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Effects on the equations of motion of the fractal structures of the geodesics of a nondifferentiable space
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http ://luth.obspm.fr/~luthier/nottale/
Laurent Nottale���CNRS���
LUTH, Paris-Meudon Observatory
References
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Nottale, L., 1993, Fractal Space-Time and Microphysics : Towards a Theory of Scale Relativity, World Scientific (Book, 347 pp.)!Chapter 5.6 : http ://luth.obspm.fr/~luthier/nottale/LIWOS5-6cor.pdf !!Nottale, L., 1996, Chaos, Solitons & Fractals, 7, 877-938. “Scale Relativity and Fractal Space-Time : Application to Quantum Physics, Cosmology and Chaotic systems”. !http ://luth.obspm.fr/~luthier/nottale/arRevFST.pdfNottale, L., 1997, Astron. Astrophys. 327, 867. “Scale relativity and Quantization of the Universe. I. Theoretical framework.” http ://luth.obspm.fr/~luthier/nottale/arA&A327.pdf!
Célérier Nottale 2004 J. Phys. A 37, 931(arXiv : quant- ph/0609161) !“Quantum-classical transition in scale relativity”. !http ://luth.obspm.fr/~luthier/nottale/ardirac.pdf !!Nottale L. & C élérier M.N., 2007, J. Phys. A : Math. Theor. 40, 14471-14498 (arXiv : 0711.2418 [quant-ph]). !“Derivation of the postulates of quantum mechanics form the first principles of scale relativity”.!!Nottale L., 2011, Scale Relativity and Fractal Space-Time (Imperial College Press 2011) Chapter 5.!!
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NON-DIFFERENTIABILITY
Fractality Discrete symmetry breaking (dt)
Infinity of geodesics
Fractal fluctuations
Two-valuedness (+,-)
Fluid-like description
Second order term in differential equations
Complex numbers
Complex covariant derivative
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Dilatation operator (Gell-Mann-Lévy method):
Taylor expansion:
Solution: fractal of constant dimension + transition:
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Dependence on scale of the length (=fractal coordinate) and of the effective fractal dimension
ln L
ln ε
trans
itionfractal
scale -independent
ln ε
trans
ition
fractal
delta
variation of the length variation of the scale dimension"scale inertia"
scale -independent
Case of « scale-inertial » laws (which are solutions of a first order scale differential equation in scale space).
= DF - DT
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« Galileo » scale transformation group Asymptotic behavior:
Scale transformation:
Law of composition of dilatations:
Result: mathematical structure of a Galileo group ––>
-comes under the principle of relativity (of scales)-
Road toward Schrödinger (1): infinity of geodesics
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––> generalized « fluid » approach:
Differentiable Non-differentiable
Road toward Schrödinger (2): ‘differentiable part’ and ‘fractal part’
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Minimal scale law (in terms of the space resolution):
Differential version (in terms of the time resolution):
Case of the critical fractal dimension DF = 2:
Stochastic variable:
Road toward Schrödinger (3): non-differentiability ––> complex numbers
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Standard definition of derivative
DOES NOT EXIST ANY LONGER ––> new definition
TWO definitions instead of one: they transform one in another by the reflection (dt <––> -dt )
f(t,dt) = fractal fonction (equivalence class, cf LN93) Explicit fonction of dt = scale variable (generalized « resolution »)
Improvement of « quantum » covariance
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Ref.: Nottale L., 2004, American Institute of Physics Conference Proceedings 718, 68-95! “The Theory of Scale Relativity : Non-Differentiable Geometry and Fractal Space- Time”. !http ://luth.obspm.fr/~luthier/nottale/arcasys03.pdf
Introduce complex velocity operator:
New form of covariant derivative:
satisfies first order Leibniz rule for partial derivative and law of composition (see also Pissondes’s work on this point)
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FRACTAL SPACE-TIME–>QUANTUM MECHANICS Covariant derivative operator
Fundamental equation of dynamics
Change of variables (S = complex action) and integration
Generalized Schrödinger equation
Ref: LN, 93-04, Célérier & LN 04,07. See also works by: Ord, Hermann, Pissondes, Dubois, Jumarie, Cresson, Ben Adda, Agop, …
Hamiltonian: covariant form
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––>
Additional energy term specific of quantum mechanics: explained here as manifestation of nondifferentiability and strong covariance
Origin of complex numbers in quantum mechanics. 1.
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Two valuedness of the velocity field ––> need to define a new product: algebra doubling A––>A2
General form of a bilinear product :
i,j,k = 1,2 ––> new product defined by the 8 numbers
Recover the classical limit ––> A subalgebra of A2 Then (a,0)=a. We define (0,1)=α and therefore only 2 coefficients are needed:
Complex numbers. Origin. 2.
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Define the new velocity doublet, including the divergent (explicitly scale-dependent) part:
Full Lagrange function (Newtonian case):
Infinite term in the Lagrangian ?
Since and
––> Infinite term suppressed in the Lagrangian provided:
QED
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General relativity: covariant derivative
Scale relativity: covariant derivative
Geodesics equation: Geodesics equation:
Newtonian approximation:
General + scale relativity
Quantum form
Three representations
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Geodesical (U,V) Generalized Schrödinger (P,θ)
Euler + continuity (P, V)
New « potential » energy:
Five representations (forms) of ScR equations (1) Fondamental eq. of dynamics/ geodesic eq.
(2) Schrödinger
(3) Fluid mechanics( P= |ψ|2, V) -> continuity + Euler + quantum potential
(4) Coupled bi-fluid (U, V)
(5) Diffusion (v+, v-) Fokker-Planck + BFP
7 significations (and measurement methods) of coefficient D
* Diffusion coefficient • Amplitude of fractal fluctuations * Generalized Compton length
Generalized de Broglie length
• Generalized thermal de Broglie length
* Heisenberg relation (x,v)
Heisenberg relation (t,v)
*Energy quantization etc…
3D isotropic harmonic oscillator
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Examples of geodesics
simulation of process dxk = vk+ dt + dξk
+
n=0 n=1
3D isotropic harmonic oscillator potential
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First excited level : simulation of the process dx = v+ dt + dξ+
3D isotropic harmonic oscillator potential
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First excited level: simulation of process dx = v+ dt + dξ+
Comparaison simulation - QM prediction: 10000 pts, 2 geodesics
Den
sity
of p
roba
bilit
y
Coordinate x
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Solutions: 3D harmonic oscillator potential 3D (constant density)
n=0 n=1
n=2 (2,0,0)
n=2 (1,1,0)
E = (3+2n) mDω
Hermite polynomials