Quantum confinement of scalar bosons in the Bonnor-Melvin spacetime: uniform magnetic field and rainbow gravity effects

TL;DR

The paper tackles the problem of relativistic scalar (Klein-Gordon) bosons in the Bonnor–Melvin spacetime augmented by rainbow gravity and a positive cosmological constant, focusing on geometric confinement into an infinite sequence of radial domains. It develops the KG equation for a uniform magnetic field in BM–RG spacetime and shows the radial part reduces to a hypergeometric form, yielding exact energy spectra and radial functions in the first confinement domain. Two rainbow-function models—Magueijo–Smolin and a loop-quantum-gravity–inspired pair—are analyzed, demonstrating Planck-energy invariance and symmetric particle/antiparticle spectra, with novel degeneracy features: a collapse of all $m$ states to $m=0$ for fixed radial quantum number, partially lifted by increasing $ ilde{ ext{const}}$ (the cosmological constant). The findings illuminate how gravity, topology, electromagnetism, and Planck-scale corrections jointly shape spectral and spatial properties of relativistic fields in curved, magnetized backgrounds, offering theoretical insights for quantum gravity phenomenology and potential analogs in astrophysical contexts.

Abstract

We present an exact analytical study of Klein-Gordon (KG) scalar bosons and antibosons confined in the Bonnor-Melvin (BM) spacetime under a uniform magnetic field, incorporating rainbow gravity (RG) corrections with a positive cosmological constant. The cosmological constant partitions spacetime into an infinite sequence of confinement domains bounded by impenetrable barriers. Within the first allowed domain, the KG equation reduces to a hypergeometric differential equation, yielding closed-form expressions for both the energy spectra and the radial wavefunctions in terms of hypergeometric polynomials. Two representative RG models, inspired by the Magueijo-Smolin framework and loop quantum gravity (LQG), produce Planck-scale bounded, symmetric particle-antiparticle spectra. A distinctive feature of the curved magnetized geometry is the collapse of all magnetic quantum states $m \neq 0$ onto the $m = 0$ level for each radial excitation, a degeneracy absent in flat spacetime. Increasing the cosmological constant partially lifts this collapse, establishing a direct link between the global spacetime curvature and the local quantum structure. Radial probability density analysis further shows that stronger magnetic fields enhance spatial localization, confining bosons into static or rotating ring-like configurations with nodal architectures that evolve systematically with quantum numbers. These findings reveal how gravitational confinement, topology, magnetic fields, and Planck-scale corrections jointly govern the spectral and spatial properties of relativistic quantum fields in curved and magnetized backgrounds.

Figures (6)

• Figure 1: The figure displays the energy levels of KG particles and antiparticles given by (\ref{['III.1.1']}). Specifically, we plot: (a) $E$ versus $\tilde{\epsilon}$ for $n=0$, $m=0,1,2,3,4$, $\Lambda=0.5$, and $\mathcal{B}_\circ=1=e$; (b) $E$ versus $\mathcal{B}_\circ$ for $n=0$, $m=0,1,2,3,4$, $\tilde{\epsilon}=0.5$, and $\Lambda=0.1$; (c) $E$ versus $\mathcal{B}_\circ$ for $n=2$, $m=0,1,2,3,4$, $\tilde{\epsilon}=0.5$, and $\Lambda=0.1$; (d) $E$ versus the cosmological constant $\Lambda$ for $n=2$, $m=0,1,2,3,4$, $\tilde{\epsilon}=0.5$, and $\mathcal{B}_\circ=4$; (e) $E$ versus $\Lambda$ for $n=0,1,2,3,4$, $m=2$, $\tilde{\epsilon}=0.5$, and $\mathcal{B}_\circ=4$; and (f) $E$ versus $\Lambda$ for $n=2$, $m=0,1,2,3,4$, $\tilde{\epsilon}=0.5$, and $\mathcal{B}_\circ=1$.
• Figure 2: The figure illustrates the energy levels of KG particles and antiparticles as given by Eq. (\ref{['III.3.1']}). Specifically, the plots show: (a) $E$ as a function of $\tilde{\epsilon}$ for $n = 0$, $m = 0, 1, 2, 3, 4$, $\Lambda = 0.5$, and $\mathcal{B}_\circ = 1 = e$; (b) $E$ as a function of $\mathcal{B}_\circ$ for $n = 0$, $m = 0, 1, 2, 3, 4$, $\Lambda = 0.1$, and $\tilde{\epsilon} = 0.5$; (c) $E$ as a function of $\mathcal{B}_\circ$ for $n = 1$, $m = 0, 1, 2, 3, 4$, $\Lambda = 0.1$, and $\tilde{\epsilon} = 0.5$; (d) $E$ as a function of the cosmological constant $\Lambda$ for $n = 0$, $m = 0, 1, 2$, $\mathcal{B}_\circ = 4$, and $\tilde{\epsilon} = 0.5$.
• Figure 3: The figure shows the energy levels for KG particles and antiparticles given by (\ref{['III.3.4']}), and we plot (a) $E$ against $\tilde{\epsilon}$ for $n=0$, $m=0,1,2,3,4$, $\Lambda=0.5$, and $\mathcal{B}_\circ=1=e$, (b) $E$ against $\mathcal{B}_\circ$ for $n=0$ and $m=0,1,2,3,4$, $\Lambda=0.1$, and $\tilde{\epsilon}=0.5$, (c) $E$ against $\mathcal{B}_\circ$ for $n=1$ and $m=0,1,2,3,4$, $\Lambda=0.1$, and $\tilde{\epsilon}=0.5$, and (d) $E$ against $\Lambda$ for $n=0$ and $m=0,1,2$, $\mathcal{B}_\circ=4$, and $\tilde{\epsilon}=0.5$, (d) $E$ against .
• Figure 4: Radial wave functions $R_{n,m}(\rho)$ in (\ref{['II.2.10']}) for quantum numbers $n = 0,1,2$ and $m = 0,1,2$ under two magnetic field strengths, $\mathcal{B}_\circ = 1$ (left column) and $\mathcal{B}_\circ = 10$ (right column). Each subplot depicts the behavior of $R_{n,m}(\rho)$ over the domain $\rho \in [0, \pi]$, illustrating how the magnetic field intensity affects the spatial distribution of scalar field modes in the magnetized BM universe.
• Figure 5: Radial probability density distributions $P_{n,m}(\rho)$ are presented for quantum states with $n = 0,1,2$ and $m = 0,1,2$, where $P_{n,m}(\rho) = \int_0^\rho \left| R_{n,m}(\rho') \right|^2 \, \rho' \, d\rho'$. The wavefunctions $R_{n,m}(\rho)$ are computed under a transverse magnetic field with effective coupling $\tilde{\mathcal{B}} = \frac{e \mathcal{B}_\circ}{2 \Lambda}$, using $e = 1$, $\mathcal{B}_\circ = 1$, and $\Lambda = 0.1$. The surface plots are mapped to Cartesian coordinates via $x = \rho \cos\phi$, $y = \rho \sin\phi$, where both the vertical axis and color scale represent the magnitude of $P_{n,m}(\rho)$.