The scaling Hamiltonian

We reconcile, at the semi-classical level, the original spectral realization of zeros of the Riemann zeta function as an ``absorption'' picture using the ad\`ele class space, with the ``emission'' semi-classical computations of Berry and Keating. We then use the quantized calculus to analyse the recent attempt of X.-J.~Li at proving Weil's positivity, and explain its limit. Finally, we propose an operator theoretic semi-local framework directly related to the Riemann hypothesis.