Making Waves
In a seminar coorganized 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 welldesigned 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."



Uniformly distributed points
in a fundamental domain for SL(2,Z).
Image courtesy of Fredrik Stromberg

 



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 physicsthe rules that govern
the motion of macroscopic objects like people and planets when their motion is chaotic,
with quantum physicsthe 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, as shown in the example to the right.
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 (see the
article by Sarnak for more details).


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 holomorphic 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, Stanford University
Photograph copyright C. J. Mozzochi,
Princeton NJ


Roman Holowinsky,
University of Toronto

 
Please address questions or comments to questions(at)aimath.org
AIM receives major funding from Fry's Electronics and the NSF.

