Path integrals on group manifolds. The representation independent propagator for general Lie groups. (English) Zbl 0916.58009

Singapore: World Scientific. xviii, 213 p. (1998).
Path integrals for physical systems moving on group manifolds (sometimes also on symmetric spaces), have become a subject of research in its own right. The present work emphasizes that the propagator for such a fictitious particle is a well defined generalized function and can be introduced independently that of the representation provided that the underlying Lie group \(G\) is real, separable, connected and simply connected. In addition, it is assumed that \(G\) has continuous, irreducible, unitary representations \(U\) that are square integrable in the sense that \[ \int_G| (U_gx,y)|^2 dg< \infty \] for suitably chosen vectors \(x\) and \(y\). A theorem of Duflo and Moore then guarantees the existence of a resolution of the identity.
After presenting a collection of standard results from algebra, functional analysis, and representation theory, the author discusses propagators in simple cases (one degree of freedom). He then studies path integrals on group manifolds and symmetric spaces. As an example, the propagator for the group SU(2) is constructed explicitly which relates to the quantized spinning top. Coherent-state propagators in the sense of Klauder are introduced and their classical limits are studied. The last chapter presents an outlook on future developments. Since the emphasis is on functional analysis on Lie groups, the physical applications are scarce in this work.


58D30 Applications of manifolds of mappings to the sciences
81S40 Path integrals in quantum mechanics
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory