A Beautiful Question
The relation between mathematics and art has been discussed in several books already. In this book Frank Wilczek answers the Question whether Nature itself is a piece of art (he always writes Question and Nature with a capital). Frank Wilczek is theoretical physicist and 2004 Nobel prize winner (together with Gross and Politzer) for work on the strong nuclear interaction force. Physical laws were originally derived from everyday observations, but once it comes down to particle physics, observations of everyday events are not helpful anymore, and experiments are scarce because they require more and more energy. In fact, the direction is reversed and the theory is now driving the experiments. Hence it is becoming less obvious what the model should be. Theoretical physics has turned to mathematics and draws its credibility from the mathematical models that show the most symmetry. Although there is symmetry breaking in practice, but that is only putting extra conditions on the model. It turns out that it is easier to start from the general symmetric solution and add extra conditions, instead of trying to lift an unsymmetric solution into the symmetric model.
Wilczek's work as a theoretical physicist is exactly in the tension field just described. He contributed to the worldwide efforts to find an overall model that should join the quantum theory of the four fundamental forces of Nature: Maxwell's electromagnetism, strong and the weak interaction and gravitation. The fourth, gravitational force is so weak that it does not fit as well as the others do. Even though, Einstein's relativity theory was the first that lifted space-time to a higher dimension and was integrated with Maxwell's equations while quantum physics was in its infancy.
Seen in this context, it is not difficult to imagine that Wilczek sees symmetry (or supersymmetry in the most recent model) as an indication of beauty, and therefore his answer to the Question he states at the beginning is whole-heartedly super-affirmative in the end. The core of his argumentation (or meditations as he calls it) is centered around his own work, but it requires to survey many other ingredients that preceded. So he walks through the whole evolution meandering from Pythagoras to Plato to Newton to Maxwell to Einstein to finally plunge into quantum dynamics, and in particular quantum chromodynamics, his own playground. In this ocean of subatomic particles the question about beauty finally gets an answer and symmetry is born like Venus from sea foam generated by the quantum waves. With this allegory of Venus, I imitate a technique that also Wilczek uses in his book. References to work of art is introduced alongside the physical and mathematical theories, and this includes music, poetry, and paintings.
Pythagoras studied music, vibrating strings, and related ratios of numbers, already relating mathematics and art. Music was part of the quadrivium alongside arithmetic, geometry, and astronomy. So Wilczek in his meditation on Pythagoras already raises a question of beauty: why certain combinations sound harmonic and others don't? That is connected with a particular discrete set of ratios in the length of the string (like quanta). Had Pythagoras pushed a bit harder, quantum theory would have emerged much earlier.
From Plato we know the platonic solids, the five symmetrical objects that inspired Kepler to use them as a model for the solar system. This shows Kepler's belief that Nature's design is one of symmetry. The model turned out to be wrong but it is an example where the beauty (i.e., the symmetry) of the mathematics inspired a model for reality. From Plato we also remember the allegory of the cave. We are limited by our senses so that we only see a two-dimensional black-and-white image of what is in reality a very colorful three-dimensional world outside. (Wilczek elaborates on the invention of perspective as an illustration of how we manage to represent reality within the limitations of a rectangular canvas.)
From Newton we have to remember his decomposition of light into the different colors of the rainbow and of course his mechanics. Light and mass need two different approaches, one being a wave and the other being related to a material object. But light is just a particular example of an electromagnetic wave. So they obey Maxwell's electromagnetic equations. These equations are another triumph of symmetry and simplicity. Clearly a beautiful theory to describe Nature. But it does not embrace gravity that acts on particles at a distance through a void intermedium. On the other hand, electromagnetism introduces fields visualized by fieldlines, forming the fluid filling the void in between particles attracting or repulsing each other.
Maxwell is one of Wilczek's heros. It is perhaps less generally known that Maxwell has also experimented a lot with colors. Although the sunlight contains an infinity of frequencies, the human eye pics up only three components (R,G,B) and we can compose all other colors by combining these in our mind. A strange saving in the design that evolution has bestowed upon us. Certain shrimps for example register many more frequencies. We see white light by adding red green and blue, but this is quite different from the white light that is reflected by white paint where pigments absorbs frequencies that are not reflected.
When it comes to quantum theory, Wilczek reflects on the eigenmodes of a string and of other music instruments. These remind us of Pythagoras' ratios of numbers.
The discussion of Einstein's theory is relatively short. It is used to introduce the importance of local (Galilean) invariance, which is a form of symmetry. This is an unusual approach, but it is central in Wilczek's view of gravitons and metric fields.
After an intermezzo on the symmetry of carbon structures (fullerenes, nanotubes, buckminsterfullerenes), Wilczek embarks on Core Theory (a term he prefers over the usual naming Standard Model which sounds to him too common, too much like a bucketful of conventions). Arriving on his own domain, he moves to a higher level of detail, to a higher gear for the reader, and to the higher dimension of quantumchromodynamics with its many particles, For the latter purpose he associates with every spacial point a property space. Each of the four fundamental forces has its own property space. He treats in some details all the quantumchromodynamics of the three forces with the particles these require as they show up in his own work, with a triumph of the supersymmetry (SUSY) and hence the answer to the Question. He links this to the mathematics of Emma Noether's (first) theorem about the conservation of energy (but he does not explain).
It should be clear that Wilczek, although he hops through the history and he is very fluent in juggling with all the electron, lepton, gluon and whateveron particles, he is primarily interested in answering his Question and hence is looking for the beauty that becomes more and more apparent as theoretical physics evolves. Therefore, this book is not the place to learn about all the mathematics and the formulas supporting this. He assumes that this is more or less known to the reader. If your theoretical physics is a bit rusty, you will need the 57 page small print supplement providing a dictionary defining and explaining the most important "terms of art" as he calls all these physical concepts. It is a factual list arranged in two columns per page from 'absorption' to 'z-particle'. There are a few references for further reading too. What Wilczek wants to communicate in this book is not the theory itself, he is not teaching a physics course, but he gives an idea of how theoretical physicists have struggled with the problem. He somehow communicates the satisfaction, the sensation, the joy, and the sense of beauty that it must have brought when they succeeded in nicely matching the three fundamental forces together, with a possibility to eventually include gravity as well.
It is very difficult to convince a reader, who is not involved in the research, of the overwhelming sensation it brings when after working very hard and meeting dead ends many times, finally the pieces of the puzzle just fall naturally into place. This is is not restricted to research in physics, but mathematicians and probably others may have experienced something similar. Like the three lines of the theory finally meet in one point on page 317 in the graph 'Why I ♡ SUSY', also the human quest for beauty and truth of all ages seem to converge here in the end. By starting at the beginning and collecting the star dust that makes up the jewel in the end (the timeline at the end of he book that takes several pages shows how long this road is and how many have contributed) Wilczek is able to convey some of his experience to the reader. He regularly makes comparisons and he illustrates some physical ideas or achievements with pieces of art that may be more familiar to the lay reader. The book is also richly illustrated with pictures and two sets of color plates, and some care has been taken by the publisher to make it stand out from the more common and discrete scientific books. The colorful cover that only peaks through a hole in the dust jacket is for example notable.
It is quite an experience to read this book, but the reader should have a keen interest in the crusade of theoretical physics currently storming the walls that border our understanding of Nature. But should not every human being be interested in the outcome that brings the truth? and should not everybody rejoice in the beauty that it brings along? Probably the answer to this question is also yes, though some may have more down-to-earth everyday worries to take care of for the moment being.