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A Sourcebook for the Worldwide Discovery of a Creative Organic Universe
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III. Ecosmos: A Revolutionary Fertile, Habitable, Solar-Bioplanet, Incubator Lifescape

E. Systems Cosmology: Fractal SpaceTimeMatter

Mittal, A. K. and Daksh Lohiya. Fractal Dust Model of the Universe Based on Mandelbrot’s Conditional Cosmological Principle and General Theory of Relativity. Fractals. 11/2, 2003. By which certain problems of previous theoretical formulations with regard to a fractal cosmos are resolved.

Murdzek, R. and O. Iftimie. The Self-Organizing Universe. Romanian Journal of Physics. 53/3-4, 2008. Alexandru Ioan Cuza University scientists contend that such nonlinear dynamics and geometries stretching across celestial reaches imply a cosmos whose own propensities proceed to structure itself. The full paper is available online as a previous arXiv posting.

The most recently completed redshift surveys, such as 2dFGRS, reveal spectacularly complex structures in galaxy distribution. These structures are described in terms of filaments, clusters and voids and are usually characterized by using fractal geometry language. In this paper it is shown that the fractal dimension of the large scale distribution of galaxies presents scaling behavior well described by a Verhulst-type law. This result is in agreement with the idea that the Universe we observe today is a self-structured system which emerges from a nonlinear self-organizing phenomenon. (Abstract, 601)

Nottale, Laurent. Scale Relativity and Fractal Space-Time. http://evodevouniverse.com/EDU2008Papers/NottaleSRTheoryApplicationsEDU2008.pdf. A lengthy paper summarized at the First International Conference on the Evolution and Development of the Universe by the CNRS physicist director, and also available at arXiv:0812.3857v1, wherein genotypes and phenotypes arise from quantum realms in a nested, self-similar sequence and procession. Luminous insights that do not yet get respect because they reveal and belong to the human genesis universe.

Nottale, Laurent. Scale-relativistic Cosmology. Chaos, Solitons and Fractals. 16/4, 2003. This paper in a special issue entitled The New Cosmology, considers the scale-invariant, fractal geometry of space-time.

Nottale, Laurent. The Relativity of All Things: Beyond Spacetime. Nashville, TN: Persistent Press, 2019. This is an English edition of a French science bestseller by a former French National Center for Scientific Research director. What is relativity. The word evokes thoughts of Einstein. What ultimately matters is the relationship between two objects, not their absolute properties. (xiii) Nottale’s studies go back to the 1980s, while this latest text braces a similar conclusion being availed via network complexity across many fields. The relational interactivity between particles, components, and entities, are equally real and altogether compose a creative ecosmos . One could cite Lee Smolin, Carlo Rovelli and many others who emphasize this universal quality. A prescient endorsement was made by Murray Gell-Mann in 1992 (search) that independent laws and principles do actually exist as they display into a fractally self-similar vitality. An example of a working usage of Nottale’s theories is Derivation of a Generalized Schrödinger Equation from the Theory of Scale Relativity by Pierre-Henri Chavanis at arXiv:1612.02323.

The statement of the existence of laws, universal by nature, is sufficient in itself. It is the logic of the world’s organization that requires it. Said otherwise, the principle of relativity is reduced to the basic postulate upon which science is founded: There exist laws of nature. (72)

Einstein himself explicitly considered that a realistic approach to the quantum problem could go through the introduction of non-differentiability in physics. In 1948, he wrote in a letter to Wolfgang Pauli: “Maybe someone will find out another possibility, provided he searches with enough perseverance.'” Laurent Nottale is very precisely this 'someone'! Read and study this wonderful theory, and its major experimental implications, which are fundamental for the future of science, and for philosophy. Charles Alunni, Director, Philosophy of Science, École Normale Supérieure

Paczuski, Myra and David Hughes. A Heavenly Example of Scale-Free Networks and Self-Organized Criticality. Physica A. 342/1-2, 2004. The Imperial College, London, mathematical physicists find even the sun’s corona and its magnetic field network to exhibit these ubiquitous complex system properties.

Palmer, Tim. Bell’s Theorem, Non-Computability and Conformal Cyclic Cosmology. arXiv:2108.10902. In this paper for Roger Penrose’s 90th birthday, he Oxford University polyphysicist (search) adds further reasons and evidence why a celestial reality may be most of all distinguished by a self-similar fractality across every expanse. See also Quantum Physics from Number Theory by TP (2209.05549) and Parametric Invariance by Mario de Oliveira (2203.07262) for further views.

This paper draws on a number of Roger Penrose's ideas such as a Conformal Cyclic Cosmology, non-computability and gravitationally induced quantum state reduction so to propose an unconventional approach to quantum gravity: Invariant Set Theory (IST). In IST, the fundamental laws of physics describe the geometry of the phase portrait of the universe as a whole: "quantum" process are associated with fine-scale fractal geometry. With this, it can explain the experimental violation of Bell Inequalities. (Abstract excerpt)

In Section 3 a theory of quantum physics is outlined based on the notion that the universe is evolving on a cosmological invariant set. In this theory, state-space trajectories have a fractal structure homeomorphic to the set of p-adic integers. Such a theory is very conceptually different from quantum mechanics - it implies that the quantum state vector has an ensemble interpretation. Rather, the essential properties of quantum physics would arise from the geometry of the invariant set of the undivided universe. (2)

Palmer, Tim. Quantum Theory and the Symbolic Dynamics of Invariant Sets. arXiv: 1210.3940. Online October 2012, this follows up a 2009 paper “The Invariant Set Hypothesis: A New Geometric Framework for the Foundations of Quantum Theory and the Role Played by Gravity,” Proceedings of the Royal Society A (465/3165). Both run some 50 pages on arXiv. The author has a doctorate in statistical physics from Oxford University. In 2009 he was at the European Centre for Medium-Range Weather Forecasts, Reading, UK, as group leader for the application of nonlinear complexity to climate studies, in 2012 at the Clarendon Laboratory, Oxford. The earlier work was extolled in the New Scientist (March 28, 2009) by Mark Buchanan as revealing a “Fractal Reality.” Palmer offers a revised fundamental basis of quantum physics via a “contextual,” self-similar, structure of space and time. The endeavor has been vetted by the Perimeter Institute in Canada, because a radical rethinking is seen as in order. See also Palmer’s note “Climate Extremes and the Role of Dynamics” in PNAS (110/5281, 2013). Such a robust mathematical articulation of a scale-invariant, repetitive cosmic emergence would add vital credibility to a natural genesis uniVerse, implying its own iterative genetic program. We include two quotes from each edition. For a 2017 edition, see A Gravitational Theory of the Quantum at arXiv:1709.00329.

A realistic measurement-free theory for the quantum physics of multiple qubits is proposed. This theory is based on a symbolic representation of a fractal state-space geometry which is invariant under the action of deterministic and locally causal dynamics. This symbolic representation is constructed from self-similar families of quaternionic operators. Using number-theoretic properties of the cosine function, the statistical properties of the symbolic representation of the invariant set are shown to be consistent with the contextual requirements of the Kochen-Specker theorem, are not constrained by Bell inequalities, and mirror the statistics of entangled qubits. These number-theoretic properties in turn reflect the sparseness of the invariant set in state space, and relate to the metaphysical notion of counterfactual incompleteness. As a result, it is proposed that the complex Hilbert Space should merely be considered a computational convenience in the light of the algorithmic intractability of the invariant set geometry, and consequently the superposed state should not be considered a fundamental aspect of physical theory. Here some elements of an alternative ‘gravitational theory of the quantum’ are proposed, based on a deterministic and locally causal theory of gravity which extends general relativity by being geometric in both space-time and state. (2012 Abstract excerpt)

Fractal invariant sets have two key properties: self-similarity and sparseness. Both of these are crucial in constructing the required symbolic representation of ID. In Section 2.2 it is shown how self-similarity provides a simple way to conceptualise one of the paradigmatic experiments in quantum physics: that of sequential selective spin measurement. In Section 3, a mathematical structure for the symbolic representation of ID is developed based on a family of self-similar quaternionic operators acting on symbol sequences: such bit-string symbol sequences are referred to here as ‘lbits’. This structure not only describes the statistics of sequential spin experiments, it describes more generally the statistical properties of multiple qubits in quantum theory. The framework for the development of such a symbolic representation is referred to generically as ‘Invariant Set Theory (IST). (2012, 3)

A new law of physics is proposed, defined on the cosmological scale but with significant implications for the microscale. Motivated by nonlinear dynamical systems theory and black-hole thermodynamics, the Invariant Set Postulate proposes that cosmological states of physical reality belong to a non-computable fractal state-space geometry I, invariant under the action of some subordinate deterministic causal dynamics D. A new interpretation is given to the standard ‘mysteries’ of quantum theory: superposition, measurement, non-locality, emergence of classicality and so on. It is proposed that heterogeneities in the fractal geometry of I are manifestations of the phenomenon of gravity. Since quantum theory is inherently blind to the existence of such state-space geometries, the analysis here suggests that attempts to formulate unified theories of physics within a conventional quantum-theoretic framework are misguided, and that a successful quantum theory of gravity should unify the causal non-Euclidean geometry of space–time with the a temporal fractal geometry of state space. (2009 Abstract excerpt)

The Invariant Set Hypothesis is framed in terms of invariance, a concept that forms the very bedrock of physics, and conjectures that states of physical reality are defined by a fractal geometry I, embedded in state space and invariant under the action of some subordinate dynamics. The hypothesis is motivated by two concepts that would not have been known to the founding fathers of quantum theory: the generic existence of invariant fractal subsets of state space for certain nonlinear dynamical systems, and the notion that the irreversible laws of thermodynamics are fundamental rather than phenomenological in describing the physics of extreme gravitational systems. Although quantum theory is unsurpassed in terms of its agreement with experimental data, it is suggested that the Invariant Set Hypothesis provides a geometric framework for a deeper understanding of the foundations of quantum physics than can be provided by quantum theory itself. (2009, 2)

Pietronero, Luciano, et al. Complexity in Cosmology. Phuan Ong, N. and Ravin Bhatt, eds. More Is Different. Princeton: Princeton University Press, 2001. By theory and observation, a growing comprehension is being achieved of a scale-invariant, self-organizing universe, which is seen as a “radically new and original” view of cosmic emergence.

Powell, Devin. Physicists Net Fractal Butterfly. Science. 501/144, 2013. A report on the experimental verification of a phenomena proposed some 30 years ago by Douglas Hofstadter that even atomic electron trajectories in quantum realms will take upon such recursive fractal topologies.

Ramos, F. M., et al. Multiscaling and Nonextensivity of Large-Scale Structures in the Universe. Physica D. 168/404, 2002. Whereby the problem of the transition from multifractal galaxies and clusters to an overall homogeneous universe is resolved by a generalized thermostatistics method.

Rohringer, Wolfgang, et al. Non-equilibrium Scale Invariance and Shortcuts to Adiabaticity in a One-dimensional Bose Gas. Nature Scientific Reviews. 5/9820, 2015. This brief note by Vienna Center for Quantum Science and Technology researchers is another indication of an inherent, universal self-similarity across every possible domain. When and how might these many disparate findings become a worldwise discovery of a creative organic universe?

We present experimental evidence for scale invariant behaviour of the excitation spectrum in phase-fluctuating quasi-1d Bose gases after a rapid change of the external trapping potential. Probing density correlations in free expansion, we find that the temperature of an initial thermal state scales with the spatial extension of the cloud as predicted by a model based on adiabatic rescaling of initial eigenmodes with conserved quasiparticle occupation numbers. Based on this result, we demonstrate that shortcuts to adiabaticity for the rapid expansion or compression of the gas do not induce additional heating. (Abstract)

A systematic understanding of non-equilibrium dynamics in many-body quantum systems is a longstanding goal, with far-reaching applicability for many different fields of physics. Ultracold atom experiments offer clean implementations of systems that are tunable, well isolated from the environment and theoretically tractable1,2. In particular, the profound understanding available for the one-dimensional (1d) Bose gas makes it an ideal test bed for quantum many-body dynamics. (1)

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