How do the non-trivial zeros of the Riemann zeta function relate to the quantum chaotic behavior of high-dimensional systems, and what implications might this have for the study of quantum eigenstate thermalization hypothesis (ETH)?

The Riemann zeta function is a fundamental object in number theory, known for its deep connection to the distribution of prime numbers. One of the most intriguing aspects of this function is its non-trivial zeros, which lie along the "critical line" in the complex plane. The Riemann Hypothesis posits that all these zeros have a real part of 1/2, though this remains unproven. Interestingly, the behavior of these zeros has been found to share striking similarities with the statistical properties of eigenvalues in quantum systems, particularly in systems exhibiting quantum chaos. Quantum chaotic systems are those that, despite being governed by deterministic laws, display unpredictable behavior akin to classical chaotic systems when viewed in the quantum regime.

The quantum eigenstate thermalization hypothesis (ETH) is a concept in statistical mechanics that seeks to explain how isolated quantum systems can exhibit thermal equilibrium behavior, despite being in a pure quantum state. According to ETH, the individual eigenstates of a quantum system should mimic the properties of a thermal ensemble in the appropriate limit. The relevance of the question about the connection between the non-trivial zeros of the Riemann zeta function and quantum chaotic behavior arises from the possibility that insights from number theory might provide new perspectives on the statistical mechanics of quantum systems. If the distribution of these zeros is related to quantum chaotic systems, it could offer a novel approach to understanding the emergence of thermal behavior in quantum systems and even further our understanding of quantum-to-classical transitions.