93 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
Gravitational backreaction in cosmological spacetimes
We develop a new formalism for the treatment of gravitational backreaction in
the cosmological setting. The approach is inspired by projective techniques in
non-equilibrium statistical mechanics. We employ group-averaging with respect
to the action of the isotropy group of homogeneous and isotropic spacetimes
(rather than spatial averaging), in order to define effective FRW variables for
a generic spacetime. Using the Hamiltonian formalism for gravitating perfect
fluids, we obtain a set of equations for the evolution of the effective
variables; these equations incorporate the effects of backreaction by the
inhomogeneities. Specializing to dust-filled spacetimes, we find regimes that
lead to a closed set of backreaction equations, which we solve for small
inhomogeneities. We then study the case of large inhomogeneities in relation to
the proposal that backreaction can lead to accelerated expansion. In
particular, we identify regions of the gravitational state space that
correspond to effective cosmic acceleration. Necessary conditions are (i) a
strong expansion of the congruences corresponding to comoving observers, and
(ii) a large negative value of a dissipation variable that appears in the
effective equations (i.e, an effective "anti-dissipation").Comment: 36 pages, latex. Extended discussion on results and on relation to
Lemaitre-Tolman-Bondi models. Version to appear in PR
Is quantum mechanics based on an invariance principle?
Non-relativistic quantum mechanics for a free particle is shown to emerge
from classical mechanics through an invariance principle under transformations
that preserve the Heisenberg position-momentum inequality. These
transformations are induced by isotropic space dilations. This invariance
imposes a change in the laws of classical mechanics that exactly corresponds to
the transition to quantum mechanics. The Schroedinger equation appears jointly
with a second nonlinear equation describing non-unitary processes. Unitary and
non-unitary evolutions are exclusive and appear sequentially in time. The
non-unitary equation admits solutions that seem to correspond to the collapse
of the wave function.Comment: 15 page
Systematic corrections to the measured cosmological constant as a result of local inhomogeneity
We calculate the systematic inhomogeneity-induced correction to the
cosmological constant that one would infer from an analysis of the luminosities
and redshifts of Type Ia supernovae, assuming a homogeneous universe. The
calculation entails a post-Newtonian expansion within the framework of second
order perturbation theory, wherein we consider the effects of subhorizon
density perturbations in a flat, dust dominated universe. Within this
formalism, we calculate luminosity distances and redshifts along the past light
cone of an observer. The resulting luminosity distance-redshift relation is fit
to that of a homogeneous model in order to deduce the best-fit cosmological
constant density Omega_Lambda. We find that the luminosity distance-redshift
relation is indeed modified, by a small fraction of order 10^{-5}. When fitting
this perturbed relation to that of a homogeneous universe, we find that the
inferred cosmological constant can be surprisingly large, depending on the
range of redshifts sampled. For a sample of supernovae extending from z=0.02
out to z=0.15, we find that Omega_Lambda=0.004. The value of Omega_Lambda has a
large variance, and its magnitude tends to get larger for smaller redshifts,
implying that precision measurements from nearby supernova data will require
taking this effect into account. However, we find that this effect is likely
too small to explain the observed value of Omega_Lambda=0.7. There have been
previous claims of much larger backreaction effects. By contrast to those
calculations, our work is directly related to how observers deduce cosmological
parameters from astronomical data.Comment: 28 pages, 3 figures, revtex4; v2: corrected comments and the section
on previous work; v3: clarified wording. References adde
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
Backreaction of superhorizon perturbations in scalar field cosmologies
It has been suggested that the acceleration of the Universe may be due to the
backreaction of perturbations to the Friedmann-Robertson-Walker background. For
a Universe dominated by cold dark matter, it is known that the backreaction of
superhorizon perturbations can not drive acceleration. We extend this result to
models with cold dark matter together with a scalar field. We show that the
scalar field can drive acceleration only via the standard mechanism of a
constant or nearly constant piece of its potential (i.e., a cosmological
constant); there is no separate mechanism involving superhorizon backreaction.
This rules out some models which have been proposed in the literature.Comment: 5 page
Tidal Dynamics in Cosmological Spacetimes
We study the relative motion of nearby free test particles in cosmological
spacetimes, such as the FLRW and LTB models. In particular, the influence of
spatial inhomogeneities on local tidal accelerations is investigated. The
implications of our results for the dynamics of the solar system are briefly
discussed. That is, on the basis of the models studied in this paper, we
estimate the tidal influence of the cosmic gravitational field on the orbit of
the Earth around the Sun and show that the corresponding temporal rate of
variation of the astronomical unit is negligibly small.Comment: 12 pages, no figures, REVTeX 4.0; appendix added, new references, and
minor changes throughout; to appear in Classical and Quantum Gravity; v4:
error in (A24) of Appendix A corrected, results and conclusions unchanged. We
thank L. Iorio for pointing out the erro
Effects of structure formation on the expansion rate of the Universe: An estimate from numerical simulations
General relativistic corrections to the expansion rate of the Universe arise
when the Einstein equations are averaged over a spatial volume in a locally
inhomogeneous cosmology. It has been suggested that they may contribute to the
observed cosmic acceleration. In this paper, we propose a new scheme that
utilizes numerical simulations to make a realistic estimate of the magnitude of
these corrections for general inhomogeneities in (3+1) spacetime. We then
quantitatively calculate the volume averaged expansion rate using N-body
large-scale structure simulations and compare it with the expansion rate in a
standard FRW cosmology. We find that in the weak gravitational field limit, the
converged corrections are slightly larger than the previous claimed 10^{-5}
level, but not large enough nor even of the correct sign to drive the current
cosmic acceleration. Nevertheless, the question of whether the cumulative
effect can significantly change the expansion history of the Universe needs to
be further investigated with strong-field relativity.Comment: 13 pages, 6 figures, improved version published in Phys. Rev.
An inhomogeneous universe with thick shells and without cosmological constant
We build an exact inhomogeneous universe composed of a central flat Friedmann
zone up to a small redshift , a thick shell made of anisotropic matter, an
hyperbolic Friedmann metric up to the scale where dimming galaxies are observed
() that can be matched to a hyperbolic Lema\^{i}tre-Tolman-Bondi
spacetime to best fit the WMAP data at early epochs. We construct a general
framework which permits us to consider a non-uniform clock rate for the
universe. As a result, both for a uniform time and a uniform Hubble flow, the
deceleration parameter extrapolated by the central observer is always positive.
Nevertheless, by taking a non-uniform Hubble flow, it is possible to obtain a
negative central deceleration parameter, that, with certain parameter choices,
can be made the one observed currently. Finally, it is conjectured a possible
physical mechanism to justify a non-uniform time flow.Comment: Version published in Class. Quantum gra
Dirac equation from the Hamiltonian and the case with a gravitational field
Starting from an interpretation of the classical-quantum correspondence, we
derive the Dirac equation by factorizing the algebraic relation satisfied by
the classical Hamiltonian, before applying the correspondence. This derivation
applies in the same form to a free particle, to one in an electromagnetic
field, and to one subjected to geodesic motion in a static metric, and leads to
the same, usual form of the Dirac equation--in special coordinates. To use the
equation in the static-gravitational case, we need to rewrite it in more
general coordinates. This can be done only if the usual, spinor transformation
of the wave function is replaced by the 4-vector transformation. We show that
the latter also makes the flat-space-time Dirac equation Lorentz-covariant,
although the Dirac matrices are not invariant. Because the equation itself is
left unchanged in the flat case, the 4-vector transformation does not alter the
main physical consequences of that equation in that case. However, the equation
derived in the static-gravitational case is not equivalent to the standard
(Fock-Weyl) gravitational extension of the Dirac equation.Comment: 27 pages, standard LaTeX. v2: minor style changes, accepted for
publication in Found. Phys. Letter
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