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 vac- uum,charge,is presented; their relationship,is studied,and,clarified. A recently developed,formulation,is applied,to a simple,one-dimensional,example,with,frac- tional charge.,This example,provides,a setting,in which,the breakdown,of the adiabatic,approximation,and,the winding,of the chiral angle,by multiples,of 27r 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 mathe- matical and physical criterion for the occurence 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.