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. The effective action also acts as a generating functional for one-particle irreducible correlation functions. The potential component of the effective action is called the effective potential, with the expectation value of the true vacuum being the minimum of this potential rather than the classical potential, making it important for studying spontaneous symmetry breaking.

It was first defined perturbatively by Jeffrey Goldstone, Abdus Salam and Steven Weinberg in 1962, while the non-perturbative definition was introduced by Bryce DeWitt in 1963 and independently by Giovanni Jona-Lasinio in 1964.

The article describes the effective action for a single scalar field, however, similar results exist for multiple scalar or fermionic fields.

Generating functionals

These generating functionals also have applications in statistical mechanics and information theory and sign conventions.

can be fully described in the path integral formalism using the partition functional

, it can be evaluated perturbatively as the sum of all connected and disconnected Feynman diagrams. It is also the generating functional for correlation functions

responsible for generating connected correlation functions

which is calculated perturbatively as the sum of all connected diagrams. Here connected is interpreted in the sense of the cluster decomposition, meaning that the correlation functions approach zero at large spacelike separations. General correlation functions can always be written as a sum of products of connected correlation functions.

The quantum effective action is defined using the Legendre transformation

is the source current, often called the classical field, defined implicitly as the solution to

that sources the scalar field. Taking the functional derivative yields

, 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.

The effective action is also the generating 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

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

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, yielding

Symmetries

Symmetries. If the classical action has a continuous symmetry

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

. With a Hamiltonian always gives the minimum of the expectation value of the energy density. 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 using

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