Holographic-Type Gravitation via Non-Differentiability in Weyl-Dirac Theory

In the Weyl-Dirac non-relativistic hydrodynamics approach, the non-linear interaction between sub-quantum level and particle gives non-differentiable properties to the space. Therefore, the movement trajectories are fractal curves, the dynamics are described by a complex speed field and the equation of motion is identified with the geodesics of a fractal space which corresponds to a Schrödinger non-linear equation. The real part of the complex speed field assures, through a quantification condition, the compatibility between the Weyl-Dirac non-elativistic hydrodynamic model and the wave mechanics. The mean value of the fractal speed potential, identifies with the Shanon informational energy, specifies, by a maximization principle, that the sub-quantum level “stores” and “transfers” the informational energy in the form of force. The wave-particle duality is achieved by means of cnoidal oscillations modes of the state density, the dominance of one of the characters, wave or particle, being put into correspondence with two flow regimes (non-quasi-autonomous and quasi-autonomous) of the Weyl-Dirac fluid. All these show a direct connection between the fractal structure of space and holographic principle.


Introduction
The General Relativity states that there is a reciprocal conditioning between geometry and matter so that the guiding mechanism is governed by the motions of the matter itself.However, the same guiding mechanism is neglected when it is used in the study of particle dynamics at microscopic scale.Such "apparent contradiction" is can be solved, for example by means of Weyl-Dirac (WD) theory [1][2][3].
After the development of general theory of relativity, Weyl extend this theory for electromagnetic processes, from the dominance of light rays for physical measurements, where the phenomena are also described geometrically.This theory had some features that not gain the general acceptance.Later, Dirac introduces some modifications which removed the theory difficulties and he made use of the theory to provide a framework to explain his large number hypothesis.
Using the GMC formalism in WD theory, important results were obtained (the particle is represented by a spherically symmetric thin-shell solution to Einstein's equations; a geometric model with conformal invariance broken in the interior space; a new possibility to consider non-local effects, when the interior curved space-time has non causal properties, such as closed time-like curves; a transfer mechanism for energy-momentum between the thin shell and the Madelung fluid; a geometric guidance condition for the bubble at microscopic scale and a Hamilton-Jacobi equation that can be directly applied to the thin shell so that the bubble could move in step with the Madelung fluid) [4][5][6].
In [7][8][9] we have shown that the wave-particle duality may be associated with a phase transition of superconducting-normal state type.More recently [10], using the hydrodynamic model of the WD theory in the non-rela-tivistic approach, we established some properties of vacuum states.
This paper analyzes the wave-particle duality in the WD non-relativistic hydrodynamics model from the perspective of the non-differentiability of motion curves of the WD non-relativistic fluid particles.The paper is structured as follows: in Section 2 the non-differentiability of the motion curves in the WD non-relativistic hydrodynamics model; in Section 3 the wave-particle duality through cnoidaloscillation modes of the states density.

Non-Differentiability of the Motion Curves in the WD Non-Relativistic Hydrodynamics
The way in which the geometry of space-time affects the dynamics of the particle in the WD theory is given by the covariant Equation [5] 2 0 where   is the covariant derivative, R is the Ricci scalar, Λ is the cosmological constant and  is the wave function associated of the particle.So, "it is considered a matter shell on a cosmological background described by the field  which is also a source of the wave function.The law of parallel transport common to this theory requires a vector to change not only in direction but also in magnitude, after transport along a closed space-time loop.This result is given by a quantum force due to both the curvature of space-time and wave function, and consequently, due to the loss of the microscopic distinguishability of the particle's trajectories" [5].Since 2  is taken to represent the probability density, Equation (1) enables the quantum mechanical interpretation of the WD theory in the sense of Bohm [11].
In the weak field approximation (WFA) [10,[12][13][14][15][16][17][18][19]] and low speeds as compared to speed of light in vacuum, the WD equation with  is reduced to the set of equations: where  is the states density, v is the ical phase S , u is the speed speed associated to class associated to state density,   is the te tial associated to space-time,   1 R is the i scalar in the WFA apch [12][13][14][15][16][17][18][19], ħ is Planck's reduced constant, c is the light speed in vacuum, 0 m s the rest mass of material "entity" and t is the classical time.Now, certain conclusions are obvious: i) Any material "entity" is in a perman t interaction with the "subquantum level" through the quantum ell as through the "perturbations" at the quantum potential as The "sub-quantum level" is identified with a non-relativistic WD fluid de ibed by the probability density and the momentum conservation laws, see (2a,b).
se equations correspond to the generalised quantum hydrodynamics model (WD nonrelativistic hydrodynamics model); iii) In space-time topology.
These equations define the standard model hydrodynamics [11]; iv) The Equation (2a) can be written under the form: This result is obtained through the followin tions: multiplication with , inte null integration constant, applying the gradient and using th on (6) w ent equation results: gration with a e relation (3f).
Let us multiply the relati ith i  and also, let us multiply the Equation (2b) with 1 0 m  .By summing them, the movem where V is the complex speed field (for s see [20][21][22][23][24]) imilar results  is the scalar potential of the complex speed and d dt is the "covariant derivative" 0 Thus, the movements of material "entity" on continuous and non-differentiable curves (fractal curve fractal dimension 2 F D  ) are proved by "activating" a sp s with ace with a special topology, i.e. the fractal space [23][24][25][26].Once such a space admitted, the following consequences result: iv1) The dynamics of physical system are described through fractal functions that depend both on space-time coordinates and on the de Broglie scale resolution.Thus the physical quantities, which define these dynamics of the physical system, are complex functions (for example the complex speed field (8a) and the pure imaginary coefficient 0 i 2m  , corresponding to the fractal-non-fractal transition [23,24]).Moreover, the real parts of physical quantities are differentiable and independent on scale resolution the imaginary parts are non-differentiable and dependent on the resolution scale; iv2) The scale resolution reflects a certain degree of non-differentiability of the movement curve; iv3) The movement operatoris identified with the "covariant derivative" , while d dt ; iv4)The use of a generalized Newton principle turns the movement Equation ( 7) into geodesics of a fractal space; iv5) Chaoticity, either by turbulence as in the non-relativistic hydrodynamics approach, either by stochasticization as in the generalized Schrödinger approach, is achieved through non-differentiability of a fractal space.Indeed, by substituting (8a,b) in (7) and using the method described in [27,28], it results: Equation ( 10) can be integrated in a universal way and yields up to an arbitrary phase factor which may be set to zero by a suitable choice of the phase of   .Thus, the nonlin cs is obtained.We note that in the WD non-relativistic hydrodynamics, ear Schrödinger type equation (NSE) as fractal space geodesi (through ln   ) is the potential of the complex speed and in GSE is n   scalar a wave function; iv6) The compatibility between the WD nonrelativistic hydrodynamics model and the wave mechanics (WM) implies, through the relation (3d) and (3e) the quantization conditions: iv7) The mean value of the fractal potential (the imaginary part of the scalar potential of the complex speed) can be identified, without a constant factor, with the Shanon informational energy [24,29,30] ln Now, accepting a maximization principle for the informational energy in the form: for constrains with radial symmetry, we get In pol

 
the space-time toogy (4), by substituting this value in the expression s the type of force "stored and "transmitted".(15) Therefore, in the WD non relativistic hydrodynamics model and space-time topology (4), the information is "stored and transmitted" by the sub-quantum level as a he choice of 0 r specifie

Oscillations Modes of the States Density
In one-dimensional case, the Equations (2a,b) and with the restriction where M is equivalent with the Mach number.Hence, through integration, is found where 1 c and 2 c are integration constants.
ssion The solution of this equation has the expre where modulus s [31], a is an amplitude and  is an average parame-value of the states density.Details on defining ters s, and a  ore t ave-particle duality is achieved through spac can be found in [31].Theref he w e-time cnoidal oscillation modes of the states density-see Figure 1.The oscillation modes are explained through m s of the elliptical function cn, non-linearity parameter ing am g othe odulus depend on rs space-time topology.Moreover, the oscillation modes are self-similar via the non-linearity parameter-see Figures 2(a)-(c), which specifies the fractal character of the space.
The self-similarity of the cnoidal modes specifies the existence of some "cloning" mechanisms (full and fractional wavefunction revivals-a wave function evolves in time to a state describable as a collection of spatially distribuited sub-wave-functions that each closely reproduces the initial wave-function shape) [32].All these show a direct connection between the fractal structure of space and holographic principle [23,24,30,33].
The space-time cnoidal oscillation modes have the following characteristics: i) Wave number ii) Phase velocity Various sequences are obtained through the following degenerations: i) For s→0, (22) reduces to the harmonic wave packages characterized by wave number characterized by wave number   (22) reduces to the harmonic wave, while for s = 1 to the soliton one.
Eliminating the amplitude a, between ( 23) and ( 24) we obtain the relation

4.
In the WD non-relativistic hydrodynamics non-linear interaction between the sub-quantum particles induces non-differentiable properties to the nt takes place on continuum and non-differentiable curves space.Thus: a) Particle moveme (fractal curves ribed by depending q r nd scale resolutio e Broglie), fractal functions.They contain a real part, differentiable and independent on the de Broglie scale and an imaginary part, fractal and dependent on the de Broglie scale.An example of this kind is given by the complex speed fiel ard d dt is replaced by the covaria v nt deri ative d dt ; d) Apply- plex speed field, the ing the covariant derivative a com particle's motion e fractal space.Thes dinger equation; e) Chaoticity, either by turbulence like in the case of hydrodynamics, or by stocasticit Sch io principle ith radial sym uantum potential gradient, a force field results.Thus, the sub-quantum level will "stor informational energy as a force; h) In lity is rea aracter, and another one through quasi-autonomous structures (soliton, soliton package, etc.) which assures dominant particle character.Moreover, the self-similarity of cnoidal oscillation modes specify a direct connection between the fractal structure of space and holographic principle, i.e. a holographic type gravitation.
to quation become a geodesic of the e are described by a non-linear Schroy like in rodinger representation, are induced by non-differentiability; f) Real part of the speed field assures through a quantification condition the compatibility between the WD non-relativistic hydrodynamics and wave mechanics; g) Average size of the fractal scalar potential of the complex speed field, without a certain constant factor, can be identified with informat nal Shanon energy.Accepting a maximization of the informational energy for constraints w metry, in a special topology, through q e" and "transfer" general, waveparticle dua lised by cnoidal oscillation modes of the states density.These are characterized by two distinct flow regimes, one by non-quasi-autonomous structures (wave, wave package, etc.) which assures dominant undulatory ch " Nova, New York, 1999.


E s are the complete elliptical grals of first and second kindof modulus s, cn is the Jacobi elliptical function of argument  inte-

3 Figure 3 .
Figure 3. Flow regimes of the no c WD fluid versus λ and non-linearity parameters.It res s the change of flow .n-relativisti ult for s 0.7 