Representation Theory of Quantized Poincare Algebra. Tensor Operators and Their Application to One-Partical Systems

TL;DR

This paper develops a representation theory for the quantized Poincaré algebra, linking it to classical representations, and constructs tensor operators and invariants useful for q-relativistic equations.

Contribution

It introduces a detailed theory of tensor operators for the $ppa$-Poincare9 algebra, including invariants and the Wigner-Eckart theorem, extending classical representation concepts.

Findings

Representations of QPA are closely related to classical Poincare9 algebra.

Explicit construction of covariant components of four-momenta and Pauli-Lubanski vector.

Proven Wigner-Eckart theorem for QPA.

Abstract

A representation theory of the quantized Poincar\'e ($\kappa$-Poincar\'e) algebra (QPA) is developed. We show that the representations of this algebra are closely connected with the representations of the non-deformed Poincar\'e algebra. A theory of tensor operators for QPA is considered in detail. Necessary and sufficient conditions are found in order for scalars to be invariants. Covariant components of the four-momenta and the Pauli-Lubanski vector are explicitly constructed.These results are used for the construction of some q-relativistic equations. The Wigner-Eckart theorem for QPA is proven.