Quantized Dissipation from the Inverse-Square Anomaly in a Non-Hermitian Klein-Gordon Field

Abstract: We construct an exactly solvable relativistic model that embeds the anomalous inverse-square interaction into a non-Hermitian Klein-Gordon field theory through a purely imaginary, scale-invariant scalar potential. The stationary field equation reduces to an inverse-square Schrodinger-type problem with a quadratic spectral parameter. Imposing a strictly outgoing boundary condition at the singularity-interpreted as irreversible absorption-selects a unique physical realization and converts the fall-to-the-center instability into a discrete, log-periodic spectrum of complex energies. The resulting decay rates exhibit universal geometric spacing, determined solely by the anomalous scaling exponent and insensitive to microscopic short-distance regularization. This structure defines an emergent kinematic energy scale that controls dissipative dynamics and provides a minimal analytic framework for studying scale anomaly, boundary-condition-induced non-Hermiticity, and quantized dissipation in relativistic open quantum systems.

• The paper demonstrates that imposing an outgoing boundary condition at the r=0 singularity transforms an instability into a log-periodic complex energy spectrum.
• It applies a non-Hermitian Klein-Gordon framework with a purely imaginary, scale-invariant potential to yield discrete, geometric decay rates independent of short-distance details.
• The work establishes an analytical link between scale anomaly, quantized dissipation, and an emergent kinematic temperature, paving the way for experimental exploration.

Quantized Dissipation and the Inverse-Square Anomaly in a Non-Hermitian Klein-Gordon Field

Overview

This paper establishes an exactly solvable relativistic quantum field theory embedding the inverse-square anomaly within a non-Hermitian Klein-Gordon framework utilizing a purely imaginary, scale-invariant scalar potential. By imposing a strictly outgoing boundary condition at the $r=0$ singularity, corresponding to an absorbing sink, the classic fall-to-the-center instability is regularized, resulting in a discrete, log-periodic spectrum of complex energies with universal geometric decay rates. The construction elucidates a direct analytical connection between scale anomaly, non-Hermiticity, and quantized dissipation in relativistic open quantum systems.

Non-Hermitian Embedding and Scale-Invariant Potential

The model involves a complex scalar field $\psi(t,\mathbf{r})$ obeying the Klein-Gordon equation in flat spacetime, with a potential $V(r) = -m + i\gamma/r$, where $\gamma \in \mathbb{R}$. This choice exactly cancels the explicit mass term and replaces it with a purely imaginary, scale-invariant mass—rendering the $r=0$ singularity a perfect absorber. This field-theoretic embedding, which is non-Hermitian, does not rely on any nonrelativistic approximation and preserves continuous scale invariance. The field equation reduces to a stationary Schrödinger-type problem with a spectral parameter $E^2$ and an attractive $-\gamma^2/r^2$ interaction.

Spectral Problem and Boundary Condition Selection

Analyzing the radial reduction reveals two square-integrable solutions near $r=0$ with behaviors $u_\ell(r) \sim r^{1/2\pm i\sigma_\ell}$. The operator ceases to be essentially self-adjoint for $\alpha_\ell > 1/4$, leading to a Hermitian ambiguity typically resolved via self-adjoint extensions. However, in the present open-system context, a physical outgoing boundary condition uniquely selects solutions corresponding to irreversible absorption—the singularity acts as a one-way probability sink. Mathematically, this fixes the relative phase and eliminates ambiguity, transforming the dynamical instability into quantized dissipation characterized by complex-conjugate pairs of resonant energies.

Log-Periodic Spectrum and Universality

The outgoing condition at the singularity quantizes the complex resonances, leading to a discrete spectrum $\psi(t,\mathbf{r})$0 ($\psi(t,\mathbf{r})$1). The decay rates consequently form a log-periodic ladder: $\psi(t,\mathbf{r})$2. Critically, this geometric spacing depends solely on the anomalous scaling parameter $\psi(t,\mathbf{r})$3 and is insensitive to short-distance regularization, reflecting universality analogous to renormalization-group limit cycles encountered in Efimov physics.

Emergent Kinematic Temperature

The log-periodic energy and decay structure defines an emergent, kinematic "Hawking-like" temperature:

$\psi(t,\mathbf{r})$4

This temperature has no dynamical (gravitational) origin but instead arises from symmetry breaking via the irreversible boundary condition.

Analogy with Black-Hole Quasi-Normal Modes

The structure of the spectrum aligns closely with black-hole QNMs, where boundary conditions (purely ingoing at the horizon, outgoing at infinity) yield a discrete set of complex decaying frequencies. In both scenarios, emergent scale invariance and irreversible boundary conditions dictate the physical spectrum, independent of detailed dynamics. However, the present construction isolates these features purely in flat spacetime without invoking gravity—highlighting the kinematical, rather than dynamical, essence of anomaly-driven dissipation.

Implications and Future Directions

The work provides a minimal analytic laboratory for studying the interplay of conformal symmetry, scale anomaly, non-Hermiticity, and quantized dissipation in relativistic contexts. The construction is amenable to realization in engineered platforms—such as photonic lattices, cold atom systems, and synthetic gauge fields—where both complex potentials and inverse-square interactions can be implemented. Unlike many open quantum system models, the dissipative hierarchy here is exactly solvable and universal, creating the possibility of direct experimental probes of anomaly-driven decay, effective temperatures, and resonance quantization.

Practically, the framework suggests extensions to fermionic fields, inclusion of gauge or spin-orbit couplings, and exploration in curved backgrounds. The explicit analytic solvability provides a benchmark for studying quantum anomalies in open relativistic systems and their ensuing dissipative phenomena, potentially informing effective descriptions of near-horizon black hole quantum dynamics, engineered dissipative quantum simulators, and theoretical treatments of non-equilibrium quantum field theory.

Conclusion

This paper rigorously demonstrates how a non-Hermitian Klein-Gordon field theory with a scale-invariant imaginary potential transforms the canonical inverse-square quantum anomaly from an instability into a discrete, quantized spectrum of dissipative resonances. The decay hierarchy and emergent kinematic temperature are universal consequences of anomalous scaling and irreversible boundary conditions, independent of regularization or microscopic detail. The construction forms a robust basis for future investigations of anomaly-driven phenomena, quantum dissipative structures, and interdisciplinary analogies with black-hole physics, both theoretically and in experimentally engineered systems.

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Citation:

"Quantized Dissipation from the Inverse-Square Anomaly in a Non-Hermitian Klein-Gordon Field" (2603.28525)