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Mathematical Horizons for Quantum Physics

(28 Jul - 21 Sep 2008)

... Jointly organized with Centre for Quantum Technologies, NUS
Partially supported by Lee Foundation and Faculty of Science, NUS

Organizing Committee ·  Confirmed Visitors · Overview · Activities · Membership Application
 

 Program Coordinator

  • Huzihiro Araki (Kyoto University)

 Organizing Committee

Co-chairs

  • Berthold Georg Englert (National University of Singapore)
  • Leong Chuan Kwek (Nanyang Technological University and
                                     National University of Singapore)

Secretary

  • Jun Suzuki (National Institute of Informatics, Japan)
  • Bess Yiyuan Fang (National University of Singapore)

 Confirmed Visitors

 Overview

Quantum theory is one of the most important intellectual developments in the early twentieth century. Since then there has been much interplay between theoretical physics and mathematics, both pure and applied. Arguably, the field of Mathematical Physics, equally at home in mathematics and in physics, emerged from John von Neumann’s seminal work on the spectral theory of linear operators in Hilbert space which was triggered by the birth of quantum theory in the mid 1920s. Yet this is just one historical example of how the mathematical insights and tools that are developed in the course of answering challenging mathematical questions arising from physical problems have contributed to the advance of both mathematics and physics. In this tradition, it is the objective of this IMS Programme to bring together Mathematicians, whose work has a bearing on quantum physics, with researchers in Mathematical Physics and Theoretical Physics, whose work will benefit from the mathematical progress. The collaboration between these scientists of different background, different expertise, and different scientific culture will bear fruit on the research of all participants by intellectual cross-fertilization.

In quantum physics, the observables are represented by (self-adjoint) linear operators on a Hilbert space, and states of the system are described as normalized positive linear functionals on an operator algebra. In the historically earliest stage, the spectrum of light emitted from atoms was explained by the spectral analysis of atomic Hamilton operators, and these investigations developed into the broad research field of Schrodinger operators. The modular theory of operator algebras brought about new contact points between mathematics and physics, which turned out to be beneficial for vast developments both in Mathematics and Theoretical Physics. Operator algebra theory became quite powerful and its applications in other branches of mathematics is described by the adjective “non-commutative.” An example is probability theory, widely used in classical physics. The non-commutative probability theory is now well developed, typically called free probability theory, which has its earliest origin in Wigner’s analysis of the spectrum of heavy atoms and is mathematically rooted in operator algebra theory.

In summary, Operator Theory and Operator Algebra Theory form the mathematical basis of Quantum Physics and provide the indispensable mathematical language for theoretical studies in Quantum Physics. Not only are they used in Quantum Physics as powerful tools, but also they are often directly influenced by problems which arise in Quantum Physics. Thus, the unifying mathematical theme of the Programme is Non-Commutative Analysis.
 

 Activities

The Programme will consist of four overlapping three-week Sessions, each devoted to a selected topic. In Session 1, the problem of bringing a given state to a target state by perturbing the interaction with a time-dependent external laser field is studied as a typical subject of quantum control. The specific form of problems in quantum control can stimulate a new development of non-commutative analysis in addition to solving physical problems. The random matrix, whose connection with quantum chaos is being studied, is a typical subject of non-commutative probability theory. (Note that probability theory and analysis are very closely related, especially free probability theory is based on operator algebra theory.)

Session 2 is devoted to operator algebras in quantum information, which is a non- commutative analysis. Equilibrium statistical mechanics has been developed with full use of operator algebra theory, giving a strong influence backward. The same is expected of the subject of Session 3, which is non-equilibrium statistical mechanics. Session 4 deals with relativistic extensions of the traditional Schrödinger operator theory when one is mainly concerned with atoms, molecules and solids on one hand, and deals with the operator algebra description of a system of infinitely many degrees of freedom when one is mainly concerned with the quantized radiation field. Both are cases of non-commutative analysis, mathematically speaking.

Programme Structure

Each Session has a Session Organizer who is in charge of defining the Session and the selection of the Discussion Leaders and the participants. At the start of each Session, there will be presentations by the Discussion Leaders to lay the groundwork. There follows an intense period of about 20 days of discussions and close collaborations among the participants. The Session ends with talks summarizing the progress accomplished and a round-table discussion defining future problems and areas of close collaboration.

Overall Programme Coordinator: Prof. Huzihiro Araki (University of Kyoto)

Session 1: Quantum Control and Dynamics

Topical Problems

  • Molecular quantum control
    • Discussion Leaders: Arne Keller (Universite Paris-Sud), and Hans-Rudolf Jauslin (Université de Bourgogne)
  • Quantum chaos
    • Discussion Leader: Stephan DeBievre (UFR de Mathématiques et Laboratoire CNRS Paul Painlevé)
  • Laser-driven models in quantum computing systems
    • Discussion Leader: Goong Chen (Texas A&M University)

Report of Session 1: PDF

Session 2: Operator Algebras in Quantum Information

  • Period: 11–31 August 2008 (weeks 3-5)
  • Organizers: Burkhard Kümmerer (Technische Universität Darmstadt), Hans Maassen (Radboud University, Nijmegen)

Topical Problems

  • Entropy in quantum channels and the problem of additivity of quantum capacity
    • Discussion Leader: Alexander Holevo (Steklov Mathematical Institute)
  • Stability of quantum algorithms in the presence of external noise
    • Discussion Leader: Mark Fannes (Katholieke Universiteit Leuven)
  • Entanglement of multipartite and infinite systems
    • Discussion Leader: Reinhard Werner (Technische Universität Braunschweig

Report of Session 2: PDF

Session 3: Non-equilibrium Statistical Mechanics

Topical Problems

  • Large deviation theory for quantum fluctuations
    • Discussion Leader: Jan Derezinski (University of Warsaw)
  • Non-equilibrium steady states
    • Discussion Leader: Claude Alain Pillet (Université du Sud Toulon-Var)

Report of Session 3: PDF

Session 4: Strongly Interacting Many-Particle Systems

  • Period: 1–21 September 2008 (weeks 6-8)
  • Organizer: Heinz Siedentop (Ludwig-Maximilians-Universität München)

Topical Problems

  • The theory of large atoms, molecules, and solids
    • Discussion Leaders: Heinz Siedentop (Ludwig-Maximilians-Universität München), Volker Bach (Johannes Gutenberg-Universität Mainz)
  • The mathematical description of the radiation field and its interaction with matter
    • Discussion Leaders: Heinz Siedentop (Ludwig-Maximilians-Universität München), Volker Bach (Johannes Gutenberg-Universität Mainz)

Report of Session 4 : PDF

Public Lectures

Title: Knot or not Knot?
Date & Time: 13 Aug 2008, 6:30pm - 7:30pm
Speaker: Burkhard Kümmerer, Technical University of Darmstadt, Germany
Venue: LT31, Block S16,
            Science Drive 1, Singapore 117543

Title: Are Quantum Computers The Next Generation Of Supercomputers?
Date & Time: 27 Aug 2008, 6:30pm - 7:30pm
Speaker: Reinhard Werner, Technische Universität Braunschweig, Germany
Venue: LT31, Block S16,
            Science Drive 1, Singapore 117543



For attendance at these activities, please complete the online registration form.

The following do not need to register:

  • Invited speakers or participants.

  • Those applying for membership with financial support.

 

 Membership Application

The Institute for Mathematical Sciences invites applications for membership for participation in the above program. Limited funds to cover travel and living expenses are available to young scientists. Applications should be received at least three (3) months before the commencement of membership. Application form is available in (MSWord|PDF|PS) format for download.

More information is available by writing to:
Secretary
Institute for Mathematical Sciences
National University of Singapore
3 Prince George's Park
Singapore 118402
Republic of Singapore
or email to imssec(AT)nus.edu.sg.

For enquiries on scientific aspects of the program, please email Jun Suzuki at physj(AT)nus.edu.sg.

Organizing Committee ·  Confirmed Visitors · Overview · Activities · Membership Application