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Weyl quantization for semidirect products. (English) Zbl 1117.81087

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

Citations:

Zbl 0883.22016
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References:

[1] Arnal, D.; Cahen, M.; Gutt, S., Representations of compact Lie groups and quantization by deformation, Acad. R. Belg. Bull. Cl. Sc. 3e série, LXXIV, 45, 123-141 (1988) · Zbl 0681.58016
[2] Arnal, D.; Cortet, J.-C., Nilpotent Fourier transform and applications, Lett. Math. Phys., 9, 25-34 (1985) · Zbl 0616.46041
[3] Arnal, D.; Cortet, J.-C.; Flato, M.; Sternheimer, D., Star-products: quantization and representations without operators, (Tirapegui, E., Field Theory, Quantization and Statistical Physics (1981), D. Reidel Publishing Company), 85-111
[4] Baguis, P., Semidirect products and the Pukansky condition, J. Geom. Phys., 25, 245-270 (1998) · Zbl 0932.22011
[5] B. Cahen, Star-représentations induites, Thèse, Université de Metz, 1992; B. Cahen, Star-représentations induites, Thèse, Université de Metz, 1992
[6] Cahen, B., Deformation program for principal series representations, Lett. Math. Phys., 36, 65-75 (1996) · Zbl 0843.22020
[7] Cahen, B., Quantification d’une orbite massive d’un groupe de Poincaré généralisé, C. R. Acad. Sci. Paris Série I, 325, 803-806 (1997) · Zbl 0883.22016
[8] Cahen, B., Quantification d’orbites coadjointes et théorie des contractions, J. Lie Theory, 11, 257-272 (2001) · Zbl 0973.22009
[9] Cahen, B., Contractions of \(SU(1, n)\) and \(SU(n + 1)\) via Berezin quantization, J. Anal. Math., 97, 83-101 (2005) · Zbl 1131.22005
[10] Cahen, M.; Gutt, S.; Rawnsley, J., Quantization on Kaehler manifolds I, Geometric interpretation of Berezin quantization, J. Geom. Phys., 7, 45-62 (1990) · Zbl 0719.53044
[11] Carinena, J.-F.; Garcia-Bondia, J.-M.; Varilly, J.-C., Relativistic quantum kinematics in the Moyal representation, J. Phys. A, 23, 901-933 (1990) · Zbl 0706.60108
[12] Cotton, P.; Dooley, A. H., Contraction of an adapted functional calculus, J. Lie Theory, 7, 147-164 (1997) · Zbl 0882.22015
[13] Folland, B., Harmonic Analysis in Phase Space (1989), Princeton University Press · Zbl 0682.43001
[14] Gotay, M., Obstructions to quantization, (J. Nonlinear Science, Mechanics: From Theory to Computation (Essays in Honor of Juan-Carlos Simo) (2000), Springer: Springer New York), 271-316 · Zbl 1041.53507
[15] Guillemin, V.; Sternberg, S., Symplectic Techniques in Physics (1984), Cambridge University Press · Zbl 0576.58012
[16] Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, vol. 34 (2001), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0993.53002
[17] Helgason, S., Groups and Geometric Analysis, Mathematical Surveys and Monographs, vol. 83 (2000), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0965.43007
[18] Hormander, L., The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math., 32, 359-443 (1979) · Zbl 0388.47032
[19] Hormander, L., The Analysis of Linear Partial Differential Operators, vol. 3 (1985), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York, (Section 18.5) · Zbl 0612.35001
[20] Kirillov, A. A., Elements of the Theory of Representations, Grundlehren der mathematischen Wissenschaften, vol. 220 (1976), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0342.22001
[21] Kirillov, A. A., Merits and demerits of the orbit method, Bull. Amer. Math. Soc., 36, 4, 433-488 (1999) · Zbl 0940.22013
[22] Kostant, B., Quantization and unitary representations, (Modern Analysis and Applications. Modern Analysis and Applications, Lecture Notes in Mathematics, vol. 170 (1970), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York), 87-207 · Zbl 0223.53028
[23] Landsman, N. P., Strict quantization of coadjoint orbits, J. Math. Phys. 39, 12, 6372-6383 (1998) · Zbl 0986.81057
[24] Rawnsley, J. H., Representations of a semi direct product by quantization, Math. Proc. Camb. Phil. Soc., 78, 345-350 (1975) · Zbl 0313.22014
[25] Simms, D. J., Lie Groups and Quantum Mechanics, Lecture Notes in Mathematics, vol. 52 (1968), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0161.24002
[26] Taylor, M. E., Noncommutative Harmonic Analysis, Mathematical Surveys and Monographs, vol. 22 (1986), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0604.43001
[27] Voros, A., An algebra of pseudo differential operators and the asymptotics of quantum mechanics, J. Funct. Anal., 29, 104-132 (1978) · Zbl 0386.47031
[28] Wildberger, N. J., Convexity and unitary representations of a nilpotent Lie group, Invent. Math., 89, 281-292 (1989) · Zbl 0684.22005
[29] Wildberger, N. J., On the Fourier transform of a compact semi simple Lie group, J. Austral. Math. Soc. A, 56, 64-116 (1994) · Zbl 0842.22015
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