This definition over multiple states is necessary because multiple different states, each of which corresponds to a particular source current, may result in the same expectation value. It can further be shown that the effective potential is necessarily a convex function . Calculating the effective potential perturbatively can sometimes yield a non-convex result, such as a potential that has two local minima. However, the true effective potential is still convex, becoming approximately linear in the region where the apparent effective potential fails to be convex. The contradiction occurs in calculations around unstable vacua since perturbation theory necessarily assumes that the vacuum is stable. For example, consider an apparent effective potential with two local minima whose expectation values and are the expectation values for the states and , respectively. Then any in the non-convex region of can also be acquired for some using However, the energy density of this state is meaning cannot be the correct effective potential at since it did not minimize the energy density. Rather the true effective potential is equal to or lower than this linear construction, which restores convexity. See also Background field method Correlation function Path integral formulation Renormalization group Spontaneous symmetry breaking References Further reading Das, A. : Field Theory: A Path Integral Approach, World Scientific Publishing 2006 Schwartz, M.D.: Quantum Field Theory and the Standard Model, Cambridge University Press 2014 Toms, D.J.: The Schwinger Action Principle and Effective Action, Cambridge University Press 2007 Weinberg, S.: The Quantum Theory of Fields: Modern Applications, Vol.II, Cambridge University Press 1996 Quantum field theory