Scalar bosonic oscillator fields in LV-wormholes

Abstract: We investigate the quantum dynamics of scalar bosonic oscillator fields propagating in a (3+1)-dimensional Lorentz-violating (LV) wormhole spacetime within a modified gravity framework. The underlying geometry, characterized by a smooth minimal-radius throat and a globally regular redshift sector, induces nontrivial curvature effects that significantly modify the spectral properties of the Klein-Gordon (KG) field. The field dynamics are formulated in the presence of a nonminimally coupled vector background of the form $\mathcal{F}_μ=(\mathcal{F}_t(x),0,0,0)$, which, under the physically motivated ansatz $\mathcal{F}_t(x)=Ω\, r(x)$, generates an effective KG-oscillator interaction intrinsically encoded by the wormhole geometry. The resulting effective potential is regular and finite at the throat, eliminating centrifugal singularities and ensuring globally well-defined propagation across the minimal-radius region. The spectral problem reduces to a confluent Heun structure, leading to conditionally exact solutions and a discrete energy spectrum governed by curvature, Lorentz-violation strength, and oscillator frequency. The associated eigenvalue structure exhibits a relativistic particle-antiparticle symmetry with curvature-induced deformation and parameter-dependent confinement. Our results demonstrate that LV wormhole spacetimes act as effective dispersive quantum gravitational media, in which spacetime topology and spontaneous Lorentz symmetry breaking jointly regulate confinement, spectral quantization, and the global evolution of scalar bosonic modes.

• The paper demonstrates that the quantum dynamics of scalar bosonic fields in a Lorentz-violating wormhole exhibit conditional exact solvability.
• It employs a modified Klein-Gordon equation with a nonminimal vector coupling to derive a Heun-type equation with discrete energy eigenvalues.
• The spectral analysis reveals that curvature-induced modifications eliminate centrifugal singularities and influence particle/antiparticle symmetry.

Scalar Bosonic Oscillator Fields in Lorentz-Violating Wormholes

Introduction and Context

The paper "Scalar bosonic oscillator fields in LV-wormholes" (2605.03366) presents a rigorous analysis of the quantum dynamics of scalar (spin-0) bosonic oscillator fields propagating in a Lorentz-violating (LV) traversable wormhole spacetime within a modified gravity framework. Lorentz symmetry, foundational in both the Standard Model and General Relativity, is here explicitly broken by a nonminimally coupled background vector field, following the lines of the Standard-Model Extension (SME) formalism and specific bumblebee-type models. The authors examine the quantum spectroscopic properties of scalar fields in this geometry, focusing on curvature-induced modifications to their spectra and the interplay between global geometric features and the background LV sector.

LV Wormhole Geometry and Background

The spacetime considered is a static, spherically symmetric, traversable wormhole in (3+1) dimensions, described by the metric: $$ds^2 = -\mathcal{A}(x)\,dt^2 + \frac{1}{\mathcal{A}(x)}\,dx^2 + r(x)^2\left(d\theta^2 + \sin^2\theta\, d\varphi^2\right)\,,$$ where $\mathcal{A}(x)$ is the redshift function and $r(x)$ encodes the areal radius, displaying a smooth minimal radius at the throat $x=0$. The Lorentz-violating deformation is introduced through the radial geometry $r(x) = \sqrt{a^2 + x^2/(1-\zeta)}$ where $a$ is the throat's minimal radius and $0 \leq \zeta < 1$ is the LV parameter. A constant-lapse $\mathcal{A}(x) = 1$ eliminates horizons, ensuring global traversability for scalar (and antiparticle) excitations. The background vector field is set as $\mathcal{F}_\mu = (\Omega\, r(x), 0, 0, 0)$, introducing an effective KG-oscillator interaction directly tied to spacetime geometry.

The combination of global regularity, absence of horizons, and the LV parameter controlling spatial deformation distinguishes this wormhole background from more conventional traversable wormholes or black hole spacetimes. The construction guarantees boundedness and global regularity of potentials felt by quantum fields at and across the throat.

Quantum Field Dynamics and Effective Potentials

Scalar field dynamics are governed by the modified Klein-Gordon (KG) equation with a nonminimal vector coupling: $$\left[ g^{\mu\nu} (\nabla_\mu - \mathcal{F}_\mu)(\nabla_\nu - \mathcal{F}_\nu) - m_\circ^2 \right]\Psi = 0\,,$$ where the nonminimal coupling naturally induces a (generalized) KG-oscillator interaction through the metric's curvature and the form of $\mathcal{A}(x)$0. Upon separation of variables and suitable transformation, the radial equation reduces to: $\mathcal{A}(x)$1 with curvature- and LV-deformed parameters: $\mathcal{A}(x)$2

The effective potential for the associated Schrödinger-like equation is

$\mathcal{A}(x)$3

Key features include:

• Absence of centrifugal singularity: The angular barrier remains bounded and smooth at $\mathcal{A}(x)$4 due to the geometry, unlike the divergent centrifugal term in flat space.
• Harmonic confinement: The quadratic $\mathcal{A}(x)$5 term persists but is LV-curvature deformed.
• Global regularity: Potential is finite everywhere, supporting traversability and smooth field evolution.

Spectral Problem and Conditional Exact Solvability

The radial equation is transformed into a confluent Heun equation. The physical requirement of global regularity, square integrability, and finite probability density at the throat restricts solutions to the class of confluent Heun polynomials, leading to conditionally exact solvability. Specifically, polynomial truncation of the Heun series is possible only for discrete sets of parameter values, enforcing algebraic constraints among $\mathcal{A}(x)$6, $\mathcal{A}(x)$7, $\mathcal{A}(x)$8, $\mathcal{A}(x)$9, and $r(x)$0: $r(x)$1 with corresponding energy eigenvalues,

$r(x)$2

The spectral structure, therefore, is strictly quantized, with discrete bound state energies that are entirely determined by the wormhole’s curvature, the strength of Lorentz violation, and oscillator frequency.

The analysis reveals a relativistic two-branch (particle/antiparticle) spectrum with explicit curvature- and LV-induced deformation:

• Increasing $r(x)$3 (throat size): Weakens curvature effects, reducing spectral deformation.
• Increasing $r(x)$4 (LV strength): Compresses levels via enhanced effective confinement.
• Excitation number $r(x)$5 and angular momentum $r(x)$6: Structured coupling with frequency and spectral shifts, especially prominent for higher $r(x)$7 or $r(x)$8.

The energy quantization critically depends on the parametric consistency dictated by the geometry and the LV sector, distinguishing these solutions from the generic KG oscillator problem.

Physical Implications and Theoretical Impact

The results demonstrate that LV wormhole spacetimes act as dispersive, quantum-gravitational “media” in which topology and spontaneous Lorentz symmetry breaking collectively regulate:

• Quantum confinement
• Spectral quantization
• Global regularity and traversability for quantum fields
• Modification and control of relativistic particle/antiparticle branch symmetry

In contrast with standard KG oscillators or fields in generic spacetimes, the spectrum here is not only quantized but also conditional on background geometric and symmetry-breaking parameters. The removal of the centrifugal singularity by geometric effects and the emergence of conditional exact solvability through the Heun structure are notable departures from conventional quantum field dynamics.

Practically, such results have implications for the study of quantum processes in exotic spacetime backgrounds, especially in scenarios where Lorentz symmetry breaking is expected at high energies, as in various quantum gravity models. The work provides a benchmark for comparing scalar quantum dynamics in traversable wormholes with and without Lorentz invariance.

Theoretically, this study highlights:

• The role of topology and symmetry breaking in quantum spectral phenomena
• The critical nature of conditional exact solvability in higher Fuchsian quantum problems
• A tight geometric-quantum connection where spectral properties are dictated by spacetime structure and fundamental symmetry parameters

Possible directions for future developments include extending the analysis to interacting fields, fermionic systems, more general LV backgrounds, or dynamical wormhole geometries, as well as quantifying the impact of these spectral shifts in observable phenomena (e.g., Hawking-like radiation, vacuum polarization, or the stability of traversable wormholes under quantum fluctuations).

Conclusion

The paper rigorously demonstrates that scalar bosonic oscillator fields propagating in a Lorentz-violating traversable wormhole background are governed by a spectrally quantized, curvature- and LV-deformed system exhibiting conditional exact solvability. The effective quantum potential eliminates centrifugal singularities and supports global traversability, with the energy spectrum and frequency determined by nontrivial correlations among quantum numbers and geometric/LV parameters. This analysis elucidates the central role of geometry and symmetry breaking in shaping quantum field spectra and provides a foundation for further studies of quantum dynamics in nontrivial, Lorentz-violating spacetimes (2605.03366).