Effective action
quantum corrected version of the classical 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. 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, with slightly different factors of
i
{\displaystyle i}
and sign conventions.
A quantum field theory with action
S
[
ϕ
]
{\displaystyle S[\phi ]}
can be fully described in the path integral formalism using the partition functional
Z
[
J
]
=
∫
D
ϕ
e
i
S
[
ϕ
]
+
i
∫
d
4
x
ϕ
(
x
)
J
(
x
)
.
{\displaystyle Z[J]=\int {\mathcal {D}}\phi e^{iS[\phi ]+i\int d^{4}x\phi (x)J(x)}.}
Since it corresponds to vacuum-to-vacuum transitions in the presence of a classical external current
J
(
x
)
{\displaystyle J(x)}
, it can be evaluated perturbatively as the sum of all connected and disconnected Feynman diagrams. It is also the generating functional for correlation functions
⟨
ϕ
^
(
x
1
)
…
ϕ
^
(
x
n
)
⟩
=
(
−
i
)
n
1
Z
[
J
]
δ
n
Z
[
J
]
δ
J
(
x
1
)
…
δ
J
(
x
n
)
|
J
=
0
,
{\displaystyle \langle {\hat {\phi }}(x_{1})\dots {\hat {\phi }}(x_{n})\rangle =(-i)^{n}{\frac {1}{Z[J]}}{\frac {\delta ^{n}Z[J]}{\delta J(x_{1})\dots \delta J(x_{n})}}{\bigg |}_{J=0},}
where the scalar field operators are denoted by
ϕ
^
(
x
)
{\displaystyle {\hat {\phi }}(x)}
. One can define another useful generating functional
W
[
J
]
=
−
i
ln
Z
[
J
]
{\displaystyle W[J]=-i\ln Z[J]}
responsible for generating connected correlation functions
⟨
ϕ
^
(
x
1
)
⋯
ϕ
^
(
x
n
)
⟩
con
=
(
−
i
)
n
−
1
δ
n
W
[
J
]
δ
J
(
x
1
)
…
δ
J
(
x
n
)
|
J
=
0
,
{\displaystyle \langle {\hat {\phi }}(x_{1})\cdots {\hat {\phi }}(x_{n})\rangle _{\text{con}}=(-i)^{n-1}{\frac {\delta ^{n}W[J]}{\delta J(x_{1})\dots \delta J(x_{n})}}{\bigg |}_{J=0},}
which is calculated perturbatively as the sum of all connected diagrams.
The public source identifies “Effective action” as quantum corrected version of the classical action. This brief keeps that definition visible, then builds a research path around Effective, action and quantum.
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