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 ($M_P \sim10^{-5}$g) featuring in any theory of quantum gravity. In particular, the Compton-Schwarzschild correspondence posits a smooth transition between the Compton wavelength ($R_C \propto 1/M$) below the Planck mass and the Schwarzschild radius ($R_{\rm S} \propto M$) above it. The duality between $R_{\rm C}$ and $R_{\rm S}$ 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 $M + βM_P^2/M$ 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 $R_{\rm S} \propto M^{1/(1+n)}$ below this scale but $R_{\rm C}$ depends on the form of the higher-dimensional wave-function. If it is spherically symmetric, then $R_{\rm C} \propto M^{-1}$, 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, $R_{\rm C} \propto M^{-1/(1+n)}$ and so duality is preserved but the Planck mass is unchanged.