非常数标量曲率背景下的Klein-Gordon粒子和Dirac粒子
Klein-Gordon and Dirac particles in non-constant scalar-curvature background
摘要：
在半无限实验室（$x > 0$）中考虑Klein-Gordon方程和Dirac方程，背景度量为$ds^2 =u^2(x) η_{μν} dx^μdx^ν$和$ds^2=-dt^2+u^2(x)η_{ij}dx^i dx^j$，背景度量为$u(x)=e^{\pm gx}$ .这些度量具有非恒定的标量曲率。的解决方案的各个方面进行了研究。对于第一个度规与$u(x)=e^{gx}$，它表明，光谱是离散的，与自旋为0的粒子的基态能量$E^2_{min}=p^2c^2 + g^2c^2\hbar^2$。对于$u(x)=e^{-gx}$，发现光谱是连续的。对于具有$u(x)=e^{-gx}$的第二种度规，每个粒子取决于其横动量，可以具有连续或离散谱。对于克莱因-戈登粒子，这个阈值横动量是$\sqrt{3}g/2$，而对于狄拉克粒子，它是$g/2$。没有针对$u(x)=e^{gx}$病例的解决方案。本文还讨论了这些度量的一些几何性质。
英文摘要：
The Klein-Gordon and Dirac equations are considered in a semi-infinite lab ($x > 0$) in the presence of background metrics $ds^2 =u^2(x) η_{μν} dx^μdx^ν$ and $ds^2=-dt^2+u^2(x)η_{ij}dx^i dx^j$ 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.
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