Deformation quantization, quantization and the Klein-Gordon equation

The aim of this paper is to give a basic introduction to deformation quantization (DQ) to physicists. We compare DQ to canonical quantization and path integral methods. It is described how certain issues such as the roles of associativity, coordinate invariance, dynamics and operator orderings are understood in the context of DQ. Convergence issues in DQ are mentioned. Additionally, we formulate the Klein-Gordon (KG) equation in DQ. Original results are discussed which include the exact construction of the Fedosov star-product on the dS and AdS spacetimes. Also, the KG equation is written down for these spacetimes.