We now see that what looks like a very simple collection of symmetries of the square table – the four rotations {90°, 180°, 270°, 0°} – actually has a lot of inner structure, or rules for how the members of the set can interact.
First of all, we can compose any two symmetries (that is, apply them one after another).
Second, there is a special symmetry, the identity. In our example, this is the rotation by 0 degrees. If we compose it with any other symmetry, we get back the same symmetry. For example,
Third, for any symmetry S, there is the inverse symmetry S′ such that the composition of S and S′ is the identity.
And now we come to the main point: the set of rotations along with these three structures comprise an example of what mathematicians call a group.
The symmetries of any other object also constitute a group, which in general has more elements – possibly, infinitely many.4
Let’s see how this works in the case of a round table. Now that we have gained some experience, we can see right away that the set of all symmetries of the round table is just the set of all possible rotations (not just by multiples of 90 degrees), and we can visualize it as the set of all points of a circle.
Each point on this circle corresponds to an angle between 0 and 360 degrees, representing the rotation of the round table by this angle in the counterclockwise direction. In particular, there is a special point corresponding to rotation by 0 degrees. It is marked on the picture below, together with another point corresponding to rotation by 30 degrees.
We should not think of the points of this circle as points of the round table, though. Rather, each point of the circle represents a particular rotation of the round table. Note that the round table does not have a preferred point, but our circle does; namely, the one corresponding to rotation by 0 degrees.5
Now let’s see if the above three structures can be applied to the set of points of the circle.
First, the composition of two rotations, by φ1 and φ2 degrees, is the rotation by φ1 + φ2 degrees. If φ1 + φ2 is greater than 360, we simply subtract 360 from it. In mathematics, this is called addition modulo 360. For example, if φ1 = 195 and φ2 = 250, then the sum of the two angles is 445, and the rotation by 445 degrees is the same as the rotation by 85 degrees. So, in the group of rotations of the round table we have
Second, there is a special point on the circle corresponding to the rotation by 0 degrees. This is the identity element of our group.
Third, the inverse of the counterclockwise rotation by φ degrees is the counterclockwise rotation by (360−φ) degrees, or equivalently, clockwise rotation by φ degrees (see the drawing).
Thus, we have described the group of rotations of the round table. We will call it the circle group. Unlike the group of symmetries of the square table, which has four elements, this group has infinitely many elements because there are infinitely many angles between 0 and 360 degrees.
We have now put our intuitive understanding of symmetry on firm theoretical ground – indeed, we’ve turned it into a mathematical concept. First, we postulated that a symmetry of a given object is a transformation that preserves it and its properties. Then we made a decisive step: we focused on the set of all symmetries of a given object. In the case of a square table, this set consists of four elements (rotations by multiples of 90 degrees); in the case of a round table, it is an infinite set (of all points on the circle). Finally, we described the neat structures that this set of symmetries always possesses: any two symmetries can be composed to produce another symmetry, there exists the identical symmetry, and for each symmetry there exists its inverse. (The composition of symmetries also satisfies the associativity property described in endnote 4.) Thus, we came to the mathematical concept of a group.
A group of symmetries is an abstract object that is quite different from the concrete object we started with. We cannot touch or hold the set of symmetries of a table (unlike the table itself), but we can imagine it, draw its elements, study it, talk about it. Each element of this abstract set has a concrete meaning, though: it represents a particular transformation of a concrete object, its symmetry.
Mathematics is about the study of such abstract objects and concepts.
Experience shows that symmetry is an essential guiding principle for the laws of nature. For example, a snowflake forms a perfect hexagonal shape because that turns out to be the lowest energy state into which crystallized water molecules are forced. The symmetries of the snowflake are rotations by multiples of 60 degrees; that is, 60, 120, 180, 240, 300, and 360 (which is the same as 0 degrees). In addition, we can “flip” the snowflake along each of the six axes corresponding to those angles. All of these rotations and flips preserve the shape and position of the snowflake, and hence they are its symmetries.*
In the case of a butterfly, flipping it turns it upside down. Since it has legs on one side, the flip is not, strictly speaking, a symmetry of the butterfly. When we say that a butterfly is symmetrical, we are talking about an idealized version of it, where its front and back are exactly the same (unlike those of an actual butterfly). Then the flip exchanging the left and the right wings becomes a symmetry. (Alternatively, we can imagine exchanging the wings without turning the butterfly upside down.)
This brings up an important point: there are many objects in nature whose symmetries are approximate. A real-life table is not perfectly round or perfectly square, a live butterfly has an asymmetry between its front and back, and a human body is not fully symmetrical. However, even in this case it turns out to be useful to consider their abstract, idealized versions, or models – a perfectly round table or an image of the butterfly in which we don’t distinguish between the front and the back. We then explore the symmetries of these idealized objects and adjust whatever inferences we can make from this analysis to account for the difference between a real-life object and its model.
This is not to say that we do not appreciate asymmetry; we do, and we often find beauty in it. But the main point of the mathematical theory of symmetry is not aesthetic. It is to formulate the concept of symmetry in the most general, and hence inevitably most abstract, terms so that it could be applied in a unified fashion in different domains, such as geometry, number theory, physics, chemistry, biology, and so on. Once we develop such a theory, we can also talk about the mechanisms of symmetry breaking – viewing asymmetry as emergent, if you will. For example, elementary particles acquire masses because the so-called gauge symmetry they obey (which will be discussed in Chapter 16) gets broken. This is facilitated by the Higgs boson, an elusive particle recently discovered at the Large Hadron Collider under the city of Geneva.6 The study of such mechanisms of symmetry breaking yields invaluable insights into the behavior of the fundamental blocks of nature.
I’d like to point out some of the basic qualities of the abstract theory of symmetry because this is a good illustration of why mathematics is important.
The first is universality. The circle group is not only the group of symmetries of a round table, but also of all other round objects, like a glass, a bottle, a column, and so forth. In fact, to say that a given object is round is the same as to say that its group of symmetries is the circle group. This is a powerful statement: we realize that we can describe an important attribute of an object (“being round”) by describing its symmetry group (the circle). Likewise, “being square” means that the group of symmetries is the group of four elements described above. In other words, the same abstract mathematical object (such as the circle group) serves many different concrete objects, and it points to universal properties that they all have in common (such as roundness).7
The second is objectivity. The concept of a group, for example, is independent of our interpretation. It means the same thing to anyone who learns it. Of course, in order to understand it, one has to know the language in which it is expressed, that is, mathematical language. But anyone can learn this language. Likewise, if you want to understand the meaning of René Descartes�
� sentence “Je pense, donc je suis,” you need to know French (at least, those words that are used in this sentence) – but anyone can learn it. However, in the case of the latter sentence, once we understand it, different interpretations of it are possible. Also, different people may agree or disagree on whether a particular interpretation of this sentence is true or false. In contrast, the meaning of a logically consistent mathematical statement is not subject to interpretation.8 Furthermore, its truth is also objective. (In general, the truth of a particular statement may depend on the system of axioms within which it is considered. However, even then, this dependence on the axioms is also objective.) For example, the statement “the group of symmetries of a round table is a circle” is true to anyone, anywhere, at any time. In other words, mathematical truths are the necessary truths. We will talk more about this in Chapter 18.
The third, closely related, quality is endurance. There is little doubt that the Pythagorean theorem meant the same thing to the ancient Greeks as it does to us today, and there is every reason to expect that it will mean the same thing to anyone in the future. Likewise, all true mathematical statements we talk about in this book will remain true forever.
The fact that such objective and enduring knowledge exists (and moreover, belongs to all of us) is nothing short of a miracle. It suggests that mathematical concepts exist in a world separate from the physical and mental worlds – which is sometimes referred to as the Platonic world of mathematics (we will talk more about that in the closing chapter). We still don’t fully understand what it is and what drives mathematical discovery. But it’s clear that this hidden reality is bound to play a larger and larger role in our lives, especially with the advent of new computer technologies and 3D printing.
The fourth quality is relevance of mathematics to the physical world. For example, a lot of progress has been made in quantum physics in the past fifty years because of the application of the concept of symmetry to elementary particles and interactions between them. From this point of view, a particle, such as an electron or a quark, is like a round table or a snowflake, and its behavior is very much determined by its symmetries. (Some of these symmetries are exact, and some are approximate.)
The discovery of quarks is a perfect example of how this works. Reading the books Evgeny Evgenievich gave me, I learned that at the root of the Gell-Mann and Ne’eman classification of hadrons that we talked about in the previous chapter is a symmetry group. This group had been previously studied by mathematicians – who did not anticipate any connections to subatomic particles whatsoever. The mathematical name for it is SU(3). Here S and U stand for “special unitary.” This group is very similar in its properties to the group of symmetries of the sphere, which we will talk about in detail in Chapter 10.
Mathematicians had previously described the representations of the group SU(3), that is, different ways that the group SU(3) can be realized as a symmetry group. Gell-Mann and Ne’eman noticed the similarity between the structure of these representations and the patterns of hadrons that they had found. They used this information to classify hadrons.
The word “representation” is used in mathematics in a particular way, which is different from its more common usage. So let me pause and explain what this word means in the present context. Perhaps, it would help if I first give an example. Recall the group of rotations of a round table discussed above, the circle group. Now imagine extending the tabletop infinitely far in all directions. This way we obtain an abstract mathematical object: a plane. Each rotation of the tabletop, around its center, gives rise to a rotation of this plane around the same point. Thus, we obtain a rule that assigns a symmetry of this plane (a rotation) to every element of the circle group. In other words, each element of the circle group may be represented by a symmetry of the plane. For this reason mathematicians refer to this process as a representation of the circle group.
Now, the plane is two-dimensional because it has two coordinate axes and hence each point has two coordinates:
Therefore, we say that we have constructed a “two-dimensional representation” of the group of rotations. It simply means that each element of the group of rotations is realized as a symmetry of a plane.9
There are also spaces of dimension greater than two. For example, the space around us is three-dimensional. That is to say, it has three coordinate axes, and so in order to specify a position of a point, we need to specify its three coordinates (x, y, z) as shown on this picture:
We cannot imagine a four-dimensional space, but mathematics gives us a universal language that allows us to talk about spaces of any dimension. Namely, we represent points of the four-dimensional space by quadruples of numbers (x, y, z, t), just like points of the three-dimensional space are represented by triples of numbers (x, y, z). In the same way, we represent points of an n-dimensional space, for any natural number n, by n-tuples of numbers. If you have used a spreadsheet program, then you have encountered such n-tuples: they appear as rows in a spreadsheet, each of the n numbers corresponding to a particular attribute of the stored data. Thus, every row in a spreadsheet refers to a point in an n-dimensional space. (We will talk more about spaces of various dimensions in Chapter 10.)
If each element of a group can be realized, in a consistent manner,10 as a symmetry of an n-dimensional space, then we say that the group has an “n-dimensional representation.”
It turns out that a given group can have representations of different dimensions. The reason elementary particles can be assembled in families of 8 and 10 particles is that the group SU(3) is known to have an 8-dimensional and a 10-dimensional representations. The 8 particles of each octet constructed by Gell-Mann and Ne’eman (like the one shown on the diagram in the previous chapter) are in one-to-one correspondence with the 8 coordinate axes of an 8-dimensional space which is a representation of SU(3). The same goes for the decuplet of particles. (But particles cannot be assembled in families of, say, 7 or 11 particles because mathematicians have proved that the group SU(3) has no 7- or 11-dimensional representations.)
At first, this was just a convenient way to combine the particles with similar properties. But then Gell-Mann went further. He postulated that there was a deep reason behind this classification scheme. He basically said that this scheme works so well because hadrons consist of smaller particles – sometimes two and sometimes three of them – the quarks. A similar proposal was made independently by physicist George Zweig (who called the particles “aces”).
This was a stunning proposal. Not only did it go against the popular belief at the time that protons and neutrons as well as other hadrons were indivisible elementary particles, these new particles were supposed to have electric charges that were fractions of the charge of the electron. This was a startling prediction because no one had seen such particles before. Yet, quarks were soon found experimentally, and as predicted, they had fractional electric charges!
What motivated Gell-Mann and Zweig to predict the existence of quarks? Mathematical theory of representations of the group SU(3). Specifically, the fact that the group SU(3) has two different 3-dimensional representations. (Actually, that’s the reason there is a “3” in this group’s name.) Gell-Mann and Zweig suggested that these two representations should describe two families of fundamental particles: 3 quarks and 3 anti-quarks. It turns out that the 8- and 10-dimensional representations of SU(3) can be built from the 3-dimensional ones. And this gives us a precise blueprint for how to construct hadrons from quarks – just like in Lego.
Gell-Mann named the 3 quarks “up,” “down,” and “strange.”11 A proton consists of two up quarks and one down quark, whereas a neutron consists of two down quarks and one up quark, as we saw on the pictures in the previous chapter. Both of these particles belong to the octet shown on the diagram in the previous chapter. Other particles from this octet involve the strange quark as well as the up and down quarks. There are also octets that consist of particles that are composites of one quark and one anti-quark.
Imagine: a seemingly esoteric mathematical theory empowered us to get to the heart of the building blocks of nature. How can we not be enthralled by the magic harmony of these tiny blobs of matter, not marvel at the capacity of mathematics to reveal the inner workings of the universe?
As the story goes, Albert Einstein’s wife Elsa remarked, upon hearing that a telescope at the Mount Wilson Observatory was needed to determine the shape of space-time: “Oh, my husband does this on the back of an old envelope.”
Physicists do need expensive and sophisticated machines such as the Large Hadron Collider in Geneva, but the amazing fact is that scientists like Einstein and Gell-Mann have used what looks like the purest and most abstract mathematical knowledge to unlock the deepest secrets of the world around us.
Regardless of who we are and what we believe in, we all share this knowledge. It brings us closer together and gives a new meaning to our love for the universe.
Love and Math Page 3