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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