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

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.Comment: 13 pages, 2 figure

A mathematical structure for the generalization of the conventional algebra

An abstract mathematical framework is presented in this paper as a unification of several deformed or generalized algebra proposed recently in the context of generalized statistical theories intended to treat certain complex thermodynamic or statistical systems. It is shown that, from mathematical point of view, any bijective function can be used in principle to formulate an algebra in which the conventional algebraic rules are generalized

Fractal geometry, information growth and nonextensive thermodynamics

This is a study of the information evolution of complex systems by geometrical consideration. We look at chaotic systems evolving in fractal phase space. The entropy change in time due to the fractal geometry is assimilated to the information growth through the scale refinement. Due to the incompleteness of the state number counting at any scale on fractal support, the incomplete normalization $\sum_ip_i^q=1$ is applied throughout the paper, where $q$ is the fractal dimension divided by the dimension of the smooth Euclidean space in which the fractal structure of the phase space is embedded. It is shown that the information growth is nonadditive and is proportional to the trace-form $\sum_ip_i-\sum_ip_i^q$ which can be connected to several nonadditive entropies. This information growth can be extremized to give power law distributions for these non-equilibrium systems. It can also be used for the study of the thermodynamics derived from Tsallis entropy for nonadditive systems which contain subsystems each having its own $q$. It is argued that, within this thermodynamics, the Stefan-Boltzmann law of blackbody radiation can be preserved.Comment: Final version, 10 pages, no figures, Invited talk at the international conference NEXT2003, 21-28 september 2003, Villasimius (Cagliari), Ital

Arrows of times, non-integer operators, self-similar structures, zeta functions and Riemann hypothesis: A synthetic categorical approach

© 2017 L & H Scientific Publishing, LLC. The authors have previously reported the existence of a morphism be- tween the Riemann zeta function and the "Cole and Cole" canonical transfer functions observed in dielectric relaxation, electrochemistry, mechanics and electromagnetism. The link with self-similar struc- tures has been addressed for a long time and likewise the discovered of the incompleteness which may be attached to any dynamics con- trolled by non-integer derivative operators. Furthermore it was al- ready shown that the Riemann Hypothesis can be associated with a transition of an order parameter given by the geometric phase at- tached to the fractional operators. The aim of this note is to show that all these properties have a generic basis in category theory. The highlighting of the incompleteness of non-integer operators considered as critical by some authors is relevant, but the use of the morphism with zeta function reduces the operational impact of this issue with- out limited its epistemological consequences

Complex fractional differential operators geometrical phase transition and Riemann Conjecture

The authors show the existence a bi univocal application between Riemann zeta functions and dynamic processes under the control of Non Integer Differential Operator. They show that, in the Fourier space, Riemann zeta function is related to hyperbolic geodesics with angles at infinity determined by the non integer parts of the power laws. The authors assert that Riemann Conjecture can be considered as a geometrical phase transition based upon the cancelation of the geometrical symmetries at infinity. A quasi self similarity of the zeta functions is associated to the self similarity of the dynamics. This characteristic assures the validity of the Riemann conjecture. © 2013 IFAC

Role of Riemann's and Goldbach's hypotheses in the behaviour of complex systems: Introduction to the concept of "sciances"

The authors have already established a bi univocal correspondence between Riemann zeta functions and dynamic processes under the control of integro-differential operator of non-integer complex order. We recall that the Riemann zeta function can then be related to hyperbolic geodesics whose angles at the boundary are determined by the real part of the power laws that define the Riemann series. It is suggested that Riemann's conjecture can be reduced to a geometrical phase transition with a reduction of the parameter of order resulting from the combination of a pair symmetries associated with a quasi-self similarity of geodetics. The well-known relationship with the set of prime numbers must be considered as the result of the local existence of stationary 'state' in the dynamics. The work is focused on the 'non stationary' behaviour of Riemann zeta function. It is shown that the main characteristic of the dynamics of complex systems may be associated to a hybridizing between a pair of states and/or processes able to give a geometrical status to the concept of the time and equally to the concept of energy. It is based on the mirror properties of complementary zeta function. It is shown also that the set of prime numbers, which controls the transitions between 'states', is the simplest form of the complexity. This analysis suggests the existence of a mathematical relationship between Riemann's and Goldbach's hypothesis. Such relationship would be the base of an extension of the principles of the science for the analysis of the complexity. According to previous proposal we name this extension that enlarges the principles of the science : sciance with 'a'