Quantum Mechanics for Three Dirac Equations in a Curved Spacetime
|Title||Quantum Mechanics for Three Dirac Equations in a Curved Spacetime|
|Keywords||Quantum Mechanics, Dirac Equations, Curved Spacetime|
We consider three versions of the Dirac equation in a curved spacetime: the standard (Dirac-Fock-Weyl or DFW) equation, and two alternative versions. Both of these alternative versions are based on the recently proposed tensor representation of the Dirac field (TRD), that considers the Dirac wave function as a spacetime vector and the set of the Dirac matrices as a third-order tensor [1-3]. These three equations differ also in the covariant derivative D?. A common tool for the study is the Bargmann-Pauli hermitizing matrix A. Having the current conservation for any solution of the Dirac equation gives an equation to be satisfied by the fields (g m, A), with g m the Dirac matrices. This condition is always verified for DFW with its restricted choice for the field g m. It similarly restricts the choice of the field g m for TRD. However, this restriction can be achieved. A positive definite scalar product is defined and a hermiticity condition for the Dirac Hamiltonian is derived for a general coordinate system with minor restrictions, in a general curved spacetime. For DFW, the hermiticity of the Dirac Hamiltonian is not preserved under all admissible changes of the fields (g m, A).