Energy Loss Reveals Hidden Order in Relativistic Quantum Fields

Mansour Haghighat and Ali Nouri at Shiraz University present an exactly solvable model incorporating an anomalous inverse-square interaction into a non-Hermitian Klein-Gordon field theory. Imposing specific boundary conditions transforms instability into a discrete spectrum of complex energies, revealing universal geometric spacing in decay rates. The model provides a minimal framework for understanding scale anomaly, non-Hermiticity, and ultimately, quantized dissipation within relativistic open quantum systems, offering new insights into how energy loss occurs at a fundamental level. Quantized decay rates reveal scale invariance in relativistic open quantum systems A logarithmic progression of decay rates has been demonstrated with unprecedented precision, rather than continuous variation. This discrete spectrum, governed by the anomalous scaling exponent γ, signifies a fundamental shift in understanding. Previous models required arbitrary parameters to maintain scale invariance, but this model achieves exact solvability by embedding an inverse-square interaction within a relativistic Klein-Gordon field theory, effectively transforming instability into predictable energy levels. The Klein-Gordon equation, a cornerstone of relativistic quantum mechanics, describes the evolution of scalar fields, fundamental entities in particle physics, and its non-Hermitian extension allows for the incorporation of dissipative effects, crucial for modelling open quantum systems that exchange energy with their environment. This contrasts with traditional Hermitian quantum mechanics which assumes closed, isolated systems. The framework establishes a connection between scale anomaly, non-Hermitian physics, and quantized dissipation, offering a minimal analytic approach to understanding energy loss in relativistic open quantum systems. Dissipative dynamics are governed by an emergent kinematic energy scale, mirroring renormalization-group behaviour observed in Efimov physics; this universality ensures insensitivity to microscopic short-distance regularization. Calculations have yielded an effective temperature, Teff, arising from the logarithmic decay spectrum of energy levels, with a value approximately equal to 1 divided by 2πkB multiplied by 1 over σlr0, where σl represents the anomalous scaling exponent and r0 is a short-distance matching scale. The emergence of an effective temperature is particularly noteworthy, as it suggests a thermal-like behaviour arising purely from quantum mechanical principles, without invoking classical thermodynamics or external heat baths. This temperature is not a measure of the system’s internal energy, but rather a characteristic scale of the decay rates, reflecting the rate at which the system loses energy to its surroundings. This emergent temperature originates purely from scale invariance and the imposed one-way boundary conditions, without requiring any gravitational dynamics. A direct parallel exists between this system and black-hole quasi-normal modes, where the inverse-square interaction mirrors the near-horizon potential influencing perturbations; both scenarios enforce a unidirectional flow due to physical irreversibility. The geometric spacing of decay rates remains unaffected by microscopic regularization details, echoing renormalization-group behaviour seen in Efimov physics, and demonstrating a strong universality. The analogy to black-hole quasi-normal modes is compelling, as it suggests that the underlying physics governing energy loss in these seemingly disparate systems may be fundamentally similar. Black holes, due to their extreme gravity, also exhibit irreversible absorption of matter and energy, leading to characteristic decay patterns described by quasi-normal modes. The fact that the same mathematical structure emerges in both systems hints at a deeper connection between quantum mechanics, gravity, and information loss. Revealing the system’s behaviour hinged on constructing an exactly solvable model. The problematic inverse-square interaction was embedded within a relativistic Klein-Gordon field theory, a mathematical framework describing the behaviour of particles moving at or near the speed of light, analogous to how Newtonian mechanics describes everyday objects. This approach reduced a complex problem to a simpler, inverse-square Schrodinger-type equation with a manageable spectral parameter, allowing isolation of the key physics. Incorporating a strictly outgoing boundary condition, representing irreversible absorption, generates a discrete spectrum of complex energies exhibiting geometric spacing determined by the anomalous scaling exponent, and the transformation into a quadratic spectral parameter enabled the isolation of key physical behaviours. The outgoing boundary condition is crucial for enforcing irreversibility, ensuring that energy flows outwards from the system and is not reflected. This condition effectively simulates the absorption of energy by the environment, leading to the decay of the quantum state. Analytical calculation of quantum energy loss via inverse-square anomaly Modelling particle decay and black-hole behaviour requires understanding how quantum systems shed energy. This work offers a surprisingly simple way to calculate that energy loss, bypassing the usual need for complex computer simulations. However, the model relies heavily on a specific, mathematically convenient ‘inverse-square anomaly’, where energy appears to vanish at a central point, and it’s not immediately clear how well these findings translate to more realistic physical scenarios. The inverse-square anomaly represents a singular point in the potential, creating a strong attractive force that leads to instability and energy loss. While mathematically tractable, this specific form of the potential may not accurately reflect the complexities of real-world interactions. Despite this simplification, the value of this work lies in providing a clear analytical pathway where previously only complex simulations existed. Consequently, scientists can now explore the fundamental connection between quantum energy loss and concepts like scale anomaly, a disruption of symmetry, without being hindered by computational difficulties. By embedding an inverse-square interaction within a non-Hermitian Klein-Gordon field theory, the approach demonstrates how instability transforms into predictable energy levels, avoiding the need for complex simulations typically used to model such behaviour. The resulting geometric spacing of decay rates, governed by the anomalous scaling exponent, suggests a fundamental connection to information transfer and opens the possibility of realising this model in platforms like photonic lattices and cold-atom systems. Scale anomaly arises when the scaling properties of a physical system change with energy, leading to deviations from expected behaviour. Understanding scale anomaly is crucial for developing a consistent theory of quantum gravity and for explaining the behaviour of systems at extremely high energies. The potential for realising this model in experimental platforms like photonic lattices and cold-atom systems is particularly exciting. Photonic lattices, created by interfering laser beams, can mimic the behaviour of quantum particles, allowing for the study of complex quantum phenomena. Similarly, cold-atom systems, where atoms are cooled to extremely low temperatures, provide a highly controllable environment for investigating quantum mechanics. These platforms offer the possibility of directly observing the quantized decay rates and verifying the predictions of the model, potentially leading to new insights into the nature of quantum dissipation and the fundamental laws governing energy loss. The research successfully demonstrated a method for transforming an unstable quantum system into one with predictable energy levels. This is significant because it provides a clear analytical solution where complex computer simulations were previously required to study energy loss and scale anomaly. By using a non-Hermitian Klein-Gordon field theory with an inverse-square interaction, scientists observed a geometric spacing in decay rates determined by the anomalous scaling exponent. The authors suggest this framework can be used to further investigate scale anomaly and dissipation in open quantum systems. 👉 More information 🗞 Quantized Dissipation from the Inverse-Square Anomaly in a Non-Hermitian Klein-Gordon Field 🧠 ArXiv: https://arxiv.org/abs/2603.28525 Tags: