Quantum mechanics - Vrije Universiteit quantum mechanics. 1. Two valuedness of the velocity field ¢â‚¬â€œ¢â‚¬â€œ>

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    Quantum mechanicsQuantum mechanics Effects 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 a nondifferentiable spacenondifferentiable space

    http ://luth.obspm.fr/~luthier/nottale/

    Laurent Nottale CNRS

    LUTH, Paris-Meudon Observatory

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    ReferencesReferences 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, 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 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”.

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    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

    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

    tra ns

    itio n

    fractal

    scale - independent

    ln ε

    tra ns

    itio n

    fractal

    de lta

    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).

    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 in another 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 operator Classical (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 of composition (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: explained here 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 in quantum mechanics. 1.quantum mechanics. 1.

    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:

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    Complex numbers. Origin. 2.Complex numbers. Origin. 2. 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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    SOLUTIONSSOLUTIONS Visualizations, 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 oscillator potentialpotential

    Animation

    Firts excited level : simulation of the process dx = v+ dt + dξ+

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    3D isotropic harmonic oscillator3D isotropic harmonic oscillator potentialpotential

    First excited level: simulation of process dx = v+ dt + dξ+

    Comparaison simulation - QM prediction: 10000 pts, 2 geodesics

    D en

    sit y

    of p

    ro ba

    bi lit

    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 geodesics Kepler central potential GM/r State 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 to theoretical distribution solution of Schrödinger equation