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.

[caption id="attachment_24154" align="aligncenter" width="480"] A demonstration of V.S. Ramachandran's mirror-technique[/caption]

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.

[caption id="attachment_24152" align="aligncenter" width="190"] The tesseract (a.k.a. hypercube)[/caption]

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.

[caption id="attachment_24151" align="aligncenter" width="162"] Robert S. Mulliken (1896-1986)[/caption]

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 a coincidental design; rather, it's very well-defined.

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?

[caption id="attachment_24149" align="aligncenter" width="640"] 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.[/caption]

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.

[caption id="attachment_24138" align="aligncenter" width="365"] Water entrainment in an estuary is an example of a natural asymptote or, in other words, a system's "yearning" for symmetry[/caption]

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.

[caption id="attachment_24137" align="aligncenter" width="275"] Jeffrey Goldstone (L) & Yoichiro Nambu[/caption]

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.

[caption id="attachment_24136" align="aligncenter" width="373"] 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.[/caption]

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...

[caption id="attachment_24131" align="aligncenter" width="560"] xkcd #849[/caption]

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.

So where does the Higgs boson come from?

(Before we begin: all uses of the term "vacuum" in this discussion do not denote the absolute nothingness we know it colloquially to be. In quantum field theory, a vacuum is a space that contains "fleeting electromagnetic waves, within which things pop in and out of existence.")

[caption id="attachment_23450" align="alignright" width="300"] "Noob! I am your father!"[/caption]

When Alderaan was annihilated by the Death Star, Obi-Wan Kenobi sensed it even though he was not in the vicinity of the Empire's fleet. He said, "I feel a great disturbance in the Force, as if millions of voices suddenly cried out in terror, and were suddenly silenced." Now, imagine a place of absolute nothingness, also called a zero-point field, within such a Force, that contains no physical particles or energy quantizations. The best way to characterize this system would be in terms of its symmetry: that, in any direction starting from the centre of this vacuum, the probability of an event occurring is equal and zero. Now, imagine sticking your finger into such a system.

The first thing that happens is the symmetry breaks. A mass excitation is said to have been introduced in the zero-point field and that, starting from the centre and moving in any direction, the chances of finding a finger in one particular direction are going to be very good. Therefore, in anticipation of the finger being introduced into the system, the equations that define the system in the beginning must be able to account for symmetry-breaking. Now, eliminate any fingers and imagine that instead of a zero-point field, there is a continuous sea of particles with an unbroken energy symmetry, i.e., the chances of finding a perturbation in any direction is zero and that there is no mass excitation in the spectrum of possible excitations.

[caption id="attachment_23452" align="aligncenter" width="305"] A radial probability-cloud showing the probability decreasing from 1 (at the centre) to 0 (moving outward)[/caption]

While the system is in this phase, instantaneously expose it to an extremely high-temperature surrounding (t→10¹⁵ K, or approaching a thousand-trillion degrees Celsius). The sea of particles begins to vibrate because of the excess energy that has been imparted to their system, but the symmetry isn't yet broken. Suddenly, at 10¹⁵ K, the symmetry breaks spontaneously—which is thermodynamic lingo for "irreversibly"—but there is still no mass excitation.

That is because of Goldstone's theorem, formulated by Jeffrey Goldstone, which states that spontaneous symmetry breaking will occur only with massless excitations. The loss of symmetry can be interpreted as a gain of perturbations, as if a long-wavelength wave has been introduced as a disturbance into the system, giving rise to a field that acquires a non-invariant expectation value, and not necessarily corresponding to a vacuum expectation.

[caption id="attachment_23453" align="aligncenter" width="600"] Illustration of a standing wave, with amplitude described along the y-axis and its frequency, along the x-axis[/caption]

Simply put, the sea of particles can be thought of as a quantum fluid, or an infinite fabric. Beyond the critical temperature, the energy excitations of the particles pushes them to fall out of order, like a group of prisoners in a prison where the conditions are steadily getting worse: there will be a riot at some point, and the condition that precipitates the riot will have induced criticality in the system of prisoners. Just like the universe started to cool a few fractions-of-a-second after the Big Bang, a rollback of the adverse conditions will induce a gradual return to normalcy in the prison—a long-wavelength, low-frequency return to square one. At the same time, the guards remain on alert for small and localized disruptions because there is a non-zero expectation value (of an event).

[caption id="attachment_23454" align="aligncenter" width="640"] A history of the Big Bang: after one-hundred-billionth of a second, the symmetry has been broken to differentiate the electromagnetic and weak forces at an energy of 1 Tev. The LHC has been recreating just this situation by smashing together two beams of protons, each of 3.5 TeV, to reunify them as the electroweak force and look for the Higgs boson.[/caption]

For order to have resulted after the symmetry is broken, an order parameter is introduced. This is because, when the system was symmetric, the uniform probability eliminated the need to characterize any variations simply because the variations weren't there. After the spontaneous symmetry breaking, the system is said to be ordered in terms of the order parameter, although this is a different kind of order: one where subsystems can interact with excitations in the background field—the Force—to give rise to frames of reference (FoR). FoRs are localizations of activity that are logical enough to become something against which other such activities can be compared. (Notice now that the variations of the order parameter in the system are symmetric.)

The gain of perturbations in the system is understood in terms of the exchange of massless particles called the Nambu-Goldstone bosons, whose existence was postulated by Goldstone and Yoichiro Nambu, and the system is said to be in the Nambu-Goldstone mode. Such bosons arise out of the perturbing energy of the surrounding heat-reservoir to mediate the perturbing force. At the same time, the symmetry-breaking should've resulted in the formation of two "pieces" from one. These are two of the four fundamental forces of nature: the electromagnetic force and the weak force, the latter being responsible for radioactivity and beta decay. They are forces that are long-term manifestations of energy-transfer.

(The other two fundamental forces are gravitation and the strong nuclear force. Gravitational forces are mediated by the Higgs boson, whose origins we're exploring, and the strong force is mediated by gluons, which bind to quarks, other gluons and anti-gluons. The strong force holds the protons and neutrons in a nucleus together.)

[caption id="attachment_23455" align="aligncenter" width="275"] Jeffrey Goldstone and Yoichiro Nambu[/caption]

Just like the Nambu-Goldstone bosons are carriers of the perturbing force, two other kinds of bosons are brought into existence to carry the electromagnetic and weak forces: these are the massless photons (for the Em force) and the initially-massless W^±/Z bosons (for the weak force). The debut of massiveness, or the possession of mass, happens when the W^± and Z bosons "eat" the Nambu-Goldstone bosons and become extremely massive, in the process largely reducing the range of the weak force (hence, the name). The mass they acquire is of the non-vanishing sort, which means they can't lose it: the consumption is permanent. The process as a whole is known as the Higgs mechanism, and here we finally come to the Higgs boson, or the God particle.

Just as the original Nambu-Goldstone bosons arose out of an interaction between the high temperature and the background vacuum—again, the Force—so did the W^± and Z bosons arise out of the interaction between the Nambu-Goldstone bosons and what is called the Higgs field: a quantum field that pervades the entire universe and could be imagined to be the Force itself. Mathematically, the Higgs field has four components, two neutral and two charged just like an electric field has two components, its strength at a point and its direction at that point. The four Higgs components are all Nambu-Goldstone bosons: the two charged ones are consumed by the W⁺ and W^- bosons and one of the neutral ones is consumed by a Z boson. The remaining neutral boson had to have been consumed by a fourth boson or the quantum field would've exhibited a disturbance quite unlike what we've ever experienced.

This fourth boson is the Higgs boson—which means the fourth fundamental force is the gravitational force. The particle's existence means the presence of a Higgs field, and the presence of a Higgs field means the consumption by W^± and Z bosons of the Nambu-Goldstone bosons, and the consumption means the occurrence of a Higgs mechanism. Ultimately, the occurrence of a Higgs mechanism means its present formulation, along with that of the electroweak symmetry, holds true, firmly attesting to the Standard Model's ability to explain the dynamics of all the known subatomic particles.

[caption id="attachment_23456" align="aligncenter" width="640"] A schematic view of the LHC[/caption]