A survey of star product geometry (Q2738432)

This paper is a survey of some geometrical aspects of star products. The star products considered here on the phase-space \({\mathbb R}^{2n}\) are obtained by exponentiating Poisson brackets. For example, the usual Poisson brackets on \({\mathbb R}^{2n}\) exponentiate to the Moyal star product \(\ast_M\) which corresponds (via Weyl correspondence) to the product of differential operators. In the case when \(n=1\), the product \(f\ast_Mg\) can be defined by an integral formula in which appears an integral kernel related to the area of a phase-space triangle. Similar considerations for chains of star products \(f_1\ast_Mf_2\ast_M\dots\ast_Mf_p\) lead to a geometrical interpretation of associativity and symmetric properties of \(\ast_M\). Other examples of star products are considered. In particular, it is shown that the Poisson brackets of simple Landau orbits in a constant magnetic field exponentiate to a star product.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00039].