Induced Representations of Tensors and Spinors of any Rank in the SHP Theory

TL;DR

This paper introduces a modified induced representation approach for relativistic particles with spin, enabling the construction of arbitrary rank spinors and tensors with invariant angular momentum decomposition, applicable to wave packets and many-body systems.

Contribution

It develops a new method to construct spinors and tensors of any rank using a modified Wigner's induced representation, including a covariant Pauli-Lubanski operator.

Findings

Constructed scalar and vector fields with transformation properties.

Developed a covariant Pauli-Lubanski operator as a Casimir of the Poincaré group.

Enabled the description of many-body relativistic systems with definite angular momentum.

Abstract

We show that a modification of Wigner's induced representation for the description of a relativistic particle with spin can be used to construct spinors and tensors of arbitrary rank, with invariant decomposition over angular momentum. In particular, scalar and vector fields, as well as the representations of their transformations, are constructed. The method that is developed here admits the construction of wave packets and states of a many body relativistic system with definite total angular momentum. Furthermore, a Pauli-Lubanski operator is constructed on the orbit of the induced representation which provides a Casimir operator for the Poincar\'e group and which contains the physical intrinsic angular momentum of the particle covariantly.