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This paper presents an exactly solvable relativistic model where a non-Hermitian Klein-Gordon field exhibits quantized dissipation driven by an inverse-square anomaly, effectively converting fall-to-the-center instability into a discrete spectrum of complex energies. By imposing strictly outgoing boundary conditions at the singularity, the study establishes a minimal analytic framework for understanding scale anomalies and dissipative dynamics in open quantum systems without relying on microscopic regularization. The work offers significant theoretical insight into how boundary-induced non-Hermiticity shapes spectral properties in relativistic field theories.
Quantized Dissipation from the Inverse-Square Anomaly in a Non-Hermitian Klein-Gordon Field
arXiv:2603.28525v2 Announce Type: replace 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.
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.
Revised version, 9 pages. Added two appendices establishing the relation between Case's self-adjoint extension and the non-Hermitian absorbing boundary condition through analytic continuation
arXiv:2603.28525 [quant-ph]
(or arXiv:2603.28525v2 [quant-ph] for this version)
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