Semiclassical Analysis and Quantization

Workshop at Cologne, 2nd-3rd of May 2008

Purpose of the workshop

Since Classical Mechanics is not valid at the atomic scale, it is necessary to use Quantum Mechanics, and in many cases, to go from the given classical system to its quantal description. This process is known as Quantization. The converse process is also very important. That is, given a quantum description of a system, we want to study the behaviour of the system when the Planck constant h is small. The regime of Quantum Mechanichs when h is small is called semiclassical regime.

A manifestation of this regime is through the spectrum of operators which represent observables. The asymptotics of the spectrum and related objects (the projection on the low-lying states) when h goes to zero involve geometric properties of the classical system.

The purpose of this workshop is to discuss some recent developments in the area. There will be ample time and opportunity for discussions.

Brice Camus
Patrick Erdelt
(Frankfurt am Main)
Laurent Charles
William Kirwin
Stefan Waldmann
Nikolai Neumaier

G. Marinescu

Brice Camus, Semi-classical spectral estimates

Abstract: In this talk I will present the semi-classical trace formula for $h$-pseudodifferential operators. This formula is a mathematically rigorous version of M. Gutzwiller trace formula which provides, in the semi-classical regime, a duality between quantum-spectrum and length-spectrum. I will emphasize on the mathematical "raison d'ętre" of this relation and try to give some extensions when the classical dynamical system admits an equilibrium. According to time allowance I can try to explain applications to eigenvectors-estimates.

Laurent Charles, Quantization of polygon spaces

Moduli spaces of polygons have been studied since the nineties for their topological and symplectic properties. Under generic assumptions these are symplectic manifolds with natural global action angle coordinates. Applying geometric quantization to the polygon spaces, one obtains invariant subspaces of tensor product of several irreducible representation of SU(2). These quantum spaces admit natural sets of commuting observables. I will explain in which sense these operators quantize the action coordinates and how we can use them to prove the asymptotics of the 6j-symbols.

Will Kirwin, Unitarity in "quantization commutes with reduction''

Abstract: Two common operations on the category of symplectic manifolds are geometric quantization and symplectic reduction. A result of Guillemin and Sternberg (1985), commonly referred to as ''quantization commutes with reduction'', states that there is a certain geometrically natural isomorphism between ''reduce then quantize'' and ''quantize then reduce''. We will review the basic facts of the case, and see that although the isomorphism is geometrically natural, it is not in general unitary (which is the canonical notion of ''equivalent'' in quantum mechanics). In this talk, I will explain in what sense unitarity can be recovered; a full understanding of the situation requires two additional ideas: the semi-classical limit and the metaplectic correction. (This is joint work with Brian Hall (U. Notre Dame)).

Stefan Waldmann, Deformation Quantization of Linear Poisson Structures

Abstract: In this talk I will briefly discuss a geometric construction of a formal star product on the dual of a vector bundle E* which quantizes in frist order a linear Poisson structure arising from a Lie algebroid structure on E. This will provide an alternative construction to known quantizations. If there is time, I will discuss the role of unimodularity and traces.

Friday, 2.05.08

11:45-13:00 Waldmann

16:00-17:15 Kirwin

Saturday, 3.05.08

10:00-11:15 Charles

11:45-13:00 Camus

Supported by  
SFB Symmetries and Universality in Mesoscopic Systems
Graduiertenkolleg Globale Strukturen in Geometrie und Analysis


George Marinescu

last modified at

23th of April 2008