On p-adic Stochastic Dynamics, Supersymmetry and the Riemann Conjecture

Abstract

We construct (assuming the quantum inverse scattering problem has a solution ) the operator that yields the zeroes of the Riemman zeta function by defining explicitly the supersymmetric quantum mechanical model (SUSY QM) associated with the p-adic stochastic dynamics of a particle undergoing a Brownian random walk . The zig-zagging occurs after collisions with an infinite array of scattering centers that fluctuate randomly. Arguments are given to show that this physical system can be modeled as the scattering of the particle about the infinite locations of the prime numbers positions. We are able then to reformulate such p-adic stochastic process, that has an underlying hidden Parisi-Sourlas supersymmetry, as the effective motion of a particle in a potential which can be expanded in terms of an infinite collection of p-adic harmonic oscillators with fundamental (Wick-rotated imaginary) frequencies $\omega_p = i log~p$ (p is a prime) and whose harmonics are $\omega_{p, n} = i log ~ p^n$. The p-adic harmonic oscillator potential allow us to determine a one-to-one correspondence between the amplitudes of oscillations $a_n$ (and phases) with the imaginary parts of the zeroes of zeta $\lambda_n$, after solving the inverse scattering problem.