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A Schrödinger Equation

Frederick Ira Moxley

Published Oct 16, 2017• 0 citations
OpenAlex

Abstract

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-M\"uller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schr\"odinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at $\sigma=1/2$.

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The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-M"uller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schr"odinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ=1/2\sigma=1/2σ=1/2.
The intimate connection between the zeros of the Riemann zeta function and the distribution of prime numbers makes this function famous beyond the borders of mathematics. Since the precise location of the non-trivial zeros (Riemann hypothesis) is still unconfirmed many efforts are directed to the investigation of the zeta function, not only with mathematical but also with physical methods. The physical approach in this thesis is inspired by the interaction of a cavity field with a two-level atom described by the Jaynes-Cummings-Paul model. However, in our case the interaction Hamiltonian depends logarithmically on the photon number operator. The time-evolved quantum states produce with appropriately chosen reference states the values of the Riemann zeta function. It is shown that in the complex half-plane with real parts larger than one, the cavity field suffices to create the zeta function whereas the analytical continuation of the zeta function can only be realized with entangled states. Moreover, the continuous Newton method is used to illustrate the behavior of the Riemann zeta function in the complex plane. This method depicts a complex function F by lines of constant phase which start at the poles of the function and lead into its zeros. In the case of the Riemann zeta function, this method would reveal if the Riemann hypothesis was violated.
The Hilbert-Pólya conjecture asserts that the imaginary parts of the nontrivial zeros of the Riemann zeta function (the Riemann zeros) are the eigenvalues of a self-adjoint operator (a quantum mechanical Hamiltonian, in the physical sense), as a promising approach to prove the Riemann hypothesis (cf.\cite{SH2011}). Instead of the eigenvalues, in this paper we consider observable-geometric phases as the realization of the Riemann zeros in a periodically driven quantum system, which were introduced in \cite{Chen2020} for the study of geometric quantum computation. To this end, we further introduce the notion of non-Abelian observable-geometric phases, involving which we give an approach to finding a physical system to study the Riemann zeros. Since the observable-geometric phases are connected with the geometry of the observable space according to the evolution of the Heisenberg equation, this sheds some light on the investigation of the Riemann hypothesis.
We propose measurements on a quantum system to realize the Riemann zeta function ζ . A single system, that is classical interference, suffices to create the Dirichlet representation of ζ . In contrast, we need measurements performed on two entangled quantum systems to extend ζ into the critical strip of complex space where the non-trivial zeros of ζ are located. As a consequence, we can view these zeros as a result of a Schrödinger cat which is by its very construction similar to, but in its details very different from, the superposition formed by two coherent states of identical amplitudes but opposite phases. This interpretation suggests that entanglement in quantum mechanics is the analogue of analytic continuation of complex analysis.
The Riemann Hypothesis (RH), one of the most profound unsolved problems in mathematics, concerns the nontrivial zeros of the Riemann zeta function. Establishing connections between the RH and physical phenomena could offer new perspectives on its physical origin and verification. Here, we establish a direct correspondence between the nontrivial zeros of the zeta function and dynamical quantum phase transitions (DQPTs) in two realizable quantum systems, characterized by the averaged accumulated phase factor and the Loschmidt amplitude, respectively. This precise correspondence reveals that the RH can be viewed as the emergence of DQPTs at a specific temperature. We experimentally demonstrate this correspondence on a five-qubit spin-based system and further propose an universal quantum simulation framework for efficiently realizing both systems with polynomial resources, offering a quantum advantage for numerical verification of the RH. These findings uncover an intrinsic link between nonequilibrium critical dynamics and the RH, positioning quantum computing as a powerful platform for exploring one of mathematics' most enduring conjectures and beyond.
In this paper, I explore the intriguing connection between hidden variables in quantum mechanics and the nontrivial zeros of the Riemann zeta function.Despite the apparent randomness in quantum phenomena, I demonstrate that deterministic hidden variables can coexist without violating Bell's inequalities, particularly in scenarios involving small angles.Using MATLAB simulations and a simplified space-time model, I illustrate this relationship, shedding light on the fundamental nature of quantum systems.This research not only deepens our understanding of quantum mechanics but also opens new avenues for investigating the Riemann zeta function's implications, potentially transforming our approach to both fields.
This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappings, and laser photons governed by electromagnetically induced transparency (EIT) to examine the existence of the RH. In considering the well-developed as Riemann zeta function, we find that the existence of RH has a corrected and self-consistent solution. Specifically, there is the only one pole at s = 1 on the complex plane for Riemann’s functions, which generalizes to all non-trivial zeros while s > 1. The essential solution is based on the BEC phases and on the nature of the laser photon(s). This work also incorporates Heisenberg commutators [ x ^ , p ^ ]=1/2 in the field of quantum mechanics. We found that a satisfactory solution for the RH would be incomplete without the formalism of Heisenberg commutators, BEC phases, and EIT effects. Ultimately, we propose the application of qubits in connection with the RH.