nonlocal Definition and 1 Threads

In field theory, a nonlocal Lagrangian is a Lagrangian, a type of functional





L


[
ϕ
(
x
)
]


{\displaystyle {\mathcal {L}}[\phi (x)]}

containing terms that are nonlocal in the fields



ϕ
(
x
)


{\displaystyle \phi (x)}

, i.e. not polynomials or functions of the fields or their derivatives evaluated at a single point in the space of dynamical parameters (e.g. space-time). Examples of such nonlocal Lagrangians might be:






L


=


1
2




(



∂

x


ϕ
(
x
)



)



2


−


1
2



m

2


ϕ
(
x

)

2


+
ϕ
(
x
)
∫



ϕ
(
y
)


(
x
−
y

)

2







d

n


y
.


{\displaystyle {\mathcal {L}}={\frac {1}{2}}{\big (}\partial _{x}\phi (x){\big )}^{2}-{\frac {1}{2}}m^{2}\phi (x)^{2}+\phi (x)\int {\frac {\phi (y)}{(x-y)^{2}}}\,d^{n}y.}







L


=
−


1
4





F



μ
ν



(

1
+



m

2



∂

2





)




F



μ
ν


.


{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}{\mathcal {F}}_{\mu \nu }\left(1+{\frac {m^{2}}{\partial ^{2}}}\right){\mathcal {F}}^{\mu \nu }.}





S
=
∫
d
t


d

d


x

[


ψ

∗



(

i
ℏ


∂

∂
t



+
μ

)

ψ
−



ℏ

2



2
m



∇

ψ

∗


⋅
∇
ψ

]

−


1
2


∫
d
t


d

d


x


d

d


y

V
(

y

−

x

)

ψ

∗


(

x

)
ψ
(

x

)

ψ

∗


(

y

)
ψ
(

y

)
.


{\displaystyle S=\int dt\,d^{d}x\left[\psi ^{*}\left(i\hbar {\frac {\partial }{\partial t}}+\mu \right)\psi -{\frac {\hbar ^{2}}{2m}}\nabla \psi ^{*}\cdot \nabla \psi \right]-{\frac {1}{2}}\int dt\,d^{d}x\,d^{d}y\,V(\mathbf {y} -\mathbf {x} )\psi ^{*}(\mathbf {x} )\psi (\mathbf {x} )\psi ^{*}(\mathbf {y} )\psi (\mathbf {y} ).}


The Wess–Zumino–Witten action.
Actions obtained from nonlocal Lagrangians are called nonlocal actions. The actions appearing in the fundamental theories of physics, such as the Standard Model, are local actions; nonlocal actions play a part in theories that attempt to go beyond the Standard Model and also in some effective field theories. Nonlocalization of a local action is also an essential aspect of some regularization procedures. Noncommutative quantum field theory also gives rise to nonlocal actions.

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