Thermal Casimir Effect of Scalar Field From Path Integral Formulation Arista Romadani (a,b*), Agus Purwanto (a), Bintoro Anang Subagyo (a), and Apriadi Salim Adam (c)
a) Department of Physics, FSAD, Sepuluh Nopember Institute of Technology, Kampus ITS Sukolilo, Surabaya, 60111, Indonesia
b) Department of Physics, Faculty of Science and Technology, Universitas Islam Negeri Maulana Malik Ibrahim Malang, Malang 65144, Indonesia
c) Research Center for Quantum Physics, National Research and Innovation Agency (BRIN), South Tangerang 15314, Indonesia
Abstract
We investigate the thermal Casimir effect of a massless scalar field confined between two parallel plates using path integral formalism at finite temperature. The path integral formulation is constructed from the Euclidean partition function through functional integration over scalar field configurations. By imposing Dirichlet boundary conditions on the plates, the vacuum modes along the confined direction become discretized, leading to the renormalized Casimir energy density through exponential cutoff regularization. We find that the renormalized Casimir energy density preserves the characteristic geometric behavior of the Casimir effect, proportional to \(d^{-4}\). Furthermore, the finite-temperature of the Casimir energy density is derived within the Matsubara formalism using exponential cutoff regularization, leading to the thermal correction of the Casimir free energy. We also analyze the thermal correction of renormalized Casimir thermodynamic quantities such as entropy and internal energy in both high and low temperature limits. We find that the thermal corrections of the renormalized Casimir entropy and internal energy exhibit a similar behavior. In the high temperature limit, thermal fluctuations dominate the system and generate blackbody-like contributions. In the low temperature limit, the quantum vacuum contribution remains dominant and the corrections become exponentially suppressed. In addition, at zero temperature, the renormalized Casimir entropy vanishes, demonstrating consistency with the third law of thermodynamics. These results provide a consistent thermodynamic description of the finite temperature Casimir effect.
