Sum or Integral? Understanding the Calculation of Casimir Effect Measurements

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SUMMARY

The discussion centers on the calculation methods used in measuring the Casimir effect, specifically the distinction between using a sum and an integral. When analyzing the space between the plates, only discrete frequency modes are present, necessitating the use of a sum. Conversely, in the vacuum energy measurement without the plates, a continuum of frequencies exists, which requires the application of an integral. This fundamental difference in frequency representation justifies the respective use of sums and integrals in the calculations.

PREREQUISITES
  • Understanding of quantum field theory concepts
  • Familiarity with the Casimir effect
  • Knowledge of discrete versus continuous mathematical representations
  • Basic calculus, particularly integration and summation techniques
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  • Research the mathematical formulation of the Casimir effect
  • Study the differences between discrete and continuous spectra in quantum mechanics
  • Learn about the implications of vacuum energy in quantum field theory
  • Explore advanced calculus techniques, focusing on integrals and series
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Physicists, researchers in quantum mechanics, and students studying advanced theoretical physics will benefit from this discussion, particularly those interested in the mathematical foundations of the Casimir effect.

epislon58
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I am reading the paper on the Casimir effect and they measure the space in between the plates using a sum and the energy of the vacuum without the plates using an intregal. Why do they use the sum and intregal, should they be switched.
 

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Between the plates, only modes with a discrete set of frequencies exist, so we sum over them. Without the plates, a continuum of frequencies is possible (all frequencies are allowed) so the sum becomes an integral.
 
Thank you sir
 

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