Nonlocal Lagrangian - Alchetron, The Free Social Encyclopedia

In field theory, a nonlocal Lagrangian is a Lagrangian, a type of functional L [ ϕ ( x ) ] containing terms that are nonlocal in the fields ϕ ( 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 = 1 2 ( ∂ x ϕ ( x ) ) 2 − 1 2 m 2 ϕ ( x ) 2 + ϕ ( x ) ∫ ϕ ( y ) ( x − y ) 2 d n y .

L = − 1 4 F μ ν ( 1 + m 2 ∂ 2 ) F μ ν .

S = ∫ d t d d x [ ψ ∗ ( i ℏ ∂ ∂ t + μ ) ψ − ℏ 2 2 m ∇ ψ ∗ ⋅ ∇ ψ ] − 1 2 ∫ d t d d x d d y V ( y − x ) ψ ∗ ( x ) ψ ( x ) ψ ∗ ( y ) ψ ( 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.

References

Nonlocal Lagrangian Wikipedia

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