Quantized Area of the Schwarzschild Black Hole: A non-Hermitian Perspective

TL;DR

The paper addresses how to reconcile Bekenstein's horizon-area quantization with quantum-mechanical oscillator formalisms and extends this to a non-Hermitian, $\mathbb{PT}$-symmetric Swanson oscillator. It first reproduces the standard area spectrum via a harmonic-oscillator mapping $A(n)=(n+\tfrac{1}{2})\omega$ with $\omega=\tfrac{8π}{m_P^2}$ and Hawking temperature $T_H=m_P^2/(8πM)$, then shows that a Swanson Hamiltonian yields a generalized spectrum $A(\alpha,\beta)=(n+\tfrac{1}{2})(\alpha+\beta+\tfrac{8π}{m_P^2})$ and modified mass, temperature, and entropy. The authors demonstrate that a logarithmic entropy correction of the form $-k\ln(A)$ can be realized with $k=1/2$ under suitable parameter choices, and they derive a bound $|k|<π/4$ arising from positivity constraints. This work provides a consistent non-Hermitian QM perspective on black-hole area quantization and tightens theoretical expectations for entropy corrections in quantum gravity regimes.

Abstract

In this work, our aim is to link Bekenstein's quantized form of the area of the event horizon to the Hamiltonian of the non-Hermitian Swanson oscillator which is known to be $\mathbb{PT}$-symmetric. We achieve this by employing a similarity transformation that maps the non-Hermitian quantum system to a scaled harmonic oscillator. Our procedure is standard and well known. We, first of all, consider the unconstrained reduced Hamiltonian which is directly expressed in terms of the Schwarzschild mass and implies a periodic character for the conjugate momentum (which represents the asymptotic time coordinate), the period being the inverse Hawking temperature. This leads to the quantization of the event-horizon area in terms of the harmonic-oscillator levels. Within the framework of the Swanson oscillator, we proceed to derive novel expressions for the Hawking temperature and the black hole entropy. Notably, the logarithmic area-correction term -(1/2)$\ln$(area) is consistent with our results whereas -(3/2) $\ln$(area) is not.