Explicit conversion from the Casimir force to Planckʹs law of radiation

The Casimir force has its origin in the finite modification of the infinite zero-point energy induced by a specific boundary condition for the spatial configuration. In terms of the imaginary-time formalism at finite temperature, the root of Planckʹs law of radiation can be traced back to the finite modification of the infinite vacuum energy induced by the periodic boundary condition in the temporal direction. We give the explicit conversion from the Casimir force to Planckʹs law of radiation, which shows an apparent correspondence between the system bounded by parallel conducting plates and the thermodynamic system. The temperature inversion symmetry and the duality relation in the thermodynamics are also discussed. We conclude that the effective temperature characterized by the spatial extension should no longer be regarded as genuine temperature.