Group-theoretical derivation of path integrals

Akira Inomata, Georg Junker

E. A. Kochetov

A path integral written in terms of the group theoretic coherent states by using the Kähler structure of the coherent state manifold with the particular emphasis on the boundary-fixing term derivation is considered herein. The path integral for a propagator of the system with Hamiltonian linear in the SU(2)/SU(1,1) generators is shown to be diagonalized by an appropriate motion in the phase space.

M. Böhm, Georg Junker

Applications of group theoretical methods in the path integral formalism of nonrelativistic quantum theory are considered. Analysis of the symmetry of the Lagrangian leads to the expansion of the short time propagator in matrix elements of unitary irreducible representations of the symmetry group. Identification of the coordinates with the group parameters transforms the path integral to integrals over the group manifold. The integration is performed using the orthogonality of the representations. Compact and noncompact rotation groups are considered, where the corresponding path integral is embedded in Euclidean and pseudo-Euclidean spaces, respectively. The unit sphere and unit hyperboloid may either be viewed as the group manifold itself or at least as a group quotient. In the first case Fourier analysis leads to an expansion in group characters. In the second case an expansion in zonal spherical functions is obtained. As examples the groups SO(n), SU(2), SO(n−1,1), and SU(1,1) are explicitly discussed. The path integral on SO(n+m) and SO(n,m) in bispherical coordinates is also treated.

Wolfgang A. Tomé

M. Böhm, Georg Junker

The path integral for the n-dimensional free particle is considered. According to the underlying symmetry, the short time propagator is expanded in zonal spherical functions of the Euclidean group G=T n ■ SO(n) with respect to the subgroup H=SO(n). The group theoretical approach to path integration, including the radial part, is explicitly demonstrated.