Relativistic wavefunctions on the Poincare group

TL;DR

This paper develops a formalism for relativistic wavefunctions on the Poincaré group, demonstrating their factorization, providing a Lagrangian approach, and explicitly constructing spin-1/2 wavefunctions, linking to quantum field theory.

Contribution

It introduces a factorization of relativistic wavefunctions on the Poincaré group and formulates their Lagrangian and field equations, connecting group harmonic analysis with quantum field theory.

Findings

Wavefunctions factorize into translation and Lorentz parts.

Explicit spin-1/2 wavefunction construction.

Relation established with quantum field theory.

Abstract

The Biedenharn type relativistic wavefunctions are considered on the group manifold of the Poincar\'{e} group. It is shown that the wavefunctions can be factorized on the group manifold into translation group and Lorentz group parts. A Lagrangian formalism and field equations for such factorizations are given. Parametrizations of the functions obtained are studied in terms of a ten-parameter set of the Poincar\'{e} group. An explicit construction of the wavefunction for the spin 1/2 is given. A relation of the proposed description with the quantum field theory and harmonic analysis on the Poincar\'{e} group is discussed.