Functional differential equation approach to the large N expansion and mean field perturbation theory

An apparent difference between formulating mean field perturbation theory for λφ⁴ field theory via path integrals or via functional differential equations when there are external sources present is shown not to exist when mean field theory is considered as the N = 1 limit of the 0(N)λφ⁴ field theory. A simple method is given for determining the 1/N expansion for the Green's functions in the presence of external sources by directly solving the functional differential equations order by order in 1/N. The 1/N expansion for the effective action Γ(φ, χ) is obtained by directly integrating the functional differential equations for the fields φ and χ ( ) in the presence of two external sources j = −δΓ/δφ, S = −δΓ/δχ.