1 Quantum mechanics Quantum mechanics Effects on the equations of motion Effects on the equations of motion of the fractal structures of the geodesics of a of the fractal structures of the geodesics of a nondifferentiable space nondifferentiable space http ://luth.obspm.fr/~luthier/nottale/ Laurent Nottale CNRS LUTH, Paris-Meudon Observatory
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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):
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