Results: General Summary

Since the consequences of Scale Relativity cover a wide range of physical domains, its results and predictions were up to now dispersed in different papers written for different communities. This post is a good occasion to collect them (in a not fully exhaustive way) and thus to provide the reader with a wide view of the abilities of the theory. The details of these results and predictions will be presented in dedicated posts.

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, the prediction of new planets in the solar system, or of the value of the cosmological constant).

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: the 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 spacetime, that implies a breaking of time reversibility at the level of our elementary description, then a doubling of information, 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 in nity of geodesics between any couple of events.



• Correspondence principle: becomes an equality, thanks to the introduction of complex momentum and energy.



• Schrödinger, Klein-Gordon and Dirac equations: demonstrated as equations of geodesics of fractal, nondifferentiable space-time. The quantum terms are implemented from 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 fractal dimension 2).



• Quantum / Classical transition: inherent to the description (since included in the solution to the simplest scale differential equation), identified with the transition from fractal (scale dependence) to nonfractal (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 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, replaces the infinite scale.



• 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 of vacuum fluctuations and is found to vary in terms of resolution as ε^-6. Therefore the quantum energy density and the cosmological energy density that manifests itself in terms of 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 the new framework. The causality problem disappears in terms of Lorentzian dilation laws; there is no need of an inflation phase, then no need to introduce an unknown unobserved 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 Λ = 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 the 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(λ_z/λ_GUT) = log(λ_z/Λ_P)/√2 ≈ 17/√2 ≈ 12.



• 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 the 'state of scale' lnε, leads in special scale relativity to general mass-charge relations of the form α ln(λ_c/Λ_P) = k/2, where k is integer, α is a coupling constant, λ_c a typical Compton scale, inversely related to a mass scale and Λ_P is the Planck length-scale.



• ElectroWeak scale: given by the mass / charge relation α_0∞ln(λ_EW/Λ_P) = 1, i.e.,
  λ_EW = Λ_Pe^4π2 = 1.397 x 10¹⁷Λ_P ≡ 123 GeV (while the v.e.v. of the Higgs field is
  174 GeV = 123√2 GeV).



• Electron scale: given by the mass / charge relation α_0eln(λ_e/Λ_P) = 1, i.e.,
  m_e = m_Pe^-3/8α_e ≈ 0.5 MeV.



• Weak boson mass ratio (value of Weinberg angle): we predict that α₂ = 2α₁ at electroweak scale, so that m_W/m_Z = (10/13)^1/2, and sin²θ = 3/13 at this scale.



• Elementary fermion mass spectrum: recovered from a cancellation effect between special scale-relativistic corrections and radiative corrections. (However, 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 (observed value: 174.3 ± 5.1 GeV).



• Values of low energy coupling constants: derived from their renormalization group equations and from the conjecture that the value 1/4π² is critical for coupling constants. We find α_e = 137.04 ± 0.03 from α_0∞ = 1/4π² and α₃(m_Z) = 0.116 ± 0.0005 from α₃(m_GUT) = 1/4π².



• Power of galaxy-galaxy correlation function: the observed value γ = 1.8 at
  ≈ 1 - 10 Mpc is explained as the result of a scale-relativistic correction to the standard value γ = 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 'quantum-gravitational' equation, holding for chaotic system on very large time scales (beyond the horizon of predictibility).



• Quantization of binary galaxies: the quantization in terms of 72/n km/s observed by Tifft and colleagues 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 linear quantization (harmonic oscillator) that accounts for the observed 'global' redshift quantization at 36 km/s.

New predictions

• Precise value of the strong coupling constant: we predict, as quoted above,
  α₃(m_Z) = 0.116 ± 0.0005, more precise than the current value, 0.118 ± 0.002.



• 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 only poorly known (80.42 ± 0.04 GeV).



• Breaking of 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 various observed cross sections of particle collisions in future high energy accelerators (LHC...) to depart from their values calculated from standard quantum mechanics (i.e., Galilean scale-relativistic laws). The departure may be expressed, to lowest order, in terms of a scale-varying effective Planck constant.



• Value of the cosmological constant: it is predicted to be Λ = 1.36 x 10^-56 cm^-2, under the assumption that the fractal-nonfractal transition for the vacuum energy density occurs at the classical radius of the electron.



• New planets in the solar system: some of the 'orbitals' predicted by the theory do not contain observed planets. For some of them this can be understood (n = 1 of the inner system is too close to the Sun, n = 7 and 10 are destroyed by resonances with Jupiter), but some others may contain objects that have up to now escaped detection (n = 2 of the inner system, at 0.185 A.U., n > 6 of the outer system).



• 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 solar system. This prediction, made in 1995, is verified on the external planetary systems discovered since then.



• 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...



• Structuration of the universe: in a similar way, galaxies in the universe are predicted by the present theory to form structures 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 connection, SU(3) being, as is well-known, the symmetry group of QCD.



• Value of power of galaxy correlation function at very large scale: in the 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 precise measurement of the cosmological constant.

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Quoted from: Nottale, L., 1996, Chaos, Solitons and Fractals, 7, 877-938
"Scale Relativity and Fractal Space-Time: Application to Quantum Physics, Cosmology and Chaotic systems"

Primeval Density Fluctuations

Another fundamental consequence of the quantum gravitational theory concerns the question of density fluctuations at the recombination epoch. In the standard approach to the problem of structure formation, no structure can be formed from a strictly constant matter density. The standard model of formation therefore assumes that the present structures originate from the gravitational growth of early fluctuations that existed at the recombination epoch. These structures have actually been observed through their temperature signature in the Cosmic Microwave Background Radiation field, though at a low level of δΤ/Τ ≈ 2 x 10^-5.

However, in the standard approach these fluctuations should themselves be understood in terms of still earlier fluctuations. But the strong isotropization that prevails during the prerecombination phase poses a difficult problem. The standard solution to this problem consists of making the postulate that the Universe would have known an inflation phase (i.e. exponential expansion) of unknown origin, which would have multiplied quantum fluctuations by an enormous factor.

We can suggest another solution to this problem in the scale-relativity framework, without any need of an inflationary phase. Indeed, as recalled in the Missing Mass Section, the fractality of space-time involves a transformation the geodesics equations in a Schrödinger-like equation. As a direct consequence, there is a tendency to form structures at any epoch: these structures are described by probability density distributions given by the square of the modulus of the probability amplitudes which are solutions of this gravitational Schrödinger equation. The classical approach, because of its deterministic and differentiable nature, predicts structures at a given epoch which are the result of an evolution from previous existing structures: these structures of the past are taken as initial conditions (in position and velocity). The new quantum-like approach is organized in a fundamentally different way. The loss of determinism of individual trajectories is compensated by a determinism of structures. At each epoch, stationary or steady-state solutions can be found in correspondence with the shape of the potential and the limiting and matching conditions. These structures do also evolve (as given by the time-dependent Poisson-Schrödinger system), in correspondence with the evolution of the environment.

Therefore one expects the occurence of quantum fluctuations (structures) at the decoupling epoch, but according to the macroscopic quantum theory based on
D = GM/2α_gc which applies at this epoch, instead of the microscopic quantum theory for which D = h/2m. No inflation is needed in the new framework to obtain a scale invariant, quantumlike, spectrum at z ≈ 1000.
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Quoted from: Nottale L., 2003, Chaos Solitons and Fractals, 16, 539
"Scale-relativistic cosmology"