Is the chiral angle related to the vacuum charge. A study in one dimension

A resolution of ambiguities that arise in relating the chiral angle to the vacuum charge is presented; their relationship is studied and clarified. A recently developed formulation is applied to a simple one-dimensional example with fractional charge. This example provides a setting in which the breakdown of the adiabatic approximation and the winding of the chiral angle by multiples of 2π can be studied. We find that the above situations do not a priori imply a change in the vacuum charge nor any ambiguity in its definition. However, a mathematical and physical criterion for the occurrence of level crossing, leading to an integer change in the charge of the vacuum, is provided. The results are given and analyzed from a physical point of view and their generality is discussed