MAA 7527 - Noncommutative Dynamical Systems

General Course Information

Textbook: Operator Algebras in Dynamical Systems, by Shoichiro Sakai
Prerequisites: Functional Analysis
Credit hours: 3
Grading System:
This is not a typical course. Each student will be required to make one presentation in class and to write one term paper - topics for both to be selected from a list provided by instructor or by mutual agreement.
Office hours: MWF, seventh period (or by appointment)

Brief Course Description

Capitalizing upon the foundations of operator algebra theory established in the courses MAA 7526 and 7527, the aim of this course is to introduce the student to noncommutative dynamical system theory in general and to applications in quantum theory in particular. The text Operator Algebras in Dynamical Systems, by Shoichiro Sakai (Cambridge Univ. Press)) reviews in the first chapter the basic concepts and theorems from the theory of C*-algebras which shall be assumed (and which have been covered in the mentioned courses, with a few exceptions which will be covered in this course in detail). I shall guide the motivated student who has not attended the previous courses quickly through the essential material. The second chapter then introduces the notion of uniformly continuous dynamical systems and studies their relation with bounded derivations. Chapter 3 looks at the situation which arises when the derivation determining the dynamics is unbounded, which is physically more common and which is mathematically more subtle. In Chapter 4 the important notions of ground states and KMS states are introduced. The text will be supplemented with additional material, as needed.

These topics will be addressed essentially at an introductory level (the exact depth of coverage will depend upon the class), since an exhaustive treatment would be technically too involved. They shall be illustrated by applications from theoretical physics, which actually motivated the initiation of this field of mathematics. No prior familiarity with quantum theory - either in the form of quantum mechanics, quantum statistical mechanics or quantum field theory - will be assumed. A heuristic introduction to the physical background will be provided in the course and shall be necessarily somewhat vague. However, the mathematics will not be vague, and the graded projects can be chosen to be purely mathematical. In other words, the mathematics was originally motivated by physics, but can be grasped on its own merits just like any other branch of mathematics.

Since I feel that this material cannot be suitably grasped by using the standard technique of assigning homework problems from a text and then giving exams, the students shall be required to carry out some class projects selected from a list I shall provide and designed both to extend the content of the course and to provide the students with the necessary active participation in the material.

Texts on Operator Algebra Theory

W.B. Arveson, An Invitation to C*_Algebras
K.R. Davidson, C*-Algebras By Example
R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras}, Volumes 1 and 2 (Volumes 3 and 4 contain problems and solutions)
V.I. Paulsen, Completely Bounded Maps and Dilations
G.K. Pedersen, C*-Algebras and Their Automorphism Groups
S. Sakai, C*-Algebras and W*-Algebras
S. Stratila and L. Zsido, Lectures on von Neumann Algebras
V.S. Sunder, An Invitation to Von Neumann Algebras
M. Takesaki, Theory of Operator Algebras, I (only one volume ever appeared)

Texts on Operator Algebras and Quantum Theory

O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Volumes 1 and 2
H. Baumgärtel, Operatoralgebraic Methods in Quantum Field Theory
H. Baumgärtel and M. Wollenberg, Causal Nets of Operator Algebras
E.B. Davies, Quantum Theory of Open Systems
G.G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory
R. Haag, Local Quantum Physics
K. Kraus, States, Effects, and Operations
G. Ludwig, Foundations of Quantum Mechanics, Volumes 1 and 2
M. Ohya and D. Petz, Quantum Entropy and Its Use
G.L. Sewell, Quantum Theory of Collective Phenomena
W. Thirring, A Course in Mathematical Physics , Volumes 3 and 4.
R.M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics

HERE is my introduction to Algebraic Quantum Theory complete with a bibliography and interesting links. (UNDER CONSTRUCTION !)

Class Presentation Topics - Only one student per topic

1. Normal States
(White)

2. Statistical Independence (Start with: S.J. Summers, Rev. Math. Phys., 2, 201-247 (1990))
(Ssembatya)

3. Arveson Spectral Theory and the Spectrum Condition for Higher-Dimensional Groups (Start with: Chapter 8 of G.K. Pedersen, C*-Algebras and Their Automorphism Groups)
(Vishwanath)

4. Spectral Subspaces Determine Representations - includes proof of Theorem 9 (Start with W. Arveson, J. Funct. Anal., 15, 217-243 (1974))

5. Arveson Spectral Theory and Representations of the Canonical Commutation Relations (Start with: W. Arveson, Proc. Symp. Pure Math., 38, 199-269 (1982))
(Chastain)

6. Continuity of Derivations on Spectral Subspaces (Start with: O. Bratteli and A. Kishimoto, Contemp. Math., 62, 403-412 (1987))
(Bereczky)

7. Completely Positive Maps and Stinespring's Theorem (Start with: E. Stormer, Acta Math., 110, 233-278 (1963) and E.B. Davies, Quantum Theory of Open Systems)
(Swearingen)

8. Characterization of Quasi-Covariant Representations of C*-Dynamical Systems (Start with: H.J. Borchers, Commun. Math. Phys., 88, 95-103 (1983))
(Florig)

9. Domains of Closed Derivations (Section 3.3 of the text.)
(Lokvancic)

10. Other Topics in Unbounded Derivation Theory

11. Dissipations (Start with G. Lindblad, Commun. Math. Phys., 48, 119-130 (1976), C.W. Thompson, Commun. Math. Phys., 62, 71-78 (1978))

Term Paper Topics - More than one student can write on the same topic

1. Completely Positive Maps and Quantum Dynamics (Start with: E. Stormer, Acta Math., 110, 233-278 (1963), G. Lindblad, Commun. Math. Phys., 48, 119-130 (1976), C.W. Thompson, Commun. Math. Phys., 62, 71-78 (1978) and E.B. Davies, Quantum Theory of Open Systems)

2. Symmetry Breaking, Goldstone's Theorem and Arveson Spectral Theory (Start with: V. Berzi, Rep. Math. Phys., 16, 293-304 (1979) and V. Berzi, Lett. Math. Phys., 5, 373-377 (1981))

3. Statistical Independence (Start with: S.J. Summers, Rev. Math. Phys., 2, 201-247 (1990))

4. Dilations and Open Systems (Start with: E.B. Davies, Quantum Theory of Open Systems)

5. Arveson Spectral Theory, Connes Spectrum and Applications (Start with: M. Takesaki, Structure of Factors and Automorphism Groups and A. Connes, Ann. Sci. E.N.S., 6, 133-252 (1973))

6. Dissipations

7. Ergodicity and Mixing Properties of Quantum Evolutions

8. Implementability of Automorphism Groups

9. Unbounded Derivations in Abelian C*-Algebras (Start with: Section 3.5 in the text)

10. Other Topics in Unbounded Derivation Theory

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Last updated on April 14, 1997