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Fractals in the Quantum Theory of Spacetime

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

The question of the foundation of quantum mechanics from first principles remains one of the main open problems of modern physics. In its currentform, it is an axiomatic theory of an algebraic nature founded upon a set of postulates, rules and derived principles. This is to be compared withEinstein's theory of gravitation, which is founded on the principle of relativity and, as such, is of an essentially geometric nature. In its framework,gravitation is understood as a very manifestation of the curvature of a Riemannian space-time.

It is therefore relevant to question the nature of the quantum space-time and to ask for a possible refoundation of the quantum theory upongeometric first principles. In this context, it has been suggested that the quantum laws and properties could actually be manifestations of a fractaland nondifferentiable geometry of space-time [52,69,71], coming under the principle of scalerelativity [53,54]. This principle extends, to...

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Abbreviations

Fractality:

In the context of the present article, the geometric property of being structured over all (or many) scales, involving explicit scale dependence which may go up to scale divergence.

Spacetime:

Inter‐relational level of description of the set of all positions and instants (events) and of their transformations. The events are defined with respect to a given reference system (i. e., in a relative way), but a spacetime is characterized by invariant relations which are valid in all reference systems, such as, e. g., the metric invariant. In the generalization to a fractal space-time, the events become explicitly dependent on resolution.

Relativity:

The property of physical quantities according to which they can be defined only in terms of relationships, not in an abolute way. These quantities depend on the state of the reference system, itself defined in a relative way, i. e., with respect to other coordinate systems.

Covariance:

Invariance of the form of equations under general coordinate transformations.

Geodesics:

Curves in a space (more generally in a spacetime) which minimize the proper time. In a geometric spacetime theory, the motion equation is given by a geodesic equation.

Quantum Mechanics:

Fundamental axiomatic theory of elementary particle, nuclear, atomic, molecular, etc. physical phenomena, according to which the state of a physical system is described by a wave function whose square modulus yields the probability density of the variables, and which is solution of a Schrödinger equation constructed from a correspondence principle (among other postulates).

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Author information

Authors and Affiliations

  1. CNRS, Paris Observatory and Paris Diderot University, Paris, France

    Laurent Nottale

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  1. Laurent Nottale
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Nottale, L. (2009). Fractals in the Quantum Theory of Spacetime. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_228

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