×

zbMATH — the first resource for mathematics

Strict deformation quantization of locally convex algebras and modules. (English) Zbl 1330.53117
Summary: In this work, various symbol spaces with values in a sequentially complete locally convex vector space are introduced and discussed. They are used to define vector-valued oscillatory integrals which allow to extend Rieffel’s strict deformation quantization to the framework of sequentially complete locally convex algebras and modules with separately continuous products and module structures, making use of polynomially bounded actions of \(\mathbb{R}^n\). Several well-known integral formulas for star products are shown to fit into this general setting, and a new class of examples involving compactly supported \(\mathbb{R}^n\)-actions on \(\mathbb{R}^n\) is constructed.

MSC:
53D55 Deformation quantization, star products
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, A.; Sternheimer, D., Deformation theory and quantization, Ann. Phys., 111, 61-151, (1978) · Zbl 0377.53025
[2] Gerstenhaber, M., On the deformation of rings and algebras, Ann. of Math., 79, 59-103, (1964) · Zbl 0123.03101
[3] Kontsevich, M., Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66, 157-216, (2003) · Zbl 1058.53065
[4] Waldmann, S., Poisson-geometrie und deformationsquantisierung. eine einführung, (2007), Springer-Verlag Heidelberg, Berlin, New York · Zbl 1139.53001
[5] Rieffel, M. A., Deformation quantization for actions of \(\mathbb{R}^d\), Mem. Amer. Math. Soc., 106, 506, 93, (1993) · Zbl 0798.46053
[6] Connes, A., Noncommutative geometry, (1994), Academic Press San Diego, New York, London · Zbl 0681.55004
[7] Gayral, V.; Gracia-Bondia, J. M.; Iochum, B.; Schucker, T.; Varilly, J. C., Moyal planes are spectral triples, Comm. Math. Phys., 246, 569-623, (2004) · Zbl 1084.58008
[8] Doplicher, S.; Fredenhagen, K.; Roberts, J. E., The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys., 172, 187-220, (1995) · Zbl 0847.53051
[9] Bahns, D.; Waldmann, S., Locally noncommutative space-times, Rev. Math. Phys., 19, 273-305, (2007) · Zbl 1127.81027
[10] Heller, J. G.; Neumaier, N.; Waldmann, S., A \(C^\ast\)-algebraic model for locally noncommutative spacetimes, Lett. Math. Phys., 80, 257-272, (2007) · Zbl 1127.53075
[11] Grosse, H.; Lechner, G., Noncommutative deformations of wightman quantum field theories, J. High Energy Phys., 09, 131, (2008) · Zbl 1245.81069
[12] Buchholz, D.; Lechner, G.; Summers, S. J., Warped convolutions, Rieffel deformations and the construction of quantum field theories, Comm. Math. Phys., 304, 95-123, (2011) · Zbl 1227.46043
[13] Heller, J. G., Lokal nichtkommutative raumzeiten und strikte deformationsquantisierung, (Fakultät für Mathematik und Physik, (2006), Physikalisches Institut, Albert-Ludwigs-Universität Freiburg), (Master’s thesis)
[14] Bieliavsky, P.; Detournay, S.; Spindel, Ph., The deformation quantizations of the hyperbolic plane, Comm. Math. Phys., 289, 2, 529-559, (2009) · Zbl 1170.53074
[15] Bieliavsky, P.; Bonneau, P.; Maeda, Y., Universal deformation formulae, symplectic Lie groups and symmetric spaces, Pacific J. Math., 230, 41-57, (2007) · Zbl 1152.22009
[16] Bieliavsky, P., Strict quantization of solvable symmetric spaces, J. Symplectic Geom., 1, 2, 269-320, (2002) · Zbl 1032.53080
[17] Bielieavky, P.; Massar, M., Oscillatory integral formulae for left-invariant star products on a class of Lie groups, Lett. Math. Phys., 58, 115-128, (2001) · Zbl 0998.53059
[18] Bieliavsky, P.; Gayral, V., Deformation quantization for actions of Kählerian Lie groups, Mem. Amer. Math. Soc., 236, 1115, (2015) · Zbl 1323.22005
[19] Kaschek, D.; Neumaier, N.; Waldmann, S., Complete positivity of rieffel’s deformation quantization by actions of \(R^d\), J. Noncommut. Geom., 3, 361-375, (2009) · Zbl 1172.53055
[20] Hörmander, L., The analysis of linear partial differential operators I, (1990), Springer
[21] Hörmander, L., Fourier integral operators. I, Acta Math., 127, 1-2, 79-183, (1971) · Zbl 0212.46601
[22] Grigis, A.; Sjöstrand, J., Microlocal analysis for differential operators. an introduction, (1994), Cambridge University Press · Zbl 0804.35001
[23] Buchholz, D.; Summers, S. J., Warped convolutions: A novel tool in the construction of quantum field theories, (Seiler, E.; Sibold, K., Quantum Field Theory and Beyond: Essays in Honor of Wolfhart Zimmermann, (2008), World Scientific), 107-121 · Zbl 1206.81072
[24] Gracia-Bondia, J. M.; Varilly, Joseph C., Algebras of distributions suitable for phase space quantum mechanics. I, J. Math. Phys., 29, 869-879, (1988) · Zbl 0652.46026
[25] Dubois-Violette, M.; Kriegl, A.; Maeda, Y.; Michor, P., Smooth *-algebras, Progr. Theor. Phys. Suppl., 144, 54-78, (2001) · Zbl 1026.46062
[26] Baumgärtel, H.; Wollenberg, M., Causal nets of operator algebras, (1992), Akademie Verlag · Zbl 0749.46038
[27] Andersson, A., Operator deformations in quantum measurement theory, Lett. Math. Phys., 104, 415-430, (2014) · Zbl 1300.81007
[28] Much, A., Quantum mechanical effects from deformation theory, J. Math. Phys., 55, (2014) · Zbl 1322.81045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.