Abstract
The novelty of the Jean Pierre Badiali last scientific works stems to a quantum approach based on both (i) a return to the notion of trajectories (Feynman paths) and (ii) an irreversibility of the quantum transitions. These iconoclastic choices find again the Hilbertian and the von Neumann algebraic point of view by dealing statistics over loops. This approach confers an external thermodynamic origin to the notion of a quantum unit of time (Rovelli Connes' thermal time). This notion, basis for quantization, appears herein as a mere criterion of parting between the quantum regime and the thermodynamic regime. The purpose of this note is to unfold the content of the last five years of scientific exchanges aiming to link in a coherent scheme the Jean Pierre's choices and works, and the works of the authors of this note based on hyperbolic geodesics and the associated role of Riemann zeta functions. While these options do not unveil any contradictions, nevertheless they give birth to an intrinsic arrow of time different from the thermal time. The question of the physical meaning of Riemann hypothesis as the basis of quantum mechanics, which was at the heart of our last exchanges, is the backbone of this note.
Highlights
Despite the unstoppable success of the technosciences based on both quantum mechanics, standard particle model and cosmological model, at least two questions must be investigated among many issues that the theories leave open [1, 2]: (i) the question of the ontological status of the time and (ii) the obsessive interrogation concerning the existence or the absence of an intrinsic “arrow of time”
The origin of these questions comes from the equivocal equivalence of the status of time in any types of mechanical formalisms
The statistical knowledge of the high dimensions system requires (i) the definition of a Liouville measure μL based on the symplectic structure of the phase space and (ii) the value of the configuration distribution ZC, dμ ∼ (1/ZC) e−βH, with β = 1/kBT related to the inverse of the temperature. This point of view is discretized in quantum mechanics (QM)
Summary
As suggested in recent studies [24, 25], a heuristic reasoning about the symmetries and automorphisms backed on Zατn led us to assume (i) that the Riemann conjecture concerning the distribution of the non-trivial zeros of zeta function ζ(s) = 0 could be validated starting from physical arguing by using self-similar properties of ζ(s) obvious from Cole and Cole impedance and recursive dynamics [21, 26], (ii) that the complex variable s = α +iφ associated to the metric of the geometry through d = 1/α accommodates, through its complex component, something of the formal nature of the concept of arrow of time and (iii) that according to the consequence of Montgomery hypothesis, QM states should be related to the set of zeros, and joined to the disappearance of above time intrinsic arrow. Both approaches should be theoretically bonded via the existence of a renormalization group over Zατn capable of compressing the scaling ambiguities characterizing the singularities of the fractional dynamics: scaling extension of figure 1 for tiling the Poincaré half plan [16, 29]
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