Feb 18, 2014 - FBI agent Fox Mulder finally gets to see some real UFOs in a se- ... colleague Dana Scully are astounded by a series of bright disks. 7 ...
Quantum Fractals - Introduction Arkadiusz Jadczyk February 18, 2014
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Introduction
This book brings together two concepts. The first is over a hundred years old - the “quantum”, while the second, “fractals”, is newer, achieving popularity after the pioneering work of Benoit Mandelbrot. Both areas of research are expanding dramatically day by day. It is somewhat amazing that quantum theory, in spite of its age, is still a boiling mystery as we see in some quotes from recent publications addressed to non–expert readers: “Heisenberg uncertainty principle stressed in new test”’: Pioneering experiments have cast doubt on a founding idea of the branch of physics called quantum mechanics. The Heisenberg uncertainty principle is in part an embodiment of the idea that in the quantum world, the mere act of measuring can affect the result. But the idea had never been put to the test, and a team writing in Physical Review Letters says ”weak measurements” prove the rule was never quite right. BBC News, September 7, 2012 [Pal12] “Will we ever understand quantum theory?”: Quantum mechanics must be one of the most successful theories in science.(....) Yet the weird thing is that no one actually understands quantum theory. BBC Future, January 25, 2013 [Bal13b] “Quantum Weirdness? It’s All in Your Mind”: ... Quantum mechanics is an incredibly successful theory but one full of strange paradoxes. ... Scientific American, June 10, 2013 [vB13] ”Proof mooted for quantum uncertainty Study confirms principles limits on measurement accuracy”: Encapsulating the strangeness of quantum mechanics is a single mathematical expression. According to every undergraduate physics textbook, the uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a subatomic particle the more precisely one knows the particles position at a given moment, the less precisely one can know the value of its momentum. But the original version of the principle, put forward by physicist Werner Heisenberg in 1927, couches quantum indeterminism in a different way as a fundamental limit to how well a detector can measure quantum properties. Heisenberg offered no direct proof for 1
this version of his principle, and expressed his ideas only informally and intuitively’, says physicist Jos Uffink of the University of Minnesota in Minneapolis. Nature, June 27, 2013 [Cow13] It’s “an inertia” of thinking, says David Deutsch, a British physicist at the University of Oxford [Deu03] in his paper on “Physics, Philosophy and Quantum Technology”, suggesting the need of a more general ”Quantum Constructor Theory.” The main issues here are “quantum information”, “quantum computing”, and “quantum cryptography”. But we do not yet understand quantum theory, its limitations, its full potential, and its connections to other areas of science. There is no doubt about the successful applications of quantum theory in technology of today and of tomorrow. Yet if we want to make the full use of this theory, we need to cure it from its diseases and apparent paradoxes that follow from its vagueness in some areas. If the solution to these problems have not been found till now - something must have been overlooked. Probably something related to “quantum measurement”. Chapter 4 of this book discusses those foundational questions which also happened to be related to the processes relevant for creation of quantum fractals. One such important problem is Heisenberg’s uncertainty and its interpretations and implications. There are precise mathematical formulas, and there are their interpretations. These interpretations do not follow from the so successful mathematical machinery - they are imposed on top of it with questionable philosophical underpinnings. My research related to quantum fractals targets that question: can we expand the formalism so as to describe the processes not yet described and predict what has not been predicted so far? Can we do it in such a way that we will still be in agreement with the numerous successful applications? Quantum fractals result from a new kind of measurement processes. I became interested in quantum fractals because they are considered “impossible” and/or irrelevant. Yet they exist within a well defined theory that describes the so far vague concept of ”measurement”. Indeed, after sufficient amount of research we find that their “impossibility” and/or irrelevance is the result of inertia and prejudices. If Quantum Theory still poses unresolved problems - it can only be because of deeply ingrained prejudices - nothing else. String theories and other “theories of everything” will not help a least in this respect. They do not even try to address the foundational problems 2
of quantum theory. As a result fundamental unresolved issues become even more obscured. There are many different interpretations of quantum theory: the standard, orthodox, Copenhagen interpretation, Bohmian, CSL (continuous spontaneous localization), GRW (Ghirardi-Rimini-Weber), transactional, Bayesian, ... The poll conducted among the participants of the conference “Quantum Physics and the Nature of Reality” held in July 2011 in Austria, and organized by Anton Zeilinger, revealed that opinions of the leading experts evenly split among different interpretations and their implications [SKZ13]. But none of these different interpretations really predicts dramatic new effects. Some physicists say “do not worry about interpretation”, just “shut up and calculate”. Louisiana State University physicist Jonathan P. Dowling, specializing in quantum computing, describes this attitude in detail using himself as an example: When it comes to interpretations of quantum theory, I am a pantheist. (...) Pantheism is not without its pitfalls, but unlike Pascals original wager, you are hedging more of your bets. Im happy to revel in the Copenhagen interpretation on Monday, Wednesday, and Friday; exploit the Many-Worlds interpretation on Tuesday, Thursday, and Saturday; and on Sunday turn in desperation to Bohm theory. Unlike Pascals wager, however, I have nothing to lose by this strategy and everything to gain. Often a perplexing quantum problem becomes completely clear to me when I switch from Copenhagen to Many Worlds. (Nothing is ever cleared up when I resort to Bohm theory, but then I usually sleep in on Sundays.) I dont believe any of them! or perhaps I believe all of them. The physicist Edwin T. Jaynes once said, So long as I can use and teach a physical theory, I dont have to believe it. Well played! [Dow13, p. 34–35] Yet interpretation does matter: it suggests to us the directions of our research. It shapes questions that we ask. It tells us what to not bother about while designing new experiments. I am convinced that new effects will come very soon from overcoming the tabus imposed by our current interpretation of Heisenberg’s uncertainty relations. Discussing these issues in detail would take a whole new book, however I discuss the main points specifically in Ch. 4.6. 3
One may wonder why I devote space to those issues that this book is really not about? The book is mainly about fractals - of a special kind. But I am first of all a physicist. Quantum Fractals emerged as a byproduct of my research. Here I am concentrating upon the fractal aspects, related methodology questions, structure and properties of these fractals. I hope that this book will encourage quantum physicists, busy with their quantum cryptographical qubits, to look into something having not only applied appeal but also aesthetical values: quantum fractals that are hidden in their qubits. I also hope that the fractal industry will expand owing to the new perspective stemming from quantum theoretical applications. The mathematical formalism developed here should have applications in those areas of research that stem from quantum concepts applied within quantum theory itself but also beyond. Since, here and there, while discussing quantum fractals I have to use precise mathematical terms that are rather specialized, the book has an Appendix where the a concise but precise definition of the most important mathematical terms is given. New questions will reveal new fields of research and cast new light on old areas. With Quantum Fractals comes challenge that will open new paths. Can we find these quantum fractals in Nature today? If so, where? This question is addressed, within the framework of Quantum Theory, in Section 4.7. But my guess is that most promising is the search for quantum fractals in the Cosmos and the study of light. Quantum Fractals have a history and, as I am convinced you will see from this book, also their bright future. Originally they emerged from my interest in the foundational problems of the still mysterious quantum theory, as well as my work on filling up its most disturbing blind spots - ”the measurement problem”. John Stewart Bell (1928–1990), the distinguished CERN quantum physicist, gave one of his papers the provocative title: “Against Measurement” [Bel90]. There he suggested banning the word “measurement” from physicist vocabulary altogether! Nevertheless Quantum theory is a theory of measurements - as it is often repeated. But what are these ”measurements”? What kind of processes are involved there? It is while trying to find answers to these questions, answers that would satisfy me, I came up with an idea of a “piecewise deterministic process” that produced quantum jumps as discrete events taking place in time. Continuous time evolution of potentialities encoded in the quantum wave function is, in this approach, interrupted by occasional discontinuities “quantum jumps” 4
- according to a precisely defined piecewise deterministic random process. The resulting process produces a picture that is similar to that observed in neuronal activity in the brain (see Fig. 1). Similar spikes can be seen,
Figure 1: Spikes and bursts, in a way similar to quantum jumps, interrupt the continuous accumulation of the electric potential in neuronal activity. URL: http://en.wikipedia.org/wiki/Neural\%20oscillation for instance, in sunspot activity and financial markets - which is intriguing enough in itself. From the mathematical description of the process it became evident to me that measuring simultaneously quantum complementary quantities is possible - in a sense that is contrary to the prevailing interpretations of Heisenberg’s uncertainty relations. Mathematics not only allows us to do what the standard quantum theory textbooks never dare to do, it also predicts the patterns - though chaotic and fractal-like. That may be the reason why there were no attempts to see these patterns experimentally, as there was no theory indicating what to look for. To look for something we must first have a theory, or at least an idea, even if only rough and approximate, perhaps even a false one, of what to look for. Now, that we have such a theory, with the piecewise-deterministic process paradigm, with quantum fractals, perhaps experiments will follow? 5
As of today, quantum fractals are just images and algorithms. They are pretty. But in the future, I am sure, we will find them all around us. And not only in the highly specialized realm of quantum microphysics, but also in macrocosmic light patterns, and also in the extensions of the quantum formalism that regulate social and psychic phenomena, processes in the brain, thought formation, etc. In the above mentioned 2011 poll [SKZ13] (see also [Bal13a]) on the foundational issues of quantum theory we find “More than two-thirds believed that there is no fundamental limit to quantum theory that it should be possible for objects, no matter how big, to be prepared in quantum superpositions like Schr¨odingers cat. So the era where quantum theory was associated only with the atomic realm appears finally over.” Certainly Quantum Fractals do have their Quantum Future, but let me return to their history. Originally quantum fractals came up as patterns formed as the results of millions of consecutive quantum jumps of a spin 1/2 quantum state during a continuous monitoring of mutually incompatible spin directions. More generally they can be also formed during continuous monitoring of position and momentum, when the wave functions jump wildly. Millions of wave function reductions would give us a fractal-like pattern on the phase space - something that is impossible according to the standard quantum mechanics (except, perhaps, for Wigner’s function, which hides the resulting chaos behind “negative probabilities”). This program has not yet been implemented, but I see no reason why it cannot be. Wave functions, or quantum state vectors are strange objects. Their ontological status is debatable. Can we really “see”, in some kind of an experiment, the patterns that they form? What are these wave functions anyway? According to some interpretations wave functions (or “pure quantum states”) are purely subjective, they represent our knowledge. As Henry Stapp puts it in his book “Mind, Matter and Quantum Mechanics” [Sta93]: “The Copenhagen interpretation is often criticized on the grounds that it is subjective, i.e., that it deals with the observers knowledge of things, rather than those things themselves. This charge arises mainly from Heisenbergs frequent use of the words ‘knowledge’ and ‘observer’. Since quantum theory is fundamentally a 6
procedure by which scientists make predictions, it is completely appropriate that it refer to the knowledge of the observer. For human observers play a vital role in setting up experiments and in noting their results.” A pure objective interpretation of quantum wave functions can hardly be spotted. Yet a subjective (or Bayesian) interpretation is not the only one possible. Wave functions can be shadows of some deeper, perhaps hyperdimensional, reality, encoding space-time-energy information about the objects, information based on the data that have been irreversibly registered and stored. Shadows can be seen - even if they are only shadows. Shadows can have shadows of their own. Shadows can be registered and analysed through the set of discrete data. Once we have a theory that tells us how to decode the data, we can reconstruct the shadows. Once we have a theory that tells us how to reconstruct from shadows of certain level shadows of a deeper level - we can reconstruct these deeper levels as well. In this book I will touch upon all these problems and describe first steps in these directions. But the mathematical formalism and algorithms of quantum fractals apply not only to quantum state vectors but also to relativistic light aberration. Therefore they can be, in principle, observed, in the sky - as there is the formal relation between the quantum spin state Bloch sphere and the heavenly sky of Special Relativity Theory. My favorite example here is somewhat out of touch with reality, but it serves the purpose of activating our imagination. To set the scene let me quote a passage from “Beyond Startrek Physics from Alien Invasions to the End of Time” by Lawrence M. Kraus [Kra98, p. 21]: The traditional notion has been that UFOs don’t behave like rockets or planes (this is, after all, what makes them UFOs). Strange lights that flit unlikely distances back and forth across the sky, like the dazzling display in Steven Spielberg’s Close Encounters of the Third Kind, are typical. More recently, in one of the early episodes of The X-Files, the ardent UFO hunter and FBI agent Fox Mulder finally gets to see some real UFOs in a secret air force installation somewhere in the Southwest (could it be Area 51?), and these vehicles do just what UFOs are supposed to do – namely, everything our own aircraft can’t. Mulder and his colleague Dana Scully are astounded by a series of bright disks 7
moving at incredible speeds through the skies above the remote base, turning at 90-degree angles on a dime. Now imagine a monkey flying such an UFO disk. The monkey operates a joystick with 24 positions - each position defines an instant maneuver that changes the direction and rotates the disk. The monkey marks on the hemispherical transparent dome the position of a fixed star - say Alpha Centauri. After each such maneuver the new mark appears on the dome - the change of the visible star position is the result of the relativistic aberration. The monkey operates the joystick randomly and is not getting tired. After about a hundred thousand of such random operations a fractal pattern starts to appear on the screen. Today the above scene, although it is supported by rigorous mathematics, is fit only for a Sci-Fi movie. The author of ”Beyond Star Trek” rightly argues that such maneuvers are impossible according to the conventional science and technology. But human beings seems to fear no challenges, and tomorrow our science and technology can be ready experiments resulting in essentially the same effect. Perhaps the most amazing of all quantum fractals is the one I have discovered after starting the work on this book - the parabolic quantum fractal stemming from the most simple formula z 7→ z + 4 and its 24 octahedral variations - you see on the cover of this book. In Fig 2 we have another, exterior view of this object. 1 Its shape reminds us the Buckminster Fuller geodesic dome - we may call it the Quantum Dome. It has huge circular “windows”. I do not have a mathematical formula for these windows, though certainly there is one - yet to be discovered. When stereographically projected from the sphere on the plane - the windows that have different sizes on the sphere acquire the same size on the plane. The Quantum Dome fractal is also “an impossible one” - according to a mathematical theorem proved by Andrew Vince [Vin13] (based on the definition of an attractor proposed by Barnsley and Vince [BV11]) we can’t get a “true fractal” from parabolic transformations such as z 7→ z + 4.2 And yet here it is, and its existence asks for a better understanding of what should be called a fractal and what not. 1
You can see it rotating on Youtube channel: http://www.youtube.com/watch?v=YNzf1o4GAAY 2 Of course I tried also other parabolic transformations such as z + 1 and z + 2, - but z + 4 produces the most pleasing pattern.
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Figure 2: Parabolic Quantum Fractal. To speculate even further3 - the Quantum Dome (or similar fractals) may also have engineering applications. Can such a design have some special optimizing material properties? Can it be used as a “fractal antenna” of some kind, the way fractal antennas are being used in sophisticated engineering cell phones. Can it be used for space-propulsion of some sort? I am having these thoughts while looking at the intriguing and unexpected shapes and design patterns. There is also a mysterious connection to Pythagorean triples and quadruples (see Ch. 2.6.13, discrete space-time structures etc. All of these deserves further research. This book may serve, I hope, as a jumping–off platform in this respect.
References [Bal13a] Philip Ball. Experts still split about what quantum theory means. poll reveals diverse views about foundational questions in physics. 3
To quote from Irving John Good, a British brilliant mathematician, who worked as a cryptologist with Alan Turing: “It is often better to be stimulating and wrong than boring and right.” [Goo62, p. 1]
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Nature News, January 2013. [Bal13b] Philip Ball. Will we ever understand quantum theory? BBC Future online article, January 2013. [Bel90]
John Bell. Against measurement. Physics World, August:33–40, 1990.
[BV11]
M. F. Barnsley and A. Vince. The chaos game on a general iterated function system. Ergodic Theory and Dynamical Systems, 31(04):1073–1079, July 2011.
[Cow13] Ron Cowen. Proof mooted for quantum uncertainty. 498(7455):419–420, June 2013.
Nature,
[Deu03] Deutsch. Physics, philosophy, and quantum technology. In Proceedings of the Sixth International Conference on Quantum Communication, Measurement and Computing, Shapiro, J.H. and Hirota, O., Eds., pages 419–426. Rinton Press, Princeton, NJ., 2003. [Dow13] Jonathan P. Dowling. Schr¨ odinger’s Killer App: Race to Build the Worlds First Quantum Computer. CRC Press, 2013. [Goo62] Irving John Good, editor. The Scientist Speculates: An Anthology of Partly–Baked Ideas. Basic Books, 1962. [Kra98] Lawrence M. Krauss. Beyond Star Trek: From Alien Invasions to the End of Time. BasicBooks, 1998. [Pal12]
Jason Palmer. Heisenberg uncertainty principle stressed in new test. BBC News Science & Environment, September 2012.
[SKZ13] Maximilian Schlosshauer, Johannes Kofler, and Anton Zeilinger. A snapshot of foundational attitudes toward quantum mechanics. preprint, jan 2013. url = http://arxiv.org/abs/1301.1069. [Sta93]
H. P. Stapp. Mind, Matter and Quantum Mechanics. Springer, 1993.
[vB13]
Hans C. von Baeyer. Quantum weirdness? it’s all in your mind. Scientific American, 308(6):46–51, June 2013. 10
[Vin13] Andrew Vince. M¨obius iterated function systems. Trans. Amer. Math. Soc., 365 (2013):491–509, 2013.
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