Twisted crossed products and magnetic pseudodifferential operators. (English) Zbl 1199.46158
Boca, Florin-Petre (ed.) et al., Advances in operator algebras and mathematical physics. Proceedings of the 2nd conference on operator algebras and mathematical physics, Sinaia, Romania, June 26–July 4, 2003. Bucharest: Theta (ISBN 973-85432-7-4). Theta Series in Advanced Mathematics 5, 137-172 (2005).
In a previous paper, M. Măntoiu and R. Purice [J. Math. Phys. 45, No. 4, 1394–1417 (2004; Zbl 1068.81043)] proposed a Weyl pseudodifferential calculus adapted to the study of quantum nonrelativistic particles in the presence of a magnetic field. The aim of the paper under review is to emphasize the connection existing between this calculus and abelian twisted \(C^*\) dynamical systems. One shows that the Weyl calculus for magnetic pseudodifferential operators can be recast in the language of twisted cross products.
The first section of the paper is intended to be a motivation for what follows and is devoted to a heuristic presentation of the case of zero magnetic field. Commutation relations, Weyl calculus, dynamical systems and crossed products, already appear in a simple form. A more systematic review of twisted dynamical systems, covariant representations and twisted crossed products is performed in the next section. Some simplifying assumptions are introduced in order to make this part more accessible to nonspecialists and, at the same time, more adapted to the scope of the paper. One also proves the existence of Schrödinger-type representations. Using the notions introduced in the second section, a generalized pseudodifferential calculus can be defined. This is done in the third section of the paper. Finally one shows that the pseudodifferential magnetic calculus is a particular case of the generalized pseudodifferential calculus we just mentioned.
We have to put into evidence the enlightening remarks and comments made along the whole paper. They contribute essentially to a deeper understanding of the objects involved and of the connections existing between them and put into evidence the advantages of authors’ point of view.
For the entire collection see [Zbl 1108.46300].
The first section of the paper is intended to be a motivation for what follows and is devoted to a heuristic presentation of the case of zero magnetic field. Commutation relations, Weyl calculus, dynamical systems and crossed products, already appear in a simple form. A more systematic review of twisted dynamical systems, covariant representations and twisted crossed products is performed in the next section. Some simplifying assumptions are introduced in order to make this part more accessible to nonspecialists and, at the same time, more adapted to the scope of the paper. One also proves the existence of Schrödinger-type representations. Using the notions introduced in the second section, a generalized pseudodifferential calculus can be defined. This is done in the third section of the paper. Finally one shows that the pseudodifferential magnetic calculus is a particular case of the generalized pseudodifferential calculus we just mentioned.
We have to put into evidence the enlightening remarks and comments made along the whole paper. They contribute essentially to a deeper understanding of the objects involved and of the connections existing between them and put into evidence the advantages of authors’ point of view.
For the entire collection see [Zbl 1108.46300].
Reviewer: Mihai Pascu (Bucharest)
MSC:
| 46L65 | Quantizations, deformations for selfadjoint operator algebras |
| 46L55 | Noncommutative dynamical systems |
| 46N50 | Applications of functional analysis in quantum physics |
| 81R15 | Operator algebra methods applied to problems in quantum theory |
