Is formal QED mathematically consistent?
  The formal structure of quantum electrodynamics consists of various elements. These include the Schroedinger equation which evolves the system forward in time, the vacuum state which is assumed to be the state with a free field energy of zero, and the principle of gauge invariance. In this paper we will examine the mathematical consistency of these elements. It will be shown that these elements, which are basic to the formal structure of the theory, are not mathematically consistent. 
Dan Solomon