Featured Research

from universities, journals, and other organizations

Mathematicians Illuminate Deep Connection Between Classical And Quantum Physics

Date:
October 17, 2008
Source:
American Institute of Mathematics
Summary:
Mathematicians have proven a significant version of the quantum unique ergodicity conjecture. The new work, based in the pure mathematics area of number theory, illuminates deep connections between classical and quantum physics in what is being hailed as one of the best theorems of the year.

Fundamental domains and zeros of cusp forms. This picture shows the zeros of a weight 500 Hecke eigenform in a particular fundamental domain for SL(2,Z). Zeev Rudnick proved that QUE implies that the zeros of the associated cusp forms also are equidistributed in the (hyperbolic) upper half-plane. So, this picture is an illustration of the result of Holowinsky and Soundararajan.
Credit: Image courtesy of Fredrik Stromberg

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 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."


Story Source:

The above story is based on materials provided by American Institute of Mathematics. Note: Materials may be edited for content and length.


Cite This Page:

American Institute of Mathematics. "Mathematicians Illuminate Deep Connection Between Classical And Quantum Physics." ScienceDaily. ScienceDaily, 17 October 2008. <www.sciencedaily.com/releases/2008/10/081010081650.htm>.
American Institute of Mathematics. (2008, October 17). Mathematicians Illuminate Deep Connection Between Classical And Quantum Physics. ScienceDaily. Retrieved July 28, 2014 from www.sciencedaily.com/releases/2008/10/081010081650.htm
American Institute of Mathematics. "Mathematicians Illuminate Deep Connection Between Classical And Quantum Physics." ScienceDaily. www.sciencedaily.com/releases/2008/10/081010081650.htm (accessed July 28, 2014).

Share This




More Matter & Energy News

Monday, July 28, 2014

Featured Research

from universities, journals, and other organizations


Featured Videos

from AP, Reuters, AFP, and other news services

Europe's Highest Train Turns 80 in French Pyrenees

Europe's Highest Train Turns 80 in French Pyrenees

AFP (July 25, 2014) Europe's highest train, the little train of Artouste in the French Pyrenees, celebrates its 80th birthday. Duration: 01:05 Video provided by AFP
Powered by NewsLook.com
TSA Administrator on Politics and Flight Bans

TSA Administrator on Politics and Flight Bans

AP (July 24, 2014) TSA administrator, John Pistole's took part in the Aspen Security Forum 2014, where he answered questions on lifting of the ban on flights into Israel's Tel Aviv airport and whether politics played a role in lifting the ban. (July 24) Video provided by AP
Powered by NewsLook.com
Creative Makeovers for Ugly Cellphone Towers

Creative Makeovers for Ugly Cellphone Towers

AP (July 24, 2014) Mobile phone companies and communities across the country are going to new lengths to disguise those unsightly cellphone towers. From a church bell tower to a flagpole, even a pencil, some towers are trying to make a point. (July 24) Video provided by AP
Powered by NewsLook.com
Algonquin Power Goes Activist on Its Target Gas Natural

Algonquin Power Goes Activist on Its Target Gas Natural

TheStreet (July 23, 2014) When The Deal's Amanda Levin exclusively reported that Gas Natural had been talking to potential suitors, the Ohio company responded with a flat denial, claiming its board had not talked to anyone about a possible sale. Lo and behold, Canadian utility Algonquin Power and Utilities not only had approached the company, but it did it three times. Its last offer was for $13 per share as Gas Natural's was trading at a 60-day moving average of about $12.50 per share. Now Algonquin, which has a 4.9% stake in Gas Natural, has taken its case to shareholders, calling on them to back its proposals or, possibly, a change in the target's board. Video provided by TheStreet
Powered by NewsLook.com

Search ScienceDaily

Number of stories in archives: 140,361

Find with keyword(s):
Enter a keyword or phrase to search ScienceDaily for related topics and research stories.

Save/Print:
Share:

Breaking News:
from the past week

In Other News

... from NewsDaily.com

Science News

Health News

Environment News

Technology News



Save/Print:
Share:

Free Subscriptions


Get the latest science news with ScienceDaily's free email newsletters, updated daily and weekly. Or view hourly updated newsfeeds in your RSS reader:

Get Social & Mobile


Keep up to date with the latest news from ScienceDaily via social networks and mobile apps:

Have Feedback?


Tell us what you think of ScienceDaily -- we welcome both positive and negative comments. Have any problems using the site? Questions?
Mobile: iPhone Android Web
Follow: Facebook Twitter Google+
Subscribe: RSS Feeds Email Newsletters
Latest Headlines Health & Medicine Mind & Brain Space & Time Matter & Energy Computers & Math Plants & Animals Earth & Climate Fossils & Ruins