Weyl quantization for semidirect products. (English) Zbl 1117.81087

Cahen, Benjamin

Differ. Geom. Appl. 25, No. 2, 177-190 (2007).

Summary: Let \(G\) be the semidirect product \(V\rtimes K\) where \(K\) is a connected semisimple non-compact Lie group acting linearily on a finite-dimensional real vector space \(V\). Let \(\mathcal O\) be a coadjoint orbit of \(G\) associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation \(\pi\) of \(G\). We consider the case when the corresponding little group \(K_0\) is a maximal compact subgroup of \(K\). We realize the representation \(\pi\) on a Hilbert space of functions on \(\mathbb R^n\) where \(n=\dim(K)-\dim(K+0)\). By dequantizing \(\pi\) we then construct a symplectomorphism between the orbit \(\mathcal O\) and the product \(\mathbb R^{2n}\times {\mathcal O}'\) where \({\mathcal O}'\) is a little group coadjoint orbit. This allows us to obtain a Weyl correspondence on \(\mathcal O\) which is adapted to the representation \(\pi\) in the sense of [B. Cahen, C. R. Acad. Sci., Paris, Sér. I 325, 803–806 (1997; Zbl 0883.22016)]. In particular we recover well-known results for the Poincaré group.

MSC:

81S10	Geometry and quantization, symplectic methods
22E70	Applications of Lie groups to the sciences; explicit representations
81R05	Finite-dimensional groups and algebras motivated by physics and their representations
81S30	Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics

Keywords:

semidirect products; unitary representations; coadjoint orbits; Weyl quantization

Citations:

Zbl 0883.22016

References:

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