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The novelty of the Jean Pierre Badiali last scientific works stems to aquantum approach based on both (i) a return to the notion of trajectories(Feynman paths) and (ii) an irreversibility of the quantum transitions. Theseiconoclastic choices find again the Hilbertian and the von Neumann algebraicpoint of view by dealing statistics over loops. This approach confers anexternal thermodynamic origin to the notion of a quantum unit of time (RovelliConnes' thermal time). This notion, basis for quantization, appears herein as amere criterion of parting between the quantum regime and the thermodynamicregime. The purpose of this note is to unfold the content of the last fiveyears of scientific exchanges aiming to link in a coherent scheme the JeanPierre's choices and works, and the works of the authors of this note based onhyperbolic geodesics and the associated role of Riemann zeta functions. Whilethese options do not unveil any contradictions, nevertheless they give birth toan intrinsic arrow of time different from the thermal time. The question of thephysical meaning of Riemann hypothesis as the basis of quantum mechanics, whichwas at the heart of our last exchanges, is the backbone of this note.

From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis