Symbols of Power

“In the land of theoretical physics, equations have always been king.”

“A less well known but much more successful example of geometry applied to physics is Murray Gell-Mann’s “eightfold way”, which is a means of organizing subatomic particles. This organization has an underlying explanation using triangles with quarks located at the vertices.”

“For the past five years, I and a group of my colleagues (including Charles Doran, Michael Faux, Tristan Hubsch, Kevin Iga, Greg Landweber and others) have been following the geometric-physics path pioneered by Kepler and Gell-Mann. The geometric objects that interest us are not triangles or octagons, but more complicated figures known as “adinkras”, a name Faux suggested.”

“The word “adinkra” is of West African etymology, and it originally referred to visual symbols created by the Akan people of Ghana and the Gyamen of Côte d’Ivoire to represent concepts or aphorisms.”

“However, the mathematical adinkras we study are really only linked to those African symbols by name. Even so, it must be acknowledged that, like their forebears, mathematical adinkras also represent concepts that are difficult to express in words. Most intriguingly, they may even contain hints of something more profound — including the idea that our universe could be a computer simulation, as in the Matrix films.”

“Moving on to the Standard Model, the set of equations used to describe the physics of quarks, leptons (the family of particles that contains the electron) and force-carrying particles like the photon (carrier of the electromagnetic force) is also largely determined by symmetry groups.”

“Such diagrams are more than pictures. In fact, it was an insight drawn from such diagrams that led Gell-Mann and George Zweig to a new understanding of nuclear matter. Gell-Mann and Zweig realized that patterns in diagrams showing families of nuclear particles meant that those particles must be made up of smaller, more fundamental particles: quarks.”

“The nuclear-particle octet diagram gets its name because there are particles associated with each of its six vertices, and two additional particles associated with its centre, hence an “octet” of particles.”

“This diagram is useful as a kind of accounting tool: in certain nuclear reactions, two or more experiments will lead to simply related results if one member of this family is replaced by another. For example, measuring how a proton is deflected from a neutron by the strong nuclear force will yield a result that is directly related to the deflection of a ?- particle from a neutron.”

“This is the power of using symmetries. When we know that certain symmetries are present in nature, we can use one experiment to predict the outcome of many others.”

“As its name implies, the theory of supersymmetry takes the idea of symmetry a step further. In the Standard Model there is a dichotomy between leptons and quarks — collectively called “matter particles” — and force-carrying particles like photons.”

“All matter particles are fermions, particles with half-integer quantum spin that obey the Pauli exclusion principle.”

“Force-carrying particles, in contrast, are bosons, which have integer spin and can violate the exclusion principle.”

“This means that not only photons but also gluons (which carry the strong nuclear force), the W and Z bosons (which carry the weak nuclear force), and even the hypothetical Higgs boson are all free to possess any allowed quantum numbers in composite systems.”

“SUSY breaks this rule that all matter particles are fermions and all carriers are bosons. It does this by relating each Standard Model particle to a new form of matter and energy called a “superpartner”. In its simplest form, SUSY states that every boson has a corresponding “super-fermion” associated with it, and vice versa. These superpartners have not yet been observed in nature, but one of the main tasks of CERN’s Large Hadron Collider (LHC) will be to look for experimental evidence of their existence. If the LHC finds these superpartners, then the Standard Model will have to be replaced by the Minimal Supersymmetric Standard Model (MSSM), or perhaps another more exotic variant.”

“It is here that adinkras might prove useful. Just as a weight-space diagram is a graphical representation that precisely encodes the mathematical relations between the members of SU(3) families, so an adinkra is a graphical representation that precisely encodes the mathematical relations between the members of supersymmetry families.”

“Now that we know a little bit about how adinkras can be used, we can begin to discuss what they look like. All adinkras are constructed by starting with squares, cubes and their higher-dimensional generalizations; these structures provide a “skeleton” that is then “decorated” by additional operations.”

“Each of these decorations has a mathematical significance, which I will discuss later. For the moment, let us just concentrate on building a simple adinkra.”

“To make a square into an adinkra, we begin by placing a white dot at one vertex (figure 2). The rules of adinkras then dictate that the two line segments connected to the white dot must have black dots at their opposite ends. This means that the final unpopulated vertex is connected to “black dot” vertices, so it must be populated by a white dot. Next, we need to assign directions to each line segment, or link. To keep track of these different directions, we assign distinct colours to each of them: all links that point in the same direction are assigned the same colour, and links that point in different directions are never assigned the same colour.”

“Then, we need to assign an “edge-parity” to each link: each coloured line can be drawn as either solid or dashed. Every two-colour closed path in an adinkra must contain an odd number of dashed links. One last rule is that white dots and black dots are never allowed to have the same vertical position; that is, no black dot in an adinkra is ever allowed to appear at the same height as a white dot.”

“There is no limit to the number of colours that may be used to construct an adinkra. As a result, higher-dimensional adinkras have a certain aesthetic appeal (figure 3).”

“As Einstein once said, “After a certain high level of technical skill is achieved, science and art tend to coalesce in aesthetics, plasticity and form.” Perhaps the “artistic” depictions shown here are an example of this.”

“But adinkras, like Gell-Mann’s octets, are not just pictures. In fact, they are in some ways rather similar to Feynman diagrams, which are the series of line drawings used to describe calculations in quantum electrodynamics.”

“The “decorated tesseract” adinkra on the left can be broken down into two separate adinkras. The author’s collaboration of mathematicians and other physicists has introduced the name “gnomoning” for this process of subtracting a smaller adinkra from larger ones. The name gnomoning was used by Euclid, the founder of geometry, to describe a plane figure obtained by removing a smaller figure that is similiar to the large one.”

“Investigating this question launched our “treasure hunt” in a completely unexpected direction: computer codes.

Modern computer and communication technologies have come to prominence by transmitting data rapidly and accurately. These data consist principally of strings of ones and zeros (called bits) written in long sequences called “words”. When these computer words are transmitted from a source to a receiver, there is always the chance that static noise in the system can alter the content of any word. Hence, the transmitted word might arrive at the receiver as pure gibberish.”

“One of the first people to confront this problem was the mathematician Richard Hamming, who worked on the Manhattan Project during the Second World War. In 1950 he introduced the idea of “error-correcting codes” that could remove or work around any un wanted changes to a transmitted signal. Hamming’s idea was for the sending computer to insert extra bits into words in a specific manner such that the receiving computer could, by looking at the extra bits, detect and correct errors introduced by the transmission process. His algorithm for the insertion of these extra bits is known as the “Hamming code”.”

“The construction of such error-correcting codes has been pursued since the beginning of the computer age and many different codes now exist. These are typically divided into families; for example, the “check-sum extended Hamming code” is a rather complicated variant of the Hamming code and it belongs to a family known as “doubly even self-dual linear binary error-correcting block codes” (an amazing mouthful!). Yet whatever family they belong to, all error-correction codes serve the same function: they are used to detect errors and allow the correct transmission of digital data.”

“How does this relate to adinkras? The middle adinkra in figure 4 is obtained by folding the image on the left of the figure. The folding involves taking pairs of the dots of the same type and “fusing them together” as if they were made of clay. In general, an adinkra-folding process will lead to diagrams where the associated equations do not possess the SUSY property. In order to ensure that this property is retained, we must carry out the fusing in such a way that white dots are only fused with other white dots, black dots with other black dots, and lines of a given colour and dashing are only joined with lines that possess the same properties. Most foldings violate this, but there is one exception — and it happens to be related to a folding that involves doubly even self-dual linear binary error-correcting block codes.”

“The part of science that deals with the transmission of data is called information theory. For the most part, this is a science that has largely developed in ways that are unrelated to the fields used in theoretical physics.”

“However, with the observation that structures from information theory — codes — control the structure of equations with the SUSY property, we may be crossing a barrier.”

“I know of no other example of this particular intermingling occurring at such a deep level. Could it be that codes, in some deep and fundamental way, control the structure of our reality? In asking this question, we may be ending our “treasure hunt” in a place that was anticipated previously by at least one pioneering physicist: John Archibald Wheeler.”

“Wheeler, who died in 2008, was an extremely well-regarded figure within physics. He served as advisor to a clutch of important physicists, including Richard Feynman, while his own work included the concept of the “S-matrix” (a mathematical tool that helps us understand Standard Model particles). Beyond the physics community, Wheeler is probably best known for coining the terms “black hole” and “wormhole”. But he also coined a slightly less familiar phrase — “it from bit” — and this is what concerns us here.”

“The idea of “it from bit” is a complex one, and Wheeler’s own description of it is probably still the best. In 1990 he suggested that “every ‘it’ — every particle, every field of force, even the space-time continuum itself — derives its function, its meaning, its very existence entirely…from the apparatus-elicited answers to yes-or-no questions, binary choices, bits”.”

“The “it from bit” principle, he continued, “symbolizes the idea that every item of the physical world has at bottom…an immaterial source and explanation: that which we call reality arises in the last analysis from the posing of yes-no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and that this is a participatory universe”.”

“When I first heard the idea of “it from bit” as a young physicist, I thought Wheeler must be crazy. The concept of a world made up of information just sounded strange, and (although I did not know it at the time) I was not the only one who thought so. However, sometimes crazy ideas turn out to be true, and Wheeler has been proved right before. As Feynman said, “When I was [Wheeler’s] student, I discovered that if you take one of his crazy ideas and you unwrap the layers of craziness from it one after another, like lifting layers off an onion, at the heart of the idea you will often find a powerful kernel of truth.” Indeed, another of Wheeler’s “crazy” ideas — his suggestion that a positron can be treated as an electron moving backwards in time — played a role in Feynman later winning a Nobel prize.”

“As for my own collaboration on adinkras, the path my colleagues and I have trod since the early 2000s has led me to conclude that codes play a previously unsuspected role in equations that possess the property of supersymmetry.”

“This unsuspected connection suggests that these codes may be ubiquitous in nature, and could even be embedded in the essence of reality.”

“If this is the case, we might have something in common with the Matrix science-fiction films, which depict a world where everything human beings experience is the product of a virtual-reality-generating computer network.”

“If that sounds crazy to you — well, you could be right. It is certainly possible to overstate mathematical links between different systems: as the physicist Eugene Wigner pointed out in 1960, just because a piece of mathematics is ubiquitous and appears in the description of several distinct systems does not necessarily mean that those systems are related to each other.”

“The number pi, after all, occurs in the measurement of circles as well as in the measurement of population distributions. This does not mean that populations are related to circles.”

“Yet for a moment, let us imagine that this alternative Matrix-style world contains some theoretical physicists, and that one of them asks, “How could we discover whether we live inside a Matrix?”. One answer might be “Try to detect the presence of codes in the laws that describe physics.” I leave it to you to decide whether Wigner’s warning should be applied to the theoretical physicists living in the Matrix — and to us.”