Mathematicians Illuminate Deep Connection Between Classical And Quantum Physics

In summary, Soundararajan and Holowinsky's proof of the QUE conjecture is a remarkable achievement with significant implications for both mathematics and physics.
  • #1
H.M. Murdock
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I thought this 'd be interesting

http://www.sciencedaily.com/releases/2008/10/081010081650.htm

Mathematicians Illuminate Deep Connection Between Classical And Quantum Physics
ScienceDaily (Oct. 17, 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 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."
 
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  • #2


I find this forum post very interesting. The work of Soundararajan and Holowinsky in solving the QUE conjecture is a significant achievement in the field of quantum chaos. This conjecture has been an important problem in understanding the connections between classical and quantum physics, and their proof provides valuable insight into the behavior of waves in different shapes and manifolds.

Their use of number theory, a branch of pure mathematics, to solve this problem also highlights the unexpected connections between seemingly unrelated fields. This further emphasizes the interdisciplinary nature of science and how different areas of study can come together to solve complex problems.

Moreover, their work has practical applications in understanding the behavior of particles in strong magnetic fields, which has important implications in various fields such as condensed matter physics and materials science.

Overall, this research adds to our understanding of the fundamental principles that govern our physical world and opens up new avenues for exploration and discovery. It is a testament to the power of collaboration and the pursuit of knowledge through both experimentation and mathematical reasoning.
 
  • #3


I find this research to be incredibly exciting and groundbreaking. The deep connection between classical and quantum physics has long been a topic of interest and study, but this new theorem provides even more evidence of the interplay between these two seemingly distinct fields. The fact that the techniques of number theory, a purely abstract branch of mathematics, were able to solve a problem in quantum chaos is truly remarkable and showcases the power of interdisciplinary collaboration in science. This research also has potential implications for our understanding of highly excited states in quantum mechanics, which could have practical applications in various fields such as chemistry and materials science. Overall, this work highlights the importance of mathematics in unraveling the mysteries of the physical world and opens up new avenues for further exploration and discovery.
 

1. What is the significance of the connection between classical and quantum physics?

The connection between classical and quantum physics allows us to better understand the fundamental laws that govern our universe. It also helps bridge the gap between the macroscopic world of classical physics and the microscopic world of quantum physics.

2. How did mathematicians discover this connection?

Mathematicians used advanced mathematical techniques, such as representation theory and number theory, to identify the underlying similarities between classical and quantum physics. They also studied the symmetries and symmetry breaking patterns in both theories.

3. How does this connection impact current research in physics?

This connection has opened up new avenues for research and has led to a deeper understanding of the underlying principles of physics. It has also allowed for the development of new mathematical tools and techniques for studying both classical and quantum systems.

4. What are some practical applications of this discovery?

The connection between classical and quantum physics has potential applications in various fields, such as cryptography, quantum computing, and energy production. It also has implications for our understanding of the universe and the fundamental laws that govern it.

5. Are there any limitations to this connection?

While the connection between classical and quantum physics has provided valuable insights, it is still an ongoing area of research and there are limitations to our current understanding. The connection may not fully explain all phenomena in the universe, and there may be other underlying principles and theories yet to be discovered.

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