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arXiv · 2605.28875

Quantum and Thermal Properties of the Klein-Gordon Inverted Harmonic Oscillator with Physical Applications

Kevin Hernández·Mustapha Maamache
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Abstract

We develop a systematic framework for the quantum and thermal properties of a Klein-Gordon scalar field subject to an inverted harmonic potential $-{1\over2} m^2\omega^2 x^2$. Starting from a non-Hermitian momentum substitution $P \to P - m\omega x$, we employ a symplectic phase-space rotation $V = \exp\!\left[-\tfrac{\pi}{8}(xp+px)\right]$ to map the system onto an analytically tractable effective harmonic oscillator evaluated at $xe^{i\pi/4}$. This allows us to define a well-regulated partition function $Z(\beta,\omega,m)$ and derive closed-form expressions for the free energy, entropy, and thermal correlation functions. We then apply this framework to three physical settings: (i) scalar field fluctuations during cosmological inflation, (ii) quantum fields near black-hole horizons, and (iii) order-parameter dynamics near second-order phase transitions in condensed matter. Our results unify previously scattered results in the literature and provide new predictions for the finite-temperature spectral density and entanglement entropy of unstable quantum systems.

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Kevin Hernández, Mustapha Maamache. 2026-05-26. Quantum and Thermal Properties of the Klein-Gordon Inverted Harmonic Oscillator with Physical Applications. https://arxiv.org/abs/2605.28875

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