A "toy" model in QFT with no lower bound to the energy

In quantum field theory, it is generally assumed that there is a lower bound to the energy, which is normally assumed to the the vacuum state. While this may be a reasonable assumption for a free field it is not necessarily the case for interacting fields. In this paper I will examine a “toy” model of a neutral scalar field interacting with a charged scalar field and show that there is no lower bound to the energy in this case.