

"Carl [Bender, above] is unquestionably one of Washington University's best teachers and mentors. He has an uncanny ability to frame difficult ideas in language that is perfectly tuned to what the student already knows," says Professor John Clark, chair of the Department of Physics. 
HardWorking Model
An innovative researcher and devoted teacher, Professor Carl Bender is changing the way physicists look at quantum mechanics.
by Janni L. Simner
Some years ago, Carl Bender's father was putting Bender's son, Michael, to bed by recounting the story of a wellknown physics problem: that of the brachistochrone, a quickest path that a bead on a wire can take to get from one point to another. The curved shape of the brachistochrone was wellknown; it had been worked out 300 years before. But as Bender listened, he suddenly realized that his father, a highschool physics teacher, was wrong.
"My father said, 'What do you mean, I'm wrong?'," recalls Bender, who is professor of physics at Washington University. "He said, 'This is a classical physics problem; it's solved.' But I said, 'That's no longer the right answer, and I'm going to find the right answer.'" Working with an undergraduate eager for a challenge, Bender updated the brachistochrone to take into account Einstein's theory of relativity.
Bender makes a habit of questioning accepted answers, and, more recently, that habit has led to work that's changing the way physicists look at the field of quantum mechanics.
Quantum mechanics—the study of the behavior of very small particles—is much newer than the brachistochrone problem, but still, its basic principles were worked out early in the 20th century. "Most of those principles are very physical," Bender explains. "But one postulate is different from all the others because it's a mathematical axiom. It's not something you can measure." That axiom states that certain aspects of quantum mechanics must be "Hermitian," meaning, among other things, that they must remain in the realm of real numbers—that is, numbers that can, whether positive or negative, rational or irrational, be found on a number line.
"But insisting quantum mechanics must be Hermitian," Bender says, "is like saying all numbers must be even. You can't give an experimental reason for it. You can tell me that aesthetically even numbers are nicer than odd ones, but you can't say, 'Odd numbers are rejected by this experiment.'" In 1998, Bender proposed an alternative, nonHermitian theory, one that allows for complex numbers—that is, numbers that exist outside the number line, in what might more accurately be termed a number plane.
To those familiar with quantum mechanics, saying Hermiticity isn't required is a little like saying the sky doesn't need to be blue. "When I give talks on this, I hear comments from people in the audience that it has to be wrong," Bender says. "But then they think of an example, and they try it out—and to their amazement, it seems to work."

Above is Professor Carl Bender's model, showing the real energy levels of a complex nonHermitian quantum theory. 
Saying a theory seems to work is different from saying that it's been proven; right now, nonHermitian quantum mechanics isn't a statement of how the universe does work, but rather it is a model of how it might work. Yet in the sciences, such working models are far from trivial. "Carl has opened up a whole new area of physics," explains Washington University colleague Claude Bernard, professor of physics. "We don't know yet whether this type of theory describes the real world, but if it does, it represents a fundamental revolution in basic physics. And even if it doesn't, it forces us to reevaluate what we thought we knew about how nature must operate."
Around the world, researchers are taking part in that
reevaluation, investigating and challenging and applying Bender's work. Along the way, they're finding possible explanations for some unanswered questions: questions such as why the universe seems to be expanding at an accelerating rate, for example, or why so much more matter has been observed than antimatter. In 2003, an entire international conference focused on nonHermitian quantum mechanics; in 2004, there were two such conferences; and in 2005, three more conferences in Italy, Turkey, and South Africa are planned. New papers appear almost daily.
"This has inspired a tremendous amount of rigorous study," Bender says. "I can't tell you how exciting that is." Bender and his graduate students have published more than 30 papers of their own since 1998, three of them in Physical Review Letters, the field's most distinguished journal. Bender recently received a Guggenheim Award for his work, as well as a grant from the United Kingdom's Engineering and Physical Sciences Research Council, both of which helped fund a year of research at London's Imperial College.
Pursuing groundbreaking work isn't new for Bender. "I could point to a long string of similarly exciting innovations and breakthroughs, all the way back to Carl's Ph.D. work," says John Clark, chair of the Department of Physics and Wayman Crow Professor of Physics.
Bender admits he finds his current work particularly compelling, though. "I'm having more fun now than I've ever had in my life," he says.
Bender believes physics should be fun, and he is committed to fostering that enthusiasm in his students. He works with three graduate students, whom he views as colleagues. They meet daily, investigating ideas and testing conclusions. "I really like collaborating," Bender says. "Discovering things together, that's a very intense relationship."
He collaborates with gifted undergraduates, too, and proudly displays the resulting publications on his Web site (including a paper, "Relativistic Brachistochrone," published in the Journal of Mathematical Physics in 1986). He says the potential for undergraduate collaboration is one of the things that makes Washington University a great school. "Undergraduates here have the chance, if they're interested and bright, of doing meaningful research—not just bottle washing, but actual research and thinking."
"Carl has opened up a whole new area of physics," explains Claude Bernard, professor of physics. "We don't know yet whether this type of theory describes the real world, but if it does, it represents a fundamental revolution in basic physics. ..."
Bender's classes are consistently popular; students recall fondly both his enthusiastic teaching and his awful physics puns. "Carl is unquestionably one of Washington University's best teachers and mentors," Clark says. "He has an uncanny ability to frame difficult ideas in language that is perfectly tuned to what the student already knows."
Bender works with students in other ways, as well: by helping to interview applicants for the College of Arts & Sciences' prestigious Compton Fellowship; by asking students to help review his current textbookinprogress, Partial Differential Equations for Scientists and Engineers (he's also the author of Advanced Mathematical Methods for Scientists and Engineers); and by being one of the coaches for the undergraduate Putnam Mathematical Competition, in which the University often ranks in the top 10.
"The Putnam problems are very tricky, but there's always an insightful solution," Bender says. "I try to teach students to think in new ways while coming up with the answers." He adds that solving clever problems with known answers is only the first step to becoming a scientist, and he encourages those students who can to take the next step as well. "With research, you don't know that there's an answer," he explains. "You have to be willing to invest a lot of energy into something that might not work. You have to be an optimist. You also have to have a nose for choosing problems that you can solve. A good research problem is the most precious thing there is."

Professor Bender (standing) works closely with both undergraduate and graduate students, whom he views as colleagues. At right, he collaborates with doctoral students Qinghai Wang (left) and Sebastian Brandt. Bender also coaches undergraduates for the Putnam Mathematical Competition, in which the University often ranks in the top 10. 
Washington U. alumnus Jade Vinson was both a student and collaborator of Bender's. "Professor Bender presented problems as if approaching them for the first time," remembers Vinson, A.B./M.A. '97, who also holds a Ph.D. from Princeton and is now a computational biologist at the Massachusetts Institute of Technology. "He explained what a sensible person should try first and why, showing the same excitement you'd expect the first time the problem was solved." In 1997, the paper Vinson coauthored with Bender won the Morgan Prize for the best undergraduate research in the country.
"Carl treats students as equals," Bernard says. "He takes their contributions very seriously. His enthusiasm and caring approach encourages them, and his quick mind tweaks good first steps into brilliant final products."
Back when he was a student himself, Bender initially avoided studying physics because he didn't want to compete with his father. "As far as I could tell, my father knew everything about physics," Bender says. But Bender was eventually drawn to the field by the promise of tackling difficult problems there. "I was very idealistic. I wanted to do something that was hard."
He received encouragement from an undergraduate teacher at Cornell University and went on to do graduate work at Harvard University. He spent time at the Institute for Advanced Study at Princeton, the Massachusetts Institute of Technology, Imperial College in London, the TechnionIsrael Institute of Technology in Haifa, and the Los Alamos National Laboratory before joining Washington University in 1977.
In addition to his work with students, Bender has served on numerous University committees and journal editorial boards over the years. He is now the editorinchief of the Journal of Physics A. "I can always count on Carl as an advocate of the highest quality and integrity," Clark says, "both in the way our department operates as a whole and in our individual approaches to teaching and research."
Right now, Bender continues to enjoy exploring the new area of quantum mechanics he's developed, and to watch others doing the same. "It's not dull," he says. "It's not more of the same. We're actually trying to think of new principles, and that's exciting."
Janni L. Simner, A.B. '89, is a freelance writer based in Tucson, Arizona.
