Linear superposition as a core theorem of quantum empiricism

TL;DR

This paper derives the quantum state as a vector space element using empirical and set-theoretic principles, establishing superposition without relying on traditional physical or philosophical assumptions.

Contribution

It provides a direct, axiomatic-free derivation of quantum superposition from minimal experimental entities and set theory, bypassing traditional interpretational issues.

Findings

Quantum states form a linear vector space

Superposition principle derived from set-theoretic ensemble creation

Quantum observables exhibit non-commutativity

Abstract

Clarifying the nature of the quantum state $|\Psi\rangle$ is at the root of the problems with insight into counter-intuitive quantum postulates. We provide a direct-and math-axiom free-empirical derivation of this object as an element of a vector space. Establishing the linearity of this structure--quantum superposition--is based on a set-theoretic creation of ensemble formations and invokes the following three principia: ($\textsf{I}$) quantum statics, ($\textsf{II}$) doctrine of the number in the physical theory, and ($\textsf{III}$) mathematization of matching the two observations with each other (quantum covariance). All of the constructs rest upon a formalization of the minimal experimental entity--the registered micro-event, detector click. This is sufficient for producing the $\mathbb C$-numbers, axioms of linear vector space (superposition principle), statistical mixtures of states, eigenstates and their spectra, and non-commutativity of observables. No use is required of the spatio-temporal concepts. As a result, the foundations of theory are liberated to a significant extent from the issues associated with physical interpretations, philosophical exegeses, and mathematical reconstruction of the entire quantum edifice.