Nambu-Goldstone bosons

The symmetry incarnations

This post was originally published on October 6, 2012. I recently rediscovered it and decided to republish it with a few updates.

Geometric symmetry in nature is often a sign of unperturbedness, as if nothing has interfered with a natural process and that its effects at each step are simply scaled-up or scaled-down versions of each other. For this reason, symmetry is aesthetically pleasing, and often beautiful. Consider, for instance, faces. Symmetry of facial features about the central vertical axis is often translated as the face being beautiful, not just by humans but also monkeys.

This is only one example of one of the many forms of symmetry’s manifestation. When it involves geometric features, it’s a case of geometrical symmetry. When a process occurs similarly both forward and backward in time, it is temporal symmetry. If two entities that don’t seem geometrically congruent at first sight rotate, move or scale with similar effects on their forms, it is transformational symmetry. A similar definition applies to all theoretical models, musical progressions, knowledge and many other fields besides.

Symmetry-breaking

One of the first (postulated) instances of symmetry is said to have occurred during the Big Bang, when the universe was born. A sea of particles was perturbed 13.75 billion years ago by a high-temperature event, setting up anecdotal ripples in their system, eventually breaking their distribution in such a way that some particles got mass, some charge, some spin, some all of them, and some none of them. This event is known as electroweak symmetry-breaking. Because of the asymmetric properties of the resultant particles, matter as we know it was conceived.

Many invented scientific systems exhibit symmetry in that they allow for the conception of symmetry in the things they make possible. A good example is mathematics. On the real-number line, 0 marks the median. On either sides of 0, 1 and -1 are equidistant from 0; 5,000 and -5,000 are equidistant from 0; possibly, ∞ and -∞ are equidistant from 0. Numerically speaking, 1 marks the same amount of something that -1 marks on the other side of 0. Characterless functions built on this system also behave symmetrically on either sides of 0.

To many people, symmetry evokes the image of an object that, when cut in half along a specific axis, results in two objects that are mirror-images of each other. Cover one side of your face and place the other side against a mirror, and what a person hopes to see is the other side of the face – despite it being a reflection. Interestingly, this technique was used by neuroscientist V.S. Ramachandran to “cure” the pain of amputees when they tried to move a limb that wasn’t there).

An illustration of V.S. Ramachandran's mirror-box technique: Lynn Boulanger, an occupational therapy assistant and certified hand therapist, uses mirror therapy to help address phantom pain for Marine Cpl. Anthony McDaniel. Caption and credit: US Navy — An illustration of V.S. Ramachandran’s mirror-box technique: Lynn Boulanger, an occupational therapy assistant and certified hand therapist, uses mirror therapy to help address phantom pain for Marine Cpl. Anthony McDaniel. Caption and credit: US Navy

Natural symmetry

Symmetry at its best, however, is observed in nature. Consider germination: when a seed grows into a small plant and then into a tree, the seed doesn’t experiment with designs. The plant is not designed differently from the small tree, and the small tree is not designed differently from the big tree. If a leaf is given to sprout from the node richest in minerals on the stem, then it will. If a branch is given to sprout from the node richest in minerals on the trunk, then it will. So is mineral-deposition in the arbor symmetric? It should be if their transportation out of the soil and into the tree is radially symmetric. And so forth…

At times, repeated gusts of wind may push the tree to lean one way or another, shadowing the leaves from against the force and keeping them form shedding off. The symmetry is then broken, but no matter. The sprouting of branches from branches, and branches from those branches, and leaves from those branches, all follow the same pattern. This tendency to display an internal symmetry is characterised as fractalisation. A well-known example of a fractal geometry is the Mandelbrot set, shown below.

An illustration of recursive self-similarity in Mandelbrot set. Credit: Cuddlyable3/Wikimedia Commons

~~If you want to interact with a Mandelbrot set, check out this magnificent visualisation by Paul Neave~~ (defunct now 🙁 ). You can keep zooming in, but at each step, you’ll only see more and more Mandelbrot sets. This set is one of a few exceptional sets that are geometric fractals.

Meta-geometry and Mulliken symbols

It seems like geometric symmetry is the most ubiquitous and accessible example to us. Let’s take it one step further and look at the meta-geometry at play when one symmetrical shape is given an extra dimension. For instance, a circle exists in two dimensions; its three-dimensional correspondent is the sphere. Through such an up-scaling, we are ensuring that all the properties of a circle in two dimensions stay intact in three dimensions, and then we are observing what the three-dimensional shape is.

A circle, thus, becomes a sphere. A square becomes a cube. A triangle becomes a tetrahedron. In each case, the 3D shape is said to have been generated by a 2D shape, and each 2D shape is said to be the degenerate of the 3D shape. Further, such a relationship holds between corresponding shapes across many dimensions, with doubly and triply degenerate surfaces also having been defined.

The three-dimensional cube generates the four-dimensional hypercube, a.k.a. a tesseract. Credit: Vitaly Ostrosablin/Wikimedia Commons, CC BY-SA 3.0

Obviously, there are different kinds of degeneracy, 10 of which the physicist Robert S. Mulliken identified and laid out. These symbols are important because each one defines a degree of freedom that nature possesses while creating entities and this includes symmetrical entities as well. So if a natural phenomenon is symmetrical in n dimensions, then the only way it can be symmetrical in n+1 dimensions also is by transforming through one or many of the degrees of freedom defined by Mulliken.

Symbol	Property
A	symmetric with respect to rotation around the principal rotational axis (one dimensional representations)
B	anti-symmetric with respect to rotation around the principal rotational axis (one dimensional representations)
E	degenerate

subscript 1	symmetric with respect to a vertical mirror plane perpendicular to the principal axis
subscript 2	anti-symmetric with respect to a vertical mirror plane perpendicular to the principal axis

subscript g	symmetric with respect to a center of symmetry
subscript u	anti-symmetric with respect to a center of symmetry

prime (‘)	symmetric with respect to a mirror plane horizontal to the principal rotational axis
double prime (”)	anti-symmetric with respect to a mirror plane horizontal to the principal rotational axis

Source: LMU Munich

Apart from regulating the perpetuation of symmetry across dimensions, the Mulliken symbols also hint at nature wanting to keep things simple and straightforward. The symbols don’t behave differently for processes moving in different directions, through different dimensions, in different time-periods or in the presence of other objects, etc. The preservation of symmetry by nature is not coincidental. Rather, it is very well-defined.

Anastomosis

Now, if nature desires symmetry, if it is not a haphazard occurrence but one that is well orchestrated if given a chance to be, why don’t we see symmetry everywhere? Why is natural symmetry broken? One answer to this is that it is broken only insofar as it attempts to preserves other symmetries that we cannot observe with the naked eye.

For example, symmetry in the natural order is exemplified by a geological process called anastomosis. This property, commonly of quartz crystals in metamorphic regions of Earth’s crust, allows for mineral veins to form that lead to shearing stresses between layers of rock, resulting in fracturing and faulting. In other terms, geological anastomosis allows materials to be displaced from one location and become deposited in another, offsetting large-scale symmetry in favour of the prosperity of microstructures.

More generally, anastomosis is defined as the splitting of a stream of anything only to reunify sometime later. It sounds simple but it is an exceedingly versatile phenomenon, if only because it happens in a variety of environments and for a variety of purposes. For example, consider Gilbreath’s conjecture. It states that each series of prime numbers to which the forward difference operator – i.e. successive difference between numbers – has been applied always starts with 1. To illustrate:

2 3 5 7 11 13 17 19 23 29 … (prime numbers)

Applying the operator once: 1 2 2 4 2 4 2 4 6 …
Applying the operator twice: 1 0 2 2 2 2 2 2 …
Applying the operator thrice: 1 2 0 0 0 0 0 …
Applying the operator for the fourth time: 1 2 0 0 0 0 0 …

And so forth.

If each line of numbers were to be plotted on a graph, moving upwards each time the operator is applied, then a pattern for the zeros emerges, shown below.

The forest of stunted trees, used to gain more insights into Gilbreath's conjecture. Credit: David Eppstein/Wikimedia Commons — The forest of stunted trees, used to gain more insights into Gilbreath’s conjecture. Credit: David Eppstein/Wikimedia Commons

This pattern is called the forest of stunted trees, as if it were an area populated by growing trees with clearings that are always well-bounded triangles. The numbers from one sequence to the next are anastomosing, parting ways only to come close together after every five lines.

Another example is the vein skeleton on a hydrangea leaf. Both the stunted trees and the hydrangea veins patterns can be simulated using the rule-90 simple cellular automaton that uses the exclusive-or (XOR) function.

Bud and leaves of Hydrangea macrophylla. Credit: Alvesgaspar/Wikimedia Commons, CC BY-SA 3.0 — Bud and leaves of *Hydrangea macrophylla*. Credit: Alvesgaspar/Wikimedia Commons, CC BY-SA 3.0

Nambu-Goldstone bosons

While anastomosis may not have a direct relation with symmetry and only a tenuous one with fractals, its presence indicates a source of perturbation in the system. Why else would the streamlined flow of something split off and then have the tributaries unify, unless possibly to reach out to richer lands? Anastomosis is a sign of the system acquiring a new degree of freedom. By splitting a stream with x degrees of freedom into two new streams each with x degrees of freedom, there are now more avenues through which change can occur.

Particle physics simplifies this scenario by assigning all forces and amounts of energy a particle. Thus, a force is said to be acting when a force-carrying particle is being exchanged between two bodies. Since each degree of freedom also implies a new force acting on the system, it wins itself a particle from a class of particles called the Nambu-Goldstone (NG) bosons. Named for Yoichiro Nambu and Jeffrey Goldstone, the presence of n NG bosons in a system means that, broadly speaking, the system has n degrees of freedom.

How and when an NG boson is introduced into a system is not yet well-understood. In fact, it was only recently that a theoretical physicist, named Haruki Watanabe, developed a mathematical model that could predict the number of degrees of freedom a complex system could have given the presence of a certain number of NG bosons. At the most fundamental level, it is understood that when symmetry breaks, an NG boson is born.

The asymmetry of symmetry

That is, when asymmetry is introduced in a system, so is a degree of freedom. This seems intuitive. But at the same time, you would think the reverse is also true: that when an asymmetric system is made symmetric, it loses a degree of freedom. However, this isn’t always the case because it could violate the third law of thermodynamics (specifically, the Lewis-Randall version of its statement).

Therefore, there is an inherent irreversibility, an asymmetry of the system itself: it works fine one way, it doesn’t work fine another. This is just like the split-off streams, but this time, they are unable to reunify properly. Of course, there is the possibility of partial unification: in the case of the hydrangea leaf, symmetry is not restored upon anastomosis but there is, evidently, an asymptotic attempt.

However, it is possible that in some special frames, such as in outer space, where the influence of gravitational forces is very weak, the restoration of symmetry may be complete. Even though the third law of thermodynamics is still applicable here, it comes into effect only with the transfer of energy into or out of the system. In the absence of gravity and other retarding factors, such as distribution of minerals in the soil for acquisition, etc., it is theoretically possible for symmetry to be broken and reestablished without any transfer of energy.

The simplest example of this is of a water droplet floating around. If a small globule of water breaks away from a bigger one, the bigger one becomes spherical quickly. When the seditious droplet joins with another globule, that globule also quickly reestablishes its spherical shape.

Thermodynamically speaking, there is mass transfer but at (almost) 100% efficiency, resulting in no additional degrees of freedom. Also, the force at play that establishes sphericality is surface tension, through which a water body seeks to occupy the shape that has the lowest volume for the correspondingly highest surface area. Notice how this shape – the sphere – is incidentally also the one with the most axes of symmetry and the fewest redundant degrees of freedom? Manufacturing such spheres is very hard.

An omnipotent impetus

Perhaps the explanation of the roles symmetry assumes seems regressive: every consequence of it is no consequence but itself all over again (i.e., self-symmetry – and it happened again). Indeed, why would nature deviate from itself? And as it recreates itself with different resources, it lends itself and its characteristics to different forms.

A mountain will be a mountain to its smallest constituents, and an electron will be an electron no matter how many of them you bring together at a location (except when quasiparticles show up). But put together mountains and you have ranges, sub-surface tectonic consequences, a reshaping of volcanic activity because of changes in the crust’s thickness, and a long-lasting alteration of wind and irrigation patterns. Bring together an unusual number of electrons to make up a high-density charge, and you have a high-temperature, high-voltage volume from which violent, permeating discharges of particles could occur – i.e., lightning.

Why should stars, music, light, radioactivity, politics, engineering or knowledge be any different?

Yoichiro Nambu, the silent revolutionary of particle physics, is dead

The Wire
July 18, 2015

Particle physics is an obscure subject for most people but everyone sat up and took notice when the Large Hadron Collider discovered the particle named after Peter Higgs in 2012. The Higgs boson propelled his name to the front pages of newspapers that until then hadn’t bothered about the differences between bosons and fermions. On the other hand, it also validated a hypothesis he and his peers had made 50 years ago and helped the LHC’s collaborations revitalise their outreach campaigns.

However, much before the times of giant particle colliders – in the late 1950s, in fact – a cascade of theories was being developed by physicists the world over with much less fanfare, and a lot more of the quiet dignity that advanced theoretical physics is comfortable revelling in. It was a silent revolution, and led in part by the mild-mannered Yoichiro Nambu, who passed away on July 5, 2015.

His work and its derivatives gave rise to the large colliders like the LHC at work today, and which might well have laid the foundations of modern particle physics research. Moreover, many of his and his peers’ accomplishments are not easily discussed the way political movements are nor do they aspire to such privileges, but that didn’t make them any less important than the work of Higgs and others.

Yoichiro Nambu also belonged to a generation that marked a resurgence in Japanese physics research – consider his peers: Yoshio Nishina, Masatoshi Koshiba, Hideki Yukawa, Sin-Itiro Tomonaga, Leo Esaki, Makoto Kobayashi and Toshihide Maskawa, to name a few. A part of the reason was a shift in Japan’s dominant political attitudes after the Second World War. Anyway, the first of Nambu’s biggest contributions to particle physics came in 1960, and it was a triumph of intuition.

There was a span of 46 years between the discovery of superconductivity (by Heike Kamerlingh Onnes in 1911) and the birth of a consistent theoretical explanation for it (by John Bardeen, Leon Cooper and John Schrieffer in 1957) because the phenomenon seemed to defy some of the first principles of the physics used to understand charged particles. Nambu was inspired by the BCS theory to attempt a solution for the hierarchy problem – which asks why gravity, among the four fundamental forces, is 10³² times weaker than the strongest strong-nuclear force.

With the help of British physicist Jeffrey Goldstone, Nambu theorised that whenever a natural symmetry breaks, massless particles called Nambu-Goldstone bosons are born under certain conditions. The early universe, around 13.75 billion years ago when it was extremely small, consisted of a uniform pond of unperturbed energy. Then, the pond was almost instantaneously heated to a temperature of 173 billion Suns, when it broke into smaller packets called particles. The symmetry was (thought to be) spontaneously broken and the event was called the Big Bang.

Then, as the universe started to cool, these packets couldn’t reunify into becoming the pond they once made up, evolving instead into distinct particles. There were perturbations among the particles and the resultant forces were mediated by what came to be called Nambu-Goldstone bosons, named for the physicists who first predicted their existence.

Yoichiro in Nambu in 2008. Source: University of Chicago

Nambu was able to use the hypothetical interactions between the Nambu-Goldstone bosons and particles to explain how the electromagnetic force and the weak nuclear force (responsible for radioactivity) could be unified into one electroweak force at higher temperatures, as well as how where the masses of protons and neutrons come from. These were (and are) groundbreaking ideas that helped scientists make sense of the intricate gears that turned then to make the universe what it is today.

Then, in 1964, six physicists (Higgs, Francois Englert, Tom Kibble, Gerald Guralnik, C.R. Hagen, Robert Brout) postulated that these bosons interacted with an omnipresent field of energy – called the Higgs field – to give rise to the strong-nuclear, weak-nuclear (a.k.a. weak) and electromagnetic forces, and the Higgs boson. And when this boson was discovered in 2012, it validated the Six’s work from 1964.

However, Nambu’s ideas – as well as those of the Six – also served to highlight how the gravitational force couldn’t be unified with the other three fundamental forces. In the 1960s, Nambu’s first attempts at laying out a framework of mathematical equations to unify gravity and the other forces gave rise to the beginnings of string theory. But in the overall history of investigations into particle physics, Nambu’s work – rather, his intellect – was a keystone. Without it, the day theorists’ latinate squiggles on paper could’ve become prize-fetching particles in colliders would’ve been farther off, the day we made sense of reality farther off, the day we better understood our place in the universe farther off.

The Osaka City University, where Nambu was a professor, announced his death on July 17, due to an acute myocardial infarction. He is survived by his wife Chieko Hida and son John. Though he was an associate professor at Osaka from 1950 to 1956, he visited the Institute for Advanced Study at Princeton in 1952 to work with Robert Oppenheimer (and meet Albert Einstein). Also, in 1954, he became a research associate at the University of Chicago and finally a professor there in 1958. He received his American citizenship in 1970.

Peter Freund, his colleague in Chicago, described Nambu as a person of incredible serenity in his 2007 book A Passion for Discovery. Through the work and actions of the biggest physicists of the mid-19th century, the book fleshes out the culture of physics research and how it was shaped by communism and fascism. Freund himself emigrated from Romania to the US in the 1960s to escape the dictatorial madness of Ceausescu, a narrative arc that is partially reflected in Nambu’s life. After receiving his bachelor’s degree from the University of Tokyo in 1942, Nambu was drafted into the army and witnessed the infamous firebombing of Tokyo and was in Japan when Hiroshima and Nagasaki were bombed.

The destructive violence of the war that Nambu studied through is mirrored in the creative energies of the high-energy universe whose mysteries Nambu and his peers worked to decrypt. It may have been a heck of a life to live through but the man himself had only a “fatalistic calm”, as Freund wrote, to show for it. Was he humbled by his own discoveries? Perhaps, but what we do know is that he wanted to continue doing what he did until the day he died.

Where does the Higgs boson come from?

When the Chelyabinsk meteor – dubbed Chebarkul – entered Earth’s atmosphere at around 17 km/s, it started to heat up due to friction. After a point, cracks already present on the chunk of rock weighing 9,000-tonnes became licensed to widen and eventually split off Chebarkul into smaller parts.

While the internal structure of Chebarkul was responsible for where the cracks widened and at what temperature and other conditions, the rock’s heating was the tipping point. Once it got hot enough, its crystalline structure began to disintegrate in some parts.

Spontaneous symmetry-breaking

About 13.75 billion years ago, this is what happened to the universe. At first, there was a sea of energy, a symmetrically uniform block. Suddenly, this block was rapidly exposed to extreme heat. Once it hit about 10¹⁵ kelvin – 173 billion times hotter than our Sun’s surface – the block disintegrated into smaller packets called particles. Its symmetry was broken. The Big Bang had happened.

The Big Bang splashed a copious amount of energy across the universe, whose residue is perceivable as the CMBR.

Quickly, the high temperature fell off, but the particles couldn’t return to their original state of perfect togetherness. The block was broken forever, and the particles now had to fend for themselves. There was a disturbance, or perturbations, in the system, and the forces started to act. Physicists today call this the Nambu-Goldstone (NG) mode, named for Jeffrey Goldstone and Yoichiro Nambu.

In the tradition of particle physics treating with everything in terms of particles, the forces in the NG mode were characterised in terms of NG bosons. The exchange of these bosons between two particles meant they were exchanging forces. Since each boson is also a particle, a force can be thought of as the exchange of energy between two particles or bodies.

This is just like the concept of phonons in condensed matter physics: when atoms part of a perfectly arranged array vibrate, physicists know they contain some extra energy that makes them restless. They isolate this surplus in the form of a particle called a phonon, and address the entire array’s surplus in terms of multiple phonons. So, as a series of restlessness moves through the solid, it’ll be like a sound wave moving through it. Simplifies the math.

Anyway, the symmetry-breaking also gave rise to some fundamental forces. They’re called ‘fundamental’ because of their primacy, and because they’re still around. They were born because the disturbances in the energy block, encapsulated as the NG bosons, were interacting with an all-pervading background field called the Higgs field.

The Higgs field has four components, two charged and two uncharged. Another, more common, example of a field is the electric field, which has two components: some strength at a point (charged) and the direction of the strength at that point (neutral). Components of the Higgs field perturbed the NG bosons in a particular way to give rise to four fundamental forces, one for each component.

So, just like in Chebarkul’s case, where its internal structure dictated where the first cracks would appear, in the block’s case, the heating had disturbed the energy block to awaken different “cracks” at different points.

The Call of Cthulhu

The first such “crack” to be born was the electroweak force. As the surroundings of these particles continued to cool, the electroweak force split into two: electromagnetic (eM) and weak forces.

The force-carrier for the eM force is called a photon. Photons can exist at different energies, and at each energy-level, they have a corresponding frequency. If a photon happens to be in the “visible range” of energy-levels, then each frequency shows itself as a colour. And so on…

The force-carriers of the weak forces are the W+, W-, and Z bosons. At the time the first W/Z bosons came to life, they were massless. We know now because of Einstein’s mass-energy equivalence that this means the bosons had no energy. How were they particulate, then?

Imagine an auditorium where an important lecture’s about to be given. You get there early, your friend is late, and you decide to reserve a seat for her. Then, your friend finally arrives 10 minutes after the lecture’s started and takes her seat. In this scenario, after your arrival, the seat was there all along as ‘friend’s seat’, even though your friend took her time to get there.

Similarly, the W/Z bosons, which became quite massive later on, were initially massless. They had to have existed when the weak force came to life, if only to account for a new force that had been born. The debut of massiveness happened when they “ate” the NG bosons – the disturbed block’s surplus energy – and became very heavy.

Unfortunately for them, their snacking was irreversible. The W/Z bosons couldn’t regurgitate the NG bosons, so they were doomed to be forever heavy and, consequently, short-ranged. That’s why the force that they mediate is called the weak force: because it acts over very small distances.

You’ll notice that the W+, W-, and Z bosons make up for only three components of the Higgs field. What about the fourth component?

Enter: Higgs boson

That’s the Higgs boson. And now, getting closer to pinning down the Higgs boson means we’re also getting closer to pinning down the Higgs mechanism as valid, a quantum mechanical formulation within which we understand the behaviours of these particles and forces. This formulation is called the Standard Model.

(This blog post first appeared at The Copernican on March 8, 2013.)

The Symmetry Incarnations – Part I

Symmetry in nature is a sign of unperturbedness. It means nothing has interfered with a natural process, and that its effects at each step are simply scaled-up or scaled-down versions of each other. For this reason, symmetry is aesthetically pleasing, and often beautiful. Consider, for instance, faces. Symmetry of facial features about the central vertical axis is often translated as the face being beautiful – not just by humans but also monkeys.

However, this is just an example of one of the many forms of symmetry’s manifestation. When it involves geometric features, it’s a case of geometrical symmetry. When a process occurs similarly both forward and backward in time, it is temporal symmetry. If two entities that don’t seem geometrically congruent at first sight rotate, move or scale with similar effects on their forms, it is transformational symmetry. A similar definition applies to all theoretical models, musical progressions, knowledge, and many other fields besides.

Symmetry-breaking

One of the first (postulated) instances of symmetry is said to have occurred during the Big Bang, when the observable universe was born. A sea of particles was perturbed 13.75 billion years ago by a high-temperature event, setting up anecdotal ripples in their system, eventually breaking their distribution in such a way that some particles got mass, some charge, some a spin, some all of them, and some none of them. In physics, this event is called spontaneous, or electroweak, symmetry-breaking. Because of the asymmetric properties of the resultant particles, matter as we know it was conceived.

Many invented scientific systems exhibit symmetry in that they allow for the conception of symmetry in the things they make possible. A good example is mathematics – yes, mathematics! On the real-number line, 0 marks the median. On either sides of 0, 1 and -1 are equidistant from 0, 5,000 and -5,000 are equidistant from 0; possibly, ∞ and -∞ are equidistant from 0. Numerically speaking, 1 marks the same amount of something that -1 marks on the other side of 0. Not just that, but also characterless functions built on this system also behave symmetrically on either sides of 0.

To many people, symmetry evokes an image of an object that, when cut in half along a specific axis, results in two objects that are mirror-images of each other. Cover one side of your face and place the other side against a mirror, and what a person hopes to see is the other side of the face – despite it being a reflection (interestingly, this technique was used by neuroscientist V.S. Ramachandran to “cure” the pain of amputees when they tried to move a limb that wasn’t there). Like this, there are symmetric tables, chairs, bottles, houses, trees (although uncommon), basic geometric shapes, etc.

mirror_technique — A demonstration of V.S. Ramachandran’s mirror-technique

Natural symmetry

Symmetry at its best, however, is observed in nature. Consider germination: when a seed grows into a small plant and then into a tree, the seed doesn’t experiment with designs. The plant is not designed differently from the small tree, and the small tree is not designed differently from the big tree. If a leaf is given to sprout from the mineral-richest node on the stem then it will; if a branch is given to sprout from the mineral-richest node on the trunk then it will. So, is mineral-deposition in the arbor symmetric? It should be if their transportation out of the soil and into the tree is radially symmetric. And so forth…

At times, repeated gusts of wind may push the tree to lean one way or another, shadowing the leaves from against the force and keeping them form shedding off. The symmetry is then broken, but no matter. The sprouting of branches from branches, and branches from those branches, and leaves from those branches all follow the same pattern. This tendency to display an internal symmetry is characterized as fractalization. A well-known example of a fractal geometry is the Mandelbrot set, shown below.

If you want to interact with a Mandelbrot set, check out this magnificent visualization by Paul Neave. You can keep zooming in, but at each step, you’ll only see more and more Mandelbrot sets. Unfortunately, this set is one of a few exceptional sets that are geometric fractals.

Meta-geometry & Mulliken symbols

Now, it seems like geometric symmetry is the most ubiquitous and accessible example to us. Let’s take it one step further and look at the “meta-geometry” at play when one symmetrical shape is given an extra dimension. For instance, a circle exists in two dimensions; its three-dimensional correspondent is the sphere. Through such an up-scaling, we’re ensuring that all the properties of a circle in two dimensions stay intact in three dimensions, and then we’re observing what the three-dimensional shape is.

A circle, thus, becomes a sphere; a square becomes a cube; a triangle becomes a tetrahedron (For those interested in higher-order geometry, the tesseract, or hypercube, may be of special interest!). In each case, the 3D shape is said to have been generated by a 2D shape, and each 2D shape is said to be the degenerate of the 3D shape. Further, such a relationship holds between corresponding shapes across many dimensions, with doubly and triply degenerate surfaces also having been defined.

Hypercube — The tesseract (a.k.a. hypercube)

Obviously, there are different kinds of degeneracy, 10 of which the physicist Robert S. Mulliken identified and laid out. These symbols are important because each one defines a degree of freedom that nature possesses while creating entities, and this includes symmetrical entities as well. In other words, if a natural phenomenon is symmetrical in n dimensions, then the only way it can be symmetrical in n+1 dimensions also is by transforming through one or many of the degrees of freedom defined by Mulliken.

rober_sanderson_mulliken — Robert S. Mulliken (1896-1986)

Anastomosis

Now, if that’s the case – if symmetry is held desirable by nature, if it is not a haphazard occurrence but one that is well orchestrated if given a chance to be – why don’t we see symmetry everywhere? Why is natural symmetry broken? Is all of the asymmetry that we’re seeing today the consequence of that electro-weak symmetry-breaking phenomenon? It can’t be because natural symmetry is still prevalent. Is it then implied that what symmetry we’re observing today exists in the “loopholes” of that symmetry-breaking? Or is it all part of the natural order of things, a restoration of past capabilities?

lhc_tunnel — One of the earliest symptoms of symmetry-breaking was the appearance of the Higgs mechanism, which gave mass to some particles but not some others. The hunt for it’s residual particle, the Higgs boson, was spearheaded by the Large Hadron Collider (LHC) at CERN.

The last point – of natural order – is allegorical with, as well as is exemplified by, a geological process called anastomosis. This property, commonly of quartz crystals in metamorphic regions of Earth’s crust, allows for mineral veins to form that lead to shearing stresses between layers of rock, resulting in fracturing and faulting. Philosophically speaking, geological anastomosis allows for the displacement of materials from one location and their deposition in another, thereby offsetting large-scale symmetry in favor of the prosperity of microstructures.

Anastomosis, in a general context, is defined as the splitting of a stream of anything only to rejoin sometime later. It sounds really simple but it is a phenomenon exceedingly versatile, if only because it happens in a variety of environments and for an equally large variety of purposes. For example, consider Gilbreath’s conjecture. It states that each series of prime numbers to which the forward difference operator has been applied always starts with 1. To illustrate:

2 3 5 7 11 13 17 19 23 29 … (prime numbers)

Applying the operator once: 1 2 2 4 2 4 2 4 6 … (successive differences between numbers)
Applying the operator twice: 1 0 2 2 2 2 2 2 …
Applying the operator thrice: 1 2 0 0 0 0 0 …
Applying the operator for the fourth time: 1 2 0 0 0 0 0 …

And so forth.

If each line of numbers were to be plotted on a graph, moving upwards each time the operator is applied, then a pattern for the zeros emerges, shown below.

This pattern is called that of the stunted trees, as if it were a forest populated by growing trees with clearings that are always well-bounded triangles. The numbers from one sequence to the next are anastomosing, only to come close together after every five lines! Another example is the vein skeleton on a hydrangea leaf. Both the stunted trees and the hydrangea veins patterns can be simulated using the rule-90 simple cellular automaton that uses the exclusive-or (XOR) function.

Nambu-Goldstone bosons

Now, what does this have to do with symmetry, you ask? While anastomosis may not have a direct relation with symmetry and only a tenuous one with fractals, its presence indicates a source of perturbation in the system. Why else would the streamlined flow of something split off and then have the tributaries unify, unless possibly to reach out to richer lands? Either way, anastomosis is a sign of the system acquiring a new degree of freedom. By splitting a stream with x degrees of freedom into two new streams each with x degrees of freedom, there are now more avenues through which change can occur.

water_entrainment — Water entrainment in an estuary is an example of a natural asymptote or, in other words, a system’s “yearning” for symmetry

Particle physics simplies this scenario by assigning all forces and amounts of energy a particle. Thus, a force is said to be acting when a force-carrying particle is being exchanged between two bodies. Since each degree of freedom also implies a new force acting on the system, it wins itself a particle, actually a class of particles called the Nambu-Goldstone (NG) bosons. Named for Yoichiro Nambu and Jeffrey Goldstone, the particle’s existence’s hypothesizers, the presence of n NG bosons in a system means that, broadly speaking, the system has n degrees of freedom.

nambu_goldstone — Jeffrey Goldstone (L) & Yoichiro Nambu

How and when an NG boson is introduced into a system is not yet a well-understood phenomenon theoretically, let alone experimentally! In fact, it was only recently that a mathematical model was developed by a theoretical physicist at UCal-Berkeley, Haruki Watanabe, capable of predicting how many degrees of freedom a complex system could have given the presence of a certain number of NG bosons. However, at the most basic level, it is understood that when symmetry breaks, an NG boson is born!

The asymmetry of symmetry

In other words, when asymmetry is introduced in a system, so is a degree of freedom. This seems only intuitive. At the same time, you’d think the axiom is also true: that when an asymmetric system is made symmetric, it loses a degree of freedom – but is this always true? I don’t think so because, then, it would violate the third law of thermodynamics (specifically, the Lewis-Randall version of its statement). Therefore, there is an inherent irreversibility, an asymmetry of the system itself: it works fine one way, it doesn’t work fine another – just like the split-off streams, but this time, being unable to reunify properly. Of course, there is the possibility of partial unification: in the case of the hydrangea leaf, symmetry is not restored upon anastomosis but there is, evidently, an asymptotic attempt.

broken_mirror — Each piece of a broken mirror-glass reflects an object entirely, shedding all pretensions of continuity. The most intriguing mathematical analogue of this phenomenon is the Banach-Tarski paradox, which, simply put, takes symmetry to another level.

However, it is possible that in some special frames, such as in outer space, where the influence of gravitational forces is weak if not entirely absent, the restoration of symmetry may be complete. Even though the third law of thermodynamics is still applicable here, it comes into effect only with the transfer of energy into or out of the system. In the absence of gravity (and, thus, friction), and other retarding factors, such as distribution of minerals in the soil for acquisition, etc., symmetry may be broken and reestablished without any transfer of energy.

The simplest example of this is of a water droplet floating around. If a small globule of water breaks away from a bigger one, the bigger one becomes spherical quickly; when the seditious droplet joins with another globule, that globule also reestablishes its spherical shape. Thermodynamically speaking, there is mass transfer, but at (almost) 100% efficiency, resulting in no additional degrees of freedom. Also, the force at play that establishes sphericality is surface tension, through which a water body seeks to occupy the shape that has the lowest volume for the correspondingly highest surface area (notice how the shape is incidentally also the one with the most axes of symmetry, or, put another way, no redundant degrees of freedom? Creating such spheres is hard!).

A godless, omnipotent impetus

Perhaps the explanation of the roles symmetry assumes seems regressive: every consequence of it is no consequence but itself all over again (self-symmetry – there, it happened again). This only seems like a natural consequence of anything that is… well, naturally conceived. Why would nature deviate from itself? Nature, it seems, isn’t a deity in that it doesn’t create. It only recreates itself with different resources, lending itself and its characteristics to different forms.

A mountain will be a mountain to its smallest constituents, and an electron will be an electron no matter how many of them you bring together at a location. But put together mountains and you have ranges, sub-surface tectonic consequences, a reshaping of volcanic activity because of changes in the crust’s thickness, and a long-lasting alteration of wind and irrigation patterns. Bring together a unusual number of electrons to make up a high-density charge, and you have a high-temperature, high-voltage volume from which violent, permeating discharges of particles could occur – i.e., lightning. Why should stars, music, light, radioactivity, politics, manufacturing or knowledge be any different?

With this concludes the introduction to symmetry. Yes, there is more, much more…

A simplification of superfluidity

“Once people tell me what symmetry the system starts with and what symmetry it ends up with, and whether the broken symmetries can be interchanged, I can work out exactly how many bosons there are and if that leads to weird behavior or not,” Murayama said. “We’ve tried it on more than 10 systems, and it works out every single time.”

– Haruki Watanabe, co-author of the paper

To those who’ve followed studies on superfluidity and spontaneous symmetry-breaking, a study by Hitoshi Murayama and Haruki Watanabe at UC, Berkeley, will come as a boon. It simplifies our understanding of symmetry-breaking for practical considerations by unifying the behaviour of supercooled matter – such as BEC and superfluidity – and provides a workable formula to derive the number of Nambu-Goldstone bosons given the symmetry of the system during a few phases!

This is the R&D article that serves as a lead-in into the issue.

This is a primer on spontaneous symmetry-breaking (and the origins of the Higgs boson).

Finally, and importantly, the pre-print paper (from arXiv) can be viewed here. Caution: don’t open the paper if you’re not seriously good at math.