Why and whence the Hilbert space in quantum theory?

TL;DR

This paper explores the foundational reasons for the emergence of Hilbert spaces in quantum theory, deriving key structures from statistical descriptions and revising classical concepts like the Pythagorean theorem within a quantum context.

Contribution

It provides a derivation of Hilbert space structures from quantum statistical principles, offering a new perspective on quantum axioms and foundational concepts.

Findings

Derivation of vector space and scalar product from quantum statistics

Reinterpretation of the Pythagorean theorem in quantum context

Discussion on the derivation of the norm topology in Hilbert spaces

Abstract

We explain why and how the Hilbert space comes about in quantum theory. The axiomatic structures of vector space, of scalar product, of orthogonality, and of the linear functional are derivable from the statistical description of quantum micro-events and from Hilbertian sum of squares $|\mathfrak{a}_1|^2+|\mathfrak{a}_2|^2+\cdots$. The latter leads (non-axiomatically) to the standard writing of the Born formula $\mathtt{f} = |\langle\psi|\varphi\rangle|^2$. As a corollary, the status of Pythagorean theorem, the concept of a length, and the 6-th Hilbert problem undergo a quantum `revision'. An issue of deriving the norm topology may no have a short-length solution (too many abstract math-axioms) but is likely solvable in the affirmative; the problem is reformulated as a mathematical one.