Generalized uncertainty principle

Generalized uncertainty principle — collective name for hypothetical generalizations uncertainty principle, which take into account the influence of gravitational interactions on the maximum achievable accuracy of measuring physical quantities.[1]

From the hypothesis of the generalized uncertainty principle it follows that there is an absolute minimum of the uncertainty of the position of any particle of the order of the Planck length.[2] Therefore, it plays an important role in the theories of quantum gravity and string theory, which assume the existence of a minimum length scale.[1]

The simplest version of the generalized uncertainty principle can be arrived at in a thought experiment Heisenberg's microscope to measure the coordinates of a particle by taking into account the gravitational interaction of a photon and an observed particle.[3][2]

In system of units, wich suppose ${\displaystyle \hbar =c=1}$, it is given by an inequality relating the uncertainty of coordinate ${\displaystyle \Delta x}$, the uncertainty of momentum ${\displaystyle \Delta p}$, and the Newtonian gravitational constant ${\displaystyle G}$:[1]

${\displaystyle \Delta x\geqslant {\frac {1}{\Delta p}}+G\Delta p}$

Other versions of the generalized uncertainty principle can be arrived at in a thought experiment to measure the area of the visible horizon of a black hole[4] or in a thought experiment with microscopic black holes.[5]