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Fractal space-time and microphysics. Towards a theory of scale relativity. (English) Zbl 0789.58003
Singapore: World Scientific. xiii, 333 p. (1993).
The author, the Director of Research at CNRS, has presented in the book his ideas on a fractal strucure of space-time, on a connection of appropriate fractal dimensions to scale properties of physical quantities and a relativity principle for scales to a renormalization group.
Chapter 1 is just a general introduction and a short account of the content of the book. In Chapter 2 the author discusses the most fundamental theories of modern physics – theory of relativity and quantum theory – and introduces main notions and themata which will go through all the book. Chapter 3 is first of all a short and clear exposition of fractal geometry beginning from single objects, lines, surfaces to a whole fractal space-time. The author considers an analysis on fractal spaces in terms of non-standard analysis by A. Robinson (1961) and discusses fractal continuous non-differentiable functions similar to the famous function to K. Weierstrass (1872). The author concludes the discussion of fractality by consideration of a notion of fractal space- time. In the following Chapter 4 the author reminds to the reader that according to the quantum mechanics in a form given by R. Feynman (1948) a quantum particle trajectory is a continuous non-differentiable scale- dependent line and deduces a fractal dimension of this line. The author considers this circumstance as a substantiation of fractality for the quantum space-time. Postulating the fractal structure of quantum space- time the author deduces in Chapter 5 the main laws and relations of quantum theory: the Schrödinger equation, the Heisenberg uncertainty principle, a wave-particle dualism, an emergence of spin from fractalization of a trajectory etc. The author considers quantum mechanics as mechanics of a fractal space-time. He discusses also fractal ideas in a domain of quantum electrodynamics and considers in a qualitative and very preliminary manner some recent experiments in high energy physics. In Chapter 6 the author formulates a principle of scale relativity as a generalization of that for the coordinates of a space- time which is described by the Lorentz transformation. He presents it in terms of renormalization group and applies it to the discussion of some problems of high energy physics, the nature of charges, masses etc. According to the author the Planck scale is universal and invariant under dilatations. Chapter 7 is dedicated to a possible application of new ideas to a cosmology and to a problem of distribution of planet radii in a solar system.
The reader has to consider the book as a “snapshot” of some new ideas arising in course of development of modern science. These ideas must be verified thoroughly by experiment before they come to a fundament of future physics or be given up.

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
81V10 Electromagnetic interaction; quantum electrodynamics
37N99 Applications of dynamical systems
83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81T17 Renormalization group methods applied to problems in quantum field theory