Ard Louis explores the concept of beauty in theoretical physics. 02/02/2023

I still remember the ethereal feeling of being invisible.  After pulling an intense all–nighter at my desk, I was walking along one of Utrecht’s famous canals, still shrouded in early morning mist. The first cafés had opened. I wanted to grab their customers and shout:  Do you not know that combining quantum mechanics and special relativity inexorably leads to antimatter? Mathematical consistency forces this upon us. This is inescapable and unbearably beautiful.  But I knew that being accosted by a random unshaven undergraduate babbling about commutation relations would not go down well. I would have to keep my intellectual vertigo to myself. The dissonance between what I had seen and could not share, and the prosaic concerns of the people around me buying their breakfast croissants made me dissociate from the scene. I felt invisible. 

The reason for the all–nighter was not an imminent deadline. Instead, I was completely floored by an argument I had encountered in an advanced quantum mechanics class at the University of Utrecht. It was first written down in 1928 by the great Cambridge mathematical physicist, Paul Dirac. The question he was trying to answer can be easily formulated: how do you describe a small particle, an electron, that is also going very fast.

To describe particles moving at extremely high velocities, you need Albert Einstein’s special relativity, which, based on the principle that the speed of light is the same in all reference frames, connects mass, time, and space. Its most famous formula, E=mc2, relates energy, E, to mass, m, and the speed of light, c.  One consequence of Einstein’s relations, for example, is that the mass of a particle moving at very high speeds will increase due to its high energy.  Although special relativity may sound esoteric and complicated, it is easier to understand than you might think. I learned the basics in secondary school, where it is routinely taught in final year physics. Einstein worked it out in 1905, his annus mirabilis. 

In 1926, Austrian physicist Erwin Schrödinger published his eponymous wave equation, which describes particles, such as the electron, that are very small. It heralded the arrival of the quantum revolution, quite possibly the greatest scientific advance of the last century, and without doubt its greatest conceptual leap. In fact, Albert Einstein’s 1921 Nobel Prize was not given for relativity, but for another 1905 paper which postulated that light is not just a wave but can also behave as a particle. In other words, light is quantised. 

Schrödinger expressed this mysterious wave–particle duality in a mathematically rigorous form. On the one hand, his equation is immensely powerful. When applied to electrons and protons, for example, it describes (nearly) all of chemistry. On the other hand, and in contrast to special relativity, it employs concepts that are so radically different from the intuitions of our daily lives that Einstein did not believe that it could be complete.  The 2022 Nobel Prize in physics was given to a trio of scientists who, in a series of heroic experiments, demonstrated that the quantum weirdness to which Einstein so vehemently objected holds without a shadow of a doubt.  Notwithstanding the conceptual enigmas, doing quantum calculations is a not as hard as one might think. I taught myself the basics in secondary school, and I think that any motivated A–level physics student should be able to do the same.

However, it had never occurred to me to ask Dirac’s deceptively simple question, which implies that an electron has simultaneously to obey the laws of quantum mechanics and the relationships of Einstein’s special relativity.  Dirac showed that satisfying these conditions at the same time only works if you added another kind of particle[1].  There is no other way. Dirac, a great believer in the “unreasonable effectiveness of mathematics“, published his equation even though its physical interpretation was unclear. 

Just four years later, Carl Anderson, a physicist at the California Institute of Technology, identified this mystery particle in cosmic rays and called it the positron.  His discovery clarified that the Dirac equation predicts an amazing kind of symmetry in the laws of nature.  Every particle of ordinary matter has an antimatter counterpart. The positron is the antimatter partner of the electron.  When a particle and its antimatter counterpart collide, they mutually annihilate and their mass gets turned into energy, following Einstein’s equation E=mc2. To get a sense of the power this would unleash, a gram of matter and antimatter reacting would generate an explosion three times as big as the atomic bomb that destroyed Hiroshima.  Luckily for us there is not that much antimatter in our universe (the reason for this asymmetry is not well understood).  In very small amounts, it even has practical uses. For example, in positron emission tomography (PET) scans, positrons are produced that collide with electrons in our bodies, emitting energy in the form of light, allowing us to image tissues. 

One of Dirac’s most famous students, the physicist/theologian John Polkinghorne recounts that when Dirac was asked about his fundamental belief, he wrote on the blackboard:

“The laws of nature should be expressed in beautiful equations.” [2]

In fact, Dirac’s equation is regularly called the most beautiful equation in physics.  My favourite way to depict it uses the compact representation shown below:

But it can be written down in many other ways.  When physicists say that an equation is beautiful, they don’t mean that the symbols on the page are arranged in an aesthetically pleasing way.  Instead, they mean that something about the way the equation organises concepts is beautiful. They often substitute the word elegant.  For the Dirac equation, the way two unrelated ideas are combined to produce an unexpected and exciting outcome is both elegant and beautiful.

While it is hard to explain exactly what physicists mean by this concept, they tend to agree on what is and what is not beautiful.  You know it when you see it.  Mathematicians have very similar notions.  The neuroscientist Semir Zeki [3] scanned the brains of 15 mathematicians while they looked at equations they had rated as beautiful, indifferent, or ugly.  When they saw the beautiful equations, the part of their brains that was activated (field A1 of the medial orbito–frontal cortex if you really want to know) was the same as when people see beautiful art or hear beautiful music.  Interestingly, the equations typically picked were very similar, even though the mathematicians came from a range of different cultural backgrounds, and may not have exhibited nearly as much agreement when looking at art.  The same part of the brain lighting up in scans is not the same as saying that the experience is identical. But it supports the testimony of countless physicists and mathematicians who have claimed that what they experience is the same kind of beauty found in the greatest art, the most spectacular natural scenes, or in the most beautiful music. 

When I first came across Dirac’s arguments, I found them too fantastical to believe.  It seems preposterous that the mathematical consistency of two utterly disparate descriptions of the world, quantum mechanics and special relativity, would require something as unearthly as antimatter.  So, with the typical hubris of youth, I spent the night feverishly trying to find a loophole in his derivation. By the first light of dawn, I had to bow to the great master.  

What drove me to wander through the streets of Utrecht though was not my shock at the beauty of Dirac’s equation itself, even though for the first time, I recognized that its beauty is at least as great as Rembrandt’s Night Watch, the Okavango Delta at sunset, or Johan Sebastian Bach’s Goldberg Variations.  Instead, it was my reaction to the amazing predictive power of Dirac’s equation.  As the great Indian physicist Subrahmanyan Chandrasekhar wrote many years later: 

“It is, indeed, an incredible fact that what the human mind, at its deepest and most profound, perceives as beautiful finds its realisation in external nature. What is intelligible is also beautiful. We may well ask: how does it happen that beauty in the exact sciences becomes recognizable even before it is understood in detail and before it can be rationally demonstrated? In what does this power of illumination consist?” [4]

What I experienced was something closer to what philosophers have called the sublime. This is the sense of beauty mixed with terror that can occur when you for the first time see Mont Blanc or Mount Everest or experience a great sea–storm.  I don’t mean the kind of terror you feel when someone points a gun at you. Rather, it is the terror of your own finitude when confronted with something much bigger and greater than yourself.  I felt like Dirac had given me an unauthorised glimpse of the transcendent; that I had gone where angels fear to tread. I had experienced the sublime, and knew I would never see the world the same way again.

Ard Louis is professor of theoretical physics at the University of Oxford

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