Discrete Radial Bound-State Sector

• The discrete radial bound-state sector is defined as a set of localized, normalizable energy eigenstates emerging from the radial component of quantum wave operators.
• Imposing boundary conditions like the Robin condition for ℓ = 0 enables self-adjoint extensions that yield a unique bound state crucial for detector models.
• A cancellation mechanism in the stress–energy tensor ensures that while the bound state localizes quantum detection, it does not affect semiclassical gravitational observables.

The discrete radial bound-state sector encompasses those energy eigenstates of quantum systems, typically governed by wave equations or relativistic field theories, that are both strictly localized (normalizable) and determined by the radial component of the Hamiltonian—often interpreted as the spectrum of quantum numbers associated with radial excitations. In modern theoretical physics, this sector is pivotal for understanding the spectral structure of quantum fields, detector models, high-energy bound states, and the foundational aspects of quantum mechanics and field theory.

1. Formulation in Quantum Field Theory and Wave Equations

The discrete radial bound-state sector is fundamentally defined through the spectral analysis of radial differential operators arising in the decomposition of quantum field equations. For instance, in relativistic scenarios such as the punctured Minkowski spacetime (where the spatial origin is excised), the scalar field’s Klein–Gordon equation reduces, in the spherically symmetric sector (ℓ = 0), to a radial operator: $$A u_{ω0} \equiv \left[ -\frac{d^2}{dr^2} + \frac{ℓ(ℓ+1)}{r^2} \right] u_{ω0} = p^2 u_{ω0}$$ where p^2 is the spectral parameter and ℓ is the angular quantum number. For ℓ > 0, the operator is typically self-adjoint; for ℓ = 0 (s-wave), essential self-adjointness fails unless an appropriate boundary condition (e.g., Robin or Dirichlet) is imposed at r = 0 (Ramos et al., 24 Sep 2025).

The imposition of a boundary condition selects a self-adjoint extension and thus defines the allowed physical spectrum, separating the sector into (i) discrete bound states and (ii) the continuum. Bound states correspond to square-integrable solutions for negative p^2 (i.e., localized radial modes).

2. Boundary Conditions and Self-Adjoint Extensions

A central mechanism for the emergence of discrete radial bound states is the specification of boundary conditions at singular points (e.g., the origin). For the punctured Minkowski example, the relevant Robin boundary condition is: $$\lim_{r \to 0^+} \left[ r\, R^{(\beta)}_\omega(r) + \beta\, (r\, R^{(\beta)}_\omega(r))' \right] = 0$$ with the extension parameter β ∈ ℝ. This condition defines a family of self-adjoint extensions for the angular momentum zero (ℓ = 0) sector, and generically allows a unique bound state with frequency

$\omega_b = \sqrt{m_0^2 - \frac{1}{\beta^2}}$

provided β m_0 > 1. The radial profile

$$R_{\text{bound}}^{(\beta)}(r) = \sqrt{\frac{1}{4\pi\beta\omega_b}}\, \frac{e^{-r/\beta}}{r}$$

ensures square-integrability on ℝ^+, conferring genuine localization. The boundary condition thus both regularizes the operator and generates the bound state without the need for additional confining potentials (Ramos et al., 24 Sep 2025).

In situations where ℓ > 0, only the Dirichlet boundary condition is permitted, and no localized bound state arises from the boundary, consistent with the centrifugal barrier.

3. Sector Decomposition and Physical Interpretation

The particle detector model in (Ramos et al., 24 Sep 2025) naturally splits the mode expansion into three spectral sectors:

• Discrete radial bound-state sector: The unique localized mode arising from the Robin extension.
• Continuum (boundary-modified) sector: Modes with positive p^2 and modified by the boundary condition for ℓ = 0.
• Standard Dirichlet sector: ℓ > 0 modes unaffected by the puncture, forming the usual continuum.

This decomposition is critical. The bound state acts as the internal degree of freedom of a localized quantum detector, with the field operator written as

$$\widehat{\Psi}_\beta(x) = \Psi_{\text{bound}}(x)\,\hat{a}_{\text{bound}} + \Psi_{\text{bound}}^*(x)\,\hat{a}_{\text{bound}}^\dagger + \text{(continuum terms)}$$

where $\hat{a}_{\text{bound}}$, $\hat{a}_{\text{bound}}^\dagger$ are canonical creation-annihilation operators for the bound mode. This identification directly links the detector’s quantum oscillator degree of freedom (as in the traditional Unruh–DeWitt paradigm) to the bound-state sector of the field (Ramos et al., 24 Sep 2025).

4. Stress–Energy Tensor and Physical Consequences

A notable feature of this construction is the cancellation of the discrete sector’s explicit contribution to macroscopic observables. In particular, when evaluating the full Green’s function: $$G(x, x') = G_{\text{bound}}(x, x') + G_\beta(x, x') + G_{\text{Dirichlet}}(x, x')$$ the singular pole from the bound state cancels appropriately with a corresponding term in $G_\beta(x, x')$. As a result, the regularized stress-energy tensor ⟨$T_{\mu\nu}$⟩ does not receive contributions from the discrete radial sector; only the boundary-induced terms survive. This ensures covariant conservation and avoids spurious backreaction (Ramos et al., 24 Sep 2025).

Physically, this means that while the discrete bound mode is essential for detector localization and quantum coherence, it leaves no net effect on semiclassical gravitational observables—the observable gravitational response is entirely governed by the boundary condition-modified sectors.

5. Generalizations to Singular Geometries and Unified Perspective

The methodology of generating discrete radial bound states through self-adjoint extension is robust and can be generalized. In backgrounds with naked singularities, such as conical or global monopole spacetimes, excising the singular region and imposing similar boundary conditions yields localized bound modes, characterized analogously by radial solutions with exponential localization and frequency determined by the geometry and extension parameter (Ramos et al., 24 Sep 2025). This provides a unified approach to implementing detector localization, mode decomposition, and rigorous spectral analysis in a broad class of singular or topologically nontrivial spacetimes.

6. Operational Significance and Relation to Detector Models

In detector physics, the discrete radial bound-state sector supplies the internal excitation degrees of freedom that probe quantum fields locally—in contrast with traditional nonrelativistic oscillator models. The relativistic framework, realized in the field-based detector, leverages bound modes that arise fundamentally from the topology and boundary conditions (not from imposed potentials), ensuring covariance and proper thermodynamic behavior in diverse geometries.

At leading order in detector-field coupling, the particle detector’s mode operator can be mapped onto the bound-mode creation operator. This establishes operational equivalence with the standard Unruh–DeWitt detector and confirms that localized detector response is intimately linked with the discrete bound-state sector.

7. Mathematical Summary

Key equations:

• Radial operator: $$A u_{ω0} \equiv \left[ -\frac{d^2}{dr^2} + \frac{ℓ(ℓ+1)}{r^2} \right] u_{ω0} = p^2 u_{ω0}$$
• Robin boundary condition: $$\lim_{r \to 0^+} \left[ r\, R^{(\beta)}_\omega(r) + \beta\, (r\, R^{(\beta)}_\omega(r))' \right] = 0$$
• Bound state solution for ℓ = 0: $$R_{\text{bound}}^{(\beta)}(r) = \sqrt{\frac{1}{4\pi\beta\omega_b}}\, \frac{e^{-r/\beta}}{r}$$
• Bound state frequency: $\omega_b = \sqrt{m_0^2 - \frac{1}{\beta^2}}$ with $\beta m_0 > 1$.

Physical roles:

• Detector degree of freedom: $$\widehat{\Psi}_\beta(x) = \Psi_{\text{bound}}(x) \hat{a}_{\text{bound}} + \Psi_{\text{bound}}^*(x)\hat{a}_{\text{bound}}^{\dagger} + ...$$
• Cancellation in stress-energy tensor: $G(x, x')$ contains cancelling poles between $G_{\text{bound}}$ and $G_\beta$.

Conclusion

The discrete radial bound-state sector embodies a rigorous framework for the existence and operational use of localized, normalizable internal degrees of freedom in quantum field theory and detector models. Its appearance is tied to boundary conditions and self-adjointness in singular domains, not to ad hoc external potentials. The sector’s key properties—mode structure, frequency localization, cancellation mechanisms in macroscopic observables—define a fully relativistic approach to field-based detector modeling, with significant implications for quantum measurement theory, curved space quantum field theory, and the interpretation of localized quantum phenomena (Ramos et al., 24 Sep 2025).