Nonlocal Lagrangian

In field theory, a nonlocal Lagrangian is a Lagrangian, a type of functional ${\displaystyle {\mathcal {L}}[\phi (x)]}$ containing terms that are nonlocal in the fields ${\displaystyle \phi (x)}$, 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:

• ${\displaystyle {\mathcal {L}}={\frac {1}{2}}{\big (}\partial _{x}\phi (x){\big )}^{2}-{\frac {1}{2}}m^{2}\phi (x)^{2}+\phi (x)\int {\frac {\phi (y)}{(x-y)^{2}}}\,d^{n}y.}$
• ${\displaystyle {\mathcal {L}}=-{\frac {1}{4}}{\mathcal {F}}_{\mu \nu }\left(1+{\frac {m^{2}}{\partial ^{2}}}\right){\mathcal {F}}^{\mu \nu }.}$
• ${\displaystyle S=\int dt\,d^{d}x\left[\psi ^{*}\left(i\hbar {\frac {\partial }{\partial t}}+\mu \right)\psi -{\frac {\hbar ^{2}}{2m}}\nabla \psi ^{*}\cdot \nabla \psi \right]-{\frac {1}{2}}\int dt\,d^{d}x\,d^{d}y\,V(\mathbf {y} -\mathbf {x} )\psi ^{*}(\mathbf {x} )\psi (\mathbf {x} )\psi ^{*}(\mathbf {y} )\psi (\mathbf {y} ).}$
• 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.