Finite temperature Casimir effect in one-dimensional scalar field with double delta-function potentials

TL;DR

This work analyzes the finite-temperature Casimir effect for a (1+1)-dimensional scalar field in the presence of two delta-function barriers, comparing canonical quantization with Lifshitz theory. Zero-temperature forces agree between the two methods, while at finite temperature the long-distance behavior differs by a factor of two, with canonical quantization predicting $F_C(a,T)\sim -\frac{T}{4d}$ (in appropriate units) and Lifshitz predicting a larger magnitude. The canonical approach yields a Casimir entropy that is positive, increases with temperature, and satisfies the third law, whereas Lifshitz formulations can produce infrared divergences or negative entropy. Overall, the canonical quantization framework provides thermodynamically consistent insights into thermal Casimir effects in this geometry and clarifies the role of long-wavelength modes in the entropy and infrared behavior.

Abstract

We investigate the finite-temperature Casimir effect for a (1+1)-dimensional scalar field interacting with a pair of delta-function potentials. We employ the canonical quantization method to compute the Casimir force and entropy, contrasting the results with those from the standard Lifshitz theory. At zero temperature, both frameworks yield identical forces. For the finite-temperature case, we find that in the long-distance limit, the Casimir force decays asymptotically as $F_C(a,T)=-T/(4a)$, with the Lifshitz theory predicting a magnitude twice as large as that from canonical quantization. Crucially, the canonical quantization method yields a physically consistent entropy that remains positive and increases with temperature. These results demonstrate the robustness of the canonical quantization approach in providing a thermodynamically sound description of the thermal Casimir effect in this system.

Figures (3)

• Figure 1: Casimir force in units of $-\hbar\gamma^2/v^3$ vs dimensionless distance $d=\gamma{a}/v^2$.
• Figure 2: Casimir force in units of $-\hbar\gamma^2/(4\pi{v^3})$ vs dimensionless distance $d=\gamma{a}/v^2$. The solid lines show results from the Lifshitz theory, Eq. (\ref{['eq34']}). The dotted lines show results from canonical quantization method, Eq. (\ref{['eq32']}).
• Figure 3: Panels (a) and (b) display the integrand and the resulting Casimir entropy, respectively, versus the separation distance for different temperatures. The inset shows a detailed view of the short-separation regime.