Feshbach-Villars Formalism for a Spin-1/2 Particle in Curved Spacetime

This study extends the Feshbach-Villars (FV) formalism to encompass spin-1/2 particles propagating in curved spacetime. Starting from the covariant Dirac equation, we derive its Hamiltonian form using a tetrad (vierbein) formulation and implement a diagonal FV transformation on the associated second-order (squared) equation.The resulting Schrödinger-type evolution equation is expressed in both block-matrix and Pauli-matrix representations, explicitly displaying gravitational and electromagnetic couplings, alongside spin-field interaction terms. We analyze this formalism in (1 + 2)-and (1 + 3)-dimensional settings and apply it to cosmic-string backgrounds (static and spinning, with and without oscillator couplings) to determine the corresponding energy spectra and exact wave functions. Although the FV reformulation is algebraically equivalent to the covariant Dirac dynamics-subject to the standard restriction that eliminates spurious solutions introduced by squaring-it offers a highly practical complementary perspective. Specifically: (i) the particle and antiparticle sectors are distinctly represented by two coupled components equipped with a simple (pseudo-)unitary inner product tied to the conserved charge; (ii) the physical distinction between stationary and static spacetimes is directly encoded in the FV operator Y, which is governed by the ADM shift vector, thus clearly diagnosing how frame-dragging modifies particle-antiparticle mixing; and (iii) the FV block structure provides a streamlined approach for solving spectral problems in topologically nontrivial geometries.