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 $z_1$, a thick shell made of anisotropic matter, an hyperbolic Friedmann metric up to the scale where dimming galaxies are observed ($z\simeq 1.7$) 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