There is no axiomatic system for the quantum theory

TL;DR

This paper demonstrates that the Hilbert space formalism of quantum theory cannot be fully axiomatized, as the probability theory of measurement outcomes conflicts with the existence of the Bloch sphere in spin-1/2 systems.

Contribution

It derives a new inequality testing the Bloch sphere's existence, revealing fundamental contradictions in the axiomatic foundations of quantum theory.

Findings

The inequality is violated by quantum measurement probabilities.

The Bloch sphere cannot be consistently incorporated into the formalism.

A contradiction exists in the axiomatic structure of quantum mechanics.

Abstract

Recently, [arXiv:0810.3134] is accepted and published. We derive an inequality with two settings as tests for the existence of the Bloch sphere in a spin-1/2 system. The probability theory of measurement outcome within the formalism of von Neumann projective measurement violates the inequality. Namely, we have to give up the existence of the Bloch sphere. Or, we have to give up the probability theory of measurement outcome within the formalism of von Neumann projective measurement. Hence it turns out that there is a contradiction in the Hilbert space formalism of the quantum theory, viz., there is no axiomatic system for the theory.