# zbMATH — the first resource for mathematics

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.

##### MSC:
 53D17 Poisson manifolds; Poisson groupoids and algebroids 53D55 Deformation quantization, star products
Full Text:
##### References:
 [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 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 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 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 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) Zürich) · Zbl 1118.58016 [25] S. Waldmann, Geometric Wave Equations, 2012, p. 279 + vi, Lecture Notes for the Lecture ‘Wellengleichgungen auf Raumzeiten’ held in Freiburg in 2008/2009 and 2009. Preprint arXiv:1208.4706. [26] Minguzzi, E.; Sánchez, M., The causal hierarchy of spacetimes, (Alekseevsky, D. V.; Baum, H., Recent Developments in Pseudo-Riemannian Geometry, ESI Lectures in Mathematics and Physics, (2008), 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 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, Mathematical Physics Studies, vol. 22, (2000), Kluwer Academic Publishers Dordrecht, Boston, London)
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.