Editing Effective action (section)

==Convexity==

[[File:effective_potential_SVG.svg|330px|thumb|right|alt=An example of a two local minima apparent effective potential and the corresponding correct effective potential which is linear in the non-convex region of the apparent potential.|The apparent effective potential <math>V_0(\phi)</math> acquired via perturbation theory must be corrected to the true effective potential <math>V(\phi)</math>, shown via dashed lines in region where the two disagree.]]

For a spacetime with volume <math>\mathcal V_4</math>, the effective potential is defined as <math>V(\phi) = - \Gamma[\phi]/\mathcal V_4</math>. With a [[Hamiltonian (quantum mechanics)|Hamiltonian]] <math>H</math>, the effective potential <math>V(\phi)</math> at <math>\phi(x)</math> always gives the minimum of the expectation value of the [[energy density]] <math> \langle \Omega|H|\Omega\rangle</math> for the set of states <math>|\Omega\rangle</math> satisfying <math>\langle\Omega| \hat \phi| \Omega\rangle = \phi(x)</math>.<ref>{{cite book|first=S.|last=Weinberg|title=The Quantum Theory of Fields: Modern Applications|publisher=Cambridge University Press|date=1995|chapter=16|volume=2|pages=72–74|isbn=9780521670548}}</ref> 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]] <math>V''(\phi) \geq 0</math>.<ref>{{cite book|last1=Peskin|first1=M.E.|author1-link=Michael Peskin|last2=Schroeder|first2=D.V.|date=1995|title=An Introduction to Quantum Field Theory|publisher=Westview Press|pages=368–369|isbn=9780201503975}}</ref>

Calculating the effective potential perturbatively can sometimes yield a non-convex result, such as a potential that has two [[Maxima and minima|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 <math>V_0(\phi)</math> with two local minima whose expectation values <math>\phi_1</math> and <math>\phi_2</math> are the expectation values for the states <math>|\Omega_1\rangle</math> and <math>|\Omega_2\rangle</math>, respectively. Then any <math>\phi</math> in the non-convex region of <math>V_0(\phi)</math> can also be acquired for some <math>\lambda \in [0,1]</math> using

:<math>
|\Omega\rangle \propto \sqrt \lambda |\Omega_1\rangle+\sqrt{1-\lambda}|\Omega_2\rangle.
</math>

However, the energy density of this state is <math>\lambda V_0(\phi_1)+ (1-\lambda)V_0(\phi_2)<V_0(\phi)</math> meaning <math>V_0(\phi)</math> cannot be the correct effective potential at <math>\phi</math> since it did not minimize the energy density. Rather the true effective potential <math>V(\phi)</math> is equal to or lower than this linear construction, which restores convexity.