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 algebra...
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irk-123456789-1570072019-06-20T01:27:06Z From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis Riot, P. A. Le Méhauté 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. Новизна останнiх наукових робiт Жана-П’єра Бадiалi бере початок з квантового пiдходу, який базується на (i) поверненнi до поняття траєкторiй (траєкторiї Фейнмана), а також на (ii) необоротностi квантових переходiв. Цi iконокластичнi варiанти знову встановлюють гiльбертiан i алгебраїчну точку зору фон Неймана, маючи справу зi статистикою за циклами. Цей пiдхiд надає зовнiшню термодинамiчну першопричину поняттю квантової одиницi часу (термальний час Ровеллi Коннеса). Це поняття, базис для квантування, виникає тут як простий критерiй розрiзнення мiж квантовим режимом i термодинамiчним режимом. Мета цiєї статтi є розкрити змiст останнiх п’яти рокiв наукових дискусiй, нацiлених з’єднати в когерентну схему як уподобання i роботи Жана-П’єра, так i роботи авторiв цiєї статтi на основi гiперболiчної геодезiї, i об’єднуючу роль дзета-функцiї Рiмана. Хоча цi варiанти не представляють жодних протирiч, тим не менше, вони породжують власну стрiлу часу, iнакшу нiж термальний час. Питання фiзичного змiсту гiпотези Рiмана як основи квантової механiки, що було в центрi наших останнiх дискусiй, є суттю цiєї статтi. 2017 Article From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis / P. Riot, A. Le Méhauté // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33001: 1–13. — Бібліогр.: 51 назв. — англ. 1607-324X PACS: 05.30.-d, 05.45.-a, 11.30.-j, 03.65.Vf DOI:10.5488/CMP.20.33001 arXiv:1710.01558 http://dspace.nbuv.gov.ua/handle/123456789/157007 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
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. |
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Article |
| author |
Riot, P. A. Le Méhauté |
| spellingShingle |
Riot, P. A. Le Méhauté From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis Condensed Matter Physics |
| author_facet |
Riot, P. A. Le Méhauté |
| author_sort |
Riot, P. |
| title |
From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis |
| title_short |
From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis |
| title_full |
From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis |
| title_fullStr |
From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis |
| title_full_unstemmed |
From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis |
| title_sort |
from the arrow of time in badiali's quantum approach to the dynamic meaning of riemann's hypothesis |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/157007 |
| citation_txt |
From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis / P. Riot, A. Le Méhauté // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33001: 1–13. — Бібліогр.: 51 назв. — англ. |
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Condensed Matter Physics |
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2025-07-14T09:21:17Z |
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2025-07-14T09:21:17Z |
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| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 3, 33001: 1–13
DOI: 10.5488/CMP.20.33001
http://www.icmp.lviv.ua/journal
From the arrow of time in Badiali’s quantum
approach to the dynamic meaning of Riemann’s
hypothesis
P. Riot1, A. Le Méhauté1,2,3
1 Franco-Quebecois Institute, 37 rue de Chaillot, 75016 Paris, France
2 Physics and Information Systems Departments, Kazan Federal University,
18–35 Kremlevskaia St., 480 000 Kazan, Tatarstan, Russia Federation
3 Materials Design, 18 rue Saisset, 92120 Montrouge, France
Received April 19, 2017, in final form June 8, 2017
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.
Key words: path integrals, fractional differential equation, zeta functions, arrow of time
PACS: 05.30.-d, 05.45.-a, 11.30.-j, 03.65.Vf
1. From algebraic analysis of quantummechanics to “irreversible” Feyn-
man paths integral
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. For example, within Newtonian vision, the observable f can be analysed algebraically using
action-integral through the Lagrangian L while Poisson brackets gives time differential representations
d f /dt = {H, f }. According to Noether theorem, the energy, referred to the Hamiltonian H, is no other
than the tag of a time-shift independence of physical laws, namely a compact commutativity. 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,
therefore 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).
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
33001-1
https://doi.org/10.5488/CMP.20.33001
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
P. Riot, A. Le Méhauté
With regard to quantum perspectives, mechanical formalism introduces (i) a thickening of the me-
chanical dot, (ii) the substitution of real variables through the spectrum of operators and (iii) an emphasis
on the role of probability. According to von Neumann, the stable core of the operator algebra required to
fit the quantum data must be based upon groupoids acting on observables. In the Heisenberg framework,
the observable f̂ (for instance the paradigmatic example of the set of the rays of materials emissions)
is represented by self-adjoint operators in Hilbert space l2(R3,C) which values can be reduced within
Born-matrix representation to a set of eigenvectors |ϕn〉 chosen in the spectrum spec( f̂ ) of the groupoid.
Energy distribution is given through the linear relations Ĥ |ϕn〉 = E |ϕn〉, where the Hamiltonian Ĥ
represents the energy self-adjoint operator. The dynamics is implemented by using the commutator:
[Ĥ, f̂ ] which replaces the Poisson bracket, namely d f̂ (t)/dt = 2πi
h [Ĥ, f̂ ]. The capability of giving cyclic
representations of von Neumann algebra (extended toWeyl non-commutative algebra for standard model)
leads to expressing the dynamics via the eigenvectors Fourier components |ψn(t)〉 = |ϕn〉 exp(−iEnt/~).
This representation is unitarily equivalent to a wavemechanics usually expressed through the Schrödinger
equation, i~ d
dtψ(r, t) = [−
~2
2m∇
2 +V(r, t)]ψ(r, t). The shift from non-linear finite to linear infinite system
must be based upon the statistics dealing with a Λ-extension of the system, through a linear and positive
forms f̂ ∈ A ΦΛ(A) = (1/ZC)Tr exp(−βHΛA). Hence, the average value of the observable f̂ ∈ A is a
trace of an exponential operator. Usually, the distribution of physical data must be given by a measure
of probability on spec(A). Thus, we cannot deal with QM without dealing with Gaussian randomness
imposed by some external thermostat. At this step, a useful notion is the notion of density matrix given
by: ρN = exp(−βH). Unfortunately, N the normalization constant suffers from all misgivings involved
in thermodynamics, by the “shaky” notion of equilibrium.
1Each item of the above visions imposes its own algebraic constraints but enforces a paradigmatic
concept of time parameter [3] as a reversible ingredient of the physics. At this step, the statistics appears
as the only loophole capable of introducing irreversibility as a path to an assumptive equilibrium state
for finite β value. Nevertheless, as shown above, this assumption requires the Λ-extension, namely, the
transfer of the operator algebra in the framework of C*-algebra in which the A-algebra of its Hermitian
elements patterns the transfer (rays) between a set of perfectly well defined states. Starting from the
notion of groupoid and from the algebra of magma upon the states and by analysing the symmetries, a
mathematician can also consider the equilibrium from a set of cyclic states Ω of f̂ , based on Gelfand,
Naimark, Segal construction (GNS construction) [5] binding quantum states and the cyclic states (cyclic
transfer which assumes a specific role of scalar operators, called M-factors). At this stage, two points of
viewmust be matched together to make the irreversibility emerge from M: (i) Tomita Takesaki’s dynamic
theory [6] extended by Connes [3, 4] and (ii) Kubo, Martin and Schwinger KMS physical principle [7].
• According to Tomita-Takesaki, if A is a vonNeumann algebra, there exists a modular automorphism
group∆ based on a sole parameter t:αt = ∆−it A∆+it which leaves the algebra invariant: dαt (A)/dt =
limΛ→∞(2πi/h)[HΛA]. There is a canonical homomorphism from the additive group of reals to the
outer automorphism group of A: B, that is independent of the choice of “faithful” state. Therefore,
〈Ω, B(αt+j A)Ω〉 = 〈Ω, (αt A)BΩ〉, where ( , ) is the inner product.
• The link with KMS physical constraint extends this abstract point of view. The dynamics expression
using the Kubo density matrix allows one to change the “shaky” hypothesis of thermodynamic
equilibrium by giving it a dynamical expression. KMS suggested to define the equilibrium from
a correlation function [(γt A)B] = [B(γt+iβhA)] allowing to associate the equilibrium with a
Hamiltonian according to γt A = exp(itH/h).A. exp(−itH/h).
The matching of both sections leads: βh = 1 which is nothing but the emergence of a thermodynamic
gauge of time while the time variable stays perfectly reversible [3].
Starting from this analysis Jean Pierre Badiali (JPB) decided the exploration of QM by using the
local irreversible transfer joined to Feynman [8] path integrals model based on an iconoclast existence
of 2D self-similar “trajectories”. While this model suffers from mathematical divergences and requires
questionable renormalisation operations, Feynman model efficiency was rapidly attested. Nevertheless,
1The extension of l2(R3, C) toward l2(R3, C2×2) shifts the second order equation onto a first order equation with observables
then based upon matrix values. This shift gives birth to the Dirac operator whose algebra founds the spineur standard model of
physics. It is clearly based on inner automorphisms and internal symmetries [3, 4].
33001-2
From the arrow of time in Badiali’s quantum approach to the dynamic meaning of Riemann’s hypothesis
many physicists still considered that Feynman integrals are meaningless because the concept of trajec-
tory should “obviously” not be relevant in QM. The discernment of JPB was to take the same trail
as Feynman, by imagining irreversible series of transition giving birth to real self-similar paths at
particles. By using a Feynman Kac transfer formula for conditional expectation of transfer, he writes
q(x0, t0, x, t) =
∫
Dx(t) exp[− 1
~ A(x0, t0; x, t)] in which the rules of transfer are based on a Newtonian
action A(x0, t0; x, t) =
∫
{ 1
2 m[ dx(s)ds ]
2 + u[x(s)]}ds, he wrote the solution required for discretizing the tra-
jectories [9]. These notions are not associated with any natural Hamiltonian and require a coarse graining
of the space-time. To overcome this constraint, JPB considered the couple of functional probabilities
φ(x, t) =
∫
dyφ0(y)q(y, t0; x, t) and φ̂(xt) =
∫
q(t, x; t1, y)φ1(y)dy with t1 > t > t0. The evolution of a
system is given by a Laplacian propagator in which φ(x, t) is bended out by geometrical potential u(x, t)
according to± ∂
∂t φ(x, t)+D∆φ(x, t) = 1
~u(x, t)φ(x, t), where D = ~/2m is the quantic expression of a diffu-
sion constant and φ(x, t) cannot be normed. These equations are neither Chapman-Kolmogorov equations
nor Schrödinger like equations. Thenceforth, which physical and geometrical meaning may we attribute
to the discreet arithmetic site on which the fractal-paths are based? How do the morphisms between states
and trajectories determine the dynamical topos? How the statistical or non-statistical regularizations rul-
ing the dynamics may smooth the experimental behavior? All these issues are open. To solve them, JPB
point of view required a new visitation of the thermodynamics and in conformity with KMS point of view,
a new definition of the equilibrium expressed via the irreversibility of the local transfer. To do this, he
considered the class of the paths reduced to loops: φ(x0, t0, x0, t − t0) and their fluctuations in energy. As-
suming an average energy U determined by a thermostat, the overall fluctuations are ruled by a deviation,
on the one hand, from the reference value U and, on the other hand, from the number of loops concerned.
As Feynman had imagined it, an entropy function: Spath = kB ln
∫
q(x0t0, x0, t − t0)dx0 can be built which
is ruled by the concept of path temperature Tpath: ~kB (1/Tpath) = τ + [U − (〈uK 〉path + 〈up〉path)]
∂τ
∂u . The
emergence of an equilibrium is figured dynamically through a critical time scale 1/βh, which possesses
a strictly quantum statistical origin merely based on loops τ = (~/kB)Tpath if it can be assumed that the
temperature of the integral of the path is none other than the usual thermodynamic temperature. From
this step, JPB finds again the Rovelli-Connes assertions regarding thermal-time [10] and he proves the
Boltzmann H-theorem. By means of subtle analysis using the duality of the couple propagators (forward
and backward dynamics), he built a complex function ψ, solution of the Schrödinger equation. The
thought of JPB appears as a subtle adventure which — inscribed in the footsteps of Richard Feynman,
and implemented from a deep knowledge of QM, thermodynamics, thermochemistry and irreversible
processes — changes the traditional point of view and builds a perspective that we have to analyze now,
from an alternative point of view which replaces the transport along the fractal trail by a transfer across
an interface, both perspectives being strongly related. In brief, 1/β provides a scale of energy which
smooths the regime of quantum fluctuations according to an uncertainty relation: ∆E = ~/∆t < 1/β
namely ∆t > β~. β~ is the value of the time defining the cut-off between quantum fluctuations and ther-
modynamic fluctuations. The propagation function imparts a quadratic form to the spatial fluctuations,
namely δx2 = (β~/∂t − 1)∂t2/m. If ∆t = β~/2 then δx2 = β~2/m = 2β~D ∼ δt, the value that, with
the reserve of taking into account the entropy constant kB, must be compared to de Broglie’s length.
Thus, the coarse graining of the time will be considered as the dual of the quadratic quantification of
space, when a length in this space can be reduced to the constraints imposed by the geometrical pattern
of non-derivable trajectories (herein with a dimension two attributed implicitly and for quantum physical
reasons to the set of Feynman paths). The aim of this note is to show that this “cognitive skeleton” does
not only give birth to thermal statistical time, but through a generalization of fractal dimension, to a
purely geometric irreversible time unit: an arrow of time.
2. Zeta function and “α-exponantiation”
In addition to B. Mandelbrot initial friendship, we owe to J.P. Badiali and Professors I. Epelboin
and P.G. de Gennes the first academic support for the development of the industrial TEISI model energy
transfer on self-similar (i.e., fractal interfaces). This was at the end of seventies shortly before the
premature death of Professor Epelboin. The purpose of this model was to explain the electrodynamic
33001-3
P. Riot, A. Le Méhauté
behavior of the lithium-ion batteries which were then at the stage of their first industrial predevelopment
[11–13]. The interpretation and the patterning of electronic and ionic transfer coupled together in 2D
layered positive materials (TiS2,NiPS3) are very similar to the JPB model. The electrode is characterized
by a fractal dimension d which, due to the symmetries of real space, must be such as d ∈ [1, 2]. When the
fractal structure of the electrode is scanned by the transfer dynamics through electrochemical exchanges,
the electrode does not behave like an Euclidean interface, as a straightforward separation between two
media, but like an infinite set of sheets of approximations normed by η(ω) or a multi-sheet manifold,
thick set of self-similar interfaces working as paralleled interfaces [14], where ω is a Fourier variable.
Each η(ω)-interface is tuned by a Fourier component of the electrochemical dynamics. The overall
exchange is ruled by a transfer of energy either supplied by a battery (discharge) or stored inside the
device (charge). The impedance of positive electrode is expressed through convolution operators coupled
with the distributions of the sites of exchanges (electrode), giving birth at macroscopic level to a class
of non-integer differential operators which take into account the laws of scaling, from quantum scales of
transfer up to the macroscopic scales of measurement. This convolution between the discreet structure of
the geometry and the dynamics must be written in Fourier space by using an extension of the Mandelbrot
like fractal measure namely Nηd = 1, into operator-algebra with N = iωτ [11]. Mainly, the model
emphasizes the concept of fractal capacity (fractance) — implicitly Choquet non-additive measure and
integrals — whose charge is ruled by the non-integer differential equation i ∼ dαU/dtα with α = 1/d
[15, 16], where U is the experimental potential. In the simplest case of the first order local transfer,
hence, for canonical transfer, the Fourier transform must be expressed through Cole and Cole type of
impedance: Zα(ω) ∼ 1/[1 + (iωτ)α] [17–20] which is a generalization of the exponential transfer turned
by convolving with the d-fractal geometry. Many other interesting expressions and forms can be found,
but being basically related to exponential operator, the canonic form appears as seminal. The model was
confirmed experimentally in the frame of many convergent experiments concerning numerous types of
batteries and dielectric devices. JPB has advised all these developments especially within controversies
and intellectual showdowns. For instance, even if energy storage is at the heart of all engineering purposes
[12, 13], the use of non-integer operators renders the model accountable of the fact that energy is no
longer a natural Noetherian invariant of the new renormalizable representation. Therefore, algebraic and
topological extensions must be considered whose results are the emergence of time-dissymmetry and
of entropic-effects. Fortunately, Zα(ω) clearly appears as a geodesic of a hyperbolic space ηd(ω) = u
v
authorizing (figure 1), on the one hand, the use of non-Euclidean metrics to establish a distance between
η(ω)-interfacial sheets and, on the other hand, the tricky algebraic and topological extension of the
dynamics, practically a dual fractional expression of the exponential. The extension to “dual α-geodesic”
(Zτnα = Zα ∪ {τn}) shown in figure 1 is able to formalize the main characteristics of a global fractional
dynamics [16] which retrieves, as we can show below, a capability to rebuild, through the addition of
entropic factors, the contextual meaning of the physical process. We qualified “α-exponantiation” (with
“a”) this new global dynamics (figure 1). This denomination integrates the phase angle ϕ(α) (figure 1),
namely the symmetries and inner dynamic automorphisms caused by the d-fractal geometry. Let us
observe that if s = α + iϕ(α) a new reference, 0 < ϕ(s) < ϕ(α) defines a compact set K able to be
considered as a base for Tomita’s shift s → s + it implemented in the complex field (see below the
N fibration). In addition, let us observe that the expression of ϕ(s) requires a referential system which
can be given either with respect to experimental data (figure 1) or with regard to an a priori referential
obtained after a π/4 rotation, supposing the use of a Laplacian paradigm in which ∆(s) (figure 2) is used
as a new expression for phase-reference in place of ϕ(s). The reason of the relevance of this duality is
an extremely deep physical meaning: according to non-integer dynamic model if the transfer process is
implemented across a Peano curve (Feynman paths, nil co-dimension, no outer operator), then α = 1/2
and ∆(s) = 0, then the overall impedance recovers an inverse Fourier transform and the dynamic measure
fits a probability. The traditional concept of energy recovers its practical relevance and the space time
relationship becomes coherent with the use of the Laplacian and Dirac operators. The time used is
the reversible time of the mechanics. Conversely, if α , 1/2, the inverse Fourier transform does not
exist and, therefore, the traditional concept of time vanishes, retrieving the mere arithmetic operator
status implemented in the TEISI model, namely N = iωτ. Time has no longer any straightforward usual
meaning. These strange conclusions about “time” as well as the issues about “energy”, left many academic
colleagues dubious late in 1970-ies, but not JPB who found in these issues many reasons for reviving
33001-4
From the arrow of time in Badiali’s quantum approach to the dynamic meaning of Riemann’s hypothesis
Figure 1. (Color online) Main characteristics of exponential transfer function convoluted by d-fractal
geometry 1 < d 6 2. If d is the rightful non integer dimension of underlined geometry (TEISI model
[11, 14]) the transfer function Zα(ω), — named Cole and Cole impedance in electrodynamics [11, 17–
19], — finds its expression in 1/d-exponantial (with “a”), a kind of degenerated time function in real
space shaped by a hyperbolic geodesic close to an exponential in Fourier space. Zα(ω) is a hyperbolic
geodesic in Fourier space. Reduced to its discrete representation in N or Q such as n = u/v or v/u
rational, ZN
α (n) appears as the basis for the definition of ζ(s) Riemann function (s = α+ it) if it expresses
a mathematical fibration (figure 2). The singular points {0, 1,∞} suggest links between 1/d-exponantial
and Teichmuller-Grothendick absolute Galois group. Adjoined with its categorical Kan extension τ:
{τn} obtained through the prime-decomposition n = ωτn ∈ N and playing the role of inverse Fourier
transform, the above representation shapes, — via the pair of geodesics building the semi-circle, — the
functional relationship between ζ(s) and ζ(1 − s̄) [21–23].
the electrochemical and electrodynamical concepts. Although these disturbing issues did stay open, the
TEISI experimental efficiency suggested that there was something deeper and more fundamental behind
the model; but which thing? . . . Our obstinacy to believe in the physical meaning of the TEISI model was
rewarded early this century, by discovering that the canonical Cole and Cole impedance is closely related
with the Riemann zeta function properties [21] and that these properties are explicitly associated to the
phase-locking of fractional differential operators. ZN
α (n), the integer discretization of the Cole and Cole
impedance Zα(ω), is characterized by the hyperbolic-dynamic metric given by (u/v) = 1/nα. Therefore,
the overall discretization of Zτnα appears as a possible grounding for the definition of both Riemann zeta
function ζ(s) and ζ(1 − s̄), where s = α + iϕ(α) ∈ C, and 1/2 6 Re s < 1 [21] with an emphasis given
to the arithmetic site {Sτnα } = ∞0 L = {Zα ∩ {τn}} (see [16], page 231). An extension of the concept of
time to complex field t ∈ R ⇒ s ∈ C is a natural result of this discretization. In addition, as suggested
in recent studies [24, 25], a heuristic reasoning about the symmetries and automorphisms backed on Zτnα
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 also 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,
but also joined to the disappearance of above time intrinsic arrow. We recall that the Riemann conjecture
states that the non-trivial zeros of the zeta function ζ(s) = 0 are such as if Re s = α = 1/2 (phase locking
for d = 2), namely, in TEISI model geometrical terms, Riemann hypothesis would be related to Peano
interfaces (2D embedded geometry without any external environment). Due to the similarity between
JPB model and TEISI model which use the irreversible transfer as test functions of the distribution of the
33001-5
P. Riot, A. Le Méhauté
Figure 2. (Color online)Dynamical meaning of Riemann’s hypothesis, phases and arrow of time: Riemann
zeta function ζ(s) is proved to be erected from a mathematical fibration ±it based on an α-exponantiel
represented in the oriented complex plan from ZN
α (ω) ∪ {τn} impedance (figure 1: direct in black and
inverse in grey) when N quantization is carried out with s = α + it. If α , 1/2, the constraint of phase
±∆ , 0 over the fibration (gauge effect) is associated to the entropic properties of the dynamics while
the phase involves a dissymmetry of the fibration parameter ±it, therefore, ζ(s) , ζ(s̄). If Riemann
hypothesis is validated, α = 1/2 value associated to the quadratic self-similarity of the set of integers
N × N = N, underlined Peano geometry. This hypothesis involves ∆ = 0 and then ζ(s) = ζ(s̄). With
respect to α-exponantiel, these properties can be expressed within a pair of vector bases: either based on
phase angle ϕ(α) = π2 (1 − α) (determinism basis) or based on ∆(α) (stochastic basis). The change from
one to the other reference must be associated to a rotation of the dynamic referential keeping all non-
linear properties α , 1/2 of α-geodesics. The ∆(α) referential is, nevertheless, the most fundamental
of both for at least two reasons: (i) The non-commutativity of successive θ-rotations (groups) in the
complex plan of ζ(s) definition can be expressed via the equation θ1 ◦ θ2 = θ2 ◦ θ1e±2i∆, therefore,
∆(α) = 0 implicitly provides a gauge condition for finding again the commutativity. Due to the phase
effect, the symmetry ζ(s) = ζ(1 − s̄), the relation which naturally leads to ζ(s) = 0 as well as to the
existence of Hilbert quantummechanics states, matching here the random distribution of primes numbers
(Montgomery hypothesis) according to the von Neumann algebra and multiplicative self-similarity of
N: N × N = N. The only solution to by-pass the resurgent symmetry of t and to create dissymmetry is
then to introduce statistics from outside via an external artefact: the thermostat (t stays symmetric but
the random fluctuations of local process rebuild the quantum time arrow through a so-called “thermal
time” [9, 10, 27, 28]); (ii) The situation is opposite if the phase moves from zeros ∆(α) , 0. The loss of
symmetry within the fibration keeps the internal non-commutativity leading to an intrinsic irreversibility
of the time which may be named “arrow of time” [2]. Herein, this arrow finds its origin in the open
status of the geometry highlighted through d the non-integer metric of the geometry which founds the
α-exponantiel dynamics [29].
sites of exchanges, the morphisms concerning the scaling and the role of the metric in this operation, leads
to guess that the theory of categories and, moreover, the theory of Topos should be hidden behind the
morphism escorted by the role of zeta function. The authors will consider in these following paragraphs
only the theory of categories.
2.1. Universality of zeta Riemann function
It is well known [30] that Riemann zeta function can be expressed by means of two distinct formula-
tions: (i) additive series ζ(s) =
∑
n∈N n−s and (ii) multiplicative series ζ(s) =
∏
p∈℘(1− p−s)−1. It is also
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From the arrow of time in Badiali’s quantum approach to the dynamic meaning of Riemann’s hypothesis
well known that any analytic function must be expressed through additive series f (s) =
∑
n∈N(ansn). A
duality exists that associates f (s) based on sn and ζ(s) based on n−s; under the reserve of the sign, there is
an inversion of the spaces occupied by the complex argument s, and by the integer n. At this step, the key
point is as follows: the dual functions can be compared using the Voronin theorem [31–33]. This theorem
states that any analytic function, for example a geodesic on a hyperbolic manifold, can be approximated
under conditions set out, by so-called universal functions. The archetype of these functions is precisely
nothing else than Riemann zeta function ζ(s), namely: for K compact in the critical band 1/2 6 α < 1
with a connex complement and for f (s) analytic continuous function in its interior without zeros on
K , ∀ε > 0, lim infT→∞(1/T) × mes{τ ∈ [0,T];max|ζ(s + it) − f (s)| < ε} > 0 if t ∈ [0,T]. The zeta
function being used as reference, the extension of the abscissa according to T →∞ leads to a “crushing”
of the analytical function on the reference ζ(s). In addition, ten years after Voronin did establish his
theorem, Bagchi demonstrated [33] that the validity of the Riemann hypothesis (RH) is equivalent to
the verification of the universality theorem of Voronin in the particular case where the function f (s) is
replaced by ζ(s), namely: ∀ε > 0, lim infT→∞(1/T) × mes{τ ∈ [0,T];max|ζ(s + it) − ζ(s)| < ε} > 0.
Therefore, Bagchi’s inequality asserts that the nexus of RH is the self-similarity of Riemann function,
the property explicitly content in its link with Zτnα [21–23]. Nevertheless, since the distribution of the
zeta zeros is unknown, it must be observed that the restriction concerning the absence of zeros inside the
compact set K does not allow one to apply the Voronin theorem to ζ(s). Therefore, if the validity of RH is
related to ζ(s) self-similarity, this property must emerge within a theoretical status at margin with respect
to the field of the analysis. Practically, this observation urges us to consider RH as a singularity in an
enlarged field of mathematical categories and that is why the authors suggested to introduce the category
theory [34–37] for handling the RH issue [22, 23]. The use of this theory is justified for at least two
reasons: (i) according to the work of Rota [38], the function ζ(s) can easily be expressed in the framework
of partially ordered sets (which forms the basis of all standard dynamics), particular cases of categories;
(ii) since the Leinster works [39, 40], self-similarity as a property of a fixed point must be easily expressed
by using the language of categories. Experts in algebraic geometries will consult with profit the reference
[41]. The reader of this note will be able to find in this essay and in the associated lectures [3], the reasons
for which some engineers search for illustrating the profound but also practical signification of the famous
hypothesis. Both approaches should be theoretically bonded via the existence of a renormalization group
over Zτnα 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].
2.2. Design of Epr -space
A category is a collection of objects (a, b, . . .) and of morphisms between these objects. Morphisms
are represented by arrows (a → b) which can physically account for structural analogies or dynamics
relations. Two axioms basically rule the theory: (i) an algebraic composition of arrows a → b → c,
pointing out in the framework of set theory the homomorphism: hom(a, c), and (ii) the identity principle
which accounts for an absence of any internal dynamics of the objects (1a: a → a). We must point out
that the compositions of “arrows” can in practice be thought of as order-structures. Within the framework
of enriched categories, there is, in addition, a close link between categories and metric-structures [34].
For instance, the additive monoid (N,+, >) may be substituted by hom(a, b), after the introduction of the
notion of a distance through a normalization of the length of the arrows. N is naturally associated to the
additive law (construction) which provides, through the monoid (N,+), an ordered list of its elements
[1, 2, 3, . . . , n, n + 1, . . .], but it is also associated through the monoid (N,×) to a distinct order structure.
The question of matching both monoids makes it possible to consider it as a mere arithmetic issue, but
the order associated to (N,×) or (N,÷) must over here, — the partition of the set N, — be defined in
the following way: p < q if and only if p divides q. The order is, therefore, only partial because, as it
is well known, any integer may be written in a unique way according to n =
∏
i prii with pi prime and
ri integer while i scans a finite collection of n ∈ N; the order appearing through the set of pi is mainly
different from the order of the set of (N,+). By taking the logarithm, one obtains log(n) =
∑
i ri × log pi
for all n. Mathematically, N with partial order is a “lattice”: any pair of elements x and y has a single
smallest upper bound, in this case, the LCM (Lowest Common Multiple) and a single GLD (Greatest
Lower Divisor). The total order structure itself also constitutes a lattice for which the operators max
33001-7
P. Riot, A. Le Méhauté
and min can be substituted by LCM and the GLD. The elements of a lattice can be quantified by
associating a value v(p) for each p in such a way that for p > q we have v(p) > v(q). According to
Aczel theorem [42–44], there is always a function possessing an inverse such that the linear ordered
discrete set of values makes it possible to match the partial order associated with the multiplicative
monoid (N,×) and the order associated to (N,+, 6). In the above particular case, this result takes the
following form: both monoids (N,+, 6) and (N,×) are in correspondence by means of a logarithm,
so that: log[LCM(p, q)] = log p + log q − log[GLD(p, q)]. The dissymmetry between construction and
partition explains the role of non-linear logarithm function, hence, the paradigm of exponential function
in the physics of close additive systems, while, conversely, the treatment of non-extensive systems stays
always an open issue. Although very elementary, these characteristics of N invite us to introduce a space
of countable infinite dimension Epr which is characterized by an orthogonal vectorial basis indexed by
the quantities log pi , where pi is any prime integer and wherein the vectors have a finite number of
integer coordinates ri , the other coordinates being reduced to zero [22, 23]. Indeed, Epr is remarkably
well adapted to a linearization of the self-similar properties expressed from the discrete framework of N.
The orthogonal character of the basic axis accounts for the fact that the set of prime numbers constitutes
an anti-chain upon the partially ordered set associated with the divisibility, hence, Q the set of rational
numbers, such as defined above. The space Epr corresponds to the positive quadrant of a Hilbert space
in which the norm of unity vector is equal to log pi . It is then easy to introduce the scaling factor using
a parameter based on the complex number −s ∈ C. At coordinate point, ri is then associated with
the coordinate −s × ri . The space obtained by applying the scaling function s may be noted as N(s)
[23]. The construction of this kind of space using the logarithmic function is all the more relevant in
that its inverse, i.e., the exponential function, can be applied in return. Therefore, the total measure of
the exponential operator can then be easily computed upon the set of integer points constrained by a
complex power law n−s ∈ N(s) on Epr for any chosen parameter s. This operation gives birth to zeta
Riemann function ζ(s) =
∑
n∈N n−s which finds, therefore, in Epr its natural mathematical sphere of
definition. The zeta Riemann function is the total measure of the exponentiation operator applied upon
the set N(s) when expressed in Epr , and ζ(s) is, therefore, merely the trace of this operator in Epr :
ζ(s) = TrEpr {exp[−s log N(s)]}.
At this step, it is interesting to confront the above analysis to quantum mechanics. For example, one
can observe that Montgomery-Odlyzko hypothesis (MOH) could be based on a specific interpretation of
Epr space. Let us remind that Montgomery considers the identity of distributions between the zeros pair
correlations of the Riemann zeta function and the eigenvalue peer correlations of the Hermitian random
matrices [25]. The conjecture asserts the possibility of regularizing divergent integrals by using a Laplace
operator whose spectra are based upon the N ordered series of vectors 0 6 λ1 6 . . . 6 λn 6 . . . < λ∞.
Then, one can define the zeta spectral function according to the equation ζλ(s) =
∑
n∈N (1/λn)s. This
function is only convergent for s ∈ R but it has an extension in the complex plane. For the hermitian
operator H, we have ζλ(s) = Tr[exp(−s log H)] while det H =
∏
n∈N λn. Therefore, with respect to ζ(s),
the description requires an introduction of the concept of energy according to log(det H) = Tr(log H).
Then, the Mellin transform of the kernel of “heat equation” can then be expressed using: ζ(s) =∫∞
0 ts−1 Tr[exp(−tH)]dt with Tr[exp(−tH)] =
∑
n∈N exp(−tλn) leading to the Riemann hypothesis. But,
being upstream of this specific problem, by highlighting the role of any partial order even for α , 1/2,
Epr founds and, within a certain meaning, generalizes the implicit assumptions in MOH. Epr overcomes
the limiting role of Laplacian operator and the role Hermitian hypothesis which implicitly and a priori
imply the additive properties of the systems concerned, or in other words, admit a priori the validity of the
Riemann hypothesis [the existence of well defined random states associated to the zeros of zeta function:
Re s = α = 1/2]. The categorical link, described above for any values of s, between (N(s),+, 6) and
(N(s),÷) [i.e., (N(s),×)] referred, respectively, to the total order (forward construction) and to the partial
order (backward partition) is well adapted for dealing with non-additive systems, namely a dynamical
conception of being a steady state of arithmetic exchanges, without any additional hypothesis regarding
Re s. Obviously, the above conception can be narrowed to additive systems or steady state if α = 1/2.
According to this overall point of view, the categorical matching between construction and partition
which gives birth to a renormalization group, might be physically expressed through gauge constraints,
namely, intrinsic automorphisms required for closing the system over itself [4, 41]. Many other essential
properties of multi-scaled systems could be unveiled by formalizing the theory from Epr (s) space, even
33001-8
From the arrow of time in Badiali’s quantum approach to the dynamic meaning of Riemann’s hypothesis
if very singular interesting properties arise when, according to Riemann hypothesis, Re s = 1/2, one
introduces additional specific symmetries in Epr such as ζ(s) = ζ(1 − s̄). In general, whatever the α
value, the function ζ(s) is the total measure of the exponentiation operator on the support space N(s) at
a certain scale s while ζ(s) is constrained by Bagchi inequality based upon a time-shift s to s + it, very
identical to the one used in Tomita and KMS relations. In order to analyse a possible analogy between
both approaches, it appears then necessary to analyze how the space N(s) behaves under the shift when
i2 = −1.
2.3. Epr fibration
Let us consider the parameter s = α + iϕ variable in a compact domain K ∈ C such that, α ∈ [1/2, 1]
and 0 6 ϕ 6 ϕmax(α) 6 π/4 (figure 2). According to Borel-Lebesgue theorem, a compact domain in C
is a closed and bounded set for the usual topology of C, directly inherited from the topology associated
with R. The K bounded character is essential for backing the reasoning based on the shift from s to s+ it.
Indeed, by choosing a parameter t ∈ N as a value sufficiently high with respect to the diameter of the
domain K , the shift from s to s + it makes it possible to create a translation of the domain K [23] with a
creation of copies of K: Kt capable of avoiding any overlapping if a relevant period t = τ is rightly chosen.
Thus, t-shifts uplift a fiber above K . In the frame of α-exponantial representation, this characteristic may
be practically applied for folding the dynamic and zeta function if, for instance, K is associated to the
field of definition of Zτnα , while Zα(ω) is used to root ζ(s) on the set {α, ϕmax(α)}. Let us observe in
advance that e±it implements the fibration by starting from the gauge-phase angle ϕmax (figure 2). This
way for understanding the fibration is equivalent to replacing the additive operation (s to s + it) by a
Cartesian product. If we now replace s with s + imτ, where m scans the countless infinite set of integers;
the reciprocal image along the base change is then the fiber product of space N(s) by a discrete straight
line defined by i × m × τ. The total space is characterized by N(s) × {i × m × τ} ' N(s) × N(s) ' N(s).
Thus, the change of the basis does not realize anything else but the bijection N × N = N, characterizing a
well-known quadratic self-similarity characteristic of the set of integers. The self-similar characteristics
of N can be approached by using a particular class of polycyclic semigroups or monoids [45–48]. They
are representable as bounded linear operators of a traditional Hilbert space, of type N(s) herein. The
change of base consists in introducing such a semigroup realizing a fibration based on the self-similarity
N(s) × N(s) = N(s), or a partition within subspaces with co-dimension 1. Each sheet corresponds to
the space above the variable s + i × m × τ0. The value of the Riemann function ζ(s + i × m × τ) is
obtained as the total measure of the exponential operator on each sheet, namely this value is a truncation
of ζ(s). This truncation is the basis of Riemann hypothesis. Let us observe that m which is obtained
after a rearrangement of the numerical featuring corresponding to the isomorphism N × N = N imposes
a distribution of points α + i × m × τ that, in the complex plan, does not mesh the total order given from
(N,+, 6). Above each complex number s ∈ K and along the fiber, an appropriate category exists in Epr
based upon both initial and terminal object N(s) [49–51] leading to the folding of the N and, therefore,
to the second different order. The “disharmony” between both orders involved by the relation N2 = N
has its equivalent in the TEISI model when the previous self-similarity is expressed by η2 × (iωτ) = 1.
The interface of transfer is then a Peano interface, where the complex variable i expresses the fibration
and ωτ = n ∈ N(s) is used for the computation of ζ(s). However, via the operator general equation
ηd ×(iωτ) = 1, this “disharmony” is notified by tagging the sign of t through a phase factor i1/d generally
different from i1/(1−d). Therefore, one should distinguish at least two cases:
First case: time symmetry and the absence of junction phase. To avoid dissymmetry of the phase at
boundary, the singularity of the phase angle must be canceled, namely, ϕ(α) = π/4, ∆ = 0 (figure 2).
The dynamics basis must be expressed through Zτn1/2(s), which gives birth to a folding of ζ(1/2 ± it). RH
becomes associated to the expression of the invariance of t under a change of sign. In terms of phase
transition, t is a parameter of order and α = 1/2 is the tag which points out a singularity of “order” within
the “disorder” ruled by α , 1/2, ζ(s) , 0. The main order which must be considered whatever s ∈ C is
naturally given through ζ(s) = 0.
Second case: existence junction phase. If we take into account the fact that TEISI relation is more
general than the quadratic one and must be considered under its general form: ηd(iωτ) = 1 namely
33001-9
P. Riot, A. Le Méhauté
η × iα(ωτ)α = 1, iα introduces a critical phase angle when fibration is implemented N ×ϕN = N. If
t is the physical time parameter, this relation proves the existence of an arrow of time emerging from
the underlying fractal geometry, if the metric of this geometry requires an environment. The main
mathematical issue revealed by the controversies is then our capability or not of reducing the fractal
dynamics to a stochastic process, namely (ωτ)α → (ω′τ′)1/2. Provided we take into account the phase
angle, the presence of ∆(s) suggests that this transformation could be rightful if a thermodynamical
free energy were considered (Legendre transform). The question which must be also addressed within
an universe characterized by an α-exponantiation with α , 1/2, namely, the disappearance of perfectly
defined Hilbert states, concerns the class of groupoids capable of replacing Hilbert-Poincaré principle.
These troublesome issues occupied the latest scientific conversations I had with Jean Pierre Badiali.
3. Pro tempore conclusion regarding an arrow of time
The definition of a concept of time requires a unit which, within a progressively restrained point of
view from R to N (or Q) should match the set [0, 1]∪]1,∞] onto [0, 1], namely a basic loop. Backed on
the TEISI model and a general α-geodesic which provides a dynamic hyperbolic meaning to Riemann
zeta functions, the use of Epr space and N self-similar category, offers the chance to understand the
ambivalence of the concept of physical time. The ambivalence, that unfolds through a complex value of
time, may be expressed using a pair of clocksHL andHG.HL is related to the additivemonoids. Indeed, as
shown from α-exponantial model, n ∈ N(s = α+ iϕ) can be associated with the scanning of a hyperbolic
distance lHL = (u/v)d = (1/n)d defined on the geodesic Zα(ω) according to the additive monoid (N,+, 6)
(figure 1). The computed hyperbolic “path integral” [21] is no other than ζ(s) =
∑
n∈N n−s. Therefore,
the evolution of the n along the geodesic is ruled by pulsing n. This reversible time can be easily extended
to Q and to R as usually done. However, by arguing the concept of time from such a dynamic context,
one reveals the existence of a second tempo on HG: t = ωτ with t ∈ N capable of being tuned to the first
clock only in the frame of von Neumann (operator) algebra, taking into account an appropriate phase
chord u/v = (iωτ)1/d. Indeed, through the fractal metric, determination of the absolute values of this
tempo and, therefore, the matching of both approaches implies the critical role of the phase ϕ(α) [or ∆(α)
if referred to 1/2-geodesic], the phase which has an impact, without any possible avoidance, on the sign
of the fibration N ×ϕN = N. The second clock HG can be tuned in upon the pulse of the first one HL,
like N×Nmust be tuned on N through a product, by adjusting the edges. Practically, two situations must
be taken into consideration:
• α = 1/2, in the frame of the dynamicmodel, the base of the fibration is the 1/2-exponantial geodesic
which is a degenerate form of the dynamics characterized by the removal of any exteriority. This
form is associated to Riemann hypothesis. The Laplace transform of non-integer operator exists.
The energy fills in its usual Noetherian meaning. The spectrum of the operator applied onN can be
built upon the set of prime numbers giving birth to the category of Hilbert eigenstates. Due to the
quadratic form, the chord of both clocks can be easily obtained. The characteristics of Laplacian
natural equation may account for this tuning which originates in the quadratic self-similar structure
ofN:N2= N also expressed in the TEISI equation iωτ.[η(ωτ)]2 ' 1. The irreversibility of the time
can only have an external origin; the thermal time unit is then nothing else than the unit of time
associated to the Gaussian spatial correlations meshed by the temperature associated to an external
thermostat, which, by locking the type of fluctuations, smooths the Peano interfacial geometry via
a stochastic process. Fortunately, for energy efficiency, the engineering of batteries is not based on
this principle.
• α ∈]1/2, 1], the dynamics is based on the incomplete α-exponantial geodesic. There is not any
natural Laplace transform for such geodesics and the spectrum over N cannot provide any simple
basis for the representation of inner automorphisms joined together in a “bundle” {τn} which
assures a completion, but an entanglement when the closing of the degrees of freedom becomes
the heart of the physical issues. Fortunately, an integral involution can be built whose minimal
expression can be based upon the hybrid complex set of couples {ζ(s) ⊕ ζ(1 − s̄); ζ(s̄) ⊕ ζ(1 − s)}
in which ⊕ expresses the disjoint sum of the basic “geodesics”. {ζ(1 − s̄); ζ(1 − s)} plays the role
33001-10
From the arrow of time in Badiali’s quantum approach to the dynamic meaning of Riemann’s hypothesis
usually devoted to the inverse Laplace transform. These couples of functions that take into account,
through s and s̄, the sign of the fibration (rotation in the complex plane of geodesics) assure the
tuning of the complex dynamics and fix the status of the time taking into account the sign of
fibration. This analytical context brings the two main issues to light.
– The question of commensurability of the couple clocks HL and HG which, as above, can be
physically tuned by using a thermal regularization (entropy production). This regularization
can be based upon a Legendre transform defined from the upper limit of the α-geodesics. This
transform is allowed by the possible thermodynamic involution between α-geodesics and 1/2-
geodesics whose equation |∆(α)|+ |ϕ(α)| = π/4 provides the insurance. This involution might
explain the dissipative auto-organizations, well known in physics as well as the existence of
some optimal values of fractal dimensions in irreversible processes, especially the critical
dimensions, da = 4/3 and dg = 7/4.
– Infinitely more meaningful is the presence of the phase angle ±∆(α) , 0 which imposes
an absolute distingue between both possible signs of the parameter of fibration and a non-
commutativity of the associated operators for folding. In this context, and exclusively in
this context, the reversibility of the cyclic operators, along the fiber must be expressed by
t1.t2 = t2.t1e±2i∆, non-commutative expression fromwhich the notion of “arrow of time” takes
on an irrefutable geometrical signification and, herein, an interfacial physical meaning. The
irreversibility is then clearly based on the freedom of a boundary phase, namely the initial
conditions, when N(×N)ϕ(α) the fibration realized the matching between construction and
partition. Intrinsic irreversibility should then originate from the boundary property. It is then
in the thermodynamic framework that the ±, namely, the difference between “future” and
“past”, must be analyzed, by assigning the emergence of time-energy to the distingue between
the work and the heat. The arrow of time justifies the practical emergence of the distingue
of HL and HG while Legendre transformation can ensure the mathematical validity of the
passage from one to the other of the notions.
These last elements very exactly summarize the content of the ultimate discussions shared with my friend
Jean Pierre Badiali. Starting from Feynman analysis, his talent had assumed that the reversibility of the
time usually required for representing the dynamics of quantum processes should be a very specific case
(closed path integrals) of amore general situation (local dissipation, open path integrals and non-extensive
set) fundamentally based on the local irreversibility and ultimately complicated by the convolution with a
set of non-differential discrete paths. The problem of the “open loops” and their non-additive properties,
will stay as an open issue for him. He assured with courage this uncomfortable position during his last
ten years of research, exploring with me all trails capable of conferring a coherence to his mechanical
approach. His visionmatched, at least partially, the main-stream choices of quantummechanics according
to which the basis of macroscopic irreversibility should be the result of a statistical scaling closure, settled
by the contact with a thermostat or an experimenter. In this paradigmatic framework, the concept of
thermal time has no other physical origins than these externalities. As we have tried to show synthetically
in this note, our last exchanges concerned the possibility of passing this option for building a hybrid point
of view using the role of zeta functions. He attempted without success to introduce this function in his
own model but he understood the deep signification of Riemann hypothesis to describe complex systems
which possess well defined internal states. We had imagined our writing a book together, titled “Issues of
Time”. The disappearance of Jean Pierre has not only suspended this project, but has left us scientifically
fatherless in front of (i) the complexity of all physical open questions, (ii) the urgency of assuring science
that should never reduce to the only technosciences and, furthermore, (iii) that the research of all truths
still hidden within a shadow preserves for ever its human dynamics.
Acknowledgements
The authors would like to thank Materials Design Inc & SARL (Dr. E. Wimmer), the Federal
University of Kazan (Prof. Dr. D. Tayurskii the Professor Abe (MIE University Japan) and the office of
33001-11
P. Riot, A. Le Méhauté
the Université du Québec à Chicoutimi, Institut Franco-Québécois in Paris (Dr. S. Raynal) for the support
of these studies. Gratefulness for the ISMANS Team, especially Laurent Nivanen, Aziz El Kaabouchi,
Alexandre Wang and François Tsobnang for 16 years of fundamental research (1994–2010) about non-
extensive systems and quantum simulation, in collaboration with Jean Pierre Badiali as member of
Scientific committee.
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https://doi.org/10.1016/0167-2738(83)90207-2
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https://doi.org/10.5890/JAND.2017.06.012
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https://doi.org/10.1007/s002200000261
http://arxiv.org/abs/1305.3933v1
https://doi.org/10.1070/IM1975v009n03ABEH001485
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Вiд стрiли часу в квантовому пiдходi Бадiалi до динамiчного
значення гiпотези Рiмана
П. Рiот1, A. лє Меоте1,2,3
1 Франко-Квебекський iнститут, Париж, Францiя
2 Вiддiли фiзики та iнформацiйних систем, Казанський федеральний унiверситет,
Казань, Татарстан, Росiйська Федерацiя
3 Проектування матерiалiв,Монруж, Францiя
Новизна останнiх наукових робiт Жана-П’єра Бадiалi бере початок з квантового пiдходу, який базується на
(i) поверненнi до поняття траєкторiй (траєкторiї Фейнмана), а також на (ii) необоротностi квантових пере-
ходiв. Цi iконокластичнi варiанти знову встановлюють гiльбертiан i алгебраїчну точку зору фон Неймана,
маючи справу зi статистикою за циклами. Цей пiдхiд надає зовнiшню термодинамiчну першопричину
поняттю квантової одиницi часу (термальний час Ровеллi Коннеса). Це поняття, базис для квантування,
виникає тут як простий критерiй розрiзнення мiж квантовим режимом i термодинамiчним режимом.Ме-
та цiєї статтi є розкрити змiст останнiх п’яти рокiв наукових дискусiй, нацiлених з’єднати в когерентну
схему як уподобання i роботи Жана-П’єра, так i роботи авторiв цiєї статтi на основi гiперболiчної геодезiї,
i об’єднуючу роль дзета-функцiї Рiмана. Хоча цi варiанти не представляють жодних протирiч, тим не мен-
ше, вони породжують власну стрiлу часу, iнакшу нiж термальний час. Питання фiзичного змiсту гiпотези
Рiмана як основи квантової механiки,що було в центрi наших останнiх дискусiй, є суттю цiєї статтi.
Ключовi слова: iнтеграли за траєкторiями, диференцiальнi рiвняння в часткових похiдних,
дзета-функцiї, стрiла часу
33001-13
http://arxiv.org/abs/1509.05576v1
https://doi.org/10.1007/BF01836453
http://www.cpt.univ-mrs.fr/~coque/EspacesFibresCoquereaux.pdf
https://doi.org/10.1007/s00233-002-7010-6
https://doi.org/10.1137/0211062
https://doi.org/10.1007/BF01110627
From algebraic analysis of quantum mechanics to ``irreversible'' Feynman paths integral
Zeta function and ``-exponantiation''
Universality of zeta Riemann function
Design of Epr-space
Epr fibration
Pro tempore conclusion regarding an arrow of time
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