The Generalized Uncertainty Principle and higher dimensions: Linking black holes and elementary particles

Black holes play an important role in linking microphysics with macrophysics, with those of the Planck mass ( M P ∼ 10 −5 g) featuring in any theory of quantum gravity. In particular, the Compton-Schwarzschild correspondence posits a smooth transition between the Compton wavelength ( R C ∝ 1/ M ) below the Planck mass and the Schwarzschild radius ( R S ∝ M ) above it. The duality between R C and R S implies a form of the Generalized Uncertainty Principle (GUP) and suggests that elementary particles may be sub-Planckian black holes. The simplest possibility is that the ADM mass has the form M+βMP2/M for some constant β and this model can be extended to charged and rotating black holes, clearly relevant to elementary particles. Another possibility is that sub-Planckian black holes may arise in loop quantum gravity and this explicitly links black holes and elementary particles. Higher dimensions may modify both proposals. If there are n extra dimensions, all with the same compactification scale, one expects R S ∝ M 1/(1+ n ) below this scale but R C depends on the form of the higher-dimensional wave-function. If it is spherically symmetric, then R C ∝ M −1 , so duality is broken and the Planck mass is reduced, allowing the possibility of TeV quantum gravity. If the wave-function is pancaked in the extra dimensions, R C ∝ M −1/(1+ n ) and so duality is preserved but the Planck mass is unchanged.