A nuclear Weyl algebra. (English) Zbl 1287.53073

Waldmann, Stefan

J. Geom. Phys. 81, 10-46 (2014).

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.

Reviewer: Mircea Crâşmăreanu (Iaşi)

MSC:

53D17	Poisson manifolds; Poisson groupoids and algebroids
53D55	Deformation quantization, star products

Keywords:

Weyl algebra; deformation quantization; star products; globally hyperbolic space-time; locally convex algebra

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