Jan Derezinski
Department of Mathematical Methods in Physics
Warsaw University

REPRESENTATIONS OF CANONICAL COMMUTATION AND ANTICOMMUTATION RELATIONS

Representations of CCR and CAR is one of classic subjects of mathematical physics, developed in the 50', 60' and 70' by Friedrichs, Wightman, Segal, Berezin, Araki, Fredenhagen and many others. Their theory is one of the most elegant and useful chapters of mathematical physics, providing a natural language for many body quantum physics and quantum field theory.

Physical applications of CCR and CAR include



Contents of the minicourse

I. Representations of CCR: definitions, Schroedinger representation, Stone-von Neumann theorem, metaplectic group.

II. Representations of CAR: definitions, spinor representation, spin group.

III. Fock spaces: bosonic and fermionic second quantization, creation and annihilation operators.

IV. CCR in Fock spaces: Fock and coherent representations, Shale theorem, Bogolubov transformations, generalized metaplectic group.

V. CAR in Fock spaces: extended Fock representation, Shale-Stinespring theorem, Bogolubov transformations, generalized spin group.

VI. Quasifree representations: bosonic and fermionic quasifree vectors and Fock vacua, Araki-Woods and Araki-Wyss representations.

Bibliography.

  1. Baez, J.C., Segal, I.E., Zhou, Z.: Introduction to algebraic and constructive quantum field theory, Princeton NJ, Princeton University Press 1991.
  2. Berezin, F. A. The Method of the Second Quantization, (Russian) 2nd ed. Nauka 1986
  3. Brattelli, O., Robinson D. W.: Operator Algebras and Quantum Statistical Mechanics, Volume 2, Springer-Verlag, Berlin, second edition 1996.
  4. Friedrichs, K. O. Mathematical aspects of quantum theory of fields, New York 1953.
  5. Shale D.: Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103, (1962) 149-167
  6. Shale, D., Stinespring, W. F.: Spinor representations of the infinite orthogonal groups, J. Math. Mech. 14 (1965) 315-322