Field on Poincare group and quantum description of orientable objects

TL;DR

This paper develops a quantum framework for relativistic orientable objects using Poincare group representations, extending classical ideas to a quantum setting with wave functions on the group, revealing new symmetry and spin properties.

Contribution

It introduces a novel quantum description of orientable objects based on wave functions on the Poincare group, generalizing Wigner's approach to relativistic systems with orientation.

Findings

Wave functions depend on Poincare group elements and orientation variables.

Four types of spinors characterize the objects, with ten quantum numbers.

Derived relativistic wave equations and analyzed their symmetry properties.

Abstract

We propose an approach to the quantum-mechanical description of relativistic orientable objects. It generalizes Wigner's ideas concerning the treatment of nonrelativistic orientable objects (in particular, a nonrelativistic rotator) with the help of two reference frames (space-fixed and body-fixed). A technical realization of this generalization (for instance, in 3+1 dimensions) amounts to introducing wave functions that depend on elements of the Poincare group $G$. A complete set of transformations that test the symmetries of an orientable object and of the embedding space belongs to the group $\Pi =G\times G$. All such transformations can be studied by considering a generalized regular representation of $G$ in the space of scalar functions on the group, $f(x,z)$, that depend on the Minkowski space points $x\in G/Spin(3,1)$ as well as on the orientation variables given by the elements $z$ of a matrix $Z\in Spin(3,1)$. In particular, the field $f(x,z)$ is a generating function of usual spin-tensor multicomponent fields. In the theory under consideration, there are four different types of spinors, and an orientable object is characterized by ten quantum numbers. We study the corresponding relativistic wave equations and their symmetry properties.