Symmetric ordering and Weyl realizations for quantum Minkowski spaces

Abstract

Symmetric ordering and Weyl realizations for non-commutative quantum Minkowski spaces are reviewed. Weyl realizations of Lie deformed spaces and corresponding star products, as well as twist corresponding to Weyl realization and coproduct of momenta, are presented. Drinfeld twists understood in Hopf algebroid sense are also discussed. A few examples of corresponding Weyl realizations are given. We show that for the original Snyder space, there exists symmetric ordering but no Weyl realization. Quadratic deformations of Minkowski space are considered, and it is demonstrated that symmetric ordering is deformed and a generalized Weyl realization can be defined.