The Scaling Hamiltonian

We first explain the link between the Berry-Keating Hamiltonian and the spectral realization of zeros of the Riemann zeta function of the first author, and why there is no conflict at the semi-classical level between the "absorption" picture of A. Connes and the semiclassical "emission" computations of M. Berry and J. Keating, while the minus sign manifests itself in the Maslov phases. We then use the quantized calculus to analyse the recent attempt of X.-J. Li at proving Weil's positivity, and understand its limit. We then propose an operator theoretic semi-local framework directly related to the Riemann Hypothesis.