Formal GNS construction and WKB expansion in deformation quantization. (English) Zbl 1166.53321

Bordemann, Martin; Waldmann, Stefan

Sternheimer, Daniel (ed.) et al., Deformation theory and symplectic geometry. Proceedings of the Ascona meeting, Switzerland, June 17–21, 1996. Dordrecht: Kluwer Academic Publishers (ISBN 0-7923-4525-8/hbk). Math. Phys. Stud. 20, 315-319 (1997).

From the text: The concept of deformation quantization has been defined and exemplified in F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz and D. Sternheimer [Ann. Phys. 111, 61–110, 111–151 (1978; Zbl 0377.53024, Zbl 0377.53025)]. The existence of star-products on every symplectic manifold has been established in M. De Wilde and P. B. A. Lecomte [ Lett. Math. Phys. 7, 487–496 (1983; Zbl 0526.58023)] and in B. V. Fedosov [J. Differ. Geom 40, No. 2, 213–238 (1994; Zbl 0812.53034)]. This gives a reasonable physical picture of the noncommutative algebra quantum observables with built-in classical limit. However, the discussion of a formal analogue of representations of the deformed algebra in some ‘Hilbert space’ seems to have been restricted to examples in the literature up to now. In [Commun. Math. Phys. 195, No. 3, 549–583 (1998; Zbl 0989.53057)] the present authors have proposed how to construct formal pre-Hilbert spaces for the deformed algebra in the same category by means of a generalized Gel’fand-Naimark-Segal (GNS) construction. We now briefly review this construction and apply this in the next section to the Weyl star-product on the cotangent bundle of \(\mathbb R^n\) (which will give back the usual Schrödinger representation together with the Weyl symmetrization rule; details thereof can be found in the authors’ paper (loc. cit) .
The third section contains new material: we describe how to incorporate the usual WKB expansion into the framework of star-products and GNS representations by means of a certain positive linear functional on the deformed algebra having support on a projectable Lagrangian submanifold graph \((dS)\) of \(T^*\mathbb R^n\). The main trick is to use a suitable form of the star-exponential \(e^{*\frac{it}{\lambda}S}\).
For the entire collection see [Zbl 0923.00023].

MSC:

53D55	Deformation quantization, star products
81Q20	Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81S10	Geometry and quantization, symplectic methods

Citations:

Zbl 0377.53025; Zbl 0377.53024; Zbl 0526.58023; Zbl 0812.53034; Zbl 0989.53057