Connecting the Dirac Equation in Flat and Curved Spacetimes via Unitary Transformation

Abstract

In this work, the extension of the symmetrically spherical Dirac equation in the context of supersymmetric quantum mechanics in curved spacetime is investigated in presence of vector and tensor potentials given by V(r) and A(r), respectively, with the line element given by $ds^2 = (1+\alpha ^2 U(r))^2(dt^2-dr^2) - r^2d\theta ^2 - r^2\sin ^2\theta d\phi ^2$, where U(r) is a scalar potential and $\alpha$ is fine structure constant. Through a unitary transformation given by $U^{'}(\eta ) = \exp (i\alpha \eta \sigma ^2/2)$ we decouple the radial wave functions and obtain an equation analogous to Schrödinger equation in the context of supersymmetric quantum mechanics when $U(r) = -\epsilon V(r)$, so we can write the potential V(r) , U(r) and A(r) in terms of the superpotential W(r) . With this, the class of shape-invariant potentials can be exact solved in curved spacetime with the coupling considered. For the particular case $\eta = 0$, we recover the problem in flat spacetime, so the unitary operator $U^{'}(\eta )$ connects the systems with shape-invariant potentials in flat and curved spacetimes. As applications, two systems were analyzed: the harmonic oscillator and the Coulomb potential.