University of Minnesota
School of Physics & Astronomy


Resurgence Theory

Aleksey Cherman
Aleksey Cherman
Aleksey Cherman

Aleksey Cherman is a post doc in elementary particle theory. His work focuses on mathematical tools that have applications throughout physics. Across all of theoretical physics, there are equations used to solve various problems in nature, but they are much too complicated to be directly solved.

"The last time I solved a problem exactly was doing my homework in grad school," Cherman says. To deal with the complicated nature of their subject, theorists have developed techniques to find approximate solutions to problems. "You figure out what is most important, and build in corrections to further describe the problem." This technique is called Perturbation Theory, and it is used throughout physics. Physicists solve a sequence of problems, building up corrections, where each correction gets harder and harder to calculate. "If we’ve set things up things correctly in the first version, then the amount of inaccuracy gets smaller and smaller as we go through the sequence." The answer is a good approximation of the real answer that the physicists would have achieved had they somehow been able to solve the original problem exactly.

Cherman says, in essence, this is what theoretical physicists do for a living—they find solvable limits for difficult problems, and then figure out corrections to those simplified limits to understand nature. Unfortunately, there are two types of situations where Perturbation Theory runs into trouble. The first and most obvious issue occurs is when physicists do not know how to find a useful simplifying assumption to make the first equation in the sequence. In elementary particle theory, the main example is the strong-coupling problem, where physicists have come up with other tools, such as numerical simulations, or techniques from string theory, to understand the behavior of particles like protons and neutrons.
Cherman’s research is devoted to the other, more subtle way that that Perturbation Theory can fail. If one carries on adding up corrections to most problems, at first, the error decreases compared to the true answer. But strangely, it is known that typically the error will eventually start to increase if one makes enough steps in this process. This growing inaccuracy problem does not present a stumbling block at the order at which theoretical physicists normally work, where the first two or three corrections will get them close enough to the real answer. Cherman says that physicists have known for a long time that if they carry on to 50 or 100 corrections, at some point-- say 75 corrections—the inaccuracy will increase. "Why are we allowed to stop? This is not a new question. We have known about it and mostly ignored it because it wasn’t clear to say."

University of Minnesota Professor Arkady Vainshtein did some important early work on this topic back in 1960s. "Everything I said till now could have been said in 1965. It’s just a practical matter to not worry about since we can’t calculate 75 corrections anyway." Cherman says that the problem of increasing inaccuracy might actually be connected to the problem of strong coupling, where physicists don’t know how to organize the calculation in a series of solvable steps.

The problem of increasing inaccuracy has not gone unnoticed by mathematicians who have been trying to deal with this for 200 years. In the mid-1980s, a French mathematician, Jean Ecalle, developed an approach called Resurgence Theory, to deal with this problem. Ecalle wrote mostly in French and most physicists did not know about his work till the last ten years, and have only been using Ecalle’s tools to solve physics problems in the last three or four years. Cherman’s collaborators, who are sprinkled around the world ("I spend a lot of time on Skype," he says), took up this technology and began applying it to problems in physics. "We’ve been trying to use these tools to fix perturbation theory." Cherman says that they are not just playing with mathematical toys, but that the answers they are getting are actually yielding answers about the physics of the systems. Cherman and his collaborators are trying to crack some notorious problems in particle physics. For example, in trying to understand behavior of quarks, gluons and neutrons, theorists run into the "renormalon divergences" which get bigger as they go on. These divergences were discovered in the late 70s, and University of Minnesota Professors Mikhail Shifman and Vainshtein came up with some techniques for handling renormalon diverences in the 1980s. With Resurgence Theory, Cherman hopes that we can get deeper insight into renormalon divergences.

Cherman says that there is an enormous amount of work to do in the application of Resurgence Theory to physics problems, from simple problems to very complex ones. "The effect of this work could be profound. We can never observe free quarks and gluons. We can only observe protons and neutrons. We can see it happening when we solve the theory numerically, but understanding the behavior of quarks and gluons in detail is one of the outstanding problems in particle theory."