342 research outputs found

### 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.Comment: 33 pages, submitted for publicatio

### Emergence of complex and spinor wave functions in scale relativity. I. Nature of scale variables

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 versio

### 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.Comment: 22 pages, 2 figures, accepted for publication in J. Phys. A: Math. &
Ge

### Emergence of complex and spinor wave functions in Scale Relativity. II. Lorentz invariance and bi-spinors

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 referenc

### 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.Comment: 24 pages, LaTe

### 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 UniverseComment: 34 pages, 42 figures. Higher quality figures adde

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