θ term in a bounded region

We analyze the physical implications of adding a topological density term θTr(F and F) to a gauge theory in a bounded region. In particular, we calculate the Casimir effect on a spherical region and we show that the result is not periodic in θ, contrary to what would be expected for a true topological density. The topological nature of the θ-term can be restored if an additional boundary term required by the Atiyah-Patodi-Singer theorem is included. Then, the periodicity is trivially restored because the resulting Casimir energy is independent of θ. The results of the present work suggest that the observable effects of the θ term could be very small even without assuming θ itself to be small.