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

Abstract

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. To this end, we consider the unconstrained reduced Hamiltonian which is directly expressed in terms of the Schwarzschild mass and implies a periodic character of the conjugate momentum (which represents the asymptotic time coordinate), the period being the inverse Hawking temperature. This leads to quantization of the event-horizon area in terms of the harmonic-oscillator levels. Next, in 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.