Path Integrals and Lie Groups

TL;DR

This paper explores how Lie groups influence path integral quantization in quantum mechanics, illustrating with examples of systems with non-compact symmetries like SO(d,1) and SU(1,1), and applying harmonic analysis techniques.

Contribution

It demonstrates the application of Lie group symmetries to path integrals in quantum systems, including non-compact groups and their harmonic analysis methods.

Findings

Path integrals can be analyzed using Lie group symmetries.

Harmonic analysis on homogeneous spaces aids in quantizing systems.

Examples include free particle motion on negatively curved spaces and harmonic oscillators.

Abstract

The roles of Lie groups in Feynman's path integrals in non-relativistic quantum mechanics are discussed. Dynamical as well as geometrical symmetries are found useful for path integral quantization. Two examples having the symmetry of a non-compact Lie group are considered. The first is the free quantum motion of a particle on a space of constant negative curvature. The system has a group SO(d,1) associated with the geometrical structure, to which the technique of harmonic analysis on a homogeneous space is applied. As an example of a system having a non-compact dynamical symmetry, the d-dimensional harmonic oscillator is chosen, which has the non-compact dynamical group SU(1,1) besides its geometrical symmetry SO(d). The radial path integral is seen as a convolution of the matrix functions of a compact group element of SU(1,1) on the continuous basis.