Spectral Zeta Function & Casimir Energy
Zero-point sum, zeta regularization, and Casimir force vs plate separation
Casimir effect: vacuum zero-point fluctuations between conducting plates create an attractive force F ∝ −ℏc·π²/(240·d⁴).
Spectral zeta regularization: the divergent sum Σₙ ωₙ is regularized via ζ(s) = Σₙ ωₙ^{−s}. Casimir energy = (ℏ/2)·ζ(−1), analytically continued from Re(s)>1.
In 1D: ζ_Riemann(−1) = −1/12 → vacuum energy ∝ −1/24d. Force = −∂E/∂d ∝ −1/d².
Left panel: mode energies (sticks) with exponential regulator e^{−εω}. Right panel: Casimir force vs separation.