  Three well-known perturbative approaches to deriving low-energy effective
theories, the degenerate Brillouin-Wigner perturbation theory (projection
method), the canonical transformation, and the resolvent methods are compared.
We use the Hubbard model as an example to show how, to fourth order in hopping
t, all methods lead to the same effective theory, namely the t-J model with
ring exchange and various correlated hoppings. We emphasize subtle technical
difficulties that make such a derivation less trivial to carry out for orders
higher than second. We also show that in higher orders, different approaches
can lead to seemingly different forms for the low-energy Hamiltonian. All of
these forms are equivalent since they are connected by an additional unitary
transformation whose generator is given explicitly. The importance of
transforming the operators is emphasized and the equivalence of their
transformed structure within the different approaches is also demonstrated.
