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Nonlocal Lagrangian

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Lagrangians with nonlocal physics
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In field theory, a nonlocal Lagrangian is a Lagrangian, a type of functional L [ ϕ ( x ) ] {\displaystyle {\mathcal {L}}} 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-{\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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