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 Riemann 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 modelled as the scattering of the particle about the infinitely many locations of the prime numbers positions. We are able then to reformulate such a 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 ω p= i log p ( p is a prime) and whose harmonics are ω p,n= i log p n . The p-adic harmonic oscillator potential allows 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 the Riemann zeta function, λ n , after solving the inverse scattering problem. We review our recent proof of the Riemann hypothesis that the non-trivial zeroes of zeta are of the form s=1/2+i λ n and the solution to the quantum inverse scattering problem.