A Majorana Relativistic Quantum Spectral Approach to the Riemann Hypothesis in (1+1)-Dimensional Rindler Spacetimes

Abstract

Following the Hilbert-P\'olya approach to the Riemann Hypothesis, we present an exact spectral realization of the nontrivial zeros of the Riemann zeta function $\zeta(z)$ with a Mellin-Barnes integral that explicitly contains it. This integral defines the spectrum of the real-valued energy eigenvalues $E_n$ of a Majorana particle in a $(1+1)$-dimensional Rindler spacetime or equivalent Kaluza-Klein reductions of $(n+1)$-dimensional geometries. We show that the Hamiltonian $H_M$ describing the particle is hermitian and the spectrum of energy eigenvalues $\{E_n\}_{n \in \mathbb{N}}$ is countably infinite in number in a bijective correspondence with the imaginary part of the nontrivial zeros of $\zeta(z)$ having the same cardinality as required by Hardy-Littlewood's theorem from number theory. The correspondence between the two spectra with the essential self-adjointness of $H_M$, confirmed with deficiency index analysis, boundary triplet theory and Krein's extension theorem, imply that all nontrivial zeros have real part $\Re ( z )=1/2$, i.e., lie on the ``critical line''. In the framework of noncommutative geometry, $H_M$ is interpreted as a Dirac operator $D$ in a spectral triple $(\mathcal{A}, \mathcal{H}, D)$, linking these results to Connes' program for the Riemann Hypothesis. The algebra $\mathcal{A}$ encodes the modular symmetries underlying the spectral realization of $\zeta (z)$ in the Hilbert space $\mathcal{H}$ of Majorana wavefunctions, integrating concepts from quantum mechanics, general relativity, and number theory. This analysis offers a promising Hilbert-P\'olya-inspired path to prove the Riemann Hypothesis.