Klein-Gordon and Dirac particles in non-constant scalar-curvature background

TL;DR

This paper analyzes Klein-Gordon and Dirac particles in specific non-constant scalar-curvature backgrounds, revealing discrete and continuous spectra depending on the metric form and particle properties, with implications for quantum field behavior in curved spacetime.

Contribution

It provides the first detailed spectral analysis of Klein-Gordon and Dirac particles in exponential scalar-curvature backgrounds with varying metric functions.

Findings

Discrete spectrum for $u(x)=e^{gx}$ with specific ground state energy.

Continuous spectrum for $u(x)=e^{-gx}$ in certain cases.

Threshold transverse-momentum values differ for Klein-Gordon and Dirac particles.

Abstract

The Klein-Gordon and Dirac equations are considered in a semi-infinite lab ($x > 0$) in the presence of background metrics $ds^2 =u^2(x) \eta_{\mu\nu} dx^\mu dx^\nu$ and $ds^2=-dt^2+u^2(x)\eta_{ij}dx^i dx^j$ with $u(x)=e^{\pm gx}$. These metrics have non-constant scalar-curvatures. Various aspects of the solutions are studied. For the first metric with $u(x)=e^{gx}$, it is shown that the spectrums are discrete, with the ground state energy $E^2_{min}=p^2c^2 + g^2c^2\hbar^2$ for spin-0 particles. For $u(x)=e^{-gx}$, the spectrums are found to be continuous. For the second metric with $u(x)=e^{-gx}$, each particle, depends on its transverse-momentum, can have continuous or discrete spectrum. For Klein-Gordon particles, this threshold transverse-momentum is $\sqrt{3}g/2$, while for Dirac particles it is $g/2$. There is no solution for $u(x)=e^{gx}$ case. Some geometrical properties of these metrics are also discussed.