Convergence of star products: An example

In this talk I will consider the flat phase space $C^n$ with its usual canonical Poisson bracket. Using the Kähler structure one can define a normal ordered (= Wick ordered) quantization for polynomials on $C^n$ and use this to define a corresponding (formal in $\hbar$) star product, the Wick star product. I will explain the crucial positivity properties of this star product which allow to show that any classical state (positive functional) is also a state in the quantum theory without quantum corrections. This (very strong) positivity property is used to define a convergence scheme for the formal power series in $\hbar$. The result will be a Frechet *-algebra of real-analytic functions on which the Wick star product converges and which contains the polynomials as a dense subspace.