Path integrals in quantum theory and some applications to homogeneous spaces

Abstract

In this work we consider the path integrals in quantum theory and some of their applications to homogeneous spaces from the semiclassical point of view. We begin with the basic principles underlying the Feynman path integral formulation of quantum mechanics and show the do;nain where the method is useful and powerful. We present a method for calculating the path integrals for quadratic Lagrangians and apply it to the example of harmonic oscillator. Semiclassical propagator given by the Van Vleck-Pauli formula is also discussed. We next handle the Hamiltonian derivation of path integral. The starting-point for this derivation is the usual operator formalism of quantum mechanics. Later on we consider path integrals on homogeneous spaces concentrating on the motion on group manifolds. It turns out that for the free motion on the group manifold the semiclassical approximation the exact solution. We thus study the path integrals for U(l) and SU(2) groups. In these cases the Propagator is calculated directly by two methods : the sum over classical paths and the stationary state expansion, which are shown explicitly to be equivalent. We finally give some remarks for motion on the hyperspheres S2n +1 and S2n. Deriving a recursion relation for the propagator of S2n +1 we claim that there is the possibility of getting the exact solution from some kind of semiclassical propagator for S2n +1.