By YANG-HUI HE
Could there be a theory that describes all of the fundamental laws of nature, a Theory of Everything? Einstein thought so but he never managed to prove it. Mathematical physicist Yang-Hui He guides us through the quest to fulfil Einstein’s dream.
As Albert Einstein lay on his deathbed in 1955, he and some of the visionaries of his time had a dream. It was a dream which Einstein had held for the last decades of his remarkable life: that there should exist a single set of equations, a single principle, which describes the fundamental laws of nature. This Theory of Everything is not as outlandish as it might first seem. By 1698, Isaac Newton had realised that the same equations governed the fall of an apple and the motions of the planets — and thus was born the unified theory of gravitation. Eighty years later, James Clerk Maxwell had established that the same equations dictated the properties of electric and magnetic fields — and thus was born the unified theory of electromagnetism.
By 1916, treading along the glorious path led by these physicists, Einstein unified gravity with space and time, formulating the so-called special and general theories of relativity. It explained, to great accuracy, the behaviour of the cosmos at a macrocosmic scale: the motions of stars and galaxies, the expansion of space and the passage of eons. By the mid-twentieth century, the proponents of quantum mechanics, notably Werner Heisenberg, Erwin Schrödinger, Paul Dirac, and others, had described the cosmos at a microcosmic scale. This ‘quantum’ description of the handful of fundamental particles, such as the photon and the electron, unifies Maxwell’s electromagnetism with the strong and weak nuclear forces, and comprises the so-called Standard Model, again verified to extraordinary accuracy. The experimental search for the last piece of this puzzle, the Higgs Boson, was a key purpose for the Large Hadron Collider at CERN.
Therefore, Einstein’s dream was rather reasonable: by the twentieth century, humanity had reduced the fundamental laws of nature into two sets of equations, from which all phenomena can, in principle, be derived: those of Einstein for gravity, characterising the macrocosmos, and those of the Standard Model, describing the microcosmos. It is only natural that the marriage of these two would be the answer, the Theory of Everything. Sadly, naive attempts to do so baffled even Einstein in his last years: an infinite number of uncancellable infinities, a technical problem known as un-renormalisability of a quantum description of gravity, became the last great hurdle.
Enter string theory. In the 1980s, theoretical physicists stumbled on an apparent way out. It constituted a paradigm shift in our understanding of the world. The theory proposed that the reason for the issue of un-renormalisability of quantum gravity was that we had inherently assumed that elementary particles were truly point-like. What is the essential size of one of these elementary particles — for example, the photon? Even the practitioners of quantum theory would have answered: zero. However, it is precisely this innocent assumption which caused the untamable infinities: by allowing interactions at a single point in space-time, energy was allowed to be concentrated at zero volume. As we recall from our childhood lessons, division by zero gives problems!
Let’s imagine we smear the point out. Extending a point, an object of zero-dimension, gives a one-dimensional object, a line, which can either be closed into a loop or open like a segment. This line is the superstring, the fundamental constituent of everything. With this single generalisation, the infinities were cured and a consistent quantum theory of gravity emerged. Indeed all particles, all forces, and, in fact, space-time itself, became different vibrational modes of the string. All of reality is reduced to a symphony of cosmic strings, resonating harmoniously to give the rhythms of space, time, and matter.
So are we done? Do we have the ultimate Theory of Everything? Not quite. The unification theory of superstrings only works in ten dimensions of space-time, rather than our familiar four (three of space and one of time). On the one hand, this is interesting: it is the first time a scientific theory has predicted the dimensionality of space-time. On the other hand, where did the six extra dimensions go? One standard answer is that they are simply too small and are curled up. It is like an ant crawling along a drinking-straw which appears to be moving only in one dimension, along the straw’s length: on closer inspection, we see that the ant can also loop around the tiny circular cross-section, and consequently detect two more dimensions.
So, how big is the string? It surely must be much smaller than the atomic scale, or any scale we have so far probed, because we have not yet seen particles manifesting as strings. This idea represents a conceptual breakthrough, and an experimental nightmare. Let us think for a moment about the fundamental constants of nature. There are really only three: the speed of light, c, determining relativity, Newton’s universal constant, G, determining gravity, and Planck’s constant, h, determining quantum physics. One important combination of these constants gives a value of length, the so-called Planck length. It is therefore natural to propose that the string be of this size. In Earth units, it is about 10–35 metres, 16 orders of magnitude smaller than anything we have ever observed, and inconceivably smaller than anything we are likely to ever directly measure.
The theoretical picture is nevertheless clear: we have tiny superstrings of Planckian size vibrating in ten dimensions, six of which are curled up to about the same magnitude. The geometry and topology of how these dimensions are curled up determine the physical observables of our large-scale four-dimensional universe.
But if we are so short of experimental evidence, why, you ask, is there so much effort invested in string theory? The one-line answer is that it is beautiful. Aesthetics aside, history has taught us a peculiar and awe-inspiring characteristic of theoretical physics: theories with mathematical elegance tend to be correct. When Maxwell added a term to make his equations look symmetric, he did so purely out of a need for beauty. That term was verified experimentally within a decade. When Dirac predicted the existence of the anti-electron, he did so to make his equation more elegant. This particle was detected shortly afterwards. Of course, we need not take Dirac’s rather extreme viewpoint that “it is more important to have beauty in one’s equations than to have them fit experiment”.
And what mathematical beauty this theory has! In an almost unprecedented manner, string theory has revolutionised pure mathematics; it has solved problems which bemused geometers, it has inspired new perspectives on algebra and number theory, it has given physical insight to the most abstruse branches of mathematics… The list goes on. Indeed, Edward Witten, the torch-bearer of string theory and the only physicist to ever win the Fields Medal, wisely said: “string theory is a piece of 21st century mathematics which accidentally fell into the 20th century.” Now, in this new century, we hope that clever experimentalists will find some ingenious way of indirectly measuring stringy effects, and that the physical theory will continue to blossom. Any scientific theory must be experimentally verifiable, and when theory precedes experiment, the best bet is often one that revolutionises mathematics. No contending theory to string theory as a Theory of Everything has done this.
Ludwig van Beethoven wrote the cryptic words “Müß es Sein?” (“Must it be so?”) on top of his last string quartet. It is our wish that to this divinely beautiful cosmic string quartet, where reality manifests as a sublime melody trembling on superstrings, our response would be a resounding “Es müß Sein!” — “It must be so!”
Yang-Hui is a Professor of Mathematics at the City University of London and a Tutor at Merton College, University of Oxford. He works on various interfaces between geometry and theoretical high energy physics and is particularly interested in aspects of algebraic geometry in application to, and interacting with, gauge theory as well as string theory.