[Paper Review] Kappa-deformed quantum field theory and Casimir effect

M. V. Cougo-Pinto, C. Farina+1 more|arXiv (Cornell University)|May 19, 2003

Quantum Electrodynamics and Casimir Effect5 references4 citations

TL;DR

This paper proposes a framework for calculating the Casimir energy in $κ$-deformed quantum field theory by generalizing the standard vacuum energy expression as the half-sum of field frequencies, now using the $κ$-deformed dispersion relation. It derives a non-perturbative expression for the Casimir energy in $κ$-deformed electrodynamics, showing that in the $κ \to \infty$ limit, the result reduces to the standard Casimir energy, validating the deformation's consistency with known physics.

ABSTRACT

We consider the quantization of a scalar kappa-deformed field up to the point of obtaining an expression for its vacuum energy. The expression is given by the half sum of the field frequencies, as in the non-deformed case, but with the frequencies obeying the kappa-deformed dispersion relation. We consider a set of kappa-deformed Maxwell equations and show that for the purpose of calculating the Casimir energy the Maxwell field, as in the non-deformed case, behaves as a pair of scalar fields. Those results provide a foundation for computing the Casimir energy starting from the the half sum of field frequencies. A method of calculation starting from this expression is briefly described.

Motivation & Objective

• To extend the standard vacuum energy expression in quantum field theory to $κ$-deformed field theories by replacing the standard dispersion relation with the $κ$-deformed one.
• To establish a foundation for computing the Casimir energy in $κ$-deformed electrodynamics using the half-sum of field frequencies as the starting point.
• To demonstrate that the Maxwell field in $κ$-deformed theory behaves like a pair of scalar fields for Casimir energy calculations, analogous to the non-deformed case.
• To derive a non-perturbative expression for the Casimir energy in $κ$-deformed electrodynamics, valid beyond the $1/\kappa$ expansion.

Proposed method

• The vacuum energy is expressed as the half-sum of field frequencies, with frequencies determined by the $κ$-deformed dispersion relation $\sinh(q\omega) = q|{\bf k}|$, where $q = 1/(2\kappa)$.
• The quantization of a $κ$-deformed scalar field is performed using a deformed differential operator $\partial_q = \frac{1}{q} \sin(q\partial_0)$, leading to a non-local, infinite-order differential equation.
• The conjugate field $\pi_q$ is defined via the deformed commutation relations, and three distinct conjugate field operators ($\pi_q$, $\pi_{2q}$, $\Pi^0$) are analyzed, all reducing to $\partial_0\phi$ in the $q \to 0$ limit.
• The Casimir energy is computed by summing over transverse modes between two parallel plates, using the deformed dispersion relation and mode quantization with appropriate boundary conditions.
• Regularization is applied using both a cutoff $\Lambda_{\parallel}$ and a small parameter $\epsilon$, followed by subtraction of the spurious self-energy term to isolate the physical Casimir energy.
• The argument principle is used to evaluate the resulting series, leading to a non-perturbative integral expression for the Casimir energy in terms of $q$ and plate separation $a$.

Experimental results

Research questions

• RQ1 How does the vacuum energy of a $κ$-deformed scalar field generalize the standard half-sum of frequencies when the dispersion relation is deformed?
• RQ2 Can the Casimir effect in $κ$-deformed electrodynamics be consistently derived from the half-sum of field frequencies using the deformed dispersion relation?
• RQ3 What is the non-perturbative form of the Casimir energy in $κ$-deformed electrodynamics, and how does it reduce to the standard result in the $\kappa \to \infty$ limit?
• RQ4 How do the different conjugate field operators ($\pi_q$, $\pi_{2q}$, $\Pi^0$) in the $κ$-deformed theory compare, and which is most suitable for vacuum energy calculations?

Key findings

• The vacuum energy in $κ$-deformed field theory is given by the half-sum of frequencies that obey the $κ$-deformed dispersion relation $\sinh(q\omega) = q|{\bf k}|$, generalizing the standard expression.
• The Casimir energy for $κ$-deformed electrodynamics is derived as a non-perturbative expression: ${\cal E}_{q}(a) = -\frac{\ell^{2}}{4\pi^{2}{a}^{3}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\int_{0}^{a/q}\! dy\left(y+\frac{1}{2n}\right)\frac{e^{-2ny}}{\sqrt{1-(qy/a)^{2}}}$.
• In the limit $q \to 0$ (i.e., $\kappa \to \infty$), the derived Casimir energy reduces to the standard result ${\cal E}(a) = -\frac{\pi^{2}\ell^{2}}{720{a}^{3}}$, confirming consistency with conventional QED.
• The Maxwell field in the $κ$-deformed theory behaves like two scalar fields for Casimir energy calculations, preserving the same mode structure and boundary conditions as in the non-deformed case.
• The regularization and subtraction procedure successfully isolates the physical Casimir energy by removing the spurious self-energy contribution from the plates.
• The use of the argument principle in the calculation allows for the evaluation of the mode sum and leads to a closed-form integral expression for the Casimir energy in the $κ$-deformed framework.