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Quantum mechanicsQuantum mechanicsEffects on the equations of motionEffects on the equations of motion
of the fractal structures of the geodesics of a of the fractal structures of the geodesics of anondifferentiable spacenondifferentiable space
http ://luth.obspm.fr/~luthier/nottale/
Laurent NottaleCNRS
LUTH, Paris-Meudon Observatory
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ReferencesReferencesNottale, L., 1993, Fractal Space-Time and Microphysics : Towards a Theory of ScaleRelativity, 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 FractalSpace-Time : Application to Quantum Physics, Cosmo- logy and Chaotic systems”.http ://luth.obspm.fr/~luthier/nottale/arRevFST.pdf
Nottale, L., 1997, Astron. Astrophys. 327, 867. “Scale relativity and Quantization of theUniverse. 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 scalerelativity”.
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Fractality Discrete symmetry breaking (dt)
Infinity ofgeodesics
Fractalfluctuations
Two-valuedness (+,-)
Fluid-likedescription
Second order termin differential equations
Complex numbers
Complex covariant derivative
NON-DIFFERENTIABILITYNON-DIFFERENTIABILITY
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Dilatation operator (Gell-Mann-Lévy method):
First order scale differential equation:First order scale differential equation:
Taylor expansion:
Solution: fractal of constant dimension + transition:
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ln L
ln ε
trans
ition
fractal
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 scaledifferential equation in scale space).
Dependence on scale of the length (=fractal coordinate)Dependence on scale of the length (=fractal coordinate) and of the effective fractal dimension and of the effective fractal dimension
= DF - DT
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Asymptotic behavior:
Scale transformation:
Law of composition of dilatations:
Result: mathematical structure of a Galileo group ––>
Galileo scale transformation groupGalileo scale transformation group
-comes under the principle of relativity (of scales)-
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Road toward SchrRoad toward Schröödinger (1):dinger (1):infinity of geodesicsinfinity of geodesics
––> generalized « fluid » approach:
Differentiable Non-differentiable
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Road toward SchrRoad toward Schröödinger (2):dinger (2):‘‘differentiable partdifferentiable part’’ and and ‘‘fractal partfractal part’’
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:
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Road toward SchrRoad toward Schröödinger (3):dinger (3):non-differentiability non-differentiability ––––> complex numbers> complex numbers
Standard definition of derivative
DOES NOT EXIST ANY LONGER ––> new definition
TWO definitions instead of one: they transform one inanother by the reflection (dt <––> -dt )
f(t,dt) = fractal fonction (equivalence class, cf LN93)Explicit fonction of dt = scale variable (generalized « resolution »)
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Covariant derivative operatorCovariant derivative operatorClassical(differentiable)part
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Improvement of Improvement of « « quantumquantum » »covariancecovariance
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 ofcomposition (see also Pissondes’s work on this point)
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Covariant derivative operator
Fundamental equation of dynamics
Change of variables (S = complex action) and integration
Generalized Schrödinger equation
FRACTAL SPACE-TIMEFRACTAL SPACE-TIME––>QUANTUM MECHANICS>QUANTUM MECHANICS
Ref: LN, 93-04, Célérier & LN 04,07. See also works by: Ord,Hermann, Pissondes, Dubois, Jumarie, Cresson, Ben Adda, Agop, …
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Hamiltonian: covariant formHamiltonian: covariant form
––>
Additional energy term specific of quantum mechanics: explainedhere as manifestation of nondifferentiability and strong covariance
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Newton
Schrödinger
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Newton
Schrödinger
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Origin of complex numbers inOrigin of complex numbers inquantum mechanics. 1.quantum mechanics. 1.
Two valuedness of the velocity field ––> need to define a newproduct: 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 areneeded:
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Complex numbers. Origin. 2.Complex numbers. Origin. 2.Define the new velocity doublet, including the divergent (explicitlyscale-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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SOLUTIONSSOLUTIONSVisualizations, simulationsVisualizations, simulations
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Geodesics stochastic diferential equationsGeodesics stochastic diferential equations
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Young hole experiment: one slitYoung hole experiment: one slit
Simulation of
geodesics
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Young hole experiment: one slitYoung hole experiment: one slit
Scale dependent simulation: quantum-classical transition
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Young hole experiment: two-slitYoung hole experiment: two-slit
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3D isotropic harmonic oscillator3D isotropic harmonic oscillator
Examples of geodesics
simulation of process dxk = vk+ dt + dξk
+
n=0 n=1
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3D isotropic harmonic oscillator3D isotropic harmonic oscillatorpotentialpotential
Animation
Firts excited level : simulation of the process dx = v+ dt + dξ+
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3D isotropic harmonic oscillator3D isotropic harmonic oscillatorpotentialpotential
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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n=0 n=1
n=2(2,0,0)
n=2(1,1,0)
E = (3+2n) mDω
Hermite polynomials
Solutions: 3D harmonic oscillator potential 3D (constant density)Solutions: 3D harmonic oscillator potential 3D (constant density)
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n=0 n=1
n=2 n=2(2,0,0) (1,1,0)
Solutions: 3D harmonic oscillator potentialSolutions: 3D harmonic oscillator potential
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Simulation of geodesicsSimulation of geodesicsKepler central potential GM/rState n = 3, l = m = n-1
Process:
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n=3
Solutions: Kepler potentialSolutions: Kepler potential
Generalized Laguerre polynomials
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Hydrogen atomHydrogen atom
Distribution obtained from one geodesical line, compared totheoretical distribution solution of Schrödinger equation