Wigner Inequalities for Test of Hypothesis of Realism and Concepts of Macroscopic and Local Realism

TL;DR

This paper introduces a new Wigner inequality for testing realism in quantum systems, derives the Leggett-Garg inequality using Kolmogorov probability axioms, and demonstrates contradictions with macroscopic realism.

Contribution

It proposes a novel Wigner inequality for realism tests, provides a rigorous derivation of the Leggett-Garg inequality, and analyzes implications for quantum versus classical realism.

Findings

New Wigner inequality suitable for quantum and quantum field systems

Rigorous derivation of Leggett-Garg inequality using Kolmogorov axioms

Contradiction between quantum evolution and macroscopic realism

Abstract

We propose a new Wigner inequality suitable for test of the hypothesis of realism. We show that this inequality is not identical neither to the well-known Wigner inequality nor to the Leggett-Garg inequality in Wigner form. The obtained inequality is suitable for test of realism not only in quantum mechanical systems, but also in quantum field systems. Also we propose a mathematically consistent derivation of the Leggett--Garg inequality in Wigner form, which was recently presented in the literature, for three and $n$ distinct moments of time. Contrary to these works, our rigor derivation uses Kolmogorov axiomatics of probability theory. We pay special attention to the construction and studies of the spaces of elementary outcomes. Basing on the the Leggett--Garg inequality in Wigner form for $n$ distinct moments of time we prove that any unitary evolution of a quantum system contradicts the concept of macroscopic realism. We show that application of the concept of macroscopic realism to any quantum system leads to ``freezing'' of the system in the initial state. It is shown that for a particle with an infinite number of observables the probability to find a pair of the observables in some defined state is zero, even if the operators of these observables commute. This fact might serve as an additional logical argument for the contradiction between quantum theory and classical realism.