Heisenberg’s picture and non commutative geometry of the semi classical limit in quantum mechanics. (English) Zbl 0705.46037
The authors propose a new framework to describe the semiclassical limit in phase space. The idea is based on the construction of “tangent groupoid” - the example of noncommutative geometry proposed by A. Connes. It is defined by glueing \(C^*\)-algebras of observables indexed by a varying Planck constant \(\hslash\) and represented in terms of the quantized action-angle variables. The power of this approach is illustrated by a comparison between perturbation expansions in classical and in quantum mechanics through a Lie formalism using Liouville operators. The obtained formal power series are related to Birkhoff expansion for \(\hslash =0\) and to Rayleigh-Schrödinger one for \(\hslash \neq 0\). As a result a rigorous estimation concerning the use of Birkhoff expansion for computing the eigenvalue spectrum of complicated systems is obtained.
Reviewer: R.Alicki
MSC:
| 46L60 | Applications of selfadjoint operator algebras to physics |
| 46L87 | Noncommutative differential geometry |
| 46N50 | Applications of functional analysis in quantum physics |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 46L55 | Noncommutative dynamical systems |
| 46L51 | Noncommutative measure and integration |
| 46L53 | Noncommutative probability and statistics |
| 46L54 | Free probability and free operator algebras |
| 60A99 | Foundations of probability theory |
Keywords:
semiclassical limit in phase space; tangent groupoid; noncommutative geometry; quantized action-angle variables; comparison between perturbation expansions in classical and in quantum mechanics through a Lie formalism using Liouville operators; Birkhoff expansionReferences:
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