Dirac fermions in a spinning conical Gödel-type spacetime

In this paper, we determine the relativistic and nonrelativistic energy levels for Dirac fermions in a spinning conical Gödel-type spacetime in ( 2 + 1 ) -dimensions, where we work with the curved Dirac equation in polar coordinates and we use the tetrads formalism. Solving a second-order differential equation for the two components of the Dirac spinor, we obtain a generalized Laguerre equation, and the relativistic energy levels of the fermion and antifermion, where such levels are quantized in terms of the radial and total magnetic quantum numbers n and m j , and explicitly depends on the spin parameter s (describes the ‘spin’), spinorial parameter u (describes the two components of the spinor), curvature and rotation parameters α and β (describes the conical curvature and the angular momentum of the spinning cosmic string), and on the vorticity parameter Ω (describes the Gödel-type spacetime). In particular, the quantization is a direct result of the existence of Ω (i.e. such quantity acts as a kind of ‘external field or potential’). We see that for m j > 0 , the energy levels do not depend on s and u ; however, depend on n , m j , α , and β . In this case, α breaks the degeneracy of the energy levels and such levels can increase infinitely in the limit 4 Ω β α → 1 . Already for m j < 0 , we see that the energy levels depends on s , u and n ; however, it no longer depends on m j , α and β . In this case, it is as if the fermion/antifermion ‘lives only in a flat Gödel-type spacetime’. Besides, we also study the low-energy or nonrelativistic limit of the system. In both cases (relativistic and nonrelativistic), we graphically analyze the behavior of energy levels as a function of Ω, α , and β for three different values of n (ground state and the first two excited states).