Compactification of spatial extra dimensions

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SUMMARY

The discussion centers on the compactification of spatial extra dimensions, specifically addressing the term 4πnR in Equation 65 of a referenced paper. The term arises from the geometry of a torus, which necessitates the inclusion of 4π in the denominator to accurately represent the separation of charges in the compactified extra dimension. In contrast, if the extra dimension were compactified on a circle, the term would indeed be 2πnR. This distinction is crucial for understanding the modifications needed in the Coulomb force law.

PREREQUISITES
  • Understanding of compactification in string theory
  • Familiarity with toroidal geometry
  • Knowledge of the Coulomb force law
  • Basic grasp of mathematical notation in physics
NEXT STEPS
  • Research the implications of toroidal geometry in string theory
  • Study the differences between compactification on a circle versus a torus
  • Examine modifications to the Coulomb force law in higher dimensions
  • Explore the mathematical foundations of compact dimensions in theoretical physics
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The discussion is beneficial for theoretical physicists, string theorists, and researchers exploring higher-dimensional models and their implications in fundamental physics.

DuckAmuck
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Looking to understand results of a paper.
Hi everyone,
I am looking at a paper on compact dimensions. Equation 65 makes sense except for the term of 4*pi*n*R in the denominator. Why is it 4*pi and not 2*pi? I cannot rationalize this. Please help. Thank you.
https://arxiv.org/ftp/hep-ph/papers/0609/0609260.pdf
 
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The Coulomb force law needs to be modified such that ##R_n## takes into account the true separation of charges. The separation in the compactified extra dimension is given by this ##4\pi n R## term which is specific to the Torus geometry. If the extra dimension was compactified on a circle you may well have a factor of ##2\pi n R## as you expected.
 

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