# Institute for Mathematical Sciences Programs & Activities

**Mathematical Horizons for Quantum Physics**

(28 July - 21 September 2008)

**Mathematical Horizons for Quantum Physics**

(28 July - 21 September 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 · Venue · Funding for Young Scientists

- Huzihiro Araki (Kyoto University)

*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)

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.

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

- Period: 28 July–17 August 2008 (weeks 1-3)
- Organizer: Goong Chen (Texas A&M University)

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

- Period: 25 August–14 September 2008 (weeks 5-7)
- Organizer: Claude Alain Pillet (Université du Sud Toulon-Var)

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:

- Those invited to participate.
- Those applying for financial support.

The Institute for Mathematical Sciences has limited funds to cover partial support for travel and living expenses for young scientists interested in participating in the program. Applications should be received at least three (3) months before the commencement of the program. Application form is available in (MSWord|PDF|PS) format for download.

For enquiries, please email us at ims(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 · Venue · Funding for Young Scientists