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Fractal Space-Time And Microphysics: Towards A Theory Of Scale Relativity
This is the first detailed account of a new approach to microphysics based on two leading (i) the explicit dependence of physical laws on scale encountered in quantum physics, is the manifestation of a fundamental principle of nature, scale relativity. This generalizes Einstein's principle of (motion) relativity to scale transformations; (ii) the mathematical achievement of this principle needs the introduction of a nondifferentiable space-time varying with resolution, i.e. characterized by its fractal properties. The author discusses in detail reactualization of the principle of relativity and its application to scale transformations, physical laws which are explicitly scale dependent, and fractals as a new geometric description of space-time.
348 pages, Paperback
First published May 1, 1993
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September 27, 2012
Revolutionary ideas. Paragraphs of more widely accessible language backed up by calculations that require more advanced understanding of physics.
Nottale offers fractal space-time as a method for establishing a relativity of scale. He defines the term fractal as divergent with decreasing scale (Nottale 1993, 20) building on Mandelbrot's definition which highlights irregularity at all scales creating a "density" described by fractal dimensionality. (Nottale 1993, 33) Divergent means to go to infinity. So scale divergence means the more one zooms in, the more detail emerges. When measuring something that is fractal, the smaller the measuring stick or scale is, the longer the measured distance will be. This is referred to as scale divergence. A second key fractal property, related to scale divergence, is non-differentiability, meaning that a line is irregular or fragmented at all scales. It is never straight.
Scale divergence emerges as Nottale generalizes space-time by removing the assumption of differentiability. Cosmology continually strives for more general descriptions of space-time. A more general description encompasses more possibilities and makes less assumptions. For example, general relativity moved to a curved space-time geometry from a less general, flat, space-time geometry. By challenging the assumption of flatness, Einstein was able to account for mass through the curvature of space-time. By challenging the assumption of differentiability at small scales, Nottale opens a possible way to integrate quantum mechanics into cosmology.
Nottale points out that, like fractal space-time, the paths of quantum particles are also continuous and non-differential. In fractal space-time the geodesics (shortest path between two points, and also the path traversed by light) are also fractal – jagged and irregular. Thus particles following fractal trajectories, like Brownian motion, are simply following the fractal space-time geodesics, the paths of least resistance.
Nottale also notes that in quantum mechanics charge and energy are divergent, or go to infinity, at decreasing scales. The divergence of energy with decreasing scale is linked to Heisenberg's Uncertainty Principle, where the more accurately one variable (distance or time) is known, the less accurately its paired variable (momentum or energy) is known. This dynamic mimics the relationship of scale divergence, in that the shrinking of one variable implies the divergence of the other. Nottale deduces that this divergence begins at the de Broglie scale, above which exists classical, differentiable, space-time and below which exists quantum fractal space-time characterized by temporal reversibility and fractal dimensionality.
Nottale's fractal space-time neatly recreates many pre-existing physical formulas through fractal re-formulation, and offers a number of promising possibilities for physically testing the theory.
Nottale offers fractal space-time as a method for establishing a relativity of scale. He defines the term fractal as divergent with decreasing scale (Nottale 1993, 20) building on Mandelbrot's definition which highlights irregularity at all scales creating a "density" described by fractal dimensionality. (Nottale 1993, 33) Divergent means to go to infinity. So scale divergence means the more one zooms in, the more detail emerges. When measuring something that is fractal, the smaller the measuring stick or scale is, the longer the measured distance will be. This is referred to as scale divergence. A second key fractal property, related to scale divergence, is non-differentiability, meaning that a line is irregular or fragmented at all scales. It is never straight.
Scale divergence emerges as Nottale generalizes space-time by removing the assumption of differentiability. Cosmology continually strives for more general descriptions of space-time. A more general description encompasses more possibilities and makes less assumptions. For example, general relativity moved to a curved space-time geometry from a less general, flat, space-time geometry. By challenging the assumption of flatness, Einstein was able to account for mass through the curvature of space-time. By challenging the assumption of differentiability at small scales, Nottale opens a possible way to integrate quantum mechanics into cosmology.
Nottale points out that, like fractal space-time, the paths of quantum particles are also continuous and non-differential. In fractal space-time the geodesics (shortest path between two points, and also the path traversed by light) are also fractal – jagged and irregular. Thus particles following fractal trajectories, like Brownian motion, are simply following the fractal space-time geodesics, the paths of least resistance.
Nottale also notes that in quantum mechanics charge and energy are divergent, or go to infinity, at decreasing scales. The divergence of energy with decreasing scale is linked to Heisenberg's Uncertainty Principle, where the more accurately one variable (distance or time) is known, the less accurately its paired variable (momentum or energy) is known. This dynamic mimics the relationship of scale divergence, in that the shrinking of one variable implies the divergence of the other. Nottale deduces that this divergence begins at the de Broglie scale, above which exists classical, differentiable, space-time and below which exists quantum fractal space-time characterized by temporal reversibility and fractal dimensionality.
Nottale's fractal space-time neatly recreates many pre-existing physical formulas through fractal re-formulation, and offers a number of promising possibilities for physically testing the theory.
April 15, 2016
It is a book that claims to have a solution to various existing problems in unification between gravitation and quantum theory. It takes insights of Feynman's discovery that quantum mechanical trajectory is fractal by nature.
I personally felt the development is a bit too rushed and too wide in scope that the book becomes confusing. Furthermore, the formalism is not convincing enough in many parts, although heuristically it is amazing that things seem to fit.
There are good reasons to believe that spacetime can indeed be fractal at microscale, especially when general relativity is far from being a quantum theory. However, I think to put this framework in action one has to do more than mathematical equation juggling and come up with stronger and more convincing framework that convinces people that this should be true feature of physical phenomena before equations are thrown.
I personally felt the development is a bit too rushed and too wide in scope that the book becomes confusing. Furthermore, the formalism is not convincing enough in many parts, although heuristically it is amazing that things seem to fit.
There are good reasons to believe that spacetime can indeed be fractal at microscale, especially when general relativity is far from being a quantum theory. However, I think to put this framework in action one has to do more than mathematical equation juggling and come up with stronger and more convincing framework that convinces people that this should be true feature of physical phenomena before equations are thrown.
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