You can't just fall in love with fractals and decide, what the Hell, space-time itself (the well from which fractals are drawn) must be fractal too. As you have no doubt discovered in your reading, all phenomena are described mathematically in physics. In order to put fractal spacetime in a context that can be tested and explored, physicists must figure out a mathematically consistent theory for fractal spacetime. Much of this article is devoted to that process. It is unavoidably complex. I know many of us are uncomfortable with differentiation (calculus) let alone variations on it. I don't think it is necessary to understand every process described here (I wouldn't want to be tested on some of them!). Instead, I think it's much more interesting to see how physicists go through the process. These kinds of mathematical journeys are at the heart of physics. They tell as much about what and how the people are thinking, as they do about the phenomenon at hand. A subtext of this series of articles is how the thinking of scientists evolves, how the process of science moves.
Fractal spacetime is a possibility in quantum mechanics. In 1948, Richard Feynman described the paths of photons as they struck a mirror and reflected off of it in a shocking and brilliant way: each photon of light actually has not just one trajectory but all possible paths to the mirror and back. What we see as a single reflected path of light is the probability amplitude for all the possible paths, even crazy, long, curved paths. Ultimately, these possibilities can be mathematically described and given a geometry. And in this geometry, each photon pathway is a continuous non-differentiable trajectory, making the geometry itself fractal, if we recall from the previous article that fractal math is continuous non-differential math. These photon trajectories can be described by a fractal dimension that jumps from nonfractal behaviour (whole integer dimensions; regular spacetime) at large everyday scales to fractal behaviour (a dimension that may exist in between whole integers) at the quantum scale of physics. These in-between fractal dimensions are strange and impossible to visualize; they remind me of Platform 93/4 at King's Cross Station in Harry Potter. This classical-to-quantum/fractal transition is thought to occur at the de Broglie scale (in the nanometer (nm) or billionth of a meter range). Above this scale, we see light reflected in ordinary straight lines as we expect it to. Below this scale, things get weird. If we could see it, reflected light would be individual photons going every which possible way, but with the vast majority of them following (fractal) trajectories that are very close to the classical trajectory.
A Classical to Quantum/Fractal Transition at the Very Small Scale
Why a transition at the de Broglie scale? Every subatomic particle has a wavelength thanks to its wave nature, which is inversely proportional to its momentum. A (1 eV) photon has a de Broglie wavelength of around 1240 nm. An electron, on the other hand, has a de Broglie wavelength about a thousand times shorter than that. Unlike the photon, it has rest mass-energy that gives it much more momentum, and much shorter wavelength. This shorter wavelength is why electron microscopes, using electrons rather than photons, have much higher resolution. What is a de Broglie wavelength?
De Broglie figured out how electron orbits inside atoms remain stable rather than crashing into the nucleus, which is exactly what would happen according to Newton's laws for gravity. De Broglie realized that (a) electrons must act like waves and (b) these waves must fit around the nucleus as whole integer waves, called standing waves. Electrons can increase their energy in an atom by moving to a new shorter wavelength standing wave. This means that electron energy comes in discrete packets and it gives electrons their observed quantum (packet-like) energy levels. It also gives a resonant scale structure to atoms. Feynman suggested that at scales smaller than these standing waves, particle behaviour transitions into continuous non-differentiable behaviour, which is characteristic of fractal behviour.
De Broglie wavelength is thought to be the large-end cut-off point for fractal quantum behaviour. There is also a smallest possible cut-off point in fractal geometry in general, and it might come as a surprise that it is not zero. Planck length is the lowest possible universal scale in physics, about 1.6 x 10-35 m. Most physicists working in this area take Planck length to be fractal geometry's lowest limit and the reasons for it are quite technical. I think it's worth noting, and quite interesting, that while the mathematics of fractals allows a fractal structure such as the Koch snowflake to have infinite magnification and therefore it can theoretically approach infinitely small segment lengths that go smaller than Planck length, spacetime fractal behaviour that is measurable in any meaningful way has a Planck length limit imposed on it. To be described by our current physical laws, fractal geometry has to follow the universal limit of meaningful scale in physics. This doesn't necessarily mean that reality has a Planck length limit; it's seems to be more of an uncomfortable compromise between old physics and new physics, at least to me.
Quantum Fractal Behaviour: From a Differential Equation to a Partial Differential Equation
This leads us to the mathematics behind fractals and the mathematics behind quantum mechanics. Can they be reconciled? In order to describe how a quantum state changes over time, physicists turn to the Schrodinger equation, formulated in 1925. This incredibly important equation is the quantum equivalent of Newton's second law, a classical law that describes the motion of a system. Newton's second law is often written as a differential equation where motion can be described as a smoothly changing system. Schrodinger's equation is written as a partial differential equation. A partial differential equation is a differential equation that contains unknown multivariable functions and their derivatives. A differential equation, in contrast, deals only with functions of a single variable and their derivatives. Because of this, a partial differential equation can offer a more complete description of a complex real system. These kinds of equations are used in many areas of physics such as electrostatics, electrodynamics, fluid flow and elasticity in solid state physics - in addition to quantum mechanics. They have a unique ability to treat distinctly different phenomena as variations on a single theme, bringing their dynamics together into a single language other words. Because of their multivariable functions, partial differential equations can also describe a system changing in multiple dimensions rather than just one, another big advantage. The cost is that they are usually far more difficult to solve than ordinary differential equations are.
Partial differential equations can be generalized as stochastic partial differential equations, a word we will explore shortly. It is this generalization that paves the way toward bringing the math of quantum mechanics and fractal geometry together.
From Brownian Motion To Fractal Quantum Mechanics
The Schrodinger equation describes both the wave function of the quantum system and the evolution of the quantum state of the system over time. Interestingly, although we think of the Schrodinger equation as being useful only for describing quantum scale behaviours, it is a partial differential equation that can be generalized to describe any physical system including macroscopic systems. The trick will be going from this kind of equation to a non-differential equation, one that truly describes the close relationships between quantum mechanics, fractal geometry and chaos.
Luckily there is a phenomenon that offers a potential bridge between fractals and quantum mechanics, and that is Brownian motion, a phenomenon that has humble origins. The name, Brownian motion, was coined in 1827 to describe the random motion of particles suspended in a liquid or a gas. When you see dust particles dancing in a beam of sunlight, you are observing this kind of motion. Below is a simulation of five yellow particles that collide with a set of 800 particles, each one leaving behind a blue trail of perfectly random motion. One red velocity vector for one yellow particle is also shown. This is a computer model of Brownian motion.
Nelson's stochastic quantum mechanics formally puts this connection between fractal mathematics and quantum mechanics together. It describes a quantum particle, such as an electron, as being subject to an underlying Brownian motion of unknown origin, which, in turn, is described by two processes called Markov-Weiner processes, one backward and one forward. Combining these processes gives the electron its wave function and it transforms Newton's dynamics into the Schrodinger equation, which describes quantum behaviour, as we saw earlier. The Schrodinger equation is a partial differential equation that describes how the quantum state of the electron, for example, changes over time. It describes its wave function in other words. If you remember at the beginning of this article, I mentioned that, according to Feynman, the (classical) reflection of light breaks down into the continuous non-differential trajectories of individual photons at the de Broglie scale. This fractal dimension at the quantum scale is also the fractal dimension of Brownian motion, described mathematically as a Markov-Weiner process.
In other words, by introducing quantum Brownian motion (the Markov-Weiner process) to the Schrodinger equation (a partial differential system), quantum mechanics can be described in terms of fractal geometry, a continuous non-differential system.
One thing that makes this Brownian motion idea so interesting is that it changes how we think of spacetime at the quantum scale. Spacetime is a bit of a mystery at this scale. General relativity doesn't seem as relevant here because gravity's influence is minuscule and the strong force, which holds the nuclei inside atoms together, becomes accessible. Spacetime, viewed at the quantum scale, appears to have a mysterious non-zero energy that is not associated with any of the fundamental forces, called vacuum energy. Nelson's stochastic quantum mechanics turns the concept of vacuum energy on its head. Vacuum energy is the energy at any point in spacetime that allows the creation of virtual particles and their antimatter twins to pop into existence. Once they form, they immediately annihilate each other and this quantum froth of activity results in changing or fluctuating vacuum energy on a very tiny, quantum, scale. Brownian motion makes the vacuum fluctuations of spacetime the driver of quantum mechanics, if we consider that the underlying Brownian motion is describing quantum fluctuation. This makes vacuum fluctuations the "Brownian motion of unknown origin" mentioned earlier. Usually, physicists think of it the other way around: quantum mechanics (the uncertainty principle) is the driver of quantum fluctuations.
These Brownian processes suggest that spacetime at the quantum level is fractal rather than flat and Minkowskian (Euclidean-like three dimensions of space plus one dimension of time), if we assume that the trajectories of Feynman's virtual photons are part of a fractal curve. Because fractal spacetime is nondifferentiable, it implies there must be an infinite number of geodesics (virtual trajectories) that the photons choose from.
A fractal interpretation of quantum mechanics means that the physical properties defining not only the virtual photon, but any particle, (properties such as mass, momentum, spin, velocity, etc.) can be defined as geometric structures of its fractal trajectory. A photon, or any particle, is no longer a point with momentum that follows a trajectory (more specifically one of its virtual trajectories recalling Feynman's work). Now a particle IS the fractal structure of its trajectory.
Quantum spin is now a purely geometrical property of the virtual trajectories of the particle. The whole infinite family of possible geodesics can be extended to describe the wave-particle nature of a particle. The probability cloud of an electron, for example, owes itself to the non-differentiability of fractal spacetime and the infinite family of geodesics that results from it.
From Fractals to Quantum Jewels
A very recent article (2013) by Natalie Wolchover, called A Jewel At The Heart Of Physics, describes an intriguingly similar geometric description of particle-particle interactions that, like fractals, challenges our current understanding of spacetime. A jewel-like geometric object, calculated by physicist Nima Arkani-Hamed and his doctoral student, Jaroslav Trnka, encodes the probabilities of outcomes for particle interactions in its volume, drastically simplifying calculations of particle interactions in the process. Click on the link above to see an approximate image of this beautiful and mesmerizing jewel. Like fractal quantum theory, this object suggests that quantum interactions are the consequence of geometry. This object, called an amplituhedron, forms the basis of a new quantum field theory that might be help researchers find a quantum theory for gravity that will seamlessly connect two mutually exclusive theories together - quantum mechanics and general relativity. This would amount to nothing less than discovering the theory of everything. Like fractal quantum theory, the probabilities associated with quantum phenomena are natural outcomes of the object's geometry.
The new object really shines in the field of high-energy physics, and that is where it was born. Calculating all the possible outcomes of even a very simple gluon-gluon collision, for example, requires the calculation of millions of different possible scattering amplitudes. This is done by running Feynman diagrams through a powerful computer program and, even with the best technology available, complex collision probabilities can be practically unsolvable. A construction that took decades to come together, the amplituhedron effectively takes all these calculations and turns them into one function. Instead of tediously plotting out millions of position-time variables, the amplituhedron couches the scattering process in terms of variables called twistors. A handful of twistor diagrams can describe very complex particle interactions. These twistor diagrams correspond to the volumes of pieces that fit together to construct the amplituhedron. The Feynman diagrams can piece the amplituhedron together too, but they are far, far less efficient.
They also found a "master" amplituhedron with an infinite number of facets. Its volume represents the total amplitude of all physical processes. Lower dimensional amplituhedra, representing the interactions of limited numbers of particles, live on the faces of this master structure. I am very eager to see how this very new theory plays out and how fractal quantum theory relates to it. Are these two different aspects of a single description?
Fractal Spacetime in General Relativity?
Fractal geometry seems to meld much more successfully into quantum mechanics than it melds into the much larger scale where relativity becomes the basic description of spacetime, and at the largest super-galactic scales, fractal geometry runs into possibly fatal trouble.
The geometry of spacetime is currently described by a set of field equations for general relativity. These equations are based on the idea that spacetime is flat (the topology) and curved to a manifold by a metric that is described by continuous (smoothly changing) differential geometry. This is how spacetime is curved by momentum and we experience this curve as gravity.
There is the hope, as I mentioned earlier on in this series, that by coming up with this kind of fractal theory of spacetime, what physicists observe as the effects of dark energy could be distortion-type effects of the underlying fractal geometry of spacetime becoming significant over great distances across space. In view of the large-scale structure of the universe (the scale of galaxy clusters and larger), some physicists such as Luciano Pietronero (1987) have attempted to model the distribution of galaxies on a fractal pattern. He claimed that a fractal dimension could be detected over a wide range of scale in the universe, one which seemed to hint that there is both randomness and hierarchal structuring at work at these very large scales in the universe. Other researchers have also examined the large-scale structure of the universe for signs of fractal geometry. An interpretation of recent cosmology data (David Hogg et al., 2005) suggests that, while the universe seems to be fairly clearly homogenous at the level of galaxy distribution (mass is smoothly spread out as evidenced by early Sloan survey results), it may exhibit a fractal dimension at distances of about 60 light years.
While fractal geometry offers an incredibly enticing new paradigm in physics, recent galaxy survey data cannot be ignored. In fact, it tears a big and problematic hole in fractal cosmology. The very same galaxy surveys (for example, the WiggleZ Survey concluded in 2012) that recently proved the existence of dark energy (the increasing expansion rate of the universe) tells us that the universe is very homogenous at very large scales. Despite some of the earlier claims described above, this homogenous galaxy distribution offers no sign of any fractal-like patterning at this scale. This means that, while you would expect some kind of galaxy clustering (a fractal pattern) at ever-larger scales, none are seen. There is no deviation from pure randomness of mass distribution, what you would expect from a homogenous universe. These observations do not support fractal theory being the answer to explain dark energy, which shows its effects especially at larger and larger scales, and they pose a problem for any universal fractal spacetime theory. The Sloan Digital Sky Survey, scanned over a decade and completed in 2013, created the largest ever three-dimensional map of galaxies in the universe, a map that would require half a million HD TV's to view its full image, which contains over a trillion pixels of information. These results together with the WMAP data definitively show that both the large-scale structure of the universe (at scales larger than 250 million light years) and the cosmic background radiation are evenly distributed, or homogenous, meaning that there is no other ordering, fractal or otherwise, of the universe at the galactic scale and larger. They do not lend support to fractal theory being the answer to explain the general relativity anomalies of dark energy or dark matter, which begin to show their effects at very large scales of spacetime.
So, is the universe fractal or not? One possible way around this is to think of the universe as snow: it is made up of fractal flakes but it transitions to a smooth and uniform sea of white as you step back. In the same way, the fractal nature of spacetime can only be observed at the quantum scale.
To describe space-time in terms of fractal geometry, the theory needs some kind of scale relativity and a continuous non-differential geometry that is completely dependent on the scale of the observation, so that at scales larger than de Broglie scale, the scaling part is dominated by the differential general relativity part. The geometry is still there underlying the theory but it's hidden at this scale. At scales smaller than de Broglie scale, the scaling part is dominated by fractal geometry instead, as the differential general relativity part becomes less relevant. The theoretical scale change from quantum to everyday to galactic is a bit like the change that happens during symmetry-breaking in gauge particle theory, or during a phase transition, except that here, the underlying scale symmetry is always intact.
I should make a distinction here between this kind of overall scale relativity and the scale invariance of fractals themselves. Fractal images reappear no matter what scale you are looking it them from. Some people call this an example of scale invariance but it is better described as closely related self-similarity.
The Fractal Advantage
The universe at its largest scale does not appear to have any fractal geometric ordering. And yet, hints at an underlying fractal nature can be seen everywhere around us. As described above, it is possible to describe a transition from the (classic) macroscopic scale of physics to quantum mechanics, where underlying fractal geometry seems to show much promise (we will see more examples of this promise in a moment). It may still be possible that fractal geometry underlies spacetime at all scales, but at macroscopic and larger scales it must be hidden from observation. Even though spacetime at these scales does not exhibit any fractal nature, fractal-like ordering seems to be at work in various biological processes, structures and in geology. What makes this kind of ordering favourable in living systems in particular? Is it a cost-saving or simplification advantage? Does it impart structures (like shells and trees for example) with greater strength or durability? A 2012 paper called Fractal Structures Do More With Less investigates possible advantages of fractal-like design and hierarchal structuring in construction. Researchers found that, for example, when more hierarchal levels are added to structures less material is needed to support a given load. Trabecular (spongy) bone is a similar architectural example taken from biology, in which evolution favours hierarchal fractal-like ordering in order to maximize strength, all while minimizing the amount of input material required as well as minimizing weight. Similarly, tree branching reveals fractal-like ordering, a quality that Leonardo da Vinci noticed and called his "rule of trees" more than 500 years ago. This kind of ordering is especially obvious after deciduous trees lose their leaves in the fall. Tree branching may offer two advantages - hydrological (this arrangement most efficiently transports sap) and structural (it increases the tree's resistance to various stresses like snow load and wind). A 2011 study, based on computer-modeled trees, suggests that fractal ordering in trees protects them against wind damage in particular. There is a great deal of current research on the appearance and roles of fractal-like ordering in nature, but perhaps the most convincing evidence for fractal ordering in matter comes from solid-state physics.
Evidence of Fractals from Solid State Physics
There is increasing evidence that fractal geometry underlies processes at the molecular, atomic and quantum scale.
For example, during phase transitions from regular conductors into superconductors, when electrons in a material organize themselves in ways that resemble bosonic (force particle) behaviour rather than the fermionic behaviour that particles of matter normally display, fractal geometry pops up. Even very good conductors experience at least some electrical resistance. However, during this phase change the wave functions of electrons spread out over the whole material in a special way that allows it to conduct electricity without any resistance at all. High-temperature superconductors are especially mysterious, because the molecular jiggling that goes on at temperatures well above absolute zero should destroy the kind of ordering that is necessary for a spread-out wave function. Physicists have recently found a clue about how this phenomenon is made possible. In 2010, physicist Antonio Bianconi discovered that oxygen atoms inside ceramic compound high-temperature superconductors appear to be in random positions and to take on complex geometries that display self-symmetry, a fractal behaviour. Larger fractals correspond with higher superconductivity temperatures. No one yet knows exactly how fractal ordering seems to stabilize wave functions and make high-temperature superconductors possible.
Ali Yazdani and his colleagues at Princeton University in the US observed a fractal pattern created when electrons interfere with one another. They observed the material gallium arsenide undergoing a phase transition under a scanning tunneling microscope (it gives you atomic-scale resolution) and found that a fractal pattern is observed as it changes from a metal into an insulator. When this happens, the wave functions of the electrons change from being shared across the whole material (metallic state) to being localized at individual atomic lattice sites (insulator state). During transition, the electron wave functions get squashed together and begin to affect each other in a complicated pattern of constructive and destructive interference and this is when a fractal pattern develops. Their results were published in 2010.
Last year (2013), physicists found the first proof of a decades-old theoretical fractal pattern called the Hofstadter Butterfly. Hofstadter, a graduate student in the 1970's, discovered that electrons confined inside a crystalline atomic lattice would race around in circles when placed in a powerful magnetic field. The motion of the circling would soon become complicated (chaotic). When plotted on a graph, the motion revealed a fractal pattern that looked like a butterfly, shown below as a computer rendering, although fractals were not known at the time.
In the diagram above, the horizontal axis is energy and the vertical axis is magnetic flux through the material. Warm and cold colours represent positive and negative values for Hall conductance (a voltage difference), respectively. Like all fractal images it shows self-similarity. Small fragments of the structure contain a (distorted) copy of the entire structure.
Pablo Jarillo-Herrero at MIT in Cambridge found that by stacking a sheet of graphene with a sheet of boron nitride and applying a magnetic field, he observed discrete changes in conductivity, stepwise jumps that corresponded to the same splitting of electron energy levels that Hofstadter observed. Wolfgang Ketterle, also at MIT, is currently trying to go a bit further by making supercooled rubidium atoms act like electrons by trapping the atoms in regularly spaced pockets and guiding them with lasers and gravity to mimic the circular motions of electrons in a magnetic field. If he succeeds, he may be able to show fractal ordering at the atomic level.
Some physicists are wondering if these kinds of fractal organization observed with electrons and atoms might offer quantum clues about why living systems tend to show a preference for fractal-type structures. It's possible that, like many natural and geologic structures, the natural world at the quantum level favours fractal structures.
Other researchers are making a possible link between string theory and fractal geometry. General relativity treats spacetime as 4-dimensional with three spatial dimensions and one time dimension. String theory predicts the existence of extra dimensions in spacetime. M-theory, for example, predicts 11 dimensions. A new possibility is that the dimensions in spacetime change with fractal scale, allowing small scales to exhibit fractal properties. For example, such a theoretical framework could describe quantum relativity, or quantum gravity in other words, where gravity at quantum scales appears fractal. This idea expands upon the idea that fractal spacetime is composed of non-integer dimensions, rather than the whole integer dimensions (at all scales) of spacetime described by Euclidean space, Minkowski space and the curved spacetime of general relativity. By giving fractal spacetime non-integer dimensions, the properties of spacetime depend on the scale of observation. In this case, the very dimensions of spacetime are scale-dependent.
Fractals Push Against the Differential Heart of Physics
Fractal geometry at the quantum level seems to be gaining momentum because there is so much promise, as well as some enticing experimental evidence coming together, as physicists try to sew together a consistent fractal theory for quantum particle behaviour. Meanwhile, research into fractal biology, geology and several other fields is taking off. But the possibility of a fractal cosmos seems far less promising. At best, a possible underlying fractal nature seems to be hidden from view.
Physicist Tom Palmer thinks that fractal geometry might be alive and well in the cosmos after all - if we look for it in the right place. He argues that each physical system around us has an invariant set, a mathematical ground state, in which it is unable to lose any more information. If you take a large star, for example, you will find that it has an enormous amount of data held within all the atoms that make it up (information like quantum spins, mass, momentum, energy state, etc.). When it starts to collapse in on itself at the end of its life, some of that data is lost. When it collapses all the way into a black hole, much more data is lost. The black hole is a minimal information ground state where no more data can be lost, and this is the invariant set which underlies that star's information. This kind of logic can be extended to the universe as a whole, and the invariant set of the universe might be fractal in nature. This approach could lead to an explanation for some of the most puzzling paradoxes in spacetime such as nonlocality - the ability of two entwined particles to communicate with each other across vast distances of space, or the ability of a single particle to exist in more than one location in space at the same time. It seems reminiscent of the Holographic approach to spacetime.
Perhaps the greatest promise offered by fractals is the possibility of creating more accurate mathematical models of how nature works. It seems, as we saw earlier, that reforming quantum mechanics into a non-differential equation opens up a whole new way of investigating quantum processes. A 1993 paper by Laurent Nottale discusses the movement away from differential math toward non-differntial math in physics. Since the time of Isaac Newton, differentiatial calculus has been used to describe most physical phenomena. There are countless examples of physical and biological processes (any phenomenon that changes over time) that are described in terms of one or more differential equations. This Wikipedia link lists many examples of differentiatial calculus at use in physics, engineering, biology and economics to describe processes such as radioactive decay, diffusion, animal population dynamics and evolutionary changes, just to name a few.
Calculus (differentiation and integration), developed in the mid 17th century, was a great breakthrough in physics because it offered a way to model continuous change in systems. Yet, there is no underlying principle in place that says the fundamental laws of physics must be differentiable. What if the basic reality of the universe is more accurately modelled using non-differentiable mathematics, and that fractal geometry underlies all physical processes even though it may be hidden at larger scales? If the universe really does have a fundamentally fractal backbone, it would mean reconstructing physical laws in terms of continuous but non-differentiable equations. Quantum mechanics, for example, becomes mechanics in non-differentiable spacetime.
Fresh new possibilities like this remind us that even established paradigm-making theories are not sacred. There may be other overlooked assumptions waiting to be questioned.
A Few Final Questions:
Fractal geometry seems to impart some kind of efficient process to particle interactions, an efficiency that nature at larger everyday scales seems to draw from. Is it the geometry itself or is there something deeper from which it draws? Is fractal geometry universal, and if so how is it hidden from view at the cosmic scale? Will fractals be the link between the macro universe and the micro universe that allows physicists to find a theory for quantum gravity?