Spin 0 and spin 1/2 quantum relativistic particles in a constant gravitational field

TL;DR

This paper analyzes the solutions of Klein-Gordon and Dirac equations for spin-0 and spin-1/2 particles in a specific gravitational field, revealing differences in energy eigenvalues related to particle spin and momentum.

Contribution

It provides a detailed comparison of relativistic quantum particles in a gravitational field, highlighting how spin affects energy spectra in this context.

Findings

Spin-1/2 particles have lower energy eigenvalues than spin-0 particles at zero transverse momentum.

Energy eigenvalues for large quantum numbers are derived and compared for both particle types.

The energy difference at zero momentum is quantified as mgħc.

Abstract

The Klein-Gordon and Dirac equations in a semi-infinite lab ($x > 0$), in the background metric $\ds^2 = u^2(x) (-\dt^2 + \dx^2) + \dy^2 + \dz^2$, are investigated. The resulting equations are studied for the special case $ u(x) = 1 + g x$. It is shown that in the case of zero transverse-momentum, the square of the energy eigenvalues of the spin-1/2 particles are less than the squares of the corresponding eigenvalues of spin-0 particles with same masses, by an amount of $mg\hbar c$. Finally, for nonzero transverse-momentum, the energy eigenvalues corresponding to large quantum numbers are obtained, and the results for spin-0 and spin-1/2 particles are compared to each other.