Nonlocal Lagrangian

{{Short description|Lagrangians with nonlocal physics}}

{{refimprove|date=December 2024}}

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

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