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Applied Sciences
Volume 16
Issue 11
10.3390/app16115246
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

The Finite-Temperature Casimir Effect in a One-Dimensional Scalar Field with Two Delta-Function Potentials

by
Xu-Feng Zhao
1,2,
Shao-Zhe Lu
1,2,
Rong-Sheng Han
1,2 and
Liang Chen
1,2,3,*
1
School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
2
Institute of Condensed Matter Physics, North China Electric Power University, Beijing 102206, China
3
Hebei Key Laboratory of Physics and Energy Technology, North China Electric Power University, Baoding 071003, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2026, 16(11), 5246; https://doi.org/10.3390/app16115246 (registering DOI)
Submission received: 12 April 2026 / Revised: 15 May 2026 / Accepted: 20 May 2026 / Published: 23 May 2026
(This article belongs to the Section Quantum Science and Technology)
Versions Notes

Abstract

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.
Keywords: Casimir effect; entropy; finite-temperature quantum field theory Casimir effect; entropy; finite-temperature quantum field theory

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MDPI and ACS Style

Zhao, X.-F.; Lu, S.-Z.; Han, R.-S.; Chen, L. The Finite-Temperature Casimir Effect in a One-Dimensional Scalar Field with Two Delta-Function Potentials. Appl. Sci. 2026, 16, 5246. https://doi.org/10.3390/app16115246

AMA Style

Zhao X-F, Lu S-Z, Han R-S, Chen L. The Finite-Temperature Casimir Effect in a One-Dimensional Scalar Field with Two Delta-Function Potentials. Applied Sciences. 2026; 16(11):5246. https://doi.org/10.3390/app16115246

Chicago/Turabian Style

Zhao, Xu-Feng, Shao-Zhe Lu, Rong-Sheng Han, and Liang Chen. 2026. "The Finite-Temperature Casimir Effect in a One-Dimensional Scalar Field with Two Delta-Function Potentials" Applied Sciences 16, no. 11: 5246. https://doi.org/10.3390/app16115246

APA Style

Zhao, X.-F., Lu, S.-Z., Han, R.-S., & Chen, L. (2026). The Finite-Temperature Casimir Effect in a One-Dimensional Scalar Field with Two Delta-Function Potentials. Applied Sciences, 16(11), 5246. https://doi.org/10.3390/app16115246

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