Making waves
Mathematicians crack quantum chaos conjecture
PALO ALTO, Calif., October 10, 2008 - In a seminar co-organized
by Stanford University and the American Institute of Mathematics,
Soundararajan announced that he and Roman Holowinsky have proven
a significant version of the quantum unique ergodicity (QUE)
conjecture. "This is one of the best theorems of the year," said Peter
Sarnak, a mathematician from Princeton who along with Zeev Rudnick from
the University of Tel Aviv formulated the conjecture fifteen years
ago in an effort to understand the connections between classical and
quantum physics. "I was aware that Soundararajan and Holowinsky were both
attacking QUE using different techniques and was astounded to find that
their methods miraculously combined to completely solve the problem,"
said Sarnak. Both approaches come from number theory, an area of pure
mathematics which recently has been found to have surprising connections
to physics.
The motivation behind the problem is to understand how waves are
influenced by the geometry of their enclosure. Imagine sound waves in
a concert hall. In a well-designed concert hall you can hear every note
from every seat. The sound waves spread out uniformly and evenly. At the
opposite extreme are "whispering galleries" where sound concentrates in
a small area.
The mathematical world is populated by all kinds of shapes, some of
which are easy to picture, like spheres and donuts, and others which
are constructed from abstract mathematics. All of these shapes have
waves associated with them. Soundararajan and Holowinsky showed that
for certain shapes that come from number theory, the waves always spread
out evenly. For these shapes there are no "whispering galleries."
Quantum chaos
The quantum unique ergodicity conjecture (QUE) comes from the area
of physics known as "quantum chaos." The goal of quantum chaos is to
understand the relationship between classical physics--the rules that
govern the motion of macroscopic objects like people and planets when
their motion is chaotic, with quantum physics--the rules that govern
the microscopic world.
"The work of Holowinsky and Soundararajan is brilliant," said physicist
Jens Marklof of Bristol University, "and tells us about the behaviour
of a particle trapped on the modular surface in a strong magnetic field."
The problems of quantum chaos can be understood in terms of billiards.
On a standard rectangular billiard table the motion of the balls is
predictable and easy to describe. Things get more interesting if the
table has curved edges, known as a "stadium." Then it turns out most
paths are chaotic and over time fill out the billiard table, a result
proven by the mathematical physicist Leonid Bunimovich.
In the quantum or microscopic setting one investigates the waves that
are associated to the billiard table. The waves often spread out
uniformly. Sometimes, however, waves concentrate along an unstable
periodic path. Physicists call this "scarring."
For the stadium system yet another interesting thing can happen, known as
a "bouncing ball mode." Bouncing ball modes were observed experimentally
and only recently proven to exist by Andrew Hassell.
In their QUE conjecture, Rudnick and Sarnak hypothesized that for a large
class of systems, unlike the stadium there are no scars or bouncing ball
states and in fact all states become evenly distributed. Holowinsky and
Soundararajan's work shows that the conjecture is true in the number
theoretic setting.
Highly excited states
The conjecture of Rudnick and Sarnak deals with certain kinds of shapes
called manifolds, or more technically, manifolds of negative curvature,
some of which come from problems in higher arithmetic. The corresponding
waves are analogous to highly excited states in quantum mechanics.
Soundararajan and Holowinsky each developed new techniques to solve a
particular case of QUE. The "waves" in this setting are known as Hecke
eigenforms. The approaches of both researchers work individually most
of the time and miraculously when combined they completely solve the
problem. "Their work is a lovely blend of the ideas of physics and
abstract mathematics," said Brian Conrey, Director of the American
Institute of Mathematics.
According to Lev Kaplan, a physicist at Tulane University, "This is a
good example of mathematical work inspired by an interesting physical
problem, and it has relevance to our understanding of quantum behavior
in classically chaotic dynamical systems."
Soundararajan's talk will take place at noon on Friday, October 10,
at the Stanford Math Department.
About the American Institute of Mathematics
The American Institute of Mathematics, a nonprofit organization,
was founded in 1994 by Silicon Valley businessmen John Fry and Steve
Sorenson, longtime supporters of mathematical research. AIM is one of
the seven mathematics institutes in the U.S. funded by the National
Science Foundation. The mission of AIM is to expand the frontiers of
mathematical knowledge through sponsoring focused research projects
and workshops and encouraging collaboration among mathematicians at all
levels. AIM currently resides in Palo Alto, California, while awaiting the
completion of its permanent headquarters in Morgan Hill, California. For
more information, visit www.aimath.org.
###
Contacts:
* Brian Conrey, Executive Director of AIM, 650.845.2071, conrey@aimath.org
* Peter Sarnak, Eugene Higgins Professor of Mathematics at Princeton
University and Chair of AIM's Scientific Board, 609.258.4200 or
609.258.4229, sarnak@math.princeton.edu