New practical methods for consistently closing truncated equations of motion

We discuss some new methods for truncating and closing the equations of motion for composite operators. Our formalism automatically generates a physical theory in the sense of having positive spectral weights. Another important advantage of our methods is that (for judiciously chosen set of operators) all averages necessary to close the formalism can be calculated self-consistently from within the theory itself. We discuss how the method can be applied to the Hubbard model in both fermionic and bosonic sectors.