Chapter

Noncompact Lie Groups and Some of Their Applications

Volume 429 of the series NATO ASI Series pp 199-224

Path Integrals and Lie Groups

  • Akira InomataAffiliated withDepartment of Physics, State University of New York at Albany
  • , Georg JunkerAffiliated withInstitut für Theoretische Physik I, Universität Erlangen-Nürnberg

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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.