A method of integration for classical and quantum equations based on the connection between canonical transformations and irreducible representations of Lie groups

TL;DR

This paper introduces a unified method for integrating classical and quantum equations on Lie groups by linking canonical transformations with irreducible Lie group representations, enabling exact solutions for geodesic flows and the Klein-Gordon equation.

Contribution

It presents a novel approach connecting canonical transformations to Lie group representations, facilitating integration of classical and quantum equations on Lie groups.

Findings

Developed a method for integrating geodesic flows on Lie groups.

Constructed exact solutions for the Klein-Gordon equation on unimodular Lie groups.

Established a theorem linking canonical transformations with irreducible Lie group representations.

Abstract

We propose a method for integrating the right-invariant geodesic flows on Lie groups based on the use of a special canonical transformation in the cotangent bundle of the group. We also describe an original method of constructing exact solutions for the Klein - Gordon equation on unimodular Lie groups. Finally, we formulate a theorem which establishes a connection between the special canonical transformation and irreducible representations of Lie group. This connection allows us to consider the proposed methods of integrating for classical and quantum equations in the framework of a unified approach.