Nonlocality

Nonlocality in quantum mechanics, refers to the property of entangled quantum states in which both the entangled states "collapse" simultaneously upon measurement of one of their entangled components, regardless of the spatial separation of the two states. This "spooky action at a distance" is the content of Bell's theorem and the EPR paradox.

See also : Principle of locality

In field theory, a nonlocal Lagrangian is a 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

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

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

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.

Incidentally, the regularization techniques used to deal with ultraviolet divergences all use nonlocal actions.

Categories: Physics | Quantum mechanics | Theoretical physics

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