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

Abstract

The Klein-Gordon and Dirac equations are considered in a semi-infinite lab ($x>0$) in the presence of background metrics $d{s}^2=u^2(x){\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$ and $d{s}^2=-d{t}^2+u^2(x){\eta }_{ij}d{x}^{i}d{x}^j$ 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 $E_{min}^2=p^2{c}^2+g^2{c}^2{\hslash }^2$ 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.

14 pages, LaTeX, to be published in Int. Jour. Mod. Phys. A