AUTHOR=Carr B. J. 

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

JOURNAL=Frontiers in Astronomy and Space Sciences

VOLUME=9

YEAR=2022

URL=https://www.frontiersin.org/journals/astronomy-and-space-sciences/articles/10.3389/fspas.2022.1008221

DOI=10.3389/fspas.2022.1008221

ISSN=2296-987X

ABSTRACT=<p>Black holes play an important role in linking microphysics with macrophysics, with those of the Planck mass (<italic>M</italic><sub>P</sub> ∼ 10<sup>−5</sup> g) featuring in any theory of quantum gravity. In particular, the Compton-Schwarzschild correspondence posits a smooth transition between the Compton wavelength (<italic>R</italic><sub>C</sub> ∝ 1/<italic>M</italic>) below the Planck mass and the Schwarzschild radius (<italic>R</italic><sub>S</sub> ∝ <italic>M</italic>) above it. The duality between <italic>R</italic><sub>C</sub> and <italic>R</italic><sub>S</sub> 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 <inline-formula id="inf1"><mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi><mml:mo>+</mml:mo><mml:mi>β</mml:mi><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">P</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>/</mml:mo><mml:mi>M</mml:mi></mml:math></inline-formula> for some constant <italic>β</italic> 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 <italic>n</italic> extra dimensions, all with the same compactification scale, one expects <italic>R</italic><sub>S</sub> ∝ <italic>M</italic><sup>1/(1+<italic>n</italic>)</sup> below this scale but <italic>R</italic><sub>C</sub> depends on the form of the higher-dimensional wave-function. If it is spherically symmetric, then <italic>R</italic><sub>C</sub> ∝ <italic>M</italic><sup>−1</sup>, 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, <italic>R</italic><sub>C</sub> ∝ <italic>M</italic><sup>−1/(1+<italic>n</italic>)</sup> and so duality is preserved but the Planck mass is unchanged.</p>