A Strategy for Proving Riemann Hypothesis

Abstract

A strategy for proving Riemann hypothesis is suggested. The vanishing of the Rieman Zeta reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian operator $D^+$ having the zeros of Riemann Zeta as its eigenvalues. The construction of $D^+$ is inspired by the conviction that Riemann Zeta is associated with a physical system allowing conformal transformations as its symmetries. The eigenfunctions of $D^+$ are analogous to the so called coherent states and in general not orthogonal to each other. The states orthogonal to a vacuum state (which has a negative norm squared) correspond to the zeros of the Riemann Zeta. The induced metric in the space ${\cal{V}}$ of states which correspond to the zeros of the Riemann Zeta at the critical line $Re[s]=1/2$ is hermitian and both hermiticity and positive definiteness properties imply Riemann hypothesis. Conformal invariance in the sense of gauge invariance allows only the states belonging to ${\cal{V}}$. Riemann hypothesis follows also from a restricted form of a dynamical conformal invariance in ${\cal{V}}$ and one can reduce the proof to a standard analytic argument used in Lie group theory.