We investigate the finite-temperature Casimir interaction between two delta-function potentials in a (1+1)-dimensional scalar field model using Lifshitz theory, canonical quantization, and the Green’s function method. The Casimir force computed from all three approaches is in complete agreement. The Casimir entropy is also broadly consistent across the three methods, with subtle differences that can be traced to the infrared logarithmic divergence in the free energy. This divergence originates from the zero-frequency term and affects the entropy but not the force. In the Lifshitz approach, regularization requires an external infrared cutoff; in canonical quantization and the Green’s function method, the cutoff is naturally related to the finite size of the physical system.
Zhao et al. (2026) studied this question.