What makes physics beautiful? We asked some top researchers

They say beauty is in the eye of the beholder – and for physicists, beauty is in numbers.

Pedro Vieira, Clay Riddell Dirac Chair in Theoretical Physics at Perimeter Institute, is currently teaching a non-credit minicourse about ‘beautiful’ papers in physics.  The course alternates between lectures on nine influential papers and student-led presentations about how these monumental papers influenced physics. 

This is Vieira’s second time running the course and his first time offering it at Perimeter. He says the course is a way to cover spectacular papers while helping students understand the language of quantum field theory.

“Quantum field theory is the base of all physics,” says Vieira. “If you know this language, then you can a study a paper about spin chains or a paper about cosmology.” By covering a selection of papers, from Steven Weinberg’s 1965 article on infrared photons and gravitons, to Pasquale Calabrese and John Cardy’s 2004 work on entanglement entropy, Vieira hopes that students will identify gaps in their knowledge.

Selecting papers for the course means filtering through many amazing works. Vieira looks for papers with big impacts on the field of physics; they must have at least 1,000 citations. About half the papers that Vieira covers are suggestions – papers that his colleagues “fell in love with.”

One theme that repeats in beautiful papers is a winding path to discovery. Vieira describes one of the papers covered in the course, a 1974 study by Kenneth G. Wilson on the confinement of quarks, as an exploration.

“He introduces this important quantity that depends on the paths of particles in spacetime. And he doesn’t know exactly yet what to do with this quantity,” says Vieira. “He doesn’t know yet what this is going to be useful for – now we do – but at the time, it was like an alien object. It was a question of keeping exploring it, and we’ll find out.” 

Adrián López-Raven, a resident PhD student at Perimeter, enjoys the course because it brings students together from across the Institute. “It’s been a remarkable exercise in community building. I see people from condensed matter there, from mathematical physics.” And it’s not just different fields of study showing up, he says. PhDs, postdoctoral students, and faculty have all sat in on the lectures. 

“It’s very hard to attract people from different communities, bring them together, and make them all get something out of it,” says López-Raven. “By looking at the foundations, it’s attracted people from different spaces.” 

So what makes a ‘beautiful paper’? 

“It’s a bit subjective,” says Vieira.

“I think when you see a beautiful paper, you know it’s a beautiful paper. It might be a bit naïve, but I do think we’re trying to touch this platonic world of truth. Sometimes we bring this piece of very pure truth, and it’s almost shocking how bright it is. It’s clean, new.”

Jacqueline Caminiti, a resident PhD Student in the course, uses the metaphor of a tree to relate beauty to physics. “If the sizes and shapes of the branches on a tree were completely randomized, the typical human probably wouldn’t find it beautiful. It would be too chaotic, it would be like white noise.”

On the other hand, she adds, if a tree was built out of perfectly ordered straight lines, we might find it a bit boring. “The thing about a tree that makes it beautiful is that you can see some structure there. But at the same time, it’s slightly beyond your understanding of why this shape is emerging the way it is.”

It’s that combination of understanding and transcendent mystery that brings beauty to physics for Caminiti. “I think my favourite example of this is Einstein’s theory of special relativity,” she says.

“Einstein’s theory takes all of these mysteries, groups them together, and resolves them by saying you shouldn’t think of space and time as different structures.” But at the same time, she adds, Einstein’s theory revealed new mysteries to grapple with, such as the fact that observers can experience time differently.

“I think if beauty is the existence of a mysterious pattern, then a beautiful equation is one that simultaneously solves and reveals such patterns,” she says. 

For Dawit Belayneh, a resident PhD student at Perimeter who’s coordinating the course, one of the key characteristics of a beautiful paper is clarity. “It needs to have significance in the sense that the paper defines a few concepts, using which you can understand a wide variety of physical phenomena.” Belayneh adds that beautiful papers have an economy of concepts, an economy of ideas.

López-Raven echoes that a beautiful paper is efficient and clean. “I think there is an exercise in data in compression,” he says. “When you understand something, you need less words to describe it.”

“Some of the great ideas in physics are exercises in unification,” López-Raven adds. “The Dirac equation is a unification between quantum mechanics and special relativity. You have these beautifully robust yet mutually incompatible structures, and the fact that you can now merge them to create further structure that encompasses the two in a consistent way – that’s beautiful.”

Telling the story of beautiful physics papers

A powerful component of the beautiful papers course is the student presentations, says Vieira. It’s up to the students to take the papers covered in his lectures and place them in a modern perspective. 

“My feeling is that they value the feedback on the presentations,” says Vieira. “I’m trying to give them concrete tips.”

Caminiti is one of the students who presented in Vieira’s class. She chose Steven Weinberg’s foundational 1979 publication on phenomenological Lagrangians as a jumping-off point.

The paper was essentially the introduction of effective field theory in particle physics, explains Caminiti. “The idea is basically a theory for building theories.”

Think back to Newton, Caminiti says, and imagine him looking at the objects around him. The approach at the time was a creative, constructivist attitude, where he would have looked at interesting objects – the apple and the earth, for example – and tried to define the force that relates them.

At the time of Weinberg’s paper, people were taking this approach in particle physics, says Caminiti. Even if they knew what objects were at play, it was a struggle to find the behaviour of these objects. “It was a very difficult calculational problem,” she adds.

Weinberg took a step back, forgot about all the objects in the problem, and instead thought of the symmetries that the objects obey, says Caminiti. The resulting paper was the birth of effective field theory in particle physics. 

For her presentation, Caminiti picked the most modern problem she could imagine that could be influenced by Weinberg’s idea: quantum gravity. She settled on a 1995 paper by John F. Donaghue, entitled Introduction to the Effective Field Theory Description of Gravity. It tackles one of the key questions in modern physics: the unification of quantum mechanics and gravitational physics.

“The idea is, can we apply Weinberg’s logic to gravity?” says Caminiti. “Can we ask: what are the symmetries that quantum gravity should respect, and then write down a Lagrangian based on those symmetries to make predictions for quantum gravity?”

Following her presentation, Caminiti, like all the other student presenters, received feedback from Vieira. The feedback ranges from directions to improve writing on a chalkboard, to advice on how to tell a story in just 10 minutes. 

“Presentations in physics often determine a lot of the impact of your work and the development of your collaborative network,” says Caminiti. It’s helpful and valuable for the students to practice their communication skills and receive tailored feedback in a safe environment, she adds. 

In the end, the beautiful papers course shows that physics isn’t all numbers: within these fundamental papers lies elements of art, mysteries of the natural world, and the opportunity for storytelling and collaboration between peers. It’s a good lesson to remember – that digging into the papers of the past can provide illumination in our modern age.