Physics:Nonlocal Lagrangian

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

• ${ℒ}=\frac{1}{2}({\partial }_x\varphi (x){)}^2-\frac{1}{2}{m}^2\varphi (x{)}^2+\varphi (x)\int \frac{\varphi (y)}{(x-y{)}^2}{d}^{n}y.$
• ${ℒ}=-\frac{1}{4}{{ℱ}}_{\mu \nu }(1+\frac{m^2}{{\partial }^2}){{ℱ}}^{\mu \nu }.$
• $S=\int dt{d}^{d}x[{\psi }^{*}(i\hslash \frac{\partial }{\partial t}+\mu )\psi -\frac{{\hslash }^2}{2m}\mathrm{\nabla }{\psi }^{*}\cdot \mathrm{\nabla }\psi ]-\frac{1}{2}\int dt{d}^{d}x{d}^{d}yV(\mathbf{𝐲}-\mathbf{𝐱}){\psi }^{*}(\mathbf{𝐱})\psi (\mathbf{𝐱}){\psi }^{*}(\mathbf{𝐲})\psi (\mathbf{𝐲}).$
• 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.