Nonlocal Lagrangian Explained

l{L}[\phi(x)] containing terms that are nonlocal in the fields \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:l{L} = (1)/(2) ( \partial _{x} \phi(x) ) ^{2} - (1)/(2) m ^{2} \phi(x) ^{2} + \phi(x) \int (\phi(y))/((x - y) ^{2}) d ^{ny.} l{L} =- (1)/(4) l{F} _{\mu\nu} \left( 1 + (m ^{2})/(\partial ^{2}) \right) l{F} ^{\mu\nu} . S = \int dt d ^{dx} \left[\psi ^{*\left(i\hbar} (\partial)/(\partial t) + \mu\right)\psi - (\hbar ^{2})/(2 m) \nabla \psi ^{* ⋅} \nabla \psi\right] - (1)/(2) \int dt d ^{dx} d ^{dy} V( y - x ) \psi ^{*(x)} \psi( x ) \psi ^{*(y)} \psi( 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.