The Weyl-Mellin quantization map for $\kappa$-Minkowski Noncommutative Spacetime

TL;DR

This paper introduces a new quantization map for $$-Minkowski spacetime using Mellin transforms, reproducing noncommutative relations and mapping functions into operators with desirable mathematical properties.

Contribution

It develops a Mellin transform-based quantization map for $$-Minkowski spacetime that reproduces the noncommutative relations and maps functions into Hilbert-Schmidt and self-adjoint operators.

Findings

The quantization map reproduces $$-Minkowski commutation relations.

It maps square integrable functions into Hilbert-Schmidt operators.

The paper discusses a deformed $*$ product with examples.

Abstract

We present a quantization of the functions of spacetime, i.e.\ a map, analog to Weyl map, which reproduces the $\kappa$-Minkowski commutation relations, and it has the desirable properties of mapping square integrable funcions into Hilbert-Schmidt operators, as well as real functions into self-adjoint operators. The map is based on Mellin transform on radial and time coordinates. The map also define a deformed $*$ product which we discuss with examples.