189 research outputs found
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
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
Second-order power spectra of CMB anisotropies due to primordial random perturbations in flat cosmological models
Second-order power spectra of Cosmic Microwave Background (CMB) anisotropies
due to random primordial perturbations at the matter dominant stage are
studied, based on the relativistic second-order theory of perturbations in flat
cosmological models and on the second-order formula of CMB anisotropies derived
by Mollerach and Matarrese. So far the second-order integrated Sachs-Wolfe
effect has been analyzed using the three-point correlation or bispectrum. In
this paper we derive the second-order term of power spectra given using the
two-point correlation of temperature fluctuations.
The second-order density perturbations are small, compared with the
first-order ones. The second-order power spectra of CMB anisotropies, however,
are not small at all, compared with the first-order power spectra, because at
the early stage the first-order integrated Sachs-Wolfe effect is very small and
the second-order integrated Sachs-Wolfe effect may be dominant over the
first-order ones. So their characteristic behaviors may be measured through the
future precise observation and bring useful informations on the structure and
evolution of our universe in the future.Comment: 11 page
Direct reconstruction of dark energy
An important issue in cosmology is reconstructing the effective dark energy
equation of state directly from observations. With so few physically motivated
models, future dark energy studies cannot only be based on constraining a dark
energy parameter space. We present a new non-parametric method which can
accurately reconstruct a wide variety of dark energy behaviour with no prior
assumptions about it. It is simple, quick and relatively accurate, and involves
no expensive explorations of parameter space. The technique uses principal
component analysis and a combination of information criteria to identify real
features in the data, and tailors the fitting functions to pick up trends and
smooth over noise. We find that we can constrain a large variety of w(z) models
to within 10-20 % at redshifts z<1 using just SNAP-quality data.Comment: 5 pages, 4 figures. v2 has added refs plus minor changes. To appear
in PR
Gravitational energy and cosmic acceleration
Cosmic acceleration is explained quantitatively, as an apparent effect due to
gravitational energy differences that arise in the decoupling of bound systems
from the global expansion of the universe. "Dark energy" is a misidentification
of those aspects of gravitational energy which by virtue of the equivalence
principle cannot be localised, namely gradients in the energy due to the
expansion of space and spatial curvature variations in an inhomogeneous
universe. A new scheme for cosmological averaging is proposed which solves the
Sandage-de Vaucouleurs paradox. Concordance parameters fit supernovae
luminosity distances, the angular scale of the sound horizon in the CMB
anisotropies, and the effective comoving baryon acoustic oscillation scale seen
in galaxy clustering statistics. Key observational anomalies are potentially
resolved, and unique predictions made, including a quantifiable variance in the
Hubble flow below the scale of apparent homogeneity.Comment: 9 pages, 2 figures. An essay which received Honorable Mention in the
2007 GRF Essay Competition. To appear in a special issue of Int. J. Mod.
Phys.
Interior spacetimes sourced by stationary differentially rotating irrotational cylindrical fluids. II. Axial pressure
In a recent series of papers new exact analytical solutions of Einstein
equations representing interior spacetimes sourced by stationary rigidly
rotating cylinders of fluids have been displayed. We have first considered a
fluid with an axially directed pressure C\'el\'erier, Phys. Rev. D 104, 064040
(2021), J. Math. Phys. 64, 032501 (2023), then a perfect fluid, J. Math. Phys.
64, 022501 (2023), followed by a fluid with an azimuthally directed pressure,
J. Math. Phys. 64, 042501 (2023), and finally a fluid where the anisotropic
pressure is radially oriented, J. Math. Phys. 64, 052502 (2023). This work is
being currently extended to the cases of differentially rotating irrotational
fluids. The results are presented in a new series of papers considering, in
turn, a perfect fluid source, arXiv:2305.11565 [gr-qc], and the same three
anisotropic pressure cases. Here, fluids with an axially directed pressure are
considered. A general method for generating new mathematical solutions to the
field equations is displayed and three classes are presented so as to exemplify
this recipe. Their mathematical and physical properties are analyzed. The first
class, named class A, whose other mathematical and physical properties
determine a standard configuration, is shown to exhibit a singular axis of
symmetry which can be considered as an awkward drawback. The second class,
class B, is free from such a singularity but appears to exhibit a negative
energy density which characterizes a rather exotic kind of matter. The third
class, class C, is the best behaved since it possesses the main properties
expected from spacetimes sourced by rather standard fluids. The three classes
are matched to an exterior Lewis-Weyl vacuum and the conditions for avoiding an
angular deficit are discussed. A comparison with the rigidly rotating fluid
case is provided.Comment: 29 pages, 0 figur
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