Definitions

# Nonlocal Lagrangian

In field theory, a nonlocal Lagrangian is a Lagrangian, a type of functional $mathcal\left\{L\right\}\left[phi\left(x\right)\right]$ which contains terms which are nonlocal in the fields $phi\left(x\right)$ i.e. which are not polynomials or functions of the fields or their derivatives evaluated at a single point in the space of dynamical parameters (eg. space-time). Examples of such nonlocal Lagrangians might be

$mathcal\left\{L\right\} = frac\left\{1\right\}\left\{2\right\}\left(partial_x phi\left(x\right)\right)^2 - frac\left\{1\right\}\left\{2\right\}m^2 phi\left(x\right)^2 + phi\left(x\right) int\left\{frac\left\{phi\left(y\right)\right\}\left\{\left(x-y\right)^2\right\} , d^ny\right\}$
$mathcal\left\{L\right\} = - frac\left\{1\right\}\left\{4\right\}mathcal\left\{F\right\}_\left\{mu nu\right\}\left(1+frac\left\{m^2\right\}\left\{partial^2\right\}\right)mathcal\left\{F\right\}^\left\{mu nu\right\}$
$S=int dt , d^dx left\left[psi^*\left(ihbar frac\left\{partial\right\}\left\{partial t\right\}+mu\right)psi-frac\left\{hbar^2\right\}\left\{2m\right\}nabla psi^*cdot nabla psiright\right]-frac\left\{1\right\}\left\{2\right\}int dt , d^dx , d^dy , V\left(vec\left\{y\right\}-vec\left\{x\right\}\right)psi^*\left(vec\left\{x\right\}\right)psi\left(vec\left\{x\right\}\right)psi^*\left(vec\left\{y\right\}\right)psi\left(vec\left\{y\right\}\right)$
The WZW 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 which attempt to go beyond the Standard Model, and also appear 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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