Symmetric ordering and Weyl realizations for quantum Minkowski spaces

TL;DR

This paper reviews symmetric ordering and Weyl realizations in quantum Minkowski spaces, discussing their mathematical structures, examples, and deformations, including Snyder space and quadratic deformations.

Contribution

It introduces generalized Weyl realizations for deformed Minkowski spaces and analyzes the existence of symmetric ordering and Weyl realizations in these contexts.

Findings

Snyder space admits symmetric ordering but no Weyl realization.

Quadratic deformations allow for deformed symmetric ordering.

Generalized Weyl realizations can be constructed for certain deformations.

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.