On padic 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 padic stochastic dynamics of a particle undergoing a Brownian random walk . The zigzagging 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 padic stochastic process, that has an underlying hidden ParisiSourlas supersymmetry, as the effective motion of a particle in a potential which can be expanded in terms of an infinite collection of padic harmonic oscillators with fundamental (Wickrotated imaginary) frequencies $\omega_p = i log~p$ (p is a prime) and whose harmonics are $\omega_{p, n} = i log ~ p^n$. The padic harmonic oscillator potential allow us to determine a onetoone 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2001
 arXiv:
 arXiv:physics/0101104
 Bibcode:
 2001physics...1104C
 Keywords:

 General Physics
 EPrint:
 14 pages