Topological effects in quantum theory

Abstract

We show that the topology of configuration space affects the quantum dynamics of the system using path integral formulation. The allowed phase factors that can appear is shown to be related to the one-dimensional representations of the fundamental group of the configuration space. For the case in which the space is homotopic to $S^1$, we consider Aharonov-Casher effect and show that it is physically equivalent to the precession of a (spin) magnetic moment in magnetic field, which implies that AC effect is really a topological effect. For Berry``s topological phase, we derive a simple constraint relation between Berry``s phases. Applying this relation to slow variable dynamics, we show that, on the average, slow variable dynamics is not affected by fast variables. The confuguration space of identical particles in 2D is multiply connected. Using braid group relations for sphere, cylinder and torus, we obtain the allowed $\theta$ values that characterize statistics.