arXiv subjects

Laurent Nottale

Publications and source records attributed to Laurent Nottale.

32 records · Page 2Linked to original sources

arXiv2010

The Evolution and Development of the Universe

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.

physics.gen-ph↗

arXiv2009

Motion equations for relativistic particles in an external electromagnetic field in scale relativity

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.

physics.gen-ph↗

arXiv2009

Quantum-like gravity waves and vortices in a classical fluid

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.

physics.gen-ph↗

arXiv2008

Scale relativity and fractal space-time: theory and applications

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.

physics.gen-ph↗

arXiv2008

Generalized quantum potentials in scale relativity

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.

physics.gen-ph↗

arXiv2007

Derivation of the postulates of quantum mechanics from the first principles of scale relativity

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.

quant-ph↗

arXiv2006

Quantum-classical transition in Scale Relativity

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.

quant-ph↗

arXiv2006

The Pauli equation in scale relativity

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.

quant-ph↗

arXiv2006

Non-Abelian gauge field theory in scale relativity

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.

hep-th↗

arXiv2005

Generalized macroscopic Schrodinger equation in scale relativity

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.

gr-qc↗

arXiv2003

On the morphogenesis of stellar flows - Application to planetary nebulae

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.

astro-ph↗

arXiv2003

Gravitational structure formation in scale relativity

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

astro-ph↗

arXiv2002

A scale-relativistic derivation of the Dirac Equation

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 Schrödinger 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 Schrödinger 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.

hep-th↗

arXiv2001

Dirac Equation in Scale Relativity

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 Schrödinger 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 Schrödinger and Klein-Gordon equations, the breaking of further symmetries, namely, the differential coordinate symmetry ($dx^μ \leftrightarrow - dx^μ$) and the parity and time reversal symmetries.

hep-th↗