A nuclear Weyl algebra. (English) Zbl 1287.53073

The objective of this work is to provide a reasonable topology for an algebraic Weyl algebra making the product continuous. The starting point is a general locally convex topology on a (possibly graded) vector space \(V\) and the algebraic version of the Weyl algebra is obtained by means of a deformation quantization of the symmetric algebra \(S(V)\) encoded in a star product. A main result is that for a real parameter \(R\geq \frac{1}{2}\) the Weyl algebra \(\mathcal{W}_{R}(V)\) is nuclear if \(V\) is nuclear. If \(V\) is strongly nuclear then a second version \(\mathcal{W}_{R}{-}(V)\) is also a strongly nuclear Weyl algebra. This general construction is applied to an example from quantum field theory corresponding to a linear field equation on a globally hyperbolic space-time manifold.


53D17 Poisson manifolds; Poisson groupoids and algebroids
53D55 Deformation quantization, star products
Full Text: DOI arXiv


[1] Buchholz, D.; Grundling, H., The resolvent algebra: a new approach to canonical quantum systems, J. Funct. Anal., 254, 11, 2725-2779 (2008) · Zbl 1148.46032
[2] Binz, E.; Honegger, R.; Rieckers, A., Field-theoretic Weyl quantization as a strict and continuous deformation quantization, Ann. Henri Poincaré, 5, 327-346 (2004) · Zbl 1088.81066
[3] Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, A.; Sternheimer, D., Deformation theory and quantization, Ann. Phys., 111, 61-151 (1978) · Zbl 0377.53025
[4] Waldmann, S., Poisson-Geometrie und Deformationsquantisierung, (Eine Einführung (2007), Springer-Verlag: Springer-Verlag Heidelberg, Berlin, New York) · Zbl 1139.53001
[5] Cuntz, J., Bivariant K-theory and the Weyl algebra, K-Theory, 35, 93-137 (2005) · Zbl 1111.19003
[6] Borchers, H. J.; Yngvason, J., Necessary and sufficient conditions for integral representations of Wightman functionals at Schwinger points, Comm. Math. Phys., 47, 3, 197-213 (1976) · Zbl 0319.46060
[7] Dito, G., Deformation quantization on a Hilbert space, (Maeda, Y.; Tose, N.; Miyazaki, N.; Watamura, S.; Sternheimer, D., Noncommutative Geometry and Physics (2005), World Scientific: World Scientific Singapore), 139-157, Proceedings of the CEO International Workshop
[8] Pflaum, M. J.; Schottenloher, M., Holomorphic deformation of Hopf algebras and applications to quantum groups, J. Geom. Phys., 28, 31-44 (1998) · Zbl 1011.17014
[9] Omori, H.; Maeda, Y.; Miyazaki, N.; Yoshioka, A., Deformation quantization of Fréchet-Poisson algebras: convergence of the Moyal product, (Dito, G.; Sternheimer, D., Conférence Moshé Flato (1999)), 233-245, Quantization, Deformations, and Symmetries [31] · Zbl 0987.53036
[10] Beiser, S.; Waldmann, S., Fréchet algebraic deformation quantization of the Poincaré disk, J. Reine Angew. Math., 57 (2011), in press. Preprint arXiv:1108.2004
[11] Beiser, S.; Römer, H.; Waldmann, S., Convergence of the Wick star product, Comm. Math. Phys., 272, 25-52 (2007) · Zbl 1203.53089
[12] Brunetti, R.; Fredenhagen, K.; Ribeiro, P. L., Algebraic Structure of Classical Field Theory I: Kinematics and Linearized Dynamics for Real Scalar Fields, 53 (2012), Preprint arXiv:1209.2148
[13] Dütsch, M.; Fredenhagen, K., Algebraic quantum field theory, perturbation theory, and the loop expansion, Comm. Math. Phys., 219, 5-30 (2001) · Zbl 1019.81041
[14] Dütsch, M.; Fredenhagen, K., Perturbative algebraic field theory, and deformation quantization, Field Inst. Commun., 30, 151-160 (2001) · Zbl 0990.81054
[15] Bursztyn, H.; Waldmann, S., On positive deformations of \({}^\ast \)-algebras, (Dito, G.; Sternheimer, D., Conférence Moshé Flato (1999)), 69-80, Quantization, Deformations, and Symmetries [31] · Zbl 0979.53098
[16] Voigt, C., Bornological quantum groups, Pacific J. Math., 235, 1, 93-135 (2008) · Zbl 1157.46041
[17] Meyer, R., Smooth group representations on bornological vector spaces, Bull. Sci. Math., 128, 2, 127-166 (2004) · Zbl 1037.22011
[18] Fedosov, B. V., Deformation Quantization and Index Theory (1996), Akademie Verlag: Akademie Verlag Berlin · Zbl 0867.58061
[19] Gerstenhaber, M., On the deformation of rings and algebras III, Ann. of Math., 88, 1-34 (1968) · Zbl 0182.05902
[20] Cuntz, J., Bivariante \(K\)-Theorie für lokalkonvexe Algebren und der Chern-Connes-Charakter, Doc. Math., 2, 139-182 (1997) · Zbl 0920.19004
[21] Jarchow, H., Locally Convex Spaces (1981), B.G. Teubner: B.G. Teubner Stuttdart · Zbl 0466.46001
[22] Omori, H.; Maeda, Y.; Miyazaki, N.; Yoshioka, A., Star exponential functions for quadratic forms and polar elements, Contemp. Math., 315, 25-38 (2002) · Zbl 1047.53057
[23] Omori, H.; Maeda, Y.; Miyazaki, N.; Yoshioka, A., Orderings and non-formal deformation quantization, Lett. Math. Phys., 82, 153-175 (2007) · Zbl 1136.53064
[24] Bär, C.; Ginoux, N.; Pfäffle, F., Wave equations on Lorentzian manifolds and quantization, (ESI Lectures in Mathematics and Physics (2007), European Mathematical Society (EMS): European Mathematical Society (EMS) Zürich) · Zbl 1118.58016
[26] Minguzzi, E.; Sánchez, M., The causal hierarchy of spacetimes, (Alekseevsky, D. V.; Baum, H., Recent Developments in Pseudo-Riemannian Geometry. Recent Developments in Pseudo-Riemannian Geometry, ESI Lectures in Mathematics and Physics (2008), European Mathematical Society (EMS): European Mathematical Society (EMS) Zürich), 299-358 · Zbl 1148.83002
[27] Peierls, R. E., The commutation laws of relativistic field theory, Proc. R. Soc. A, 214, 143-157 (1952) · Zbl 0048.44606
[28] Haag, R., Local Quantum Physics (1993), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0843.46052
[29] Hollands, S.; Wald, R. M., Axiomatic quantum field theory in curved spacetime, Comm. Math. Phys., 293, 1, 85-125 (2010) · Zbl 1193.81076
[30] Brunetti, R.; Fredenhagen, K.; Verch, R., The generally covariant locality principle—a new paradigm for local quantum field theory, Comm. Math. Phys., 237, 31-68 (2003) · Zbl 1047.81052
[31] (Dito, G.; Sternheimer, D., Conférence Moshé Flato 1999. Quantization, Deformations, and Symmetries. Conférence Moshé Flato 1999. Quantization, Deformations, and Symmetries, Mathematical Physics Studies, vol. 22 (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London) · Zbl 0949.00040
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