Effective action
In quantum field theory, the quantum effective action is a modified expression for the classical action taking into account quantum corrections while ensuring that the principle of least action applies, meaning that extremizing the effective action yields the equations of motion for the vacuum expectation values of the quantum fields.It was first defined perturbatively by Jeffrey Goldstone and Steven Weinberg in 1962,[1] while the non-perturbative definition was introduced by Bryce DeWitt in 1963[2] and independently by Giovanni Jona-Lasinio in 1964.[3] The article describes the effective action for a single scalar field, however, similar results exist for multiple scalar or fermionic fields.These generating functionals also have applications in statistical mechanics and information theory, with slightly different factors ofcan be fully described in the path integral formalism using the partition functional Since it corresponds to vacuum-to-vacuum transitions in the presence of a classical external current, it can be evaluated perturbatively as the sum of all connected and disconnected Feynman diagrams.[4] Here connected is interpreted in the sense of the cluster decomposition, meaning that the correlation functions approach zero at large spacelike separations.The quantum effective action is defined using the Legendre transformation ofis the source current for which the scalar field has the expectation value, often called the classical field, defined implicitly as the solution to As an expectation value, the classical field can be thought of as the weighted average over quantum fluctuations in the presence of a currentTaking the functional derivative of the Legendre transformation with respect to, the above shows that the vacuum expectation value of 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.1PI diagrams are connected graphs that cannot be disconnected into two pieces by cutting a single internal line.means that there are a number of very useful relations between their correlation functions., is the inverse of the 1PI two-point correlation function A direct way to calculate the effective actionperturbatively as a sum of 1PI diagrams is to sum over all 1PI vacuum diagrams acquired using the Feynman rules derived from the shifted actionappears in any of the propagators or vertices is a place where an externalThis 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 configurationare not automatically symmetries of the quantum effective actionIf the classical action has a continuous symmetry depending on some functionalalways gives the minimum of the expectation value of the energy density[7] 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[8] 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.is equal to or lower than this linear construction, which restores convexity.
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