Functional evaluation of the effective potential

By use of the path-integral formulation of quantum mechanics, a series expansion for the effective potential is derived. Each order of the series corresponds to an infinite set of conventional Feynman diagrams, with a fixed number of loops. As an application of the formalism, three calculations are performed. For a set of $n$ self-interacting scalar fields, the effective potential is computed to the two-loop approximation. Also, all loops are summed in the leading-logarithmic approximation when $n$ gets large. Finally, the effective potential for scalar, massless electrodynamics is derived in an arbitrary gauge. It is found that the potential is gauge-dependent, and a specific gauge is exhibited in which all one-loop effects disappear.