Theoretical physics and the art of the abstract
New Scientist 01 March 2011 by Giovanni Vignale
Great theories draw on the same creative roots as great poetry or fiction, but the constraints of fact add something unique
There is a story called The Little Prince by the French writer Antoine de Saint-Exupéry that I find deeply inspiring. A pilot crash-lands in a desert and meets himself, thinly disguised as a young boy (the eponymous little prince) from another planet. The pilot had been a gifted child artist but had lost faith when he drew a long shape with a central hump. The adults had seen it as a hat, never allowing that it could be what the child intended: a python that has swallowed an elephant. Following the little prince through various strange encounters, we eventually learn that "whether it's a house, or the stars, or the desert, what makes them beautiful is invisible".
This quote struck me as a good introduction to my favourite science, theoretical physics - and as an explanation of the rather obscure-sounding title of my book, The Beautiful Invisible, on which this essay is based. For a long time I had wanted to write a book on the unique nature of theoretical physics.
Many people see science as dry number-crunching that manages to lose the hidden beauty of the world in what, nearly a century ago, the writer Robert Musil called "an orgy of matter-of-factness after centuries of theology". Theoretical physics, however, emerges at the heart of physics as the modern science of the invisible, a modern form of theology.
Unlike much practical science, theoretical physics provides us with a description of reality on a very abstract, mathematical level. It was precisely the love of the abstract that attracted me to physics when I too was a young boy. But then, why not do mathematics, philosophy, or even theology proper? I have often wondered. My conclusion is that I chose theoretical physics because of its sensuality, and because it was the best compromise between escaping reality and diving into it.
In mathematics you know exactly what you are talking about, because you have created the objects of your study. The non-existence of the perfect platonic circle is the best guarantee of its existence as an abstract object of study.
In theoretical physics, however, you never know exactly what you are talking about: you are creating a fictional world to model a real world that you don't understand.
But it is fiction constrained by fact, fiction that must lift itself up in the air together with the hard facts on which it is based.
René Magritte Le château des pyrenées
It is fiction like the castle in René Magritte's famous painting - a massive rock hovering over the sea that ought to collapse but is in fact suspended by the mysterious power of its internal consistency.
Throughout the book, I emphasise the artificiality of the concepts of theoretical physics. This is not the artificiality of platonic ideas, which are supposed to represent "true" reality as opposed to the illusions coming from the senses. Nevertheless, the concepts of theoretical physics may be illusions insofar as they are products of imagination: they shine on the dust of facts like rainbows on water droplets.
Looking at reality through the lens of creative imagination is, I believe, the essence of theoretical physics. It is certainly not turning one's back on reality like the self-absorbed Newton of William Blake's painting. Rather, it is an effort to see further and more clearly, to connect to an obscure reality "behind the glass", like the fictional portrayal of John Nash in the movie A Beautiful Mind, who writes his mathematical formulae on a windowpane.
Perhaps the most important creative tool at a theorist's disposal is the notion of pushing a familiar idea to its limits in order to generate new concepts, which in turn become building blocks for new theories. Consider a supposedly "elementary" concept such as speed, or velocity. It hardly exists in nature. You can see that an object moves a certain distance in a certain time, and from this you conclude it has an average speed of distance divided by time. But ask yourself what the speed is at a given instant and you quickly discover that what it is that we are really averaging when we calculate the average speed is not so straightforward.
The challenge is to define speed when there is not yet a distance. This difficulty led the Greek philosopher Zeno to conclude that an arrow in flight is, at each particular instant of time, at rest - and that therefore motion must be an illusion. In theoretical physics the paradox is solved by going to the limit, where the time interval tends to zero. This was the beginning of differential calculus and one of Newton's most important contributions to mathematics and physics.
Limits are constraints, imposed to narrow a problem and make it more solvable. In practice, they underlie all physical theories and define their range. Consider classical Newtonian mechanics, which is based on the assertion that a particle will move forever with uniform motion, without changing direction, unless compelled to do so by external forces.
This is fine for particles in empty space, but suppose you want to describe the motion of a particle immersed in a viscous liquid. You don't want to keep track of the complicated and uninteresting motions of zillions of molecules in the liquid: you want to stay focused on that one particle of interest. The way to do this is to introduce a "drag force" that only acts on the particle you are interested in, so you can ignore the complex interactions between particles. That gives you an effective, everyday theory to use. So the making of effective theories is another creative tool for the theorist. An effective theory, such as Aristotelian mechanics, can be generated from a more fundamental theory, Newtonian mechanics, by systematically discarding information. You do not attempt to describe exactly all that there is to describe: you just focus on the aspect of the phenomenon of interest to you and disregard the rest. Isn't that rather like the art of the novelist, glossing over the excruciatingly boring details of the daily lives of their protagonists?
A striking demonstration of the power of limits comes from the concept of spontaneously broken symmetry. This occurs when matter settles into an an ordered state which is less symmetrical than the physical law that describes it. Think of water molecules settling into a crystal of ice. Unlike a liquid or a vapour, the crystal is not "translationally invariant": it looks different at different points. The fundamental laws of physics, on the other hand, are translationally invariant. In ice, the symmetry is broken.
In theoretical physics, broken symmetry is obtained by letting the number of particles, N, in the system tend to infinity on a finite, theoretically imposed timescale, T. By imposing an artificial limit, the system gets "stuck" in the ordered state. The symmetry of the underlying physical laws re-emerges if we interchange the order of the limits, that is, let T tend to infinity while N remains finite. Over a long timescale the ice crystal can move as a whole, restoring the equivalence of all points in space.
The fixed view is the point of view of humans; the second, the infinity view, of God. We are trapped in a world of fixtures, of social conventions, of immutable truths - an ordered world.
But, as the philosopher and essayist Ralph Waldo Emerson wrote: "Permanence is but a word of degrees... The law dissolves the fact and holds it fluid." That annoying feature of the natural world, irreversibility, provides another example of broken symmetry. The fundamental laws of physics are time-reversible, yet our reality is blatantly irreversible. How come? You might say, well, Newtonian mechanics is wrong. But no: starting from the Newtonian mechanics of atoms, and by pushing the theory to its limits, the 19th-century physicist Ludwig Boltzmann constructed an effective theory describing how particles arrange themselves over time. Irreversibility emerges as a natural feature of this theory.
To many scientists, such a feat of emergence was unacceptable. It had to imply that either Boltzmann's mathematics or the atomic theory of matter was wrong, or perhaps both. But now we understand that effective theories can be quite different from the more fundamental theories on which they are based. Effective theories look at reality on a different level: they answer different questions.
The kind of creativity that builds theories strongly resembles artistic creativity, especially at the spiritual level - that is, in the loving attitude of the artist or scientist towards his or her own work. But there are huge differences in the way the creative work plays in society. Artists are present in their work as individuals. They do not need to explain facts, they are not answerable to other artists, and they can appeal directly to the public. Artists are also allowed to rediscover known things, for in their craft even a slight change in perspective, a different intonation, makes a big difference.
Theoreticians, however, like all scientists, must make sure their work, while distinctly original, fits well with known facts and accepted theories so that it can become part of a system of knowledge in which their own individuality will be lost.
This loss of visibility can be painful, particularly when he or she feels they are not getting due credit. This is why all over the world scientists go to great lengths to make sure their work is properly cited by colleagues, and that they cite their colleagues in return. In the arts, this would be totally superfluous: imagine Dostoevsky pausing in mid-flow to say "...and, as Tolstoy first realised, happy families are all alike".
Here, however, in a book or an essay, my individuality cannot help but be on display. My unorthodoxies are also there for all to see. The first of them is that the book is meant to be not only a book about physics, but a literary experiment in stitching together physics, literature and the arts. The second unorthodoxy is my openly non-realistic attitude toward theory, emphasising idealisations and limits.
Just as to many people the origin of life would be inexplicable without a creator, so to most physicists the success of a theory would be inexplicable without an objective reality behind it.
I think otherwise. Theoretical physics is a risky business. A famous vignette by the Argentinian cartoonist Quino, showing God seated on a cloud and having a good laugh at a book entitled Laws of Physics, makes the point very nicely.
A sceptical attitude (by which I mean the realisation that the connection between theory and reality is always tentative) is as healthy in theoretical physics as it is in life. Further, it helps to remind us that no theory can be final. Strictly speaking, there are no final theories, only effective theories with a more or less defined range of validity.
All working theorists know very well what I mean. They may disagree with this or that philosophical point, but they all realise that, when they do theory, they do not deal with nature as it is, but with an idealisation of nature - a tangential reality.
That said, in my book I retell the story of Eklavya from the Mahabharata, which I think underscores the merits of the abstract. Eklavya aspires to study archery under the great guru Drona, but Drona rejects him because Eklavya belongs to a low caste whose members are not allowed to become warriors. Undaunted, Eklavya builds a clay idol of Drona (the abstract) and, training assiduously under its spell, manages to become the greatest archer in the world. This exemplifies my view of the abstract as a liberating force, which helps us leave behind petty social conventions and narrow-minded formalities.
One of the most important concepts of theoretical physics is that there are many different ways of representing things, all of them legitimate and equivalent. And yet, even as one changes one's point of view, there are fundamental quantities and principles that do not change. These invariants provide a superior connection through which all different representations are linked to one another. Invariants in physics include the speed of light, which is the same to all observers; the length of space-time intervals; and the spin of a particle.
The existence of invariants is arguably the most important fact of theoretical physics.
Einstein's famous equation E = mc2 follows from the fact that the conservation of momentum must hold good in all reference frames. But, in a broader sense, invariants between races and cultures also make communication and understanding between people possible. They provide the basis for that most precious of social virtues: tolerance.
Cracking big problems will always require a large dose of creativity. There is no doubt that the next breakthrough in theoretical physics, whether it comes from string theory or from condensed matter theory, will make full use of all the creative tools currently available - especially abstract mathematics - and perhaps new ones that we cannot imagine.
Contemporary string theory, with all its limitations, falls squarely within the great tradition, initiated by physicist Paul Dirac, of following where the beautiful mathematics takes us. However,
I do not believe we will ever attain a final and unassailable solution to the riddle of the universe. The solution of one problem always creates new problems.
There can be no clearer demonstration of this than Dirac's discovery of the relativistic equation for the quantum-mechanical electron. At the height of his scientific creativity, Dirac set out to resolve the contradiction that pitted two great theories of his time, special relativity and quantum mechanics, against each other. Making creative use of abstract algebra, he constructed a beautiful equation for the electron wave. This equation treated space and time symmetrically, predicted the existence of the spin of the electron, and explained the optical spectra of atoms. This was an extraordinary achievement by any measure, yet it was to set off a chain of events that ultimately undermined the very concept of the "quantum particle" on which quantum mechanics had been built. Dirac had started from a single particle, but in the attempt to pin it down to one point he found that he had to generate many more particles (some of them actually antiparticles). The inevitable conclusion was that
...a single particle does not exist in the literal sense, but only as a metaphor to describe a far more complex state of infinitely many particles and antiparticles - that is, a quantum field.
Like Macbeth, Dirac managed to win and lose his battle at the same time. In the same spirit, for all our creativity, I expect the battle for the ultimate theory of the universe will eventually be won, but also, somehow, lost.
Giovanni Vignale is Curators' Professor of Physics at the University of Missouri - Columbia. He has published more than 160 papers on many-body theory and density-functional theory of electronic systems. His book The Beautiful Invisible is published next month by Oxford University Press