the fields extremize the quantum effective action rather than the classical action. This is nothing more than the principle of least action in the full quantum field theory. The reason for why the quantum theory requires this modification comes from the path integral perspective since all possible field configurations contribute to the path integral, while in classical field theory only the classical configurations contribute. The effective action is also the generation functional for one-particle irreducible (1PI) correlation functions. 1PI diagrams are connected graphs that cannot be disconnected into two pieces by cutting a single internal line. Therefore, we have with being the sum of all 1PI Feynman diagrams. The close connection between and means that there are a number of very useful relations between their correlation functions. For example, the two-point correlation function, which is nothing less than the propagator , is the inverse of the 1PI two-point correlation function Methods for calculating the effective action A direct way to calculate the effective action perturbatively as a sum of 1PI diagrams is to sum over all 1PI vacuum diagrams acquired using the Feynman rules derived from the shifted action . This works because any place where appears in any of the propagators or vertices is a place where an external line could be attached. This is very similar to the background field method which can also be used to calculate the effective action. Alternatively, the one-loop approximation to the action can be found by considering the expansion of the partition function around the classical vacuum expectation value field configuration , yielding Symmetries Symmetries of the classical action are not automatically symmetries of the quantum effective action . If the classical action has a continuous symmetry depending on some functional then this directly imposes the constraint This identity is an example of a Slavnov–Taylor identity. It is identical to the requirement that the effective action is invariant under the symmetry transformation This symmetry is identical to the original symmetry for the important class of linear symmetries For non-linear functionals the two symmetries generally differ because the average of a non-linear functional is not equivalent to the functional of an average. Convexity For a spacetime with volume , the effective potential is defined as . With a Hamiltonian , the effective potential at always gives the minimum of the expectation value of the energy density for the set of states satisfying .