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Register a new userConvergence of the Gutt Star Product
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In this work we consider the Gutt star product viewed as an associative deformation of the symmetric algebra S^bullet(g) over a Lie algebra g and discuss its continuity properties: we establish a locally convex topology on S^bullet(g) such that the Gutt star product becomes continuous. Here we have to assume a mild technical condition on g: it has to be an Asymptotic Estimate Lie algebra. This condition is e.g. fulfilled automatically for all finite-dimensional Lie algebras. The resulting completion of the symmetric algebra can be described explicitly and yields not only a locally convex algebra but also the Hopf algebra structure maps inherited from the universal enveloping algebra are continuous. We show that all Hopf algebra structure maps depend analytically on the deformation parameter. The construction enjoys good functorial properties.
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We recall some of the fundamental achievements of formal deformation quantization to argue that one of the most important remaining problems is the question of convergence. Here we discuss different approaches found in the literature so far. The recent developments of finding convergence conditions are then outlined in three basic examples: the Weyl star product for constant Poisson structures, the Gutt star product for linear Poisson structures, and the Wick type star product on the Poincare disc.
Given a locally convex vector space with a topology induced by Hilbert seminorms and a continuous bilinear form on it we construct a topology on its symmetric algebra such that the usual star product of exponential type becomes continuous. Many properties of the resulting locally convex algebra are explained. We compare this approach to various other discussions of convergent star products in finite and infinite dimensions. We pay special attention to the case of a Hilbert space and to nuclear spaces.
The choice of a star product realization for noncommutative field theory can be regarded as a gauge choice in the space of all equivalent star products. With the goal of having a gauge invariant treatment, we develop tools, such as integration measures and covariant derivatives on this space. The covariant derivative can be expressed in terms of connections in the usual way giving rise to new degrees of freedom for noncommutative theories.
A bilinear form on a possibly graded vector space $V$ defines a graded Poisson structure on its graded symmetric algebra together with a star product quantizing it. This gives a model for the Weyl algebra in an algebraic framework, only requiring a field of characteristic zero. When passing to $mathbb{R}$ or $mathbb{C}$ one wants to add more: the convergence of the star product should be controlled for a large completion of the symmetric algebra. Assuming that the underlying vector space carries a locally convex topology and the bilinear form is continuous, we establish a locally convex topology on the Weyl algebra such that the star product becomes continuous. We show that the completion contains many interesting functions like exponentials. The star product is shown to converge absolutely and provides an entire deformation. We show that the completion has an absolute Schauder basis whenever $V$ has an absolute Schauder basis. Moreover, the Weyl algebra is nuclear iff $V$ is nuclear. We discuss functoriality, translational symmetries, and equivalences of the construction. As an example, we show how the Peierls bracket in classical field theory on a globally hyperbolic spacetime can be used to obtain a local net of Weyl algebras.
In this note we classify invariant star products with quantum momentum maps on symplectic manifolds by means of an equivariant characteristic class taking values in the equivariant cohomology. We establish a bijection between the equivalence classes and the formal series in the second equivariant cohomology, thereby giving a refined classification which takes into account the quantum momentum map as well.
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