Linear superposition as a core theorem of quantum empiricism

Abstract

Clarifying the nature of the quantum state $|\Psi ⟩$ 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: ($\text{I}$) quantum statics, ($\text{II}$) doctrine of the number in the physical theory, and ($\text{III}$) 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.

Final version; 84 pages (major improvements), no figures. Russian version is available upon email-request

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