Nonlocal Lagrangian

In <a href="/facts/Field_theory_(physics)/FJdAKeRA">field theory</a>, a nonlocal Lagrangian is a <a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a>, a type of <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functional</a> 
 
 
 
 
 
 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. <a href="/facts/Space-time/uRIN1x4b">space-time</a>). Examples of such nonlocal Lagrangians might be:

<ul><li>
 
 
 
 
 
 L
 
 
 =
 
 
 1
 2
 
 
 
 
 (
 
 
 
 ∂
 
 x
 
 
 ϕ
 (
 x
 )
 
 
 
 )
 
 
 
 2
 
 
 −
 
 
 1
 2
 
 
 
 m
 
 2
 
 
 ϕ
 (
 x
 
 )
 
 2
 
 
 +
 ϕ
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 x
 )
 ∫
 
 
 
 ϕ
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 (
 x
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 )
 
 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.}
 
</li>
<li>
 
 
 
 
 
 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 }.}
 
</li>
<li>
 
 
 
 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} ).}
 
</li>
<li>The <a href="/facts/Wess%25E2%2580%2593Zumino%25E2%2580%2593Witten_model/WHNuzt64">Wess–Zumino–Witten action</a>.</li></ul>
<a href="/facts/Action_(physics)/NhWS2Mts">Actions</a> obtained from nonlocal Lagrangians are called nonlocal actions. The actions appearing in the fundamental theories of physics, such as the <a href="/facts/Standard_Model/LhVGf9ow">Standard Model</a>, are local actions; nonlocal actions play a part in theories that attempt to go beyond the Standard Model and also in some <a href="/facts/Effective_field_theory/oQ2nWoko">effective field theories</a>. Nonlocalization of a local action is also an essential aspect of some <a href="/facts/Regularization_(physics)/mzFLfUgl">regularization</a> procedures. <a href="/facts/Noncommutative_quantum_field_theory/cxz7kckl">Noncommutative quantum field theory</a> also gives rise to nonlocal actions.

Nonlocal Lagrangian open-in-new

Nonlocal Lagrangian