Thermal Casimir effect in $κ$-Minkowski space-time

TL;DR

This work analyzes the finite-temperature Casimir effect between parallel plates in κ-Minkowski space-time by constructing a κ-deformed scalar Lagrangian from the undeformed κ-Poincaré algebra, and evaluating the partition function with Matsubara frequencies under Dirichlet boundary conditions. The authors derive κ-deformed corrections to the Casimir free energy, pressure, entropy, and internal energy, showing that non-commutativity enhances the attractive force while maintaining thermodynamic consistency, including the Nernst theorem, and they identify an upper bound a ≤ 10^{-18} m with observable effects when a/L ≤ 10^{-12}. They also obtain a κ-deformed Stefan-Boltzmann law, revealing a leading T^4 term with a negative T^6 correction due to space-time non-commutativity. Overall, the study provides a rigorous framework for incorporating κ-deformation into finite-temperature quantum field theory and highlights potential experimental signatures in low-temperature, sub-micron Casimir setups.

Abstract

We study the finite temperature Casimir effect for parallel plates in the $κ$-Minkowski space-time. Using the Matsubara formalism and imposing the Dirichlet boundary conditions on a massless $κ$-scalar field, we compute the $κ$-deformed corrections to thermal Casimir free energy, pressure, entropy, and internal energy. Our results demonstrate that space-time non-commutativity enhances the attractive nature of the thermal Casimir force while preserving thermodynamic consistency; the system satisfies the Nernst theorem and laws of thermodynamics remain intact in $κ$-deformed space-time. Our analysis yields an upper bound on the deformation parameter as $a\leq10^{-18}m$. Furthermore, our results indicate that non-commutative effects become experimentally observable in Casimir effect studies when the ratio of the non-commutative scale to plate separation satisfies $a/L\leq 10^{-12}$. We also obtain the expression for Stefan-Boltzmann's law in $κ$-Minkowski space-time.

Figures (7)

• Figure 1: Variation of $\mathcal{F}^{ren~(a)}$ against $1/\tilde{\beta}$ for certain fixed values of $a/L$: (i) $a/L=0.25\times 10^{-12}$(red), $a/L=0.5\times 10^{-12}$(blue), $a/L=0.75\times 10^{-12}$(green), $a/L=1\times 10^{-12}$(purple). Here we obtain this plot by evaluating the summation in Eq.(\ref{['a18']}) with a truncation of $m=500$
• Figure 2: Plots show the variation of $\mathcal{P}^{(a)}$ against $1/\tilde{\beta}$ for certain fixed values of $a/L$: (i) $a/L=0.25\times 10^{-12}$(red), $a/L=0.5\times 10^{-12}$(blue), $a/L=0.75\times 10^{-12}$(green), $a/L=1\times 10^{-12}$(purple). To obtain this, we evaluating the summation in Eq.(\ref{['a19']}) with a truncation of $m=500$
• Figure 3: Plots show the variation of $\mathcal{S}^{(a)}_T$ against $1/\tilde{\beta}$ for certain fixed values of $a/L$: (i) $a/L=0.25\times 10^{-12}$(red), $a/L=0.5\times 10^{-12}$(blue), $a/L=0.75\times 10^{-12}$(green), $a/L=1\times 10^{-12}$(purple). This plot is obtained by evaluating the summation in Eq.(\ref{['a18']}) with a truncation of $m=500$
• Figure 4: Variation of $\mathcal{S}^{low~(a)}_T$ against $1/\tilde{\beta}$ in the low temperature limit for certain $a/L$ values: (i) $a/L=0.25\times 10^{-12}$(red), $a/L=0.5\times 10^{-12}$(blue), $a/L=0.75\times 10^{-12}$(green), $a/L=1\times 10^{-12}$(purple)
• Figure 5: Plot of $\mathcal{F}_{bb}$ against $1/{\beta}$ for certain $a$ values: (i) $a=0$ (red), $a=0.15$ (blue), $a=0.30$ (green), $a=0.45$ (purple)