Fractal space-time: a geometric analogue of relativistic quantum mechanics

Ord, G. N.

doi:10.1088/0305-4470/16/9/012

Cited by 196 publications

(132 citation statements)

References 4 publications

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“…L = L(ε) ε→0 → ∞. This theorem naturally leads to the proposal that the concept of fractal space-time [15] [16] [10] [5] is the geometric tool adapted to the research of such a new description.…”

Section: Introductionmentioning

confidence: 99%

Scale relativity and gauge invariance

Nottale

2001

Chaos, Solitons & Fractals

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The theory of scale relativity extends Einstein's principle of relativity to scale transformations of resolutions. It is based on the giving up of the axiom of differentiability of the space-time continuum. As a consequence, spacetime becomes fractal, i.e., explicitly resolution-dependent. The requirement that this geometry satisfies the principle of scale relativity leads to introduce scale laws having a Galilean form (constant fractal dimension), then a logLorentzian form. In this framework, the Planck length-time scale becomes a minimal impassable scale, invariant under dilations. Then we attempt to construct a generalized scale relativity which includes scale-motion coupling. In this last framework, one can reinterpret gauge invariance as scale invariance on the internal resolutions. This approach allows one to set new constraints in the standard model, concerning in particular the Higgs boson mass, which we find to be √ 2mW = 113.73 ± 0.06 GeV in a large class of models.

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Section: Introductionmentioning

confidence: 99%

Scale relativity and gauge invariance

Nottale

2001

Chaos, Solitons & Fractals

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“…Such a filiation is also claimed by Ord in his own fractal spacetime description [5]. Feynman's path integral decomposes the wavefunction in terms of the sum over all possible paths of equiprobable elementary wavefunction e iS cl , where S cl is the classical action.…”

Section: Discussionmentioning

confidence: 57%

“…Moreover one can show that D F = 2 plays the role of a critical dimension, since for other values one still obtains a Schrödinger form for the equation of dynamics, but which keeps an explicit scale dependence [4]. Therefore, low energy quantum physics can be identified with the case where the fractal dimension of space paths is 2 (and relativistic quantum physics with the same fractal dimension for spacetime paths [3,5]), in agreement with Feynman's characterization of the asymptotic scaling quantum domain as w ∝ (dt/τ ) −1/2 [1]. However, this is not the last word, since the loss of determinism on a fractal space has still another fundamental consequence.…”

Section: The Complex State Functionmentioning

confidence: 99%

Derivation of the postulates of quantum mechanics from the first principles of scale relativity

Nottale¹,

Célérier²

2007

J. Phys. A: Math. Theor.

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phone:1 (0)1 45 07 74 03 2 (0)1 45 07 75 38 February 2, 2008Abstract Quantum mechanics is based on a series of postulates which lead to a very good description of the microphysical realm but which have, up to now, not been derived from first principles. In the present work, we suggest such a derivation in the framework of the theory of scale relativity. After having analyzed the actual status of the various postulates, rules and principles that underlie the present axiomatic foundation of quantum mechanics (in terms of main postulates, secondary rules and derived 'principles'), we attempt to provide the reader with an exhaustive view of the matter, by both gathering here results which are already available in the literature, and deriving new ones which complete the postulate list.

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“…This translation in scale is viewed in the space reference system using the scale law (17) by integrating the ordinary differential equation between E 0 and E 1 . We obtain…”

Section: Translation In Scalementioning

confidence: 99%

“…Nottale's fundamental idea is to give up the hypothesis of differentiability of the space-time continuum. Previous works in this direction have been done by Ord 1 [17]. As a consequence, one must consider continuous but non-differentiable objects, i.e., a non-differentiable manifold.…”

Section: Introductionmentioning

confidence: 99%