Nonlocal Lagrangian

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

$\mathcal{L} = \frac{1}{2}(\partial_x \phi(x))^2 - \frac{1}{2}m^2 \phi(x)^2 + \phi(x) \int{\frac{\phi(y)}{(x-y)^2} \, d^ny}$

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

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

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 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.