Exact Solutions for the Kemmer Oscillator in 1+1 Rindler Coordinates

TL;DR

The paper addresses exact solutions for the Kemmer equation describing spin-1 bosons in $1+1$-dimensional Rindler spacetime, revealing how uniform acceleration and the Unruh-type physics modify vector fields. By implementing Dirac-oscillator coupling via $p \to p - i M \omega B \vec{r}$ with $B=\gamma^{0}\otimes\gamma^{0}$, they obtain a closed-form spectrum where the acceleration parameter defines a characteristic length $l=\left(\frac{2}{M^{2}a}\right)^{1/3}$ and enters the energies as $E_n^2 = \frac{M^{2}}{4} + \frac{1}{l^{2}} \left\{ \frac{3\pi}{4}\left(2n-\frac{1}{2}\right) \right\}^{2/3}$. In the Minkowski limit $a\to 0$, the spectrum reduces to the standard Kemmer oscillator, confirming consistency with flat-spacetime results. The results provide a tractable framework for quantum field theory in curved spacetime and analogue gravity platforms, delivering exact benchmarks for acceleration-induced effects on spin-1 fields.

Abstract

This work presents exact solutions of the Kemmer equation for spin-1 particles in $(1+1)$-dimensional Rindler spacetime, motivated by the need to understand vector bosons under uniform acceleration, including non-inertial effects and the Unruh temperature, which distinguish them from spin-0 and spin-1/2 systems. Starting from the free Kemmer field in an accelerated reference frame, we establish eigenvalue equations resembling those of the Klein--Gordon equation in Rindler coordinates. By introducing the Dirac oscillator interaction through a momentum substitution, we derive an exact closed-form spectrum for the Kemmer oscillator, revealing how the acceleration parameter modifies the characteristic length, shifts the discrete energy spectrum, and lifts degeneracies. In the Minkowski limit $a\to 0$, the standard Kemmer oscillator spectrum is recovered, ensuring consistency with flat-spacetime results. These findings provide a tractable framework for analyzing acceleration-induced effects, with implications for quantum field theory in curved spacetime, quantum gravity, and analogue gravity platforms.

Figures (3)

• Figure 1: Energy Levels $E$ of the one-dimensional Kemmer oscillator in Rindler spacetime as a function of quantum number $n$ for various acceleration Parameters $a$ .
• Figure 2: Energy spectrum $E$ as a function of the acceleration parameter $a$ for the Kemmer oscillator in 1+1 Rindler spacetime, with quantum numbers $n=0,1,2,3,4$ ($a<\sqrt{2}\omega)$. Solid lines represent the positive energy branch, while dashed lines indicate the negative branch. The condition $a<\sqrt{2}\omega$ ensures the physical admissibility of the spectrum, as derived from Eq. (92) in Section V, preventing singularities in the denominator of the energy expression (Eq. (91)).
• Figure 3: Energy spectrum $E$ as a function of the characteristic length $l=(2/(M^{2}a))^{1/3}$ for the Kemmer oscillator in 1+1 Rindler spacetime, with quantum numbers $n=0to4$ ($a<\sqrt{2}\omega$). Solid lines denote the positive energy branch, and dashed lines the negative branch. The inverted x-axis accounts for the inverse relationship between $l$ and $a$, reflecting the Rindler metric's spatial coordinate transformation (Eq. (19), Section III). This visualization highlights the acceleration-induced modification of the system's spatial scale, a critical finding for understanding near-horizon kinematics and spectral properties in curved spacetime.