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We have long ago derived a theoretical relation between the mass of the electron and the fine structure constant \cite{Nottale1994}, which writes to lowest order $\alpha \ln (m_{\mathbb{P}}/m_e) = 3/8$ (where $m_{\mathbb{P}}$ is the Planck mass). We suggest the existence of a similar relation valid for neutrinos, $\alpha \ln ({m_{\mathbb{P}}}/{m_\nu} ) =1/2$. From this relation, we theoretically predict a lightest neutrino mass $m_{\nu} = m_{\mathbb{P}}\exp (-\alpha^{-1}/2 )=0.0214$ eV. The masses of the two heavier neutrinos, $0.0231$ eV and $0.0552$ eV, can then be obtained from experimental results of neutrino oscillations., Comment: 12 pages, 2 figures

Using the formalism of stochastic embedding developed by [J. Cresson, D. Darses, J. Math. Phys. 48, 072703 (2007)], we study how the dynamics of the classical Newton equation for a force deriving from a potential is deformed under the assumption that this equation can admit stochastic processes as solutions. We focus on two denitions of a stochastic Newton's equation called dierential and variational. We rst prove a stochastic virial theorem which is a natural generalization of the classical case. The stochasticity modies the virial relation by adding a potential term called the induced potential which corresponds in quantum mechanics to the Bohm potential. Moreover, the dierential stochastic Newton equation naturally provides an action functional which sat-ises a stochastic Hamilton-Jacobi equation. The real part of this equation corresponds to the classical Hamilton-Jacobi equation with an extra potential term corresponding to the induced potential already observed in the stochastic virial theorem. The induced potential has an explicit form depending on the density of the stochastic processes solutions of the stochastic Newton equation. It is proved that this density satises a nonlinear Schr{\"o}dinger equation. Applying this formalism for the Kepler potential, one proves that the induced potential coincides with the ad-hoc ''dark potential'' used to recover a at rotation curve of spiral galaxies. We then discuss the application of the previous formalism in the context of spiral galaxies following the proposal and computations given by [D. Da Rocha and L. Nottale, Chaos, Solitons and Fractals, 16(4):565-595, 2003] where the emergence of the ''dark potential'' is seen as a consequence of the fractality of space in the context of the Scale relativity theory.

In order to study the internal dynamics of actual galaxy pairs, we need to derive the probability distribution function (PDF) of true 3D (orbital) intervelocities and interdistances between pair members from their observed (projected) values, and of the pair masses from Kepler's third law. Our Isolated Galaxy Pair Catalog (IGPC) of 13114 pairs \cite{Nottale2018a} is used here for this research. The algorithms of statistical deprojection elaborated in \cite{Nottale2018b} are applied to these observational data. We derive the orbital velocity PDFs for the whole catalog and for several selected subsamples. The interdistance PDF is deprojected and compared to analytical profiles which are expected from semi-theoretical arguments. The PDF of deprojected pair orbital velocities is characterized by the existence of a main probability peak around $\approx 150$ km.s$^{-1}$ for all subsamples of the IGPC as well as for the UGC pair catalog \cite{Chamaraux2016}. The interdistance PDFs of both the projected and deprojected data are described at large distances by the same power law with exponent $\approx -2$. The whole distributions, including their cores, are fairly fitted by King profiles. The mass deprojection yields a mass/luminosity ratio for the pairs of $M/L=(30 \pm 5)$ in Solar units. The orbital velocity probability peak is observed at the same value, $\approx 150$ km/s, as the main exoplanet velocity peak, which points toward a possible universality of Keplerian structures, whatever the scale. The pair $M/L$ ratio is just 5 times the standard ratio for luminous matter, which does not require the existence of non-baryonic dark matter in these systems., Comment: 24 pages, 19 figures, submitted for publication

The anomalous magnetic moment of the muon is one of the most precisely measured quantities in physics. Its experimental value exhibits a $4.2 \, \sigma$ discrepancy $\delta a_\mu=(251 \pm 59) \times 10^{-11}$ with its theoretical value calculated in the standard model framework, while they agree for the electron. The muon theoretical calculation involves a mass-dependent contribution which comes from two-loop vacuum polarization insertions due to electron-positron pairs and depends on the electron to muon mass ratio $x=m_e/m_\mu$. In standard quantum mechanics, mass ratios and inverse Compton length ratios are identical. This is no longer the case in the special scale-relativity framework, in which the Planck length-scale is invariant under dilations. Using the renormalization group approach, we differentiate between the origin of $ \ln x$ logarithmic contributions which depend on mass, and $x$ linear contributions which we assume to actually depend on inverse Compton lengths. By defining the muon constant $\mathbb{C}_\mu=\ln(m_\mathbb{P}/m_\mu)$ in terms of the Planck mass $m_\mathbb{P}$, the resulting scale-relativistic correction writes $\delta a_\mu= -\alpha^2 \, (x \:\ln^3 x)/(8 \; \mathbb{C}_\mu^2)$, where $\alpha$ is the fine structure constant. Its numerical value, $(230 \pm 16) \times 10^{-11}$, is in excellent agreement with the observed theory-experiment difference., Comment: 8 pages, 1 figure, Improved and updated version, account of new experimental results

The proton size has been found, with a $6\,\sigma$ statistical significance, to be larger by 4% when it is measured relatively to the electron than to the muon [Pohl2010,Antognini2013]. We solve this proton radius puzzle by accounting for the relativity of the proton scale. The proton to electron and proton to muon scale ratios are obtained by direct measurement, but their comparison requires a conversion to electron reference which is currently made by assuming the usual law of scale ratio composition, $\rho_ {pe}=\rho_ {p\mu} \times \rho_ {\mu e}$. Using instead the special scale relativistic law $\ln \rho_ {pe}=(\ln\rho_ {p\mu}+\ln\rho_ {\mu e})/(\ln\rho_ {p\mu} \ln\rho_ {\mu e}/(\ln\rho_ {\mathbb{P} e})^2$, where $\mathbb{P}$ denotes the Planck length-scale, the two determinations of the proton radius, showing now a ratio $1.009 \pm 0.008$, recover their agreement within about $1\sigma$. The proton radius puzzle therefore provides one with a highly significant test of the special scale relativity theory., Comment: 4 pages

We develop a new formalism for the study of turbulence using the scale relativity framework (applied in $v$-space according to de Montera's proposal). We first review some of the various ingredients which are at the heart of the scale relativity approach (scale dependence and fractality, chaotic paths, irreversibility) and recall that they indeed characterize fully developped turbulent flows. Then we show that, in this framework, the time derivative of the Navier-Stokes equation can be transformed into a macroscopic Schr\"odinger-like equation. The local velocity PDF is given by the squared modulus of a solution of this equation. This implies the presence of null minima $P_v(v_i)\approx 0$ in this PDF. We also predict a new acceleration component in Lagrangian representation, $A_q=\pm D_v \: d \ln P_v/dv$, which is therefore expected to diverge in these minima. Then we check these theoretical predictions by data analysis of available turbulence experiments: (1) Empty zones are in effect detected in observed Lagrangian velocity PDFs. (2) We give a direct proof of the existence of the new acceleration component by directly identifying it in the data of a laboratory turbulence experiment. (3) It precisely accounts for the bursts and calm periods of the intermittent acceleration observed in experiments. (4) Moreover, the shape of the acceleration PDF can be analytically predicted from $A_q$, and this theoretical PDF precisely fits the experimental data, including the large tails. (5) Finally, numerical simulations of this new process allow us to recover the observed autocorrelation functions of acceleration magnitude and the exponents of structure functions., Comment: 40 pages, 23 figures

The present paper is devoted to the construction of a catalog of isolated galaxy pairs extracted from the HyperLEDA extragalactic database. The radial velocities of the galaxies in the pairs are in the range $[3000,16000]$ km.s$^{-1}$. In order to get an unbiased pair catalog as complete as possible, we have limited the absolute magnitude of the galaxies to $M \leq-18.5$). The criteria used to define the isolated galaxy pairs are the following: 1) Velocity criterion: radial velocity difference between the pair members $\Delta V<500$ km.s$^{-1}$; 2) Interdistance criterion: projected distance between the members $r_p<1$ Mpc; 3) Reciprocity criterion: each member is the closest galaxy to the other one, which excludes multiplets; 4) Isolation criterion: we define a pair as isolated if the ratio $\rho=r_3/r_p$ of the projected distance of the pair to its closest galaxy (this one having a velocity difference lower than 500 km.s$^{-1}$ with respect to the pair) and the members projected interdistance $r_p$ is larger than 2.5. We have searched for these closest galaxies first in HyperLEDA M-limited source catalog, then in the full one. We have managed not to suppress the small number of pairs having close-by but faint dwarf galaxy companions. The galaxy pair catalog lists the value of $\rho$ for each isolated pair. This method allows the user of the catalog to select any isolation level (beyond the chosen limit $\rho>2.5$). Our final catalog contains 13114 galaxy pairs, of which 57\% are fairly isolated with $\rho>5$, and 30 \% are highly isolated with $\rho \geq 10$., Comment: 13 pages, 6 figures, catalog of 13114 galaxy pairs available electronically. arXiv admin note: text overlap with arXiv:1611.02618

This article is the second in a series of two presenting the Scale Relativistic approach to non-differentiability in mechanics and its relation to quantum mechanics. Here, we show Schroedinger's equation to be a reformulation of Newton's fundamental relation of dynamics as generalized to non-differentiable geometries in the first paper \cite{paper1}. It motivates an alternative interpretation of the other axioms of standard quantum mechanics in a coherent picture. This exercise validates the Scale Relativistic approach and, at the same time, it allows to identify macroscopic chaotic systems considered at time scales exceeding their horizon of predictability as candidates in which to search for quantum-like structuring or behavior., Comment: 17 pages, no figures, submitted to Phys. Rev. E This article was merged with another which will be replaced by the combined version

The present paper is devoted to the construction of a catalog of isolated galaxy pairs from the Uppsala Galaxy Catalog (UGC), using accurate radial velocities. The UGC lists 12921 galaxies to declination larger than -2 deg 30 min and is complete to an apparent diameter of 1 arcmin. The criteria used to define the isolated galaxy pairs are the following: 1) Velocity criterion: radial velocity difference between the members lower than 500 km/s; 2) Interdistance criterion: projected distance between the members smaller than 1 Mpc; 3) Reciprocity criterion: each member is the closest galaxy to the other one, which excludes multiplets; 4) Isolation information: the catalog lists the ratio between the projected distance to the closest UGC galaxy (having a velocity difference smaller than 500 km/s) and the pair members interdistance, thus allowing one to choose any isolation criterion (beyond the chosen limit 2.5). In addition, we have accounted for the small diameter bias by searching for CGCG galaxies in the pair environment and used the same isolation criterion. A peculiar investigation has allowed to gather very accurate radial velocities for pair members, from high quality HI and optical measurements (median uncertainty on velocity differences 10 km/s). Our final catalog contains 1005 galaxy pairs. Then we give some global properties of the pair catalog. We display the histograms of the radial velocity differences between the pair members and of their projected interdistances (median 0.29 Mpc). Finally, we provide an estimate of the contamination by cosmological false "pairs", which is about 10 percent up to a velocity difference of 380 km/s, beyond which all pairs are probably false., Comment: 28 pages, 7 figures, catalog available in VizieR

We review recent theoretical developments, which suggest that a set of shared principles underpin macroscopic quantum phenomena observed in high temperature super conducting materials, room temperature coherence in photosynthetic processes and the emergence of long range order in biological structures. These systems are driven by dissipative systems, which lead to fractal assembly and a fractal network of charges (with associated quantum potentials) at the molecular scale. At critical levels of charge density and fractal dimension, individual quantum potentials merge to form a charged induced macroscopic quantum potential, which act as a structuring force dictating long range order. Whilst the system is only partially coherent (i.e. only the bosonic fields are coherent), within these processes many of the phenomena associated with standard quantum theory are recovered, with macroscopic quantum potentials and associated forces having their equivalence in standard quantum mechanics. We establish a testable hypothesis that the development of structures analogous to those found in biological systems, which exhibit macroscopic quantum properties, should lead to increased critical temperatures in high temperature superconducting materials. If the theory is confirmed it opens up a new, systematic, ab initio approach to the structural development of these types of materials., Comment: 10 pages, 4 figures

Based on laboratory based growth of plant-like structures from inorganic materials, we present new theory for the emergence of plant structure at a range of scales dictated by levels of ionization (charge density), which can be traced directly back to proteins transcribed from genetic code and their interaction with external sources of charge (such as CO2) in real plants. Beyond a critical percolation threshold, individual charge induced quantum poten- tials (driven by dissipative systems) merge to form a complex, interconnected geometric web, creating macroscopic quantum potentials, which lead to the emergence of macroscopic quantum processes. The assembly of molecules into larger, ordered structures operates within these charge-induced coherent bosonic fields, acting as a structuring force in competition with exterior potentials. Within these processes many of the phenomena associated with standard quantum theory are recovered, including quantization, non-dissipation, self-organization, confinement, structuration conditioned by the environment, environmental fluctuations leading to macroscopic quantum decoherence and evolutionary time described by a time dependent Schrodinger-like equation, which describes models of bifurcation and duplication. The work provides a strong case for the existence of quintessence-like behaviour, with macroscopic quantum potentials and associated forces having their equivalence in standard quantum mechanics and gravitational forces in general relativity. The theory offers new insight into evolutionary processes in structural biology, with selection at any point in time, being made from a wide range of spontaneously emerging potential structures (dependent on conditions), which offer advantage for a specific organism. This is valid for both the emergence of structures from a prebiotic medium and the wide range of different plant structures we see today., Comment: 47 pages, 19 figures

This article motivates and presents the scale relativistic approach to non-differentiability in mechanics and its relation to quantum mechanics. It stems from the scale relativity proposal to extend the principle of relativity to resolution-scale transformations, which leads to considering non-differentiable dynamical paths. We first define a complex scale-covariant time-differential operator and show that mechanics of non-differentiable paths is implemented in the same way as classical mechanics but with the replacement of the time derivative and velocity with the time-differential operator and associated complex velocity. With this, the generalized form of Newton's fundamental relation of dynamics is shown to take the form of a Langevin equation in the case of stationary motion characterized by a null average classical velocity. The numerical integration of the Langevin equation in the case of a harmonic oscillator taken as an example reveals the same statistics as the stationary solutions of the Schrodinger equation for the same problem. This motivates the rest of the paper, which shows Schrodinger's equation to be a reformulation of Newton's fundamental relation of dynamics as generalized to non-differentiable geometries and leads to an alternative interpretation of the other axioms of standard quantum mechanics in a coherent picture. This exercise validates the scale relativistic approach and, at the same time, it allows to envision macroscopic chaotic systems observed at resolution time-scales exceeding their horizon of predictability as candidates in which to search for quantum-like dynamics and structures., Comment: 30 pages, 4 figures

A new, theoretical approach to macroscopic quantum coherence and superconductivity in the p-type (hole doped) cuprates is proposed. The theory includes mechanisms to account for e-pair coupling in the superconducting and pseudogap phases and their inter relations observed in these materials. Electron pair coupling in the superconducting phase is facilitated by local quantum potentials created by static dopants in a mechanism which explains experimentally observed optimal doping levels and the associated peak in critical temperature. By contrast, evidence suggests that electrons contributing to the pseudogap are predominantly coupled by fractal spin waves (fractons) induced by the fractal arrangement of dopants. On another level, the theory offers new insights into the emergence of a macroscopic quantum potential generated by a fractal distribution of dopants. This, in turn, leads to the emergence of coherent, macroscopic spin waves and a second associated macroscopic quantum potential, possibly supported by charge order. These quantum potentials play two key roles. The first involves the transition of an expected diffusive process (normally associated with Anderson localization) in fractal networks, into e-pair coherence. The second involves the facilitation of tunnelling between localized e-pairs. These combined effects lead to the merger of the super conducting and pseudo gap phases into a single coherent condensate at optimal doping. The underlying theory relating to the diffusion to quantum transition is supported by Coherent Random Lasing, which can be explained using an analogous approach. As a final step, an experimental program is outlined to validate the theory and suggests a new approach to increase the stability of electron pair condensates at higher temperatures., Comment: 39 pages, 4 figures

We review in this paper the use of the theory of scale relativity and fractal space-time as a tool particularly well adapted to the possible development of a future genuine theoretical systems biology. We emphasize in particular the concept of quantum-type potentials, since in many situations the effect of the fractality of space -- or of the underlying medium -- amounts to the addition of such a potential energy to the classical equations of motion. Various equivalent representations -- geodesic, quantum, fluid mechanical, stochastic -- of these equations are given, as well as several forms of generalized quantum potentials. Examples of their possible intervention in high critical temperature superconductivity and in turbulence are also described, since some biological processes may be analog in some aspects to these physical phenomena. These potential energy extra contributions could have emerged in biology from the very fractal nature of the medium, or from an evolutive advantage, since they involve spontaneous properties of self-organization, morphogenesis structuration and multi-scale integration., Comment: 24 pages

Owing to the non-differentiable nature of the theory of Scale Relativity, the emergence of complex wave functions, then of spinors and bi-spinors occurs naturally in its framework. The wave function is here a manifestation of the velocity field of geodesics of a continuous and non-differentiable (therefore fractal) space-time. In a first paper (Paper I), we have presented the general argument which leads to this result using an elaborate and more detailed derivation than previously displayed. We have therefore been able to show how the complex wave function emerges naturally from the doubling of the velocity field and to revisit the derivation of the non relativistic Schr\"odinger equation of motion. In the present paper (Paper II) we deal with relativistic motion and detail the natural emergence of the bi-spinors from such first principles of the theory. Moreover, while Lorentz invariance has been up to now inferred from mathematical results obtained in stochastic mechanics, we display here a new and detailed derivation of the way one can obtain a Lorentz invariant expression for the expectation value of the product of two independent fractal fluctuation fields in the sole framework of the theory of Scale Relativity. These new results allow us to enhance the robustness of our derivation of the two main equations of motion of relativistic quantum mechanics (the Klein-Gordon and Dirac equations) which we revisit here at length., Comment: 24 pages, no figure; very minor corrections to fit the published version: a few typos and a completed reference

One of the main results of Scale Relativity as regards the foundation of quantum mechanics is its explanation of the origin of the complex nature of the wave function. The Scale Relativity theory introduces an explicit dependence of physical quantities on scale variables, founding itself on the theorem according to which a continuous and non-differentiable space-time is fractal (i.e., scale-divergent). In the present paper, the nature of the scale variables and their relations to resolutions and differential elements are specified in the non-relativistic case (fractal space). We show that, owing to the scale-dependence which it induces, non-differentiability involves a fundamental two-valuedness of the mean derivatives. Since, in the scale relativity framework, the wave function is a manifestation of the velocity field of fractal space-time geodesics, the two-valuedness of velocities leads to write them in terms of complex numbers, and yields therefore the complex nature of the wave function, from which the usual expression of the Schr\"odinger equation can be derived., Comment: 36 pages, 5 figures, major changes from the first version, matches the published version

We present a new step in the foundation of quantum field theory with the tools of scale relativity. Previously, quantum motion equations (Schr\"odinger, Klein-Gordon, Dirac, Pauli) have been derived as geodesic equations written with a quantum-covariant derivative operator. Then, the nature of gauge transformations, of gauge fields and of conserved charges have been given a geometric meaning in terms of a scale-covariant derivative tool. Finally, the electromagnetic Klein-Gordon equation has been recovered with a covariant derivative constructed by combining the quantum-covariant velocity operator and the scale-covariant derivative. We show here that if one tries to derive the electromagnetic Dirac equation from the Klein-Gordon one as for the free particle motion, i.e. as a square root of the time part of the Klein-Gordon operator, one obtains an additional term which is the relativistic analog of the spin-magnetic field coupling term of the Pauli equation. However, if one first applies the quantum covariance, then implements the scale covariance through the scale-covariant derivative, one obtains the electromagnetic Dirac equation in its usual form. This method can also be applied successfully to the derivation of the electromagnetic Klein-Gordon equation. This suggests it rests on more profound roots of the theory, since it encompasses naturally the spin-charge coupling., Comment: 14 pages, no figures

This document is the Special Issue of the First International Conference on the Evolution and Development of the Universe (EDU 2008). Please refer to the preface and introduction for more details on the contributions. Keywords: acceleration, artificial cosmogenesis, artificial life, Big Bang, Big History, biological evolution, biological universe, biology, causality, classical vacuum energy, complex systems, complexity, computational universe, conscious evolution, cosmological artificial selection, cosmological natural selection, cosmology, critique, cultural evolution, dark energy, dark matter, development of the universe, development, emergence, evolution of the universe evolution, exobiology, extinction, fine-tuning, fractal space-time, fractal, information, initial conditions, intentional evolution, linear expansion of the universe, log-periodic laws, macroevolution, materialism, meduso-anthropic principle, multiple worlds, natural sciences, Nature, ontology, order, origin of the universe, particle hierarchy, philosophy, physical constants, quantum darwinism, reduction, role of intelligent life, scale relativity, scientific evolution, self-organization, speciation, specification hierarchy, thermodynamics, time, universe, vagueness., Comment: 355 pages, Special Issue of the First International Conference on the Evolution and Development of the Universe (EDU 2008) (online at: http://evodevouniverse.com/wiki/Conference_2008) To be published in Foundations of Science. Includes peer-reviewed papers, commentaries and responses (metadata update)

Klein-Gordon and Dirac equations are the motion equations for relativistic particles with spin 0 (so-called scalar particles) and 1/2 (electron/positron) respectively. For a free particle, the Dirac equation is derived from the Klein-Gordon equation by taking its square root in a bi-quaternionic formalism fully justified by the first principles of the scale relativity theory. This is no more true when an external electro-magnetic field comes into play. If one tries to derive the electro-magnetic Dirac equation in a manner analogous to the one used when this field is absent, one obtains an additional term which is the relativistic analogue of the spin-magnetic field coupling term encountered in the Pauli equation, valid for a non-relativistic particle with spin 1/2. There is however a method to recover the standard form of the electro-magnetic Dirac equation, with no additional term, which amounts modifying the way both covariances involved here, quantum and scale, are implemented. Without going into technical details, it will be shown how these results suggest this last method is based on more profound roots of the scale relativity theory since it encompasses naturally the spin-charge coupling., Comment: 5 pages, 1 figure, to appear in the proceedings of the Rencontres TRANS-ERI-COD 2009, held in Avignon (France), 15-17 June 2009

We have recently proposed a new general concept of macroscopic quantum-type experiment. It amounts to transform a classical fluid into a quantum-type fluid by the application of a quantum-like potential, either directly in a stationary configuration, or through a retro-active loop to simulate the time evolution. In this framework, the amplitude of the quantum potential depends on a macroscopic generalization of the Planck constant, which can be changed during the experiment, therefore simulating a quantum to classical transition. The experiment is exemplified here by an application of this concept to gravity waves at the surface of an incompressible liquid in a basin of finite height, with particular emphasis on the quantized vortex. We construct a complex wave function with the height of the fluid in the basin as its square modulus and the velocity potential as its phase. This wave function is solution of a nonlinear Schrodinger equation typical of superfluids. The quantum potential is therefore defined here in terms of the square root of the fluid height. We suggest two methods for applying this quantum-like potential to the fluid: (i) by the action of a force on the surface (wind, blower, pressure, field, etc...); (ii) by a curvature of the basin ground. In this last case the ground profile yields the quantum potential itself, while usually only the quantum force is accessible, so that such an experiment is expected to provide one with a macroscopic model of a quantum-type vacuum energy. These results may also be relevant to the study of freak waves, which have already been described by nonlinear Schrodinger equations., Comment: 23 pages, 4 figures

In the first part of this contribution, we review the development of the theory of scale relativity and its geometric framework constructed in terms of a fractal and nondifferentiable continuous space-time. This theory leads (i) to a generalization of possible physically relevant fractal laws, written as partial differential equation acting in the space of scales, and (ii) to a new geometric foundation of quantum mechanics and gauge field theories and their possible generalisations. In the second part, we discuss some examples of application of the theory to various sciences, in particular in cases when the theoretical predictions have been validated by new or updated observational and experimental data. This includes predictions in physics and cosmology (value of the QCD coupling and of the cosmological constant), to astrophysics and gravitational structure formation (distances of extrasolar planets to their stars, of Kuiper belt objects, value of solar and solar-like star cycles), to sciences of life (log-periodic law for species punctuated evolution, human development and society evolution), to Earth sciences (log-periodic deceleration of the rate of California earthquakes and of Sichuan earthquake replicas, critical law for the arctic sea ice extent) and tentative applications to system biology., Comment: 63 pages, 14 figures. In : First International Conference on the Evolution and Development of the Universe,8th - 9th October 2008, Paris, France

We first recall that the system of fluid mechanics equations (Euler and continuity) that describes a fluid in irrotational motion subjected to a generalized quantum potential (in which the constant is no longer reduced to the standard quantum constant hbar) is equivalent to a generalized Schrodinger equation. Then we show that, even in the case of the presence of vorticity, it is also possible to obtain, for a large class of systems, a Schrodinger-like equation of the vectorial field type from the continuity and Euler equations including a quantum potential. The same kind of transformation also applies to a classical charged fluid subjected to an electromagnetic field and to an additional potential having the form of a quantum potential. Such a fluid can therefore be described by an equation of the Ginzburg-Landau type, and is expected to show some superconducting-like properties. Moreover, a Schrodinger form can be obtained for the fluctuating rotational motion of a solid. In this case the mass is replaced by the tensor of inertia, and a generalized form of the quantum potential is derived. We finally reconsider the case of a standard diffusion process, and we show that, after a change of variable, the diffusion equation can also be given the form of a continuity and Euler system including an additional potential energy. Since this potential is exactly the opposite of a quantum potential, the quantum behavior may be considered, in this context, as an anti-diffusion., Comment: 33 pages, submitted for publication

Quantum mechanics is based on a series of postulates which lead to a very good description of the microphysical realm but which have, up to now, not been derived from first principles. In the present work, we suggest such a derivation in the framework of the theory of scale relativity. After having analyzed the actual status of the various postulates, rules and principles that underlie the present axiomatic foundation of quantum mechanics (in terms of main postulates, secondary rules and derived `principles'), we attempt to provide the reader with an exhaustive view of the matter, by both gathering here results which are already available in the literature, and deriving new ones which complete the postulate list., Comment: 30 pages, no figure

The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows us to recover quantum mechanics as mechanics on a non-differentiable (fractal) spacetime. The Schrodinger and Klein-Gordon equations are demonstrated as geodesic equations in this framework. A development of the intrinsic properties of this theory, using the mathematical tool of Hamilton's bi-quaternions, leads us to a derivation of the Dirac equation within the scale-relativity paradigm. The complex form of the wavefunction in the Schrodinger and Klein-Gordon equations follows from the non-differentiability of the geometry, since it involves a breaking of the invariance under the reflection symmetry on the (proper) time differential element (ds <-> - ds). This mechanism is generalized for obtaining the bi-quaternionic nature of the Dirac spinor by adding a further symmetry breaking due to non-differentiability, namely the differential coordinate reflection symmetry (dx^mu <-> - dx^mu) and by requiring invariance under parity and time inversion. The Pauli equation is recovered as a non-relativistic-motion approximation of the Dirac equation., Comment: 28 pages, no figure

In standard quantum mechanics, it is not possible to directly extend the Schrodinger equation to spinors, so the Pauli equation must be derived from the Dirac equation by taking its non-relativistic limit. Hence, it predicts the existence of an intrinsic magnetic moment for the electron and gives its correct value. In the scale relativity framework, the Schrodinger, Klein-Gordon and Dirac equations have been derived from first principles as geodesics equations of a non-differentiable and continuous spacetime. Since such a generalized geometry implies the occurence of new discrete symmetry breakings, this has led us to write Dirac bi-spinors in the form of bi-quaternions (complex quaternions). In the present work, we show that, in scale relativity also, the correct Pauli equation can only be obtained from a non-relativistic limit of the relativistic geodesics equation (which, after integration, becomes the Dirac equation) and not from the non-relativistic formalism (that involves symmetry breakings in a fractal 3-space). The same degeneracy procedure, when it is applied to the bi-quaternionic 4-velocity used to derive the Dirac equation, naturally yields a Pauli-type quaternionic 3-velocity. It therefore corroborates the relevance of the scale relativity approach for the building from first principles of the quantum postulates and of the quantum tools. This also reinforces the relativistic and fundamentally quantum nature of spin, which we attribute in scale relativity to the non-differentiability of the quantum spacetime geometry (and not only of the quantum space). We conclude by performing numerical simulations of spinor geodesics, that allow one to gain a physical geometric picture of the nature of spin., Comment: 22 pages, 2 figures, accepted for publication in J. Phys. A: Math. & Gen

Gauge field theory is developed in the framework of scale relativity. In this theory, space-time is described as a non-differentiable continuum, which implies it is fractal, i.e., explicitly dependent on internal scale variables. Owing to the principle of relativity that has been extended to scales, these scale variables can themselves become functions of the space-time coordinates. Therefore, a coupling is expected between displacements in the fractal space-time and the transformations of these scale variables. In previous works, an Abelian gauge theory (electromagnetism) has been derived as a consequence of this coupling for global dilations and/or contractions. We consider here more general transformations of the scale variables by taking into account separate dilations for each of them, which yield non-Abelian gauge theories. We identify these transformations with the usual gauge transformations. The gauge fields naturally appear as a new geometric contribution to the total variation of the action involving these scale variables, while the gauge charges emerge as the generators of the scale transformation group. A generalized action is identified with the scale-relativistic invariant. The gauge charges are the conservative quantities, conjugates of the scale variables through the action, which find their origin in the symmetries of the ``scale-space''. We thus found in a geometric way and recover the expression for the covariant derivative of gauge theory. Adding the requirement that under the scale transformations the fermion multiplets and the boson fields transform such that the derived Lagrangian remains invariant, we obtain gauge theories as a consequence of scale symmetries issued from a geometric space-time description., Comment: 24 pages, LaTeX

The scale transformation laws produce, on the motion equations of gravitating bodies and under some peculiar assumptions, effects which are anologous to those of a "macroscopic quantum mechanics". When we consider time and space scales such that the description of the trajectories of these bodies (planetesimals in the case of planetary system formation, interstellar gas and dust in the case of star formation, etc...) is in the shape of non-differentiable curves, we obtain fractal curves of fractal dimension 2. Continuity and non-differentiability yield a fractal space and a symmetry breaking of the differential time element which gives a doubling of the velocity fields. The application of a geodesics principle leads to motion equations of Schrodinger-type. When we add an outside gravitational field, we obtain a Schrodinger-Poisson system. We give here the derivation of the Schrodinger equation for chaotic systems, i.e., with time scales much longer than their Lyapounov chaos-time., Comment: 4 pages, oral communication in the framework of the 4eme Semaine de l'Astrophysique Francaise, Paris, 2004

A large class of stellar systems (e.g., planetary nebulae (PNe), supernova envelopes, LBV stars, young stars in formation) shows structures in their accretion/ejection phase that have similar characteristics. In particular, one currently observes for these objects equatorial discs, axial ejections and stable bipolar shells. However these simple shapes, which are expected to be solutions of standard hydrodynamical equations, are not yet fully understood. In this paper, we suggest a new form of these equations that takes into account the fractality and the irreversibility of particle motion in such processes. Then we study in this framework a general infall or ejection motion in a central spherical potential. From the stationary solutions allowed by this new hydrodynamical system, we deduce a specific distribution of matter density, described in terms of probability density for ejection angles. A global classification of predicted shapes, depending on the values of conservative quantities such as (E^2,L^2,L_z), is given. These results are compared with the available observational data, and allows us to theoretically predict the possible existence of more complicated structures and of correlations between observable variables, which could be checked by future observations., Comment: 12 pages, 78 small figures. Misprint corrected

In the framework of the theory of scale relativity, we suggest a solution to the cosmological problem of the formation and evolution of gravitational structures on many scales. This approach is based on the giving up of the hypothesis of differentiability of space-time coordinates. As a consequence of this generalization, space-time is not only curved, but also fractal. In analogy with Einstein's general relativistic methods, we describe the effects of space fractality on motion by the construction of a covariant derivative. The principle of equivalence allows us to write the equation of dynamics as a geodesics equation that takes the form of the equation of free Galilean motion. Then, after a change of variables, this equation can be integrated in terms of a gravitational Schrodinger equation that involves a new fundamental gravitational coupling constant, alpha_{g} = w_{0}/c. Its solutions give probability densities that quantitatively describe precise morphologies in the position space and in the velocity space. Finally the theoretical predictions are successfully checked by a comparison with observational data: we find that matter is self-organized in accordance with the solutions of the gravitational Schrodinger equation on the basis of the universal constant w_{0}=144.7 +- 0.7 km/s (and its multiples and sub-multiples), from the scale of our Earth and the Solar System to large scale structures of the Universe, Comment: 34 pages, 42 figures. Higher quality figures added

An anomalous constant acceleration of (8.7 \pm 1.3) x 10^-8 cm/s^2 directed toward the Sun has been discovered by Anderson et al. in the motion of the Pioneer 10/11 and Galileo spacecrafts. In parallel, the WMAP results have definitively established the existence of a cosmological constant Lambda=1/ L_U^2, and therefore of an invariant cosmic length-scale L_U=(2.72 \pm 0.10) Gpc. We show that the existence of this invariant scale definitively implements Mach's principle in Einstein's theory of general relativity. Then we demonstrate, in the framework of an exact cosmological solution of Einstein's field equations which is valid both locally and globally, that the definition of inertial systems ultimately depends on this length-scale. As a consequence, usual local coordinates are not inertial, so that the motion of a free body of speed v is expected to contain an additional constant acceleration a_P=v^2(\surd3 L_U), which is, using the WMAP five years results, (6.02 \pm 0.34) x 10^-8 cm/s^2 when v \approx c. Such an effect is too small to contribute significantly to the Pioneer acceleration (since v_Pioneer \approx 12 km/s << c), but could be possibly observed in a dedicated space mission., Comment: 12 pages, updated version, added references, result changed

The application of the theory of scale relativity to microphysics aims at recovering quantum mechanics as a new non-classical mechanics on a non-derivable space-time. This program was already achieved as regards the Schr\"odinger and Klein Gordon equations, which have been derived in terms of geodesic equations in this framework: namely, they have been written according to a generalized equivalence/strong covariance principle in the form of free motion equations $D^2x/ds^2=0$, where $D/ds$ are covariant derivatives built from the description of the fractal/non-derivable geometry. Following the same line of thought and using the mathematical tool of Hamilton's bi-quaternions, we propose here a derivation of the Dirac equation also from a geodesic equation (while it is still merely postulated in standard quantum physics). The complex nature of the wave function in the Schr\"odinger and Klein-Gordon equations was deduced from the necessity to introduce, because of the non-derivability, a discrete symmetry breaking on the proper time differential element. By extension, the bi-quaternionic nature of the Dirac bi-spinors arises here from further discrete symmetry breakings on the space-time variables, which also proceed from non-derivability., Comment: 13 pages, accepted for publication in Electromagnetic Phenomena, Special issue dedicated to the 75th anniversary of the discovery of the Dirac equation

The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows to recover quantum mechanics as mechanics on a non-differentiable (fractal) space-time. The Schr\"odinger and Klein-Gordon equations have already been demonstrated as geodesic equations in this framework. We propose here a new development of the intrinsic properties of this theory to obtain, using the mathematical tool of Hamilton's bi-quaternions, a derivation of the Dirac equation, which, in standard physics, is merely postulated. The bi-quaternionic nature of the Dirac spinor is obtained by adding to the differential (proper) time symmetry breaking, which yields the complex form of the wave-function in the Schr\"odinger and Klein-Gordon equations, the breaking of further symmetries, namely, the differential coordinate symmetry ($dx^{\mu} \leftrightarrow - dx^{\mu}$) and the parity and time reversal symmetries., Comment: 33 pages, 4 figures, latex. Submitted to Phys. Rev. D