# Quantum Fractals: From Heisenberg's Uncertainty to Barnsley's Fractality

## Abstract

Starting with numerical algorithms resulting in new kinds of amazing fractal patterns on the sphere, this book describes the theory underlying these phenomena and indicates possible future applications. The book also explores the following questions: What are fractals? How do fractal patterns emerge from quantum observations and relativistic light aberration effects? What are the open problems with iterated function systems based on Mobius transformations? Can quantum fractals be experimentally detected? What are quantum jumps? Is quantum theory complete and/or universal? Is the standard interpretation of Heisenberg's uncertainty relations accurate? What is Event Enhanced Quantum Theory and how does it differs from spontaneous localization theories? What are the possible applications of quantum fractals?

## Supplementary resource (1)

... Such a generalization is necessary in considering quantum measurements because the state transformation associated with a given measurement outcome cannot be defined on the states with zero probability of producing this outcome, see (2.3). IFSs acting on the set of pure quantum states have been examined in the framework of Event Enhanced Quantum Theory (EEQT) [12,13,14]; in particular, IFSs acting on the Bloch sphere were investigated in [31,35], see also [32,33,34]. A generalization to systems that act in the space of all quantum states was proposed in [39]. ...

... then A is a homography on PΘ ∼ CP 1 . Homographies on CP 1 are isomorphic to the group of Möbius maps, which are orientation preserving conformal automorphisms of the Riemann sphere: the homography [z] → [W z] on CP 1 induced by a non-singular matrix W = a b c d corresponds to the Möbius transformation C z → az+b cz+d ∈Ĉ, see[1,4,7,33,46]. Based on the number and type of fixed points, non-identity Möbius maps are classified into three types (see alsoFig. ...

We classify the Markov chains that can be generated on the set of quantum states by a unitarily evolving 3-dim quantum system (qutrit) that is repeatedly measured with a projective measurement (PVM) consisting of one rank-2 projection and one rank-1 projection. The dynamics of such a system can be visualized as taking place on the union of a Bloch ball and a single point, which correspond to the respective projections. The classification is given in terms of the eigenvalues of the 2x2 matrix that describes the dynamics arising on the Bloch ball, i.e., on the 2-dim subsystem. We also express this classification as the partition of the numerical range of the unitary operator that governs the evolution of the system. As a result, one can easily determine which of the eight possible chain types can be generated with the help of any given unitary.

... if c = 0 then f (∞) = ∞. Quantum fractals are patterns generated by iterated function systems, with place dependent probabilities, of Mobius transformations on spheres or on more general projective spaces [101]. Platonic quantum fractals for a qubit was described in this work. ...

... Each vertex of Platonic solids serves as an attraction point. Thus a solid with v vertices will define a SQIFS (Standard Quantum Iterated Function System) based on v Mobius transformations [101]. ...

There are three important types of structural properties that remain unchanged under the structural transformation of condensed matter physics and chemistry. They are the properties that remain unchanged under the structural periodic transformation-periodic properties. The properties that remain unchanged under the structural multi scale transformation-fractal properties. The properties that remain unchanged under the structural continuous deformation transformation-topological properties. In this paper, we will describe some important methods used so far to characterize the fractal properties, including the theoretical method of calculating the fractal dimension, the renormalization group method, and the experimental method of measuring the fractal dimension. Multiscale fractal theory method, thermodynamic representation form and phase change of multiscale fractal, and wavelet transform of multiscale fractal. The development of the fractal concept is briefly introduced: negative fractal dimension, complex fractal dimension and fractal space time. New concepts such as balanced and conserved universe, the wormholes connection to the whiteholes and blackholes for universes communication, quantum fractals, platonic quantum fractals for a qubit, new manipulating fractal space time effects such as transformation function types, probabilities of measurement, manipulating codes, and hiding transformation functions are also discussed. In addition, we will see the use of scale analysis theory to stimulate the elements on the fractal structure: the research on the dynamics of fractal structure and the corresponding computer simulation and experimental research. The novel applications of fractals in integrated circuits are also discussed in this paper.

... Quantum jumps caused by such monitoring lead to fractal patterns on the Bloch sphere -quantum fractals. The generation mechanism and properties have been described in the monograph [21]. In the present work we move from simple qubits to infinite dimensional Hilbert spaces, but restricting our attention to the, still manageable, finite dimensional manifold of squeezed states. ...

... (72) For S ∈ Sp c we have |c(S, A)| = 1 owing to the relations (21). In general, for nontrivial semigroup elements, the formulas (72) are important since they determine state-dependent probabilities of exciting monitoring devices whose action on quantum states is reflected by quantum jumps implemented by the semigroup operators. ...

We describe the action of the symplectic group on the homogeneous space of squeezed states (quantum blobs) and extend this action to the semigroup. We then extend the metaplectic representation to the metaplectic (or oscillator) semigroup and study the properties of such an extension using Bargmann-Fock space. The shape geometry of squeezing is analyzed and noncommuting elements from the symplectic semigroup are proposed to be used in simultaneous monitoring of noncommuting quantum variables - which should lead to fractal patterns on the manifold of squeezed states.

... Quantum jumps caused by such monitoring lead to fractal patterns on the Bloch sphere -quantum fractals. The generation mechanism and properties have been described in the monograph [21]. In the present work we move from simple qubits to infinite dimensional Hilbert spaces, but restricting our attention to the, still manageable, finite dimensional manifold of squeezed states. ...

... (72) For S ∈ Sp c we have |c(S, A)| = 1 owing to the relations (21). In general, for nontrivial semigroup elements, the formulas (72) are important since they determine state-dependent probabilities of exciting monitoring devices whose action on quantum states is reflected by quantum jumps implemented by the semigroup operators. ...

We describe the action of the symplectic group on the homogeneous space of squeezed states (quantum blobs) and extend this action to the semigroup. We then extend the metaplectic representation to the metaplectic (or oscillator) semigroup and study the properties of such an extension using Bargmann-Fock space. The shape geometry of squeezing is analyzed and noncommuting elements from the symplectic semigroup are proposed to be used in simultaneous monitoring of noncommuting quantum variables - which should lead to fractal patterns on the manifold of squeezed states.

... In physics, fractal geometry has been employed to model physical phenomena in non-integer spaces [13,14]. Furthermore, fractal patterns have been experimentally detected, from quantum observations to relativistic light aberration effects [15]. ...

This paper introduces the concept of Fractal Frenet equations, a set of differential equations used to describe the behavior of vectors along fractal curves. The study explores the analogue of arc length for fractal curves, providing a measure to quantify their length. It also discusses fundamental mathematical constructs, such as the analogue of the unit tangent vector, which indicates the curve's direction at different points, and the analogue of curvature vector or fractal curvature vector, which characterizes its curvature at various locations. The concept of torsion, describing the twisting and turning of fractal curves in three-dimensional space, is also explored. Specific examples, like the fractal helix and the fractal snowflake, illustrate the application and significance of the Fractal Frenet equations.

... The significance of fractals extends beyond mathematics into various scientific domains, encompassing biology, chemistry, earth sciences, physics, and technology. In physics, fractal geometry has been employed to model physical phenomena in non-integer spaces, and experimental evidence of fractal patterns has been observed from quantum phenomena to relativistic light effects [15][16][17][18][19]. Cantor Tartans and Menger sponge fractals serve as tools for simulating various forms of fractal porous materials, each with distinct complexities. ...

This paper provides a framework for understanding and analyzing non-differentiable fractal manifolds. By introducing specialized mathematical concepts and equations, such as the Metric Tensor, Curvature Tensors, Analogue Arc Length, and Inner Product, it enables the study of complex patterns that exhibit self-similarity across different scales and dimensions. The Analogue Geodesic and Einstein Field Equations, among others, offer practical applications in physics, highlighting the relevance and potential of fractal geometry.

... These patterns provide insights into the underlying structure and behavior of quantum systems, shedding light on the intricate dynamics and relationships at play in the microscopic world. The detection of fractal patterns through relativistic light aberration further enhances our understanding of the fundamental nature of quantum phenomena [18]. ...

In this paper, we present a generalization of diffusion on fractal combs using fractal calculus. We introduce the concept of a fractal comb and its associated staircase function. To handle functions supported on these combs, we define derivatives and integrals using the staircase function. We then derive the Fokker-Plank equation for a fractal comb with dimension α, incorporating fractal time, and provide its solution. Additionally, we explore α-dimensional and (1+α)-dimensional Brownian motion on fractal combs with drift and fractal time. We calculate the corresponding mean square displacement for these processes. Furthermore, we propose and solve the heat equation on an α-dimensional fractal comb space. To illustrate our findings, we include graphs that showcase the specific details and outcomes of our results.

... Symmetry of fractals appears naturally in many areas ranging from pure mathematics to physics to computer graphics, e.g., [17,22,31,35]. This paper addresses in a detailed manner one very particular yet prevalent class of noncontractive IFSs with symmetries. ...

An iterated function system (IFS) can be enriched with an isometry in such a way that the resulting fractal set has prescribed symmetry. If the original system is contractive, then its associated self-similar set is an attractor. On the other hand, the enriched system is no longer contractive and therefore does not have an attractor. However, it posses a self-similar set which, under certain conditions, behaves like an attractor. We give a rigorous procedure which relates a given enriched IFS to a contractive one. Further, we link this procedure to the Lasota–Myjak theory of semiattractors, and so via invariant measures to probabilistic iterated function systems. The chaos game algorithm for enriched IFSs is discussed. We illustrate our main results with several examples which are related to classical fractals such as the Sierpiński triangle and the Barnsley fern.

... Fractal dimension was introduced to quantitatively describe the geometric features of the fractal structure. This theory has been widely used in many disciplines such as physics, chemistry, biology, and geosciences [2,3,4,5,6,7,8,9,10,11,12,13]. Which opened up a whole new field of research. ...

There are three important types of structural properties that remain unchanged under the structural transformation of condensed matter physics and chemistry. They are the properties that remain unchanged under the structural periodic transformation-periodic properties. The properties that remain unchanged under the structural multi scale transformation-fractal properties. The properties that remain unchanged under the structural continuous deformation transformation-topological properties. In this paper, we will describe topological properties and characterizations in three layers: the first layer is intuitive concept, characterizations and applications, the second layer is logical physics understanding of topological properties, characterizations and applications, and the third layer is the nature of topological properties and its power. Duality and trinity are viewed as intrinsically topological objects and are recognized as common knowledge shared among human society activity, mathematics, physics, chemistry, biology and many other kinds of nature science and technology. Some important methods used so far to characterize the topological properties, including topological index, topological order, topological invariant, topology class and the topology partition are discussed. The theories of molecular topology, topological quantum matter including topological insulators, topological metal and topological superconductors and topological quantum computing are reviewed. The development of the topological duality connection between the qubit and singularity via topological space time is briefly introduced. We will see the use of iterated function systems (IFS)to simulate the connection between singularities and their qubit control codes. The novel applications of topology in integrated circuits technology are also discussed in this paper.

... Further "fractionalization" of optical beams attracts much attention in both theoretical and numerical studies of both linear [20] and nonlinear [21,22,23] Schrödinger equations. Another important achievement relates to the fabrication and exploration of optical [24,25] and quantum fractals [26,27]. Then the fractional Laplacian (1.1) plays important role in both quantum and wavediffusion processes [27] (see Sec 2). ...

This paper addresses issues surrounding the concept of fractional quantum mechanics, related to lights propagation in inhomogeneous nonlinear media, specifically restricted to a so called gravitational optics. Besides Schr\"odinger Newton equation, we have also concerned with linear and nonlinear Airy beam accelerations in flat and curved spaces and fractal photonics, related to nonlinear Schr\"odinger equation, where impact of the fractional Laplacian is discussed. Another important feature of the gravitational optics' implementation is its geometry with the paraxial approximation, when quantum mechanics, in particular, fractional quantum mechanics, is an effective description of optical effects. In this case, fractional-time differentiation reflexes this geometry effect as well.

... Further "fractionalization" of optical beams attracts much attention in both theoretical and numerical studies of both linear [20] and nonlinear [21,22,23] Schrödinger equations. Another important achievement relates to the fabrication and exploration of optical [24,25] and quantum fractals [26,27]. Then the fractional Laplacian (1.1) plays important role in both quantum and wavediffusion processes [27] (see Sec 2). ...

This paper addresses issues surrounding the concept of fractional quantum mechanics, related to lights propagation in inhomogeneous nonlinear media, specifically restricted to a so-called gravitational optics. Besides Schrödinger–Newton equation, we have also concerned with linear and nonlinear Airy beam accelerations in flat and curved spaces and fractal photonics, related to nonlinear Schrödinger equation, where impact of the fractional Laplacian is discussed. Another important feature of the gravitational optics’ implementation is its geometry with the paraxial approximation, when quantum mechanics, in particular, fractional quantum mechanics, is an effective description of optical effects. In this case, fractional-time differentiation reflexes this geometry effect as well.

... 1,13,21 Nonetheless, in many interesting cases, while theoretically possible, the coding technique does not seem feasible. 12,27 Additionally, those systems which admit a coding map are essentially contractive. 22 Various attempts to cross the contractivity barrier were the subject of several papers, e.g., Refs. ...

We prove that the random iteration algorithm works for strict attractors of infinite iterated function systems. The system is assumed to be compactly branching and nonexpansive. The orbit recovering an attractor is generated by a deterministic process and the algorithm is always convergent. We also formulate a version of the random iteration for uncountable equicontinuous systems.

... Suggestions of some authors about the possibility of using QM beyond the microscopic scale, see [11], provide some hope of getting positive result. ...

We show that the complex number structure of the probability allows to express explicitly the relationship between the energy function H and the Laplace principle of equal ignorance (LPEI). This nonlinear relationship reflecting the measurement properties of the considered systems, together with the principle of causality and Newton principle separating the dynamics from initial conditions, lead to the linear Schrodinger equation with the Max Born interpretation, for micro and macro systems!

This paper provides a framework for understanding and analyzing non-differentiable fractal manifolds. By introducing specialized mathematical concepts and equations, such
as the Metric Tensor, Curvature Tensors, Analogue Arc Length, and Inner Product, it enables the study of complex patterns that exhibit self-similarity across different scales and dimensions. The Analogue Geodesic and Einstein Field Equations, among others,
offer practical applications in physics, highlighting the relevance and potential of fractal geometry.

We study a damped kicked top dynamics of a large number of qubits (N→∞) and focus on an evolution of a reduced single-qubit subsystem. Each subsystem is subjected to the amplitude damping channel controlled by the damping constant r∈[0,1], which plays the role of the single control parameter. In the parameter range for which the classical dynamics is chaotic, while varying r we find the universal period-doubling behavior characteristic to one-dimensional maps: period-2 dynamics starts at r_{1}≈0.3181, while the next bifurcation occurs at r_{2}≈0.5387. In parallel with period-4 oscillations observed for r≤r_{3}≈0.5672, we identify a secondary bifurcation diagram around r≈0.544, responsible for a small-scale chaotic dynamics inside the attractor. The doubling of the principal bifurcation tree continues until r≤r_{∞}∼0.578, which marks the onset of the full scale chaos interrupted by the windows of the oscillatory dynamics corresponding to the Sharkovsky order. Finally, for r=1 the model reduces to the standard undamped chaotic kicked top.

Preprint of the article is at https://arxiv.org/abs/2004.11057
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We give a systematic account of iterated function systems (IFS) of weak contractions of different types (Browder, Rakotch, topological). We show that the existence of attractors and asymptotically stable invariant measures, and the validity of the random iteration algorithm (‘chaos game’), can be obtained rather easily for weakly contractive systems. We show that the class of attractors of weakly contractive IFSs is essentially wider than the class of classical IFSs' fractals. On the other hand, we show that, in reasonable spaces, a typical compact set is not an attractor of any weakly contractive IFS. We explore the possibilities and restrictions to break the contractivity barrier by employing several tools from fixed point theory: geometry of balls, average contractions, remetrization technique, ordered sets, and measures of noncompactness. From these considerations it follows that while the existence of invariant sets and invariant measures can be assured rather easily for general iterated function systems under mild conditions, to establish the existence of attractors and unique invariant measures is a substantially more difficult problem. This explains the central role of contractive systems in the theory of IFSs.

This chapter discusses the generative role of early childhood play in human development. A complex dynamical systems theory perspective can help to better understand this generative process. The sensitive dependence on initial conditions; the equivalence of different surface manifestations with underlying processes; and dynamic phase transitions become a template for apprehending young children’s play and learning, and their interface with relevantly dynamic systems of educational implementation. Connections are drawn with a dynamic themes curriculum approach.

В работе обсуждаются математические проблемы и внутренние противоречия в гл. 5 монографии «Теория физического вакуума» Г.И. Шипова. Отдельное внимание уделено секциям 5.4 и 5.5, где неправильно используется формализм подвижного репера и дифференциальных форм Картана для рассмотрения пространства-времени геометрии абсолютного параллелизма с кручением. Указывается на схожие или те же самые проблемы в других публикациях того же автора. Перечислены найденные математические противоречия, и указан правильный путь решения предмета обсуждения.

The paper discusses mathematical problems and inconsistencies in Ch. 5 of the monograph "A Theory of Physical Vacuum" by G. I. Shipov. Particular attention is paid to sections 5.4 and 5.5 , where Cartan's formalism of moving frames and differential forms is improperly employed for the study of spacetime absolute parallelism geometry with torsion. Similar or identical problems in other publications of the same author are are pointed out. Mathematical inconsistencies found are listed, and the proper way of addressing the subject is indicated.

We define fractal continuations and the fast basin of the IFS and investigate
which properties they inherit from the attractor. Some illustrated examples are
provided.

We give a survey of some results within the convergence theory for iterated random functions with an emphasis on the question of uniqueness of invariant proba- bility measures for place-dependent random iterations with nitely many maps. Some problems for future research are pointed out.

We present a quantum-like (QL) model in that contexts (complexes of e.g. mental, social, biological, economic or even political conditions) are represented by complex probability amplitudes. This approach gives the possibility to apply the mathematical quantum formalism to probabilities induced in any domain of science. In our model quantum randomness appears not as irreducible randomness (as it is commonly accepted in conventional quantum mechanics, e.g., by von Neumann and Dirac), but as a consequence of obtaining incomplete information about a system. We pay main attention to the QL description of processing of incomplete information. Our QL model can be useful in cognitive, social and political sciences as well as economics and artificial intelligence. In this paper we consider in a more detail one special application -- QL modeling of brain's functioning. The brain is modeled as a QL-computer.

The race is on to construct the first quantum code breaker, as the winner will hold the key to the entire Internet. From international, multibillion-dollar financial transactions to top-secret government communications, all would be vulnerable to the secret-code-breaking ability of the quantum computer. Written by a renowned quantum physicist closely involved in the U.S. government’s development of quantum information science, Schrödinger’s Killer App: Race to Build the World’s First Quantum Computer presents an inside look at the government’s quest to build a quantum computer capable of solving complex mathematical problems and hacking the public-key encryption codes used to secure the Internet. The “killer application” refers to Shor’s quantum factoring algorithm, which would unveil the encrypted communications of the entire Internet if a quantum computer could be built to run the algorithm. Schrödinger’s notion of quantum entanglement-and his infamous cat-is at the heart of it all. The book develops the concept of entanglement in the historical context of Einstein’s 30-year battle with the physics community over the true meaning of quantum theory. It discusses the remedy to the threat posed by the quantum code breaker: quantum cryptography, which is unbreakable even by the quantum computer. The author also covers applications to other important areas, such as quantum physics simulators, synchronized clocks, quantum search engines, quantum sensors, and imaging devices. In addition, he takes readers on a philosophical journey that considers the future ramifications of quantum technologies. Interspersed with amusing and personal anecdotes, this book presents quantum computing and the closely connected foundations of quantum mechanics in an engaging manner accessible to non-specialists. Requiring no formal training in physics or advanced mathematics, it explains difficult topics, including quantum entanglement, Schrödinger’s cat, Bell’s inequality, and quantum computational complexity, using simple analogies.

The law of track formation in cloud chambers is derived from the Liouville equation with a simple Lindblad's type generator that describes coupling between a quantum particle and a classical,
continuous medium of two-state detectors. Piecewise deterministic random process (PDP) corresponding to the Liouville equation is derived. The process consists of pairs (classical event, quantum
jump), interspersed with random periods of continuous (in general, non-linear) Schrodinger-type evolution. The classical events are flips of the detectors-they account for tracks. Quantum jumps are shown, in the simplest, homogeneous case, to be identical to those in the early spontaneous localization model of Ghirardi, Rimini and Weber (GRW). The methods and results of the present paper allow for an elementary derivation and numerical simulation of particle track formation and provide an additional perspective on GRW's proposal.