A lack of mathematical proofs and derivations for equations purporting to describe a host of quantum mechanical phenomena has not stopped physicists from making important advances. But there is still a real importance — and a great intellectual challenge — in providing those proofs and derivations. An equation becomes unassailably true only when a mathematician shows that it is. Xuwen Chen is dedicated to providing that rigorous degree of certainty.
One particularly high-profile phenomenon is called a Bose-Einstein condensate. First proposed in 1924 by the Indian physicist Satyendra Nath Bose, for whom bosons are named — and yes, that Einstein — the condensates are a form of matter in which atoms align with the same energy and in an unusually compact space. Despite its historic namesakes, the exotic and extremely cold new state of matter was only realized experimentally in 1995, a feat that was rewarded with the 2001 Nobel Prize in physics.
But here’s the take that Chen, the Tamarkin Assistant Professor of Mathematics, has on the nonlinear Schroedinger equation, the key equation involved: “It’s supposed to describe the physical phenomenon called Bose-Einstein condensation, but there is not a rigorous mathematical proof about that. What I’m doing is giving a mathematical proof.”
Achieving that proof, building on the work of Harvard Professor H.T. Yau, was the subject of Chen’s doctoral thesis, which he completed in May at the University of Maryland.
Chen doesn’t doubt the results in the lab, of course. It’s just that mathematicians aren’t the type to figure that if it walks like a nonlinear Schrodinger equation, and talks like a nonlinear Schrodinger equation, and acts like a nonlinear Schrodinger equation, it must be a nonlinear Schrodinger equation.
“The experiment matches, certainly, but mathematics and physics theory are more rigorous,” he said. “The validity has to be checked carefully.”
One demands that level of rigor because the evidence from an experiment may fit an equation, but only in an approximate way.
Say an instrument measures a temperature of .001 degrees. Relying just on an experiment might convince a naive scientist that the relevant temperature should be exactly that value. But maybe the predicted temperature is really .000798 degrees and the instrument is just not sensitive enough to report it. When instruments reach the limits of their precision, only mathematical proof can show what an equation would really predict.
“Logically, if we study something, we should say it is valid, not just that it looks valid,” he said.
That’s true everywhere, but after graduation Chen still particularly wanted to come to Brown to pursue his work. He’s keen to work in the areas of “many-body systems” and “mean-field approximation” with Walter Strauss and Justin Holmer in mathematics and Yan Guo in applied mathematics.
“Everybody in this area would like to work with them,” he said. “It’s a great opportunity.”
It might have seemed a remote opportunity when Chen was growing up Guangzhou, China, near Hong Kong. He attended college at Hunan University earning a B.S. in pure and applied mathematics in 2007. When he started, however, he was a chemistry major. Only later did he realize he liked math — and the chance to determine the validity of equations — much more.
As someone who had the chance to find what he truly loves through education, Chen is excited to teach as well as do research at Brown.
“Teaching can really inspire me,” he said. “I have benefited from the time and effort of a great many teachers, and I owe my students that same time and effort.”
Although that’s a statement that’s intuitively valid, it probably cannot be proved rigorously.