Deformation quantization in quantum mechanics and quantum field theory.

*(English)* Zbl 1039.53104
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 11-41 (2003).

The concept of deformation quantization and its role in quantum physics is reviewed. The deformation approach is compared with the Hilbert space and Feynman’s path integral approaches to quantum mechanics and illustrated for the case of a simple harmonic oscillator, for an oscillator coupled to an external source, and for a quantum field theory of scalar bosons. A remarkable formula of

*A. S. Cattaneo* and

*G. Felder* [Commun. Math. Phys. 212, 591–611 (2000;

Zbl 1038.53088)] which relates Kontsevich’s star product to the expectation value of a product of functions on a Poisson space is also indicated.

##### MSC:

53D55 | Deformation quantization, star products |

81T70 | Quantization in field theory; cohomological methods |