Triviality from the Exact Renormalization Group

Using the exact renormalization group, it is shown that no physically acceptable non-trivial fixed points, with positive anomalous dimension, exist for (i) O(N) scalar field theory in four or more dimensions, (ii) non-compact, pure Abelian gauge theory in any dimension. It is then shown, for both theories in any dimension, that otherwise physically acceptable non-trivial fixed points with negative anomalous dimension are non-unitary. In addition, a very simple demonstration is given, directly from the exact renormalization group, that should a critical fixed point exist for either theory in any dimension, then the n-point correlation functions exhibit the expected behaviour.