Differential and linear integral operators representing polynomial covariance commutation relations in $C^\infty$

TL;DR

This paper constructs representations of polynomial covariance commutation relations using pairs of differential and integral operators in smooth function spaces, including conditions and examples.

Contribution

It introduces new representations of polynomial covariance commutation relations with specific conditions on operators and provides illustrative examples.

Findings

Constructed operator representations satisfying covariance relations

Derived conditions on kernels and coefficients of operators

Provided explicit examples of such operator pairs

Abstract

Representations of polynomial covariance commutation relations by pairs of linear integral and differential operators are constructed in the space of infinitely continuously differentiable functions. Representations of polynomial covariance commutation relations by pairs consisting of a differential and linear integral operator are considered including conditions on kernels and coefficients of operators, and examples.