen. The resulting theory is a purely mathematical theory, but it leads to quantum mechanics by letting the variables be physical variables. Other applications of the main theory are also considered. The mathematical proofs are mostly deferred to the Appendix.