(detailed)

Updated: 22 February 2001. Adapted from Sec. 9 of the review paper CSF 7,877 (1996).

As a conclusion of this review, let us achieve one of the aims of the present contribution, that is to give a summary of the various results and theoretical predictions of the new theory. Since the consequences of scale relativity cover a wide range of physical domains, these results and predictions were up to now dispersed in different papers written for different communities. This review paper is a good occasion to collect them (in a not fully exhaustive way, since some recently obtained results are still in preparation), and thus to provide the reader with a wider view of the abilities of the theory.

Let us first remark that the various results of a theory may be classified according to different "levels":

(i) There are "conceptual" results, namely contributions of a theory in understanding previously misunderstood general facts or in solving general problems (for example, in our case, understanding of the origin of the complex nature of the wave function; reconciling quantum physics with the relativistic approach).

(ii) There are numerical, quantified results, i.e., theoretical predictions of already measured quantities that had still no theoretical explanation (for example, prediction of the GUT and electroweak scales in particles physics, prediction of the value of the power of the galaxy-galaxy autocorrelation function in cosmology).

(iii) There are finally pure theoretical predictions, either of new still unobserved phenomena, or of the still unknown value of measurable quantities. These "blind" predictions play a special role in testing a theory, since they are the key to its falsifiability (for example, our prediction of preferential distances for new planets in extra-solar systems, of the value of the cosmological constant, and of deviations from standard quantum mechanics at high energy >>100 GeV).

Note that some results may fall in two or three of these items, since a numerical theoretical prediction may agree with some already measured experimental result, but remain more precise. The blind prediction is only about the additional unknown figures in this case (example: our prediction of the low energy strong coupling constant, or of the m_z/m_w mass ratio). Some conceptual progress may also have a numerical counterpart (example: the solution of the vacuum energy density problem that also allows us to get an estimate of the cosmological constant).

Let us review these various kinds of consequences in the present case of the theory of scale relativity.

Conceptual results:

*Complex nature of wave function: consequence of nondifferentiability of space-time, that implies a breaking of time reversibility at the level of our elementary description, then a two-valuedness of the time derivative, of which complex numbers are the simplest representation. Time reversibility is recovered in terms of a complex process that combines the forward and backward ones. The wave function is the complex action.

*Probabilistic nature of quantum theory: consequence of nondifferentiability and fractal nature of space-time, that implies an infinity of geodesics between any couple of events.

*Correspondence principle: becomes an equality, thanks to the fact that momentum and energy become themselves complex.

*Schrödinger, Klein-Gordon equations: demonstrated as equations of geodesics on a fractal, nondifferentiable space-time. The quantum terms are implemented from the use of a scale-covariant derivative, and find their origin in a mixing of the effect of the complex representation (consequence of nondifferentiability) and of new second order terms in differential equations (consequence of the fractal dimension 2 of geodesics).

*Quantum / Classical transition: inherent to the description (since included in the solution to our simplest scale differential equation), identified with the transition from fractal behavior (scale dependence) to nonfractal behavior(scale independence).

*Divergence of masses and charges: solved by the new length-scale / mass-scale relation in special scale-relativity; the solution is linked to the new physical meaning of the Planck length-scale.

*Nature of Planck scale: becomes a minimal, impassable scale, invariant under dilations, that plays for scale-laws the same role as played by the velocity of light for motion-laws and replaces the zero point as concerns its physical behavior.

*Nature and quantization of electric charge: the charge is understood as the conservative quantity that comes from the new scale symmetry. Its quantization is a consequence of the limitation on resolutions ratios implied by the new invariant nature of the Planck scale.

*Origin of mass discretization of elementary particles: we have suggested that the masses of elementary fermions were of QED origin, and that their discretization was a consequence of charge being quantized.

*Nature of the cosmological constant: inverse of the square of a maximal, impassable length-scale L, invariant under dilation in the new special-scale-relativistic framework, replaces the infinite scale but keeps its physical properties.

*Vacuum energy density problem: the energy density is explicitly scale-dependent, so that the Planck energy density does not apply at cosmological scales. The energy density is computed as gravitational self-energy density of vacuum fluctuations and is found to vary in terms of resolution as r^-6. Therefore the quantum energy density and the cosmological energy density that manifests itself in terms of a cosmological constant become compatible.

*Large number coincidence: explained from the above calculation of self-energy density and from the introduction of the maximal invariant length-scale L.

*Problems of Big-Bang theory : many problems encountered by the standard Big-Bang theory are automatically resolved in our new framework. The horizon/causality problem disappears in the framework of Lorentzian dilation laws; there is no need of an inflation phase, then no need to introduce an arbitrary scalar field to drive it; the age of the universe becomes compatible with that of globular clusters thanks to the introduction of a positive cosmological constant L = 1/L²; the problem of the seed of density fluctuations and of the formation and evolution of structures in the universe is resolved in terms of our Schrödinger-like gravitational equation, that yields structures even in uniform density, without any need for initial fluctuations.

Quantified results:

*GUT scale: becomes in special scale-relativity the Planck mass-scale (that now differs from the Planck length-scale); given by log(l_z/l_GUT) = log(l_z/L_P) / 2^1/2 ~17 / 2^1/2 ~12. Namely, this length-scale is 10¹² times smaller than the Z boson scale: this corresponds to an energy-scale of 10¹⁴ GeV in the non-scale-relativistic standard model, but to 10¹⁹ GeV in special scale-relativity.

*Mass-charge relations: our interpretation of the charges of fundamental interactions as eigenvalues of the dilation operator acting on resolutions (in other words, as conservative quantities arising from the scale symmetries), of gauge invariance as scale invariance on resolution transformations, and of the"arbitrary" gauge function as a "relative state of scale" lnr(x,y,z,t),now function of coordinates, leads in special scale-relativity to general mass-charge relations of the form a ln(l_c/L_P) = k/2, where k is integer, a is a coupling constant, l_c a Compton scale, inversely related to a mass scale (l_c = h/mc), and whereL_P is the Planck length-scale.

*ElectroWeak scale: given by the mass / charge relation a_0oo ln(l_EW/L_P) = 1, i.e., l_EW = L_P e^4p2 =1.397 x 10¹⁷ L_P ~123 GeV (while the vacuum expectation value of the Higgs field is 174 GeV= 123 x 2^1/2 GeV).

*Electron scale: given by the predicted mass / charge relation a_0eln(l_e/L_P) = 1, i.e., m_e = m_P exp(-3/8a_e) ~0.5 MeV.

*Weak boson mass ratio (value of weak mixing angle): arguments linked to our mass-charge relations suggest that a₂ = 2 a₁ at electroweak scale, so that m_W / m_Z = (10/13)^1/2, and sin²q = 3/13 at this scale.

*Elementary fermion mass spectrum: recovered from a cancellation effect between special scale-relativistic corrections and radiative corrections. (However, except concerning the muon mass, this is still a model, not a totally constrained theory, because an unknown free parameter remains in this generation mechanism).

*Top quark mass: predicted by the above mechanism to fall just beyond the W/Z mass, at 150 ± 50 GeV (experimental value: 174 ± 17 GeV).

*Values of low energy coupling constants: derived from their renormalization group equations and from the conjecture that the value 1/4p² is critical for coupling constants. We find a_e = 137.00 ± 0.10 from the bare couplinga_0oo = 1/4p² , and a₃(m_Z) = 0.1155 ± 0.0002 from a₃(m_GUT) = 1/4p².

*Power of galaxy-galaxy correlation function: the observed value g =1.8 at ~1-10 Mpc is explained as the result of a scale-relativistic correction to the standard value g = 2.

*Structuration of the Solar System: the observed distribution of mass, angular momentum, eccentricities and positions of planets in the Solar System is accounted for by the solution to our "quantum-gravitational" equation written for a Kepler potential, (holding for large time scales, beyond ahorizon of predictability).

*Quantization of binary galaxies: the quantization in terms of 144/n km/s observed by Tifft and other authors in the velocity difference of galaxies in pairs is also predicted by the same approach (Kepler potential).

*Global redshift quantization of galaxies: when applied to uniform density, this method predicts a quantization in terms of the modes of the 3-dimensional isotropic harmonic oscillator that accounts for the observed "global" redshift quantization at 36 km/s (Tifft, Guthrie & Napier).

New predictions:

*Precise value of the strong coupling constant: we predict, as quoted above, a₃(m_Z) = 0.1155 ± 0.0005, in agreement within error bars with and more precise than the current experimental value, 0.121 ± 0.007 (1995). This value has now been improved to 0.118 ± 0.003 (PDG 2000), and remains in agreement with the theoretical prediction.

*Precise value of the weak bosons mass ratio: we predict its exact value m_W /m_Z = (10/13)^1/2 (up to small radiative corrections), while the W mass is presently (1995) only poorly known (80.2 ± 0.2 GeV). Its knowledge has recently been improved: m_W = 80.42 ± 0.04 GeV (PDG 2000).

*Lifetime of the proton: in GUT theories, the lifetime of the proton is predicted to be of the order of m_x⁴/m_p⁵ , where m_x is the unification mass and m_p is the proton mass. Since, in the framework of special scale relativity, unification is expected at the Planck mass scale, the predicted lifetime becomes > 10⁴² years. Therefore the simplest unification group, SU(5), is no longer rejected, at least as a partial symmetry that does not include gravitation.

*Deviations from standard quantum mechanics at high energy: scale-relativistic "corrections" will rapidly increase for energies larger than ~100 GeV, since they are no longer cancelled by the appearance of new elementary charged fermions, as happens in the domain 0.5 MeV (electron energy) to 174 GeV (top energy). Provided no new cancellation of electroweak origin takes place above ~100 GeV, we expect the experimental results of particle collisions in future high energy accelerators (LHC...) to depart from their values calculated from standard quantum mechanics (i.e., from Galilean scale-relativistic laws). The departure may be expressed, to lowest order, in terms of an effective (Planck constant / mass) ratio that would vary with scale as [1 + ln³(l/r) / 2 ln²(l/l_P)], where l is the top quark Compton scale, and l_P the Planck length-scale.

*Value of the cosmological constant: it is predicted to be L = 1.36 10^-56 cm^-2, i.e. W_L=0.36 h^-2 (=0.7±0.2 for H₀=70±10 km/s.Mpc) under the assumption that the fractal-nonfractal transition for the vacuum energy density occurs at about 70 MeV. This is the scale of the classical radius of the electron (70.02 MeV), but also the 6 flavor QCD scale (66±10 MeV) and even better, it is nothing else but the effective Compton length of quarks in the lightest meson (m_p/2=69.78 MeV): this value could therefore arise from the quark-hadron transition in the primeval universe. This prediction is now confirmed: the reduced cosmological constant has been recently found to be W_L=0.7±0.2 by several independant and complementary measurements (SNe's, Boomerang, gravitational lensing).

*New bodies in the solar system: some of the "orbitals" predicted by the theory do not contain observed planets (Mercury ranks n=3 in the inner system). However, they may contain objects or material that have up to now escaped detection: e.g., dust in n = 1 of the inner system at 0.043 AU (a small body would be evaporated, since too close from the Sun); asteroids in n = 7 and 10 (some families exist at these distances, in spite of the destructive resonances with Jupiter); small planet in n = 2 of the inner system, at ~0.17 A.U. and in n > 6 of the outer system (Kuiper belt). This last prediction is now confirmed.

*Universal structure of external planetary systems: we predict that the planetary systems that are expected to be discovered in the near future around nearby stars will be described by the same hydrogen-like orbitals as in our own solar system. In particular, we expec some planets to lie at 0.043 AU (n=1) and 0.17 AU (n=2) from their star in the solar-type case. (This prediction is now confirmed).

*Position and velocity structures of stars and stellar associations in our Galaxy: we predict that the velocity and position distribution of stars in the Galaxy will not be at random, but instead "quantized" according to our general "Schrödinger-gravitational" equation. This applies in particular to multiple star systems, to associations and zones of star formation, etc... The expected morphologies include single and double objects, but also chains and quartets, as frequently observed in such zones.

*Structuration of the universe: in a similar way, we expect the formation of structures in the universe at every epochs according to the SU(3) group, that is the symmetry group of the 3-dimensional harmonic oscillator. This is an example of a microscopic-macroscopic (structural) connection, SU(3) being, as is well-known, the symmetry group of QCD.

These various predictions about the formation of universal gravitational structures at all scales (in particular, in velocity space) are now confirmed on several systems, from the planetary scale to the extragalactic scales.

*Value of power of galaxy correlation function at very large scale: in our special scale-relativistic theory, the exponent of the galaxy-galaxy correlation function is no longer constant, but varies with scale. While its value is ~ 1.8 at a scale of ~10 Mpc, we predict that it will fall to ~1.5 at 100 Mpc, then decrease even farther. A precise determination of its variation with resolution would yield a new precise measurement of the cosmological constant. This prediction is now confirmed: several groups have found that the power is indeed smaller at large scale (i.e., the fractal dimension D=3-g is larger and becomes 2 in the mean at scales of several hundred Mpc).