Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis

Abstract

The Riemann Hypothesis states that the Riemann zeta function $\zeta (z)$ admits a set of ``non-trivial'' zeros that are complex numbers supposed to have real part $1/2$. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. Hilbert and Pólya suggested that the Riemann Hypothesis could be solved through the mathematical tools of physics, finding a suitable Hermitian or unitary operator that describe classical or quantum systems, whose eigenvalues distribute like the zeros of $\zeta (z)$. A different approach is that of finding a correspondence between the distribution of the $\zeta (z)$ zeros and the poles of the scattering matrix $S$ of a physical system. Our contribution is articulated in two parts: in the first we apply the infinite-components Majorana equation in a Rindler spacetime and compare the results with those obtained with a Dirac particle following the Hilbert-Pólya approach showing that the Majorana solution has a behavior similar to that of massless Dirac particles and finding a relationship between the zeros of zeta end the energy states. Then, we focus on the $S$-matrix approach describing the bosonic open string scattering for tachyonic states with the Majorana equation. Here we find that, thanks to the relationship between the angular momentum and energy/mass eigenvalues of the Majorana solution, one can explain the still unclear point for which the poles and zeros of the $S$-matrix of an ideal system that can satisfy the Riemann Hypothesis, exist always in pairs and are related via complex conjugation. As claimed in the literature, if this occurs and the claim is correct, then the Riemann Hypothesis could be in principle satisfied, tracing a route to a proof.

25 pages, 1 figure

References in corpus (9)

• Physics of the Riemann Hypothesis
• The H=xp model revisited and the Riemann zeros
• The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime
• From anyons to Majoranas
• Jost function, prime numbers and Riemann zeta function
• On real solutions of the Dirac equation for a one-dimensional Majorana particle
• Search for Double Charged Particles as Direct Test for Dark Atom Constituents
• On the $\nu$-zeros of the modified Bessel function $K_{i\nu }(x)$ of positive argument
• Asymptotics of a sum of modified Bessel functions with non-linear argument

Cited by in corpus (2)

• Analyzing Riemann's hypothesis
• Quantum collapse as undecidable proposition in an Everettian multiverse